Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21)

An Axiom System for Feedback

Tomasz W ˛as∗ , Oskar Skibski University of Warsaw {t.was, o.skibski}@mimuw.edu.pl

Abstract 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), (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 ∈ 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 . 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 is defined as follows: 1 X EVv(G) = c(u, v) · EVu(G). (1) A = (au,v)u,v∈V , where au,v = c(v, u) if (v, u) ∈ E and λ (u,v)∈Γ−(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 ∈ 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 |V |). 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-, 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 e∈Γv (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 ∈ 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 ω = (ω(0), . . . , ω(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 (ω(i), ω(i + 1)) ∈ E for every i ∈ {0, . . . , k − 1} and k ≥ 1. its direct predecessors to their number. Formally, for a decay The walk is said to start at ω(0) and end at ω(k) and the factor a ∈ R≥0, Katz centrality is defined as follows: length, |ω|, of a walk is defined to be k. The set of all walks   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) + b(v). (2) − of v and v is called a successor of u. If the length of this (u,v)∈Γv (G) walk is one, i.e., (u, v) ∈ 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 ∈ 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. Formally, Katz and a subset B ⊆ A, function f with the domain restricted to prestige is defined as follows: set B will be denoted by fB and to set A \ B: by f−B. If B X c(u, v) contains one element, i.e., B = {a}, we will skip parenthesis KPv(G) = + KPu(G). (3) − degu (G) and simply write fb and f−b. Also, for a constant x, we define (u,v)∈Γv (G) x · f as follows: (x · f)(a) = x · f(a) for every a ∈ A. Fur- thermore, for two functions with possibly different domains, Similarly to Eigenvector centrality, this system of equations f : A → X, f 0 : B → X, we define f + f 0 : A ∪ B → X does not imply a unique solution. We will discuss our nor- as follows: (f + f 0)(a) = f(a) + f 0(a) if a ∈ A ∩ B, malization condition in the next section. Katz prestige is usu- (f + f 0)(a) = f(a) if a ∈ A \ B and (f + f 0)(a) = f 0(a) if ally defined only for strongly connected graphs. In our paper, we relax this assumption and consider sums of strongly con- a ∈ B \A for every a ∈ A∪B. In particular, (b−v +2bv) are KP node weights obtained from b by doubling weight of node v. nected graphs; we denote the class of all such graphs by G . 0 0 0 0 0 For two graphs, G = (V, E, b, c),G = (V ,E , b , c ) 1Note that we cannot allow for the sum of arbitrary graphs. It is with V ∩ V 0 = ∅, their sum G + G0 is defined as follows: because if one of the graphs has a smaller principle eigenvalue, then G + G0 = (V ∪ V 0,E ∪ E0, b + b0, c + c0). Eigenvector centralities of all its nodes would be zero.

444 Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21)

