diff --git "a/D9AyT4oBgHgl3EQfSPfV/content/tmp_files/load_file.txt" "b/D9AyT4oBgHgl3EQfSPfV/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/D9AyT4oBgHgl3EQfSPfV/content/tmp_files/load_file.txt" @@ -0,0 +1,782 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf,len=781 +page_content='Selection of Centrality Measures Using Self-Consistency and Bridge Axioms Pavel Chebotarev∗ Moscow Institute of Physics and Technology 9 Inststitutskii per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', Dolgoprudny, Moscow Region, 141700 Russia January 3, 2023 Abstract We consider several families of network centrality measures induced by graph ker- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Self-consistency and Bridge axioms that appeared earlier in the literature turn out to be closely related to two of these families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We obtain a necessary and suffi- cient condition of Self-consistency, a sufficient condition of the Bridge axiom, indicate specific measures that satisfy these axioms and show that under some additional con- ditions they are incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is also shown that PageRank centrality violates most conditions under consideration, and has a property that, according to some authors, is hardly imaginable for a centrality measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Adopting such conditions as the Self- consistency or Bridge axioms allows one to dramatically reduce the length of a survey for selecting the most appropriate centrality measures in the culling method proposed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Keywords: network | centrality measure | axiomatic approach | self-consistency | bridge axiom | PageRank 1 Introduction The number of network centrality measures studied in the literature exceeds 400 [2] and many new measures appear every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This diversity needs to be structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The main means of structuring it is to establish a correspondence between the measures and their properties, some of which can be considered as normative conditions or axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The purpose of this paper is to advance this work by studying two natural axiomatic conditions, namely, the Self- consistency and Bridge axioms, which are closely related to special classes of kernel-based centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We establish a sufficient condition of the Bridge axiom, a necessary and sufficient condition of Self-consistency, and indicate centralities, some of which are well known and others are new, that satisfy these axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Very often, centrality is identified with structural importance [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, there are concepts of importance that are not reducible to centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Say, in a chain of moving people modeled by a path graph, the most important actors may be the leader and the trailer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', ∗pavel4e@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='00084v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='soc-ph] 31 Dec 2022 the least central end elements of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Moreover, the central elements of such a chain may not be of particular importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, the importance of nodes in networks is not necessarily manifested through centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Anyway, each point centrality measures some structural capital of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It turns out that the types of capital accounted for by the centralities that satisfy the Bridge axiom on the one hand and by centralities satisfying the conjunction of Self-consistency and Monotonicity on the other hand are different, and therefore these conditions are incompatible, provided that Equivalence is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, the Bridge axiom is incompatible with Transit monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' PageRank is a centrality measure that attracts a lot of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this paper, we show that it does not satisfy the most of the conditions under consideration and give an explanation of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' After introducing the basic notation in Section 2, in Section 3 we consider several families of centralities associated with graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In Section 4, the Bridge and Self-consistency axioms are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Section 5 presents a sufficient condition of the Bridge axiom as well as a number of measures that satisfy it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In Section 6, we prove a necessary and sufficient condition of Self-consistency and present centralities that satisfy it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In Section 7, simple general properties of centrality measures are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axioms of Monotonicity and Transit monotonicity are considered in Section 8 and we prove that the addition of these axioms is sufficient to ensure the properties of Section 7 and to form conditions incompatible with the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the concluding Section 9, we propose some interpretations of the results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2 Notation Let G = (V, E) be an undirected graph with node set V = V (G) and edge set E = E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The order of G is |V | = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Graph nodes will be denoted by letters u, v, w, ui, vi, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', numbers 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , or names: Medici, Pazzi, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We consider graphs with n > 1, without loops and multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since some centrality measures under study are applicable only to connected graphs, we confine ourselves to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Nodes u and v of G are neighbors iff {u, v} ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let Nu denote the set of neighbors of node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The adjacency matrix of G is denoted by A = A(G) = (auv)n×n: auv = 1 when u and v are neighbors and auv = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let ρ(A) be the spectral radius of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The degree du of a node u is the number of neighbors of u: du = |Nu|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The vector of node degrees is d = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , dn)T = A1, where 1 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , 1)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A leaf is a node that has exactly one neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Nodes u and v are equivalent in G if there exists an automorphism of G that takes u to v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' in this case we write u ∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Laplacian matrix of G is L = diag(A1) − A, where diag(x) is the diagonal matrix with vector x on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2 The union of graphs G = G1 ∪ G2 (not necessarily disjoint) is defined by: V (G) = V (G1) ∪ V (G2) and E(G) = E(G1) ∪ E(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Given a graph G, a centrality measure (or