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An Axiom System for Feedback Centralities Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21) An Axiom System for Feedback Centralities Tomasz W ˛as∗ , Oskar Skibski University of Warsaw {t.was, o.skibski}@mimuw.edu.pl Abstract Centrality General axioms EC _ EM CY _ BL Eigenvector LOC ED NC EC CY In recent years, the axiomatic approach to central- Katz LOC ED NC EC BL ity measures has attracted attention in the literature. Katz prestige LOC ED NC EM CY However, most papers propose a collection of ax- PageRank LOC ED NC EM BL ioms dedicated to one or two considered centrality measures. In result, it is hard to capture the differ- Table 1: Our characterizations based on 7 axioms: Locality (LOC), ences and similarities between various measures. Edge Deletion (ED), Node Combination (NC), Edge Compensation In this paper, we propose an axiom system for four (EC), Edge Multiplication (EM), Cycle (CY) and Baseline (BL). classic feedback centralities: Eigenvector central- ity, Katz centrality, Katz prestige and PageRank. ties, while based on the same principle, differ in details which We prove that each of these four centrality mea- leads to diverse results and often opposite conclusions. sures can be uniquely characterized with a subset In recent years, the axiomatic approach has attracted at- of our axioms. Our system is the first one in the lit- tention in the literature [Boldi and Vigna, 2014; Bloch et al., erature that considers all four feedback centralities. 2016]. This approach serves as a method to build theoret- ical foundations of centrality measures and to help in mak- ing an informed choice of a measure for an application at 1 Introduction hand. In the axiomatic approach, the measure is character- The question how to assess the importance of a node in a ized by a set of simple properties, called axioms. A number network has puzzled scientists for decades [Boldi and Vigna, of papers use the axiomatic approach to characterize feedback 2014]. While the first methods, called centrality measures, centralities. Katz prestige was considered by Palacios-Huerta have been proposed in social science in 1950s, in the last two and Volij [2004] and by Altman and Tennenholtz [2005]. decades centrality analysis has become an actively developed Kitti [2016] proposed algebraic axiomatization of Eigenvec- field in computer science, physics and biology [Brandes and tor centrality. Dequiedt and Zenou [2014] and W ˛asand Skib- Erlebach, 2005; Newman, 2005]. As a result, with a plethora ski [2018] proposed joint axiomatization of Eigenvector and of measures proposed in the literature, the choice of a central- Katz centralities. Recently, W ˛asand Skibski [2020] proposed ity measure is harder than ever. an axiomatization of PageRank. Feedback centralities form an especially appealing class of While these results are a step in the right direction, most centrality measures. These measures assess the importance of papers focus only on one or two feedback centralities. In re- a node recursively by looking at the importance of its neigh- sult, each paper proposes a collection of axioms dedicated for bors or, in directed graphs, direct predecessors. Such an as- the considered centrality, but poorly fitted to other measures. sumption is desirable in many settings, e.g., in citation net- As a consequence, these characterizations based on different works where a citation from a better journal value more [Pin- axioms do not help much in capturing the differences and ski and Narin, 1976] or in the World Wide Web where a link similarities between various centrality concepts. from a popular website can significantly increase the popu- In this paper, we propose an axiom system for four classic larity of our page [Kleinberg, 1999]. feedback centralities. Our system consists of seven axioms. Chronologically the first feedback centralities were Katz Locality, Edge Deletion, Node Combination are general ax- centrality and Katz prestige, proposed by Katz [1953]. Ar- ioms satisfied by all four centralities. Edge Compensation guably, the most classic feedback centrality is Eigenvector and Edge Multiplication concern modification of one node centrality proposed by Bonacich [1972]. In turn, the most and its incident edges. Cycle and Baseline specify centrali- popular feedback centrality is PageRank designed for Google ties in simple borderline graphs. For this set of axioms, we search engine [Page et al., 1999]. These four classic centrali- show that each of four feedback centralities is uniquely char- acterized by a set of 5 axioms: 3 general ones, one one-node- ∗Contact Author modification axiom and one borderline axiom (see Table 1). 