PageRank. Page et al. [1999] was proposed to modify Katz Hence, Katz prestige is the stationary distribution of the pro- prestige by adding a basic importance to each node. In this cess multiplied by the sum of node weights. way, for a decay factor a ∈ [0, 1), PageRank is uniquely de- Let us turn our attention to the parallel process with param- fined for all graphs as follows: eter a. The amount of entity in node v at time t equals:   t−1 a X Y a X c(u, v) a wv,G(t) = b(ω(0))· (a · c(ω(i), ω(i + 1))) . PRv (G) = a + PRu(G)+b(v). (4) ω∈Ωt(G):ω(t)=v i=0 − degu (G) (u,v)∈Γv (G) Now, if a < 1/λ, Katz centrality is the total amount of entity 2.3 Walk Interpretations of Feedback Centralities in node v through the whole process: In this section we show how PageRank’s random-walk inter- ∞ a X a pretation from [W ˛asand Skibski, 2020] can be extended for Kv (G) = wv,G(t) (7) all four feedback centralities. These interpretations result in t=0 unique definitions of Eigenvector centrality and Katz prestige This sum converges, because a < 1/λ. Now, if a = 1/λ, the consistent with Katz centrality and PageRank. sum does not converge. In such a case, Eigenvector centrality Consider a spread of some entity (money, virus, informa- can be obtained as the average amount of entity in node v: tion, probability of visit etc.) through a network. At the be- T 1/λ ginning, at time t = 0, each node has some initial amount X wv,G(t) EVv(G) = lim (8) equal to its node weight. Then, in each step the whole en- T →∞ T tity is multiplied by a scalar a, which is a parameter of the t=0 process, and then moved to direct successors. Here, we con- It can be shown that Equations (5)–(8) indeed satisfy recur- sider two variants of the process: distributed and parallel. In sive equations from Equations (1)–(4). Hence, from now on, the distributed process, the entity is spread among the outgo- we will use Equations (5)–(8) as the definitions of all four ing edges. In the parallel process, the entity is duplicated and centralities. moved along all edges. Specifically, when the entity is moved along an edge (u, v) it is either 3 Axioms • multiplied by c(u, v)/ deg+(G) (distributed process), or In this section, we present seven axioms used in our axiomatic u characterization. • multiplied by c(u, v) (parallel process). All centrality measures except for PageRank are defined As an example, consider a surfer on the World Wide Web. only for a subclass of all graphs. Hence, for them we will Node weights correspond to the probability that the surfer consider restricted versions of our axioms. Specifically, an starts surfing from a specific page and weights of edges rep- axiom restricted to class G∗ is obtained by adding an assump- resent the number of links from one page to another. A link tion that all graphs appearing in the axiom statement belong is chosen by the surfer uniformly at random which means the to G∗. In this way, we obtain a weaker version of the axiom. process is distributed. Finally, parameter a is the probability Most of our axioms are invariance axioms. They identify that the surfer stops surfing altogether. In result, the “entity” simple graph operations that do not affect centralities of all or is the probability that the surfer visits a specific page at some most nodes in a graph. The last two axioms serve as a border- time. line: they specify centralities in very simple graphs. We use Consider the distributed process with parameter a. The three axioms proposed in the axiomatization of PageRank by amount of entity in node v at time t equals: W ˛asand Skibski [2020]. Instead of the remaining three ax- ioms, we use Locality and Node Combination which are more t−1 meaningful for considered classes of graphs. a X Y a · c(ω(i), ω(i + 1)) pv,G(t) = b(ω(0)) · + . Before we proceed, let us introduce an operation of propor- degω(i)(G) ω∈Ωt(G):ω(t)=v i=0 tional combining of two nodes used in one of the axioms. Pro- portional combining differs from a simple merging of nodes Now, if a < 1, PageRank is the total amount of entity in node as it preserves the significance of outgoing edges. Take a v through the whole process: centrality measure F , graph G = (V, E, b, c) and two nodes ∞ F u, w ∈ V . Graph Cu→w(G) is a graph obtained in two steps: a X a PRv (G) = pv,G(t) (5) • scaling weights of outgoing edges of u and w propor- t=0 tionally to their centralities: multiplying weights of out- We know that this sum converges, because a < 1. Now, if going edges of u by Fu(G)/(Fu(G) + Fw(G)) and w a = 1, the sum does not converge, as the amount of entity in by Fw(G)/(Fu(G) + Fw(G)); and the network does not change over time. In such a case we can • merging node u into node w, i.e., deleting node u, trans- look at the average amount of entity in node v. In this way ferring its incoming and outgoing edges to node w and we get the definition of Katz prestige: adding the weight of node u to node w. See Figure 1 for an illustration. T p1 (t) X v,G We are now ready to present the first three axioms satisfied KPv(G) = lim (6) T →∞ T t=0 by all feedback centralities.