centrality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' sometimes, point centrality) f attaches a real number f(v) to each node v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, f depends on G, however, for simplicity we do not reflect this dependence in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In most cases G is fixed, and when it is not, we explicitly specify the graph to which centrality applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Formally, for a fixed graph G, a centrality on G is a function f : V (G) → R+ ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It associates a non- negative real number f(v) with every node v ∈ V (G) based only on the graph structure [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Various conceptions of centrality are quite diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this regard, there is no generally accepted definition of centrality that would semantically distinguish it from other types of point structural measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' On some attempts to make such a distinction, see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' When a centrality measure f(·) on G is fixed, we will write u ≻ v, u ⪰ v, and u ∼= v as short versions of f(u) > f(v), f(u) ≥ f(v), and f(u) = f(v), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Moreover, if, for instance, V = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , 7}, then ({1, 6}, {2, 3, 4}, 5, 7) is an example of centrality ranking of nodes 1 to 7 in which f(1) = f(6) > f(2) = f(3) = f(4) > f(5) > f(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3 Centrality measures induced by graph kernels In this section, we consider several families of centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let d(u, v) be the shortest path distance [8] between nodes u and v in a graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the length of a shortest path between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Two popular1 distance based centrality measures are the [Shortest path] Closeness [10,11] f(u) = � � v∈V d(u, v) �−1 (1) and [Shortest path] Eccentricity [10,12] f(u) = (max v∈V d(u, v))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (2) General classes of Closeness and Eccentricity centralities are defined by (1) and (2) with d(u, v) being arbitrary distances for graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the literature, several classes of such distances and, more generally, dissimilarity measures have been proposed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', [13,14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Substituting them in (1) and (2) provides centralities whose properties may vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Most of the alternative distances and dissimilarity measures are defined via graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let us consider several of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The parametric Katz [15] kernels (also referred to as Walk [16] or Neumann diffusion [17] kernels) are defined as P Walk(t) = ∞ � k=0 (tA)k = (I − tA)−1 (3) 1For example, in the recent study [9], the authors come to the conclusion that in the infection source identification problem “a combination of eccentricity and closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' generally performs better than several state-of-the-art source identification techniques, with higher accuracy and lower average hop error.” 3 with 0 < t < (ρ(A))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Communicability kernels [18,19] are P Comm(t) = ∞ � k=0 (tA)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' = exp(tA), t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Two other classes of kernels are defined similarly via the Laplacian matrix L = L(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Forest kernels or regularized Laplacian kernels [20,21] are P For(t) = (I + tL)−1, where t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (4) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Heat kernels are the Laplacian exponential diffusion kernels [22] P Heat(t) = ∞ � k=0 (−tL)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' = exp(−tL), t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Schoenberg’s theorem [23,24], if matrix P = (puv) is a kernel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', is positive semidef- inite), then it produces a Euclidean distance d(u, v) by means of the transformation d(u, v) = � 1 2(puu + pvv − puv − pvu) � 1 2, u, v ∈ V, (5) where factor 1 2 determines the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, all Walk, Communicability, Forest, and Heat kernels with appropriate parameters t provide distances that can be substituted in (1) and (2) to obtain Closeness and Eccentricity centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We will denote them by Closeness(Kernel) and Eccentricity(Kernel) with the corresponding kernels substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Furthermore, if Pn×n = (puv) determines a proximity measure (which means that for any x, y, z ∈ V, pxy + pxz − pyz ≤ pxx, and the inequality is strict whenever z = y and y ̸= x), then [25] transformation d(u, v) = 1 2(puu + pvv − puv − pvu), u, v ∈ V (6) provides a distance function that satisfies the axioms of a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Forest kernel with any t > 0 produces a proximity measure, while kernels in the remaining three families do so when t is sufficiently small [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The centralities obtained from a Proximity measure by transformation (6) and substitution of the resulting distance into (1) and (2) will be denoted by Closeness∗(Proximity) and Eccentricity∗(Proximity), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Moreover, if P represents a strictly positive transitional measure on G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', pxy pyz ≤ pxz pyy for all nodes x, y, and z, with pxy pyz = pxz pyy whenever every path in G from x to z visits y), then transformation ˆpuv = ln puv, u, v ∈ V produces [13,26] a proximity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this case, (6) applied to ˆP = (ˆpuv) reduces to d(u, v) = 1 2(ln puu + ln pvv − ln puv − ln pvu) (7) 4 and generates [13] a cutpoint additive distance d(u, v), viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', such a distance that d(u, v) + d(v, w) = d(u, w) whenever v is a cutpoint between u and w in G (or, equivalently, whenever all paths connecting u and w visit v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The centralities obtained from anyTransitional Measure by transformation (7) and substitution of the resulting distance into (1) and (2) will be denoted by Closeness∗(logTransitionalMeasure) and Eccentricity∗(logTransitionalMeasure), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since the Walk and Forest kernels determine [26] strictly positive transitional measures, transformation (7) applied to them generates cutpoint additive distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Substituting them into (1) and (2) produces Closeness∗(logForest), Closeness∗(logWalk) and the corresponding Eccentricity∗(·) centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, based on the above results, we define Closeness and Eccentricity centrality mea- sures obtained by substituting the: Forest kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Heat kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' logarithmic Forest kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' logarithmic Walk kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' logarithmic Heat kernel, and logarithmic Communicability kernel transformed by (5) or (6) into (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' These centralities were used in the survey proposed in [1] with parameter t = 1 for the Forest, Heat, and Communicability kernels and t = (ρ(A) + 1)−1 for the Walk kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' While the above measures are promising kernel-based centralities, they do not exhaust all kernels and transformations [14,17] that can be used to obtain such measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' To mention some alternative constructions, note that every distance on graph nodes can be integrated in the p-Means framework [27] or in the framework developed in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Closeness(Forest) centrality was examined in [29] with the conclusion that “forest distance centrality has a better discriminating power than alternate metrics such as be- tweenness, harmonic centrality, eigenvector centrality, and PageRank.” Along with this, the authors note that the order of node importance induced by forest distances on some simple graphs is consistent with their intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In addition to the above approaches, kernels and similarity/proximity measures can be used to obtain centralities directly, without transformations into distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' An example of such measures is the Estrada subgraph centrality [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This index of a graph node u is equal to the diagonal entry pComm uu of the Communicability kernel, so we denote it by Communicability(Kii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, Walk(Kii) is the measure f(u) = pWalk uu , u ∈ V determined by the diagonal entries of the Walk kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' One more type of centrality measures is constructed by summing the non-diagonal entries of the rows of a kernel matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We consider the measures of this kind Communicability(Kij) and Walk(Kij) defined by f(u) = � v̸=u pComm uv and f(u) = � v̸=u pWalk uv , u ∈ V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Finally, Total communicability [30] is obtained by summing all row entries of the Commu- nicability kernel: f(u) = � v∈V pComm uv ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' it can be described [31] in terms of “potential gain,” as well as the corresponding Walk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 5 The existence of hundreds of types and subtypes of centralities compounded by the existence of infinite families of them highlights the need for powerful tools for comparing centrality measures and choosing the most appropriate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The axiomatic approach is indispensable in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 4 Axioms of Bridge and Self-consistency The axioms considered in this section determine the relation between the centrality values of two nodes in a graph of a special structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As mentioned above, the measures under study assign centrality to nodes based solely on the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Equivalence axiom is a partial embodiment of this idea (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' [32, axiom A3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom E (Equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If u, v ∈ V (G) and u ∼ v, then f(u) = f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' All measures under consideration satisfy axiom E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' it will be assumed by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Among the most appealing axioms characterizing various classes of “reasonable” centrality measures are those of an ordinal nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Such axioms allow one to compare the centrality of some nodes, but they do not determine specific computational algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In other words, they are not fingerprints of particular centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Positive responsiveness is a type of axiom, which is of primary importance in many axiomatic constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The template of these axioms is as follows: “an increase in input (making a node more central from some point of view) leads to an increase in output (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', raises its centrality).” Now we present two axioms of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the next two sections, we will find centrality measures that satisfy them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Recall that a bridge in a graph is an edge whose deletion increases the number of graph’s connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The following axiom [33] relates the centrality of the endpoints of any bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom B (Bridge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If edge {u, v} is a bridge in G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the removal of {u, v} from E(G) separates G into two connected components (with node sets Vu ∋ u and Vv ∋ v), then |Vu| < |Vv| ⇔ f(u) < f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A strengthening of this axiom is the Ratio property [34], which holds when under the same premise, f(w) > 0 for all w ∈ V and f(u)/f(v) = |Vu|/|Vv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The idea of the second axiom is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We assume that the vector of centrality values of the neighbors of any node u carries a lot of information about the centrality of u itself (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Consistency in [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A more specific form of this idea is that “the higher the centrality values of a node’s neighbors, the higher the centrality of the node itself.” This is in line with the justification given by Bonacich and Lloyd [36] to the Eigenvector centrality, a measure satisfying (Section 6) the axiom we are going to introduce: “The eigen- vector is an appropriate measure when one believes that actors’ status is determined by those with whom they are in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This conception of importance or centrality makes sense in a variety of circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Social status rubs off on one’s associates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Receiving information 6 from knowledgeable sources adds more to one’s own knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, eigenvectors can give weird and misleading results when misapplied.” The final step in refining this concept leads to the axiom of Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the case of directed graphs that express paired comparisons, it appeared in [37–39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' for undirected graphs, in [40, 41] under the name of Structural consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It strengthens Preservation of neighborhood-inclusion [42], whose directed version goes back to Preservation of cover relation [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom S (Self-consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If for u, v ∈ V, there is a bijection between Nu to Nv such that every element of Nu is, according to f(·), no more central than the corresponding element of Nv, then f(u) ≤ f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If “no more” is actually “less” at least once, then f(u) < f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Both the Bridge and Self-consistency axioms belong to