443 Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21) 2 Preliminaries 2.2 Feedback Centralities A centrality measure is a function F that given a graph G = In this paper, we consider directed weighted graphs with node (V; E; b; c) and a node v 2 V returns a real value, denoted by weights and possible self-loops. Fv(G). This value, called a centrality of a node, is assumed to be non-negative and represents the importance of node v in 2.1 Graphs graph G. The class of feedback centralities aims to assess the im- A graph is a quadruple, G = (V; E; b; c), where V is the set portance of a node by looking at the importance of its direct of nodes, E is the set of ordered pairs of nodes called edges predecessors. We consider four classic feedback centralities. and b and c are node and edge weights: b : V ! R≥0 and Eigenvector centrality. According to Eigenvector central- c : E ! R>0. We assume that node weights are non-negative and edge weights are positive. The set of all possible graphs ity [Bonacich, 1972], the importance of a node is proportional is denoted by G. to the total importance of its direct predecessors: For a graph G, the adjacency matrix is defined as follows: 1 X EVv(G) = c(u; v) · EVu(G): (1) A = (au;v)u;v2V , where au;v = c(v; u) if (v; u) 2 E and λ (u;v)2Γ−(G) au;v = 0, otherwise. A real value r is an eigenvalue of a v V matrix A if there exists a non-zero vector x 2 R such that This system of recursive equations has multiple solutions. Ax = rx; such vector x is called an eigenvector. The princi- Hence, some additional normalization condition is usually pal eigenvalue, denoted by λ, is the largest eigenvalue. assumed to make a solution unique (e.g., the sum of central- An edge (u; v) is an outgoing edge for node u and an in- ities of all nodes is assumed to be 1 or jV j). In this paper, coming edge for node v. For v, the set of its incoming edges is we use a normalization more consistent with other feedback − + denoted by Γv (G) and outgoing edges by Γv (G). The total centralities—we discuss it in the next section. The Eigenvec- weight of outgoing edges, called the out-degree, is denoted tor centrality is usually defined only for strongly connected + + P by deg (G): deg (G) = + c(e). For any x, graph graphs. We relax this assumption by considering also sums v v e2Γv (G) G is x-out-regular if the out-degree of every node equals x: of strongly connected graphs with the same principal eigen- + 1 EV degv (G) = x for every v 2 V . A graph is out-regular if it is value. We denote the class of all such graphs by G . x-out-regular for some x. Katz centrality. Katz [1953] proposed an alternative to A walk is a sequence of nodes ! = (!(0);:::;!(k)) such Eigenvector centrality that adds a basic importance to each that every two consecutive nodes are connected by an edge: node. This shifts the emphasis from the total importance of (!(i);!(i + 1)) 2 E for every i 2 f0; : : : ; k − 1g and k ≥ 1. its direct predecessors to their number. Formally, for a decay The walk is said to start at !(0) and end at !(k) and the factor a 2 R≥0, Katz centrality is defined as follows: length, j!j, of a walk is defined to be k. The set of all walks 0 1 of length k will be denoted by Ωk(G). If there exists a walk a X a that starts in u and ends in v, then u is called a predecessor Kv (G) = a @ c(u; v) · Ku(G)A + b(v): (2) − of v and v is called a successor of u. If the length of this (u;v)2Γv (G) walk is one, i.e., (u; v) 2 E, then nodes are direct predeces- v v For a fixed a, Katz centrality is uniquely defined for graphs sors/successors. For node , the set of predecessors of is K(a) denoted by P (v) and the set of successors of v by S(v). The with λ < 1=a. We denote the class of such graphs by G . graph is strongly connected if there exists a walk between ev- Katz prestige. In Eigenvector and Katz centralities, the ery two nodes, i.e., if S(v) = V for every node v 2 V . whole importance of a node is “copied” to all its direct suc- A strongly connected graph such that every node has ex- cessors. In turn, in Katz prestige [Katz, 1953], also called actly one outgoing edge is called a cycle graph. Seeley index or simplified PageRank, a node splits its impor- Let us introduce some shorthand notation that we will use tance equally among its successors. Hence, the importance of throughout the paper. For an arbitrary function f : A ! X predecessors is divided by their out-degree.
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