445 Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21)

a c 1 1 2 xa c xa c 3 a 3 b + 3 c b 1 xb xb d u w 3 d w xd u w d u w e f e + g f e f e/x f g g g

Figure 1: Graph G (on the left) and the corresponding graph Figure 2: Graphs considered in Edge Multiplication (on the left) and F Cu→w(G) (on the right) assuming Fu(G) = 1 and Fw(G) = 2. Edge Compensation (on the right) obtained from G from Figure 1.

Locality (LOC): For every graph G = (V, E, b, c) [ ] 0 0 0 0 0 0 Edge Multiplication, proposed in W ˛asand Skibski, 2020 , and graph G = (V ,E , b , c ) s.t. V ∩ V = ∅: states that multiplying weights of all outgoing edges of node 0 Fv(G + G ) = Fv(G) for v ∈ V. u by some constant does not affect centralities. This means that the absolute weight of edges does not matter, as long as Edge Deletion (ED): For every graph G = the proportion to other outgoing edges of a node is the same. (V, E, b, c) and edge (u, w) ∈ E: This property is satisfied by both PageRank and Katz pres-

Fv(V,E\{(u, w)}, b, c) = Fv(G) for v ∈ V \S(u). tige. However, it is not satisfied by Eigenvector and Katz centralities, as it increase the importance of modified edges x Node Combination (NC): For every graph G = times. For them, we propose a similar axiom: Edge Compen- + (V, E, b, c) and nodes u, w ∈ V s.t. degu (G) = sation. In this axiom, not only weights of outgoing edges of + + degw(G) = degs (G) for every s ∈ S(u) ∪ S(w): u are multiplied by a constant, but at the same time weights of incoming edges and the node itself are divided by the same F (CF (G)) = F (G) for v ∈ V \{u, w} v u→w v constant. Edge Compensation states that this operation de- F and Fw(Cu→w(G)) = Fu(G) + Fw(G). creases the importance of u x times, but at the same time Locality and Edge Deletion are standard axioms from the does not affect the importance of other nodes. See Figure 2 literature. Locality, proposed in [Skibski et al., 2019], states for an illustration. that the centrality of a node depends solely on the part of the Finally, the last two axioms concern simple borderline graph a node is connected to. In other words, removing part cases. of the graph not connected to a node does not affect its cen- Baseline (BL): For every graph G = (V, E, b, c) trality. Edge Deletion, proposed in [W ˛asand Skibski, 2020] and an isolated node v ∈ V it holds F (G) = b(v). for PageRank, states that removing an edge from the graph v does not affect nodes which cannot be reached from the start Cycle (CY): For every out-regular cycle graph G = P of the edge. Node Combination is a new axiom. Assume (V, E, b, c) it holds Fv(G) = u∈V b(u)/|V | for two nodes u, w ∈ V and their successors have the same out- every v ∈ V . degree, but possibly different centralities. Node Combination [ ] states that in a graph obtained from proportional combining Baseline, proposed in W ˛asand Skibski, 2020 , states that of u into w, the centrality of w is the sum of centralities of the centrality of a node with no incident edges is equal to both nodes and centralities of other nodes do not change. This its baseline importance: its node weight. Baseline is satis- property is characteristic for feedback centralities which as- fied by PageRank and Katz centrality. However, it does not sociate a benefit from an incoming edge with the importance make sense for strongly connected graphs. Cycle, proposed of a node this edge comes from. We note that PageRank, Katz as the borderline case for Eigenvector centrality and Katz prestige and Katz centrality satisfy also the axiom without the prestige, considers the simplest strongly connected graph: a assumption about equal out-degrees of successors. However, cycle. Specifically, if weight of all edges in a cycle graph are it is necessary for Eigenvector centrality. equal, then centralities of all nodes are also equal. Moreover, Our next two axioms concern a modification of one node: Cycle normalizes the sum of centralities to be equal to the its weight and weights of its incident edges. sum of node weights. See Figure 3 for an illustration. As we will show, these seven axioms are enough to obtain Edge Multiplication (EM): For every graph G = the axiomatizations of all four feedback centralities. (V, E, b, c), node u ∈ V , constant x > 0:

Fv(V, E, b, c + +x·c + ) = Fv(G) for v ∈ V. x −Γu (G) Γu (G) v v Edge Compensation (EC): For every graph G = x x (V, E, b, c), node u ∈ V , constant x > 0, 0 0 b = b−u + bu/x and c = c  + −Γu (G)\{(u,u)} x x c − /x + c + · x: Γu (G)\{(u,u)}) Γu (G)\{(u,u)} 0 0 Fv(V, E, b , c ) = Fv(G) for v ∈ V \{u} Figure 3: Graphs considered in Baseline (on the left) and Cycle (on 0 0 the right). and Fu(V, E, b , c ) = Fu(G)/x.

446 Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21)

x 1 G 2 G0 13 G00 G∗ x x x x 0 x 0 v1 v2 v1 v2 v1 v2 v1 v2 v3 v4 v2 v3 x 1 3 2 x 13 13 x x x x 2 2 2 2 000 0 v5 v3 v5 13 13 v3 v5 v3 v4 v4 x x 1 3 x 0 00 00 00 x v4 13 v4 13 v4 v5 v v v v x 1 x 4 x 3 x 2

Figure 4: The construction of a cycle graph G∗ from which G can be obtained using proportional combining.

4 Main Results Clearly, the total impact of outgoing edges of every node is In this section, we present our main results: We show that equal to its Katz prestige. Also, from the Katz prestige recur- if a centrality satisfies three general axioms (LOC, ED and sive equation (Equation (3)), we see that the total impact of NC), one of the one-node-modification axioms (EM or EC) incoming edges of every node is also equal to its Katz pres- and one of the borderline axioms (BL or CY) then it must be tige. We will use this fact later on. What is also important im- one of the four feedback centralities. pacts are rational; let N be the least common multiple of all denominators of edge impacts. For graph G we have N = 13. The full proofs can be found in the extended version of the 00 paper.2 Here, we present the main ideas behind them. Based on the notion of impact we create a G (a First, consider Katz prestige and Eigenvector centrality. graph in which multiple edges exist between a pair of nodes). This multigraph is obtained by replacing edge (u, v) from the KP Theorem 1. A centrality measure defined on G satisfies original graph by N · I(u, v) edges. For example, in our sam- LOC, ED, NC, EM and CY if and only if it is Katz prestige ple graph edge (v4, v1) with impact I(v4, v1) = 2/13 is re- (Equation (6)). placed by two edges from v4 to v1. Now, the key observation Theorem 2. A centrality measure defined on GEV satisfies is that every node in the multigraph G00 has the same number LOC, ED, NC, EC and CY if and only if it is Eigenvector of incoming and outgoing edges (as the total impact of its in- centrality (Equation (8)). coming edges equals the total impact of its outgoing edges). Hence, from Euler’s theorem, we know that in G00 there ex- We begin by showing that Katz prestige and Eigenvector ists an Euler cycle—a walk that visits every edge exactly once centrality are equal for strongly connected out-regular graphs. and is a cycle: starts and ends in the same node. Then, we prove that both centralities satisfy the correspond- Graph G∗ in Figure 4 is a cycle graph corresponding to ing axioms. Hence, it remains to prove the uniqueness of both 00 0 some Euler cycle in graph G . Here, nodes vi, vi,... corre- axiomatizations. spond to node v in G00. We define node weights in a way Let F be a centrality measure defined on GEV or GKP that i that nodes corresponding to node vi sum up to b(vi). The satisfied LOC, ED, NC and CY. First, we show that F is equal construction of the cycle graph is complete. to both centralities for strongly connected out-regular graphs. Let us determine the number of nodes in G∗. In graph G00 We divide it into two steps: ∗ node vi has N ·KPvi (G) outgoing edges. Hence, in G there

• First, we consider strongly connected out-regular graphs are N