the class of positive responsive- ness axioms, however, the positivity requirement in the premise of Self-consistency is not objective: it reduces to positivity in terms of f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This implies that when f(·) satisfies ax- iom S and the values of ¯f(·) are ordered oppositely to those of f(·), then ¯f(·) also satisfies S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Consequently, the sole axiom S allows in some cases to conclude that f(u) = f(v), but never that f(u) > f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In particular, if f(u) = f(v) for all u, v ∈ V, then f(·) satisfies S for any graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, Self-consistency is usually combined with other axioms indicating how centrality is related to the graph structure itself rather than to the neighbors’ centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the following two sections, we present several results on the centrality measures that satisfy the Bridge or Self-consistency axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 5 Centrality measures satisfying the Bridge axiom In the statements of this section, the notion of a cutpoint additive distance and the Close- ness∗(logForest) and Closeness∗(logWalk) measures are those introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Connectivity centrality [34] of vertex u is equal to the number of permutations π = (π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , π|V |) of V (G) such that π1 = u and for every j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , |V | − 1}, the induced subgraph of G with node set {π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , πj} is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Any Closeness centrality of the form (1) such that the corresponding distance d(·, ·) is cutpoint additive satisfies axiom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For any connected G, consider a Closeness centrality f(u) = �� v∈V d(u, v) �−1, where d(·, ·) is a cutpoint additive distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let {u, v} be a bridge in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since v is a cutpoint between u and any node w ∈ Vv∖{v}, it holds that (f(u))−1 = � w∈V (G) d(u, w) = � w∈Vu d(u, w) + � w∈Vv d(u, w) = � w∈Vu d(u, w) + |Vv| d(u, v) + � w∈Vv d(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 7 Figure 1: A tree on which Betweenness violates axiom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, (f(v))−1 = � w∈Vv d(v, w) + |Vu| d(v, u) + � w∈Vu d(u, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Hence (f(u))−1 − (f(v))−1 = (|Vv| − |Vu|) d(u, v), consequently, f(u) < f(v) ⇔ (f(v))−1 < (f(u))−1 ⇔ |Vu| < |Vv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, f(·) satisfies the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Shortest path Closeness, Connectivity, Closeness∗(logWalk), and Closeness∗(logForest) centralities satisfy the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The fulfilment of the Bridge axiom for the Shortest path Closeness is due to Skibski and Sosnowska [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Alternatively, it follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Bridge axiom holds for Connectivity since this centrality measure satisfies the stronger Ratio property [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Walk (3) and Forest (4) kernels represent [26] strictly positive transitional measures on any connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, definition (7) transforms [13] them into cutpoint additive distances d(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Lemma 1 this implies that the Closeness centralities corresponding to these distances, namely, the Closeness∗(logWalk) and Closeness∗(logForest) centralities, satisfy the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, other strictly positive transitional measures [26] and cutpoint additive distances also produce centralities that satisfy the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is worth noting that the Betweenness centrality [44] satisfies the Bridge axiom for many graphs, however, generally this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The simplest graph on which Be- tweenness violates this axiom is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Here, axiom B requires that the centralities of nodes 0 and 5 are equal since |V0| = |V5|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, the Betweenness centrality of node 0 is higher than that of node 5, as 0 lies on the shortest path from 1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 6 Centrality measures satisfying Self-consistency To formulate a necessary and sufficient condition of Self-consistency, we introduce two defi- nitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 8 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A function ϕ : Mk → R, where Mk = {M : 0 < |M| < k}, M being a multiset2 of real numbers, will be called a scoring function if ϕ(M) is strictly increasing in any element of M, while the remaining elements, including those equal to the varying one, are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A centrality vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T assigned to a connected graph G with V (G) = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n} (xu = f(u), u ∈ V (G), where f is the corresponding centrality measure) has a monotonic neighborhood representation if there exists a scoring function ϕ such that x satisfies the system of equations xu = ϕ({xw : w ∈ Nu}), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (8) In Definition 2, {xw : w ∈ Nu} is the multiset of the components of x that correspond to the neighbors of node u in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If a centrality vector has a monotonic neighborhood represen- tation, then finding this vector reduces to solving the system (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A centrality measure on G satisfies Self-consistency if and only if the centrality vector this measure attaches to G has a monotonic neighborhood representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Suppose that the centrality vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T associated with G has a monotonic neighborhood representation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let the premise of Self-consistency be true for nodes u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Consider the equations (8) corresponding to u and v: xu = ϕ({xw : w ∈ Nu}), (9) xv = ϕ({xw : w ∈ Nv}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (10) Since there is a bijection that maps each element of Nu to an element of Nv with a greater or equal centrality, step by step replacing in (9) the xw value of each element of Nu by the x component of the corresponding element of Nv and using the definition of monotonic neighborhood representation, we get a growth or preservation of the value of ϕ(·) at each step, yielding the value xv in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This implies that xu ≤ xv, or, stronger, xu < xv whenever xw has been strictly increased at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, Self-consistency is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Conversely, suppose that a centrality measure on G is Self-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let us construct a scoring function ϕ(·) that provides a monotonic neighborhood representation of the centrality vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T associated with G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' First, we set ϕ({xw : w ∈ Nu}) def = xu for all u ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Whenever {xw : w ∈ Nu} = {xw : w ∈ Nv} for some u, v ∈ V, Self-consistency implies xu = xv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the above definition of ϕ(·) on the set of multisets P = {{xw : w ∈ Nu}, 1 ≤ u ≤ n} ⊂ Mk is not contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, we defined the function ϕP(·) on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Now, to obtain a monotonic neighborhood representation of x, it suffices to extend ϕP(·) to the entire set Mk (k = max{|Nu|, 1 ≤ u ≤ n}) of multisets of real numbers in such a way that the resulting ϕ(·) is strictly increasing on Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2A finite multiset is an equivalence class of vectors such that two vectors z and z′ are equivalent whenever z′ can be obtained from z by permuting its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As distinct from a set, a multiset may contain several copies of the same element, as the components of a vector may be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 9 By the definition of a scoring function, the strict increase of ϕ(·) is required with respect to the following preorder ≽ on Mk: for X, Y ∈ Mk, X ≽ Y ⇔ [there is a bijection between X to Y such that every element of Y does not exceed the corresponding element of X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The condition of strict increase reduces to the implication [X ≽ Y and Y ̸≽ X] ⇒ ϕ(X) > ϕ(Y ), since the second necessary implication [X ≽ Y and Y ≽ X] ⇒ ϕ(X) = ϕ(Y ) is trivial as its premise implies X = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Observe that the preorder ≽ has a numerical [utility] representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This means that there exists a function u: Mk → R such that for all X, Y ∈ Mk, X ≻ Y ⇒ u(X) > u(Y ), where, by definition, X ≻ Y ⇔ [X ≽ Y and Y ̸≽ X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Indeed, u(X) can be defined, say, as the sum of the elements of multiset X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then X ≻ Y ⇒ u(X) > u(Y ) and so u(·) is a numerical representation of ≽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Self-consistency, ϕP(·) strictly increases on P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', ϕP(·) is a numerical representation of ≽P, the restriction of ≽ to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since ≽ has a numerical representation, it follows from [45, Theorem 1] that ϕP(·) has a strictly increasing extension to Mk if and only if ϕP(·) is gap- safe increasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', is weakly increasing and for any X, Y ∈ Mk ∪ {−∞, +∞}, Y ≻ X implies inf{ϕP(Z) : Z ≽ Y, Z ∈ P} > sup{ϕP(Z) : X ≽ Z, Z ∈ P}, (11) where, by convention, sup ∅ = −∞ and inf ∅ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' To prove that ϕP(·) is gap-safe increasing, first observe that since P is finite, sup and inf in (11) can be replaced by max and min, respectively, under the convention that max ∅ = −∞ and min ∅ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then, if the [multi]sets on the left-hand and right-hand sides of (11) are both nonempty, then for any Z′′ and Z′ minimizing ϕP(Z) on the left and maximizing ϕP(Z) on the right, respectively, Z′′ ≽ Y ≻ X ≽ Z′ holds, and by the “mixed” strict transitivity3 of ≽, Z′′ ≻ Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Self-consistency this implies ϕP(Z′′) > ϕP(Z′) and (11) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Otherwise, if some multiset in (11) is empty, then we have +∞ on the left or/and −∞ on the right, in a possible combination with a finite number on one of the sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In all these cases, (11) is valid, hence ϕP(·) is gap-safe increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, by [45, Theorem 1], ϕP(·) can be extended to Mk so that its extension ϕ(·) is a strictly increasing function and therefore, provides a monotonic neighborhood representation of the centrality vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The extension of ϕP(·) to Mk can be made, in particular, using the approach proposed in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The following propositions involve five centrality measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' we now recall their definitions using the notation introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For a connected graph G of order n, vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T presents: the Walk centrality [15] if x = ∞ � k=1 (tA)k1 = ((I − tA)−1 − I)1, (12) where t ∈ R is a parameter such that 0 < t < (ρ(A))−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3This means that for any X, Y, Z ∈ Mk, Z ≽ Y ≻ X ⇒ Z ≻ X and Y ≻ X ≽ Z ⇒ Y ≻ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 10 the Bonacich centrality [46] with real parameters α and β > 0 if x satisfies the system of equations xu = � w∈Nu (α + βxw), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (13) the Generalized Degree centrality [47] if x satisfies the system of equations (I + εL)x = d, (14) where ε > 0 is a real parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' the Eigenvector centrality [48,49] if x is positive and satisfies the equation Ax = ρ(A)x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (15) the PageRank centrality [50] if x is positive and satisfies the equation4 x = � αAT(diag(A1))−1 + (1 − α)J � x, (16) where J = 1 n11T, while α ∈ R is the “teleportation” parameter such that 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Generalized Degree, Walk, Eigenvector, and Bonacich centralities sat- isfy Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since for any u, du = |Nu|, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (14) can be written in component form as xu(1 + ε|Nu|) − ε � w∈Nu xw = |Nu|, u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n, which is equivalent to xu = (1 + ε|Nu|)−1 � w∈Nu (1 + εxw), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (17) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (17) is a monotonic neighborhood representation of vector x, therefore, by Lemma 2, the Generalized Degree centrality satisfies Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It follows from (12) that (I − tA)x = td, from which xu = t � w∈Nu (1 + xw), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (18) Since for any t > 0, (18) is a monotonic neighborhood representation of x, Lemma 2 implies that the Walk centrality satisfies Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A component form of (15) is xu = (ρ(A))−1 � w∈Nu xw, u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n, (19) 4In the case of simple graphs considered in this paper, AT = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 11 which is a monotonic neighborhood representation of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Hence, by Lemma 2, the Eigenvector centrality satisfies Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The equations (13) of the Bonacich centrality provide a monotonic neighborhood representation of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Lemma 2, these centralities satisfy Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It follows from the comparison of (18) and (13) that the Walk centralities are the Bonacich centralities with α = β = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' To prove that a centrality measure satisfies Self-consistency, it suffices to find its mono- tonic neighborhood representation, as we did, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', for the Walk centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Disproving the hypothesis of the Self-consistency of some measure reduces to giving a refuting example, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', an appropriate pair of nodes in some network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Here, among others, the famous network of Florentine ruling families (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2) can be of help, as we show in Lemma 3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Figure 2: Marriage network of the Florentine ruling families of the 15th century (without the isolated Pucci family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let f(·) be a centrality measure on a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We say that two arrays (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , uk) and (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , vk) of the nodes of G are f(·) order equivalent iff for any i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , k}, sign(f(ui) − f(uj)) = sign(f(vi) − f(vj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If a centrality measure f(·) satisfies axiom S, then for the Florentine families graph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2, the following arrays of nodes are f(·) order equivalent: (a) (Tornabuoni, Albizzi) and (Ridolfi, Ginori);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (b) (Bischeri, Peruzzi) and (Guadagni, Castellani);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (c) (Bischeri, Castellani) and (Guadagni, Barbadori);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 12 Lamberteschi Ginori Guadagni Albizzi Bischeri Tornabuoni Acciaiuoli Medici Ridolfi Strozzi Peruzzi Salviati Barbadori Castellahi Pazzi(d) (Peruzzi, Castellani) and (Bischeri, Barbadori);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (e) (Tornabuoni, Ridolfi) and (Guadagni, Strozzi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (f) (Barbadori, Salviati) and (Castellani, Pazzi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (g) (Ginori, Aciaiuoli, Pazzi, Lamberteschi) and (Albizzi, Medici, Salviati, Guadagni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (a) Observe that Tornabuoni and Albizzi have three neighbors each, and they share two neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, by S, the relation between them is the same as the relation between the remaining neighbors, Ridolfi and Ginori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (b) Bischeri and Peruzzi are adjacent and have a common neighbor Strozzi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' in addition, Bischeri has a neighbor Guadagni, while Peruzzi has a neighbor Castellani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Due to S, the relation between Bischeri and Peruzzi coincides with that between Guadagni and Castellani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Indeed, it is easy to see that the edge {Bischeri, Peruzzi} cannot correct the violation of Self-consistency that may occur in the absence of this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This completes the proof of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The remaining parts are proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The following proposition demonstrates that Lemma 3 can be quite useful in proving that certain measures violate Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Walk(Kii), Communicability(Kii), Closeness(Forest), Closeness(Heat), Closeness∗(logWalk), Closeness∗(logCommunicability), Closeness∗(logForest), and Close- ness∗(logHeat) centralities violate axiom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2, Walk(Kii) and Communicability(Kii) provide a central- ity ranking in which Peruzzi ≻ Bischeri, but Guadagni ≻ Castellani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, by part (b) of Lemma 3, these measures violate Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Measures Closeness(Forest), Close- ness∗(logWalk), Closeness∗(logCommunicability), and Closeness∗(logHeat) provide rankings in which Ridolfi ≻ Tornabuoni, but Guadagni ≻ Strozzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, by part (e) of Lemma 3, these measures violate Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Measures Closeness(Heat), and Closeness∗(logForest) pro- vide rankings in which Castellani ≻ Peruzzi, but Bischeri ≻ Barbadori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, by part (d) of Lemma 3, these measures violate Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 7 On core intuition behind centrality The best example of a “central” node is the center of a star of order more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A star of order n is a graph with one node (the center) having degree n − 1 and n − 1 nodes of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The edges of a star are sometimes called rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As Freeman [51] noted, “one general intuitive theme seems to have run through all the earlier thinking about point centrality in social networks: the point at the center of a star [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='] is the most central possible position.” Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We say that a centrality measure on a star G with n ≥ 3 nodes satisfies the star condition if it assigns maximum centrality to the center of this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For an example of a centrality measure that violates the star condition, see [1, Section 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 13 Self-consistency is a strong axiom, however, as was noted, it is not comprehensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' One of its features is that it only applies to nodes of the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, it does not imply the star condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As distinct from it, the Bridge axiom implies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' On a star with two or more rays, any centrality measure that satisfies axiom B also satisfies the star condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This is true as each ray of a star is a bridge, and among the components formed after its removal, the component containing a leaf is smaller than that containing the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, axiom B does not imply that the centrality of all leaves of a star is the same, which is immediate from Self-consistency (or from axiom E, as the leaves are equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Roy and Tredan [6], trying to capture the intuition underlying the concept of centrality claim that for a path graph with nodes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n, where each node u such that 1 < u < n is linked to u − 1 and u + 1, it is (converting to our notation) “hard to imagine a centrality f such that, given a node u (u ̸= n+1 2 ), we have f(u) ̸∈ [f(u − 1), f(u + 1)].” Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let G be a path graph where each node u such that 1 < u < n is linked to u − 1 and u + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A centrality measure f on G is said to satisfy the Roy-Tredan (RT) condition if for any node u, u ̸= n+1 2 ⇒ f(u) ∈ [f(u − 1), f(u + 1)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' path centripetal condition if the centrality of a node strictly increases with increasing shortest path distance from the nearest leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Obviously, the path centripetal condition is generally stronger than the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 5 states that the path centripetal condition is fulfilled for all centralities that satisfy axioms B and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For a path graph, any centrality measure that satisfies axioms B and E also satisfies the path centripetal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let f(·) satisfy axioms B and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Consider the path graph 1—2—· · · —n, where “—” denotes an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let 1 ≤ u = v − 1 < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Suppose that v ≤ n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then {u, v} ∈ E is a bridge and by axiom B, f(u) < f(v), since |Vu| < |Vv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Hence for such u and v, the centrality strictly increases with increasing distance from the nearest leaf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The case of u ≥ n+1 2 is considered similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Finally, if u − 1 = n − v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', u and v have the same distance from the nearest leaf, then u ∼ v and by axiom E, f(u) = f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is all the more remarkable that PageRank, one of the most popular5 centrality measures, according to Roy and Tredan, is “hard to imagine” as it violates the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the path graph 1—2—3—4—5, the PageRank centrality f PR α (·) with any parameter α violates the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Namely, f PR α (2) > f PR α (1) and f PR α (2) > f PR α (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 5According to [52], “PageRank centrality is probably the most well-known and frequently used measure.” 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the path graph 1—2—3—4—5, let us search the solution of (16) in the form x = (x1, x2, x3, x2, x1)T, where x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then the first two equations of (16) have the form α 2 x2 + 1 − α 5 (2 + 2x2 + x3) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' α + α 2 x3 + 1 − α 5 (2 + 2x2 + x3) = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The solution of this system is: x2 = 2(−α2 + 2α + 4)−1(3α + 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' x3 = 2(−α2 + 2α + 4)−1(α2 + 2α + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since both differences x2 − x1 = α(α + 4)(−α2 + 2α + 4)−1 and x2 − x3 = 2α(1 − α)(−α2 + 2α + 4)−1 are strictly positive for all α ∈ (0, 1), PageRank centralities with all appropriate parameter values violate the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Namely, x2 > x1 and x2 > x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Node 3 of the 1—2—3—4—5 path can be considered as its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It follows from the proof of Proposition 6 that PageRank never assigns maximum centrality to this center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It can be shown that PageRank centrality also violates the RT condition on paths with n > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is worth noting that if the user considers the Self-consistency or Bridge axiom as an indispensable property of a centrality measure, then this leads to a dramatic reduction of the set of candidate measures (see [1], where the corresponding reduced surveys for choosing the most appropriate centrality measure are shown in Figures 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 8 Combinations with monotonicity axioms In this section, we focus on edge-monotonicity conditions, which, as well as the Self- consistency and Bridge axioms, belongs to the class of positive responsiveness axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is proved that together with axiom E (and once S) they imply the star and path centripetal conditions and contradict axiom B, while PageRank violates axioms B, S, and Transit mono- tonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The edge-monotonicity axioms of this section involve two graphs: an original graph G0 and a graph G obtained from G0 by adding an extra edge (extra edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' These axioms restrict a universal centrality measure fG(·) operating on any connected graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The word “universal” in the formulations of Propositions 7 to 10 is implied, not explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom M (Monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Suppose that u, v ∈ V (G0), fG0(u) ≤ fG0(v), u ̸= v, and G = G0 ∪ G′ ̸= G0, where V (G′) = {v, w}, E(G′) = {{v, w}}, and w ̸= u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then fG(u) < fG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' According to Monotonicity, if u is no more central than v and a new edge not adjacent to u is attached to v, then v becomes or remains more central than u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similar axioms called Adding rank monotonicity and Strict rank-monotonicity have been proposed in [47] and [53] (for directed graphs), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='2 of Dynamic monotonic- ity in [54] is the corresponding condition for directed graphs representing paired comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Monotonicity together with axiom E imply the star condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 15 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For a star with two or more rays, any centrality measure that satisfies axioms E and M also satisfies the star condition and assigns the same centrality to all leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By E, the centrality of the two nodes of a 1-ray star is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By M, adding one more node adjacent to the “center” of the 1-ray star makes the centrality of the center greater than the same centrality of the leaves, and attaching additional leaves preserves this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Transit monotonicity is a natural strengthening of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom T (Transit monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If u, v ∈ V (G0), fG0(u) ≤ fG0(v), u ̸= v, G = G0 ∪ G′ ̸= G0, and any path in G from a node of G′ to u contains v, then fG(u) < fG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' According to Transit monotonicity, if u is no more central than v and v is a cutpoint between the new edges and u, then v becomes or remains more central than u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Together with E, Transit monotonicity implies the path centripetal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For a path graph, any centrality measure that satisfies axioms E and T also satisfies the path centripetal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By E, the conclusion holds for the 1—2 path graph on two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Assume that it holds for the path graph 1—· · · —2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , k}, f(i) ≤ f(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Attaching a new node 2k +1 and the edge {2k, 2k +1} provides the path graph 1—· · · —(2k +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since any path in the new graph from 2k + 1 to i contains i + 1, axiom T implies f(i) < f(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, the centrality of the nodes i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , k + 1} of the new graph strictly increases with the increase of the shortest path distance from the nearest leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This is also the case for the remaining nodes i ∈ {k + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , 2k + 1} by axiom E, since for them i ∼ (2k + 2 − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, the conclusion of Proposition 8 is true for the 1—· · · —(2k + 1) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Adding node 2k + 2 and edge {2k + 1, 2k + 2} to it, we similarly derive that this conclusion also holds for the resulting 1—· · · —(2k + 2) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This completes the proof by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As a corollary of Proposition 8 we obtain that PageRank centrality violates axiom T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Moreover, it does not satisfy axioms B and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The PageRank centrality with any parameter α violates axioms T, B, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' PageRank centrality violates axiom T since otherwise, by Proposition 8, PageRank, obeying axiom E, satisfies the path centripetal condition and therefore the RT condition, which is not true by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Proposition 5, axioms B and E imply the path centripetal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, PageRank centrality similarly violates axiom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the path graph 1—2—3—4—5, node 2 has neighbors 1 and 3, 3 has neighbors 2 and 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' by Proposition 6, for any α ∈ (0, 1), f PR α (2) > f PR α (1) and f PR α (4) = f PR α (2) > f PR α (3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the neighbors of 3 have higher centrality values than the corresponding neighbors of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this case, axiom S requires f PR α (3) > f PR α (2), which is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, axiom S is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' On some other peculiarities of the PageRank centrality, see [1, Section 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We conclude by showing that under Equivalence, the conjunction of M and S is incom- patible with axiom B, and so is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 16 Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If a centrality measure satisfies axioms E, M, and S or axioms E and T, then it violates axiom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let a universal centrality measure satisfy axioms E and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the graph G in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3a, fG(4) = fG(3) by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the graph G0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3b, fG0(4) = fG0(3) by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Observe that G = G0 ∪ G′, where V (G′) = {0, 1} and E(G′) = {{0, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (a) (b) Figure 3: (a) A graph G on which axiom B is incompatible with E&M&S as well as with E&T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (b) G0, a subgraph of G used in the proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Assume that this universal centrality measure satisfies axioms M and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By E and M, fG(0) = fG(1) > fG(2) = fG(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, by S, fG(4) > fG(3), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Now assume that, instead of M&S, this centrality measure satisfies axiom T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since all paths in G from 0 or 1 to 3 contain 4, by T, fG(4) > fG(3) holds, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axioms B and T are incompatible under Equivalence for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Suppose that {u, v} is a bridge in G and |Vu| = |Vv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then B implies f(u) = f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, if the restriction of E(G) to Vu is sparse, while its restriction to Vv is dense, then T requires f(u) < f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The logic of axiom S is similar to that of T in terms of transferring influences, however, S is not “grounded” as it does not require any direct effect of density on centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the conjunction M&S, axiom M provides this “grounding.” 9 Discussion Each point centrality measures some structural capital of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' According to the Bridge axiom, one end-node of a bridge is more central than the other if and only if the removal of the bridge leaves the first one in a greater (in terms of the number of nodes) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this sense, the corresponding capital is node-based: it does not depend on the density of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Self-consistency states that a node’s capital increases with the capital of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By the Monotonicity axiom, edges incident to a node contribute to its capital, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the corresponding capital is locally edge-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The conjunction of the Self-consistency and Monotonicity makes this impact of edges global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As a result, this conjunction turns out to be incompatible (under Equivalence) with the node-based Bridge axiom (Proposition 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, by the same proposition, the Bridge axiom is incompatible with the Transit monotonicity axiom, which postulates the edge nature of the capital globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 17 5 4 3 2 05 4 3 2 0 GoAn additional subject of this paper is the properties of the PageRank centrality measure related to the main topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It turns out that this measure violates most of the conditions we consider and even has a property that, according to some authors, “is hard to imag- ine” for a measure of centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The reason for this is the stochastic normalization used in PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the path graph 1—2—3—4—5 used in Proposition 6, nodes 2 and 4 have maximum PageRank centrality as they are linked to the leaves: these links receive a maxi- mum weight of 1, since normalization does not change them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This maximum weight can be interpreted as the specific importance of these links for the leaves, and not for the nodes 2 and 4, which profit from this weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is this counterintuitive normalization that violates the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The axioms of Self-consistency and Bridge are quite strong, so the adoption of either of them dramatically reduces the set of centrality measures under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This fact is used in [1], where the “culling” method for determining the most appropriate centrality measure is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This method consists in compiling and completing a survey that allows the user to find a measure that matches their underlying concept of centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the framework of this method, adopting a certain axiom results in compiling a shorter survey on the set of measures that satisfy this axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In [1], the surveys reduced to the measures obeying the Self-consistency or Bridge axioms are shown in Figures 7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Acknowledgement The author thanks Anna Khmelnitskaya and Dmitry Gubanov for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Chebotarev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Gubanov, How to choose the most appropriate centrality measure?' metadata={'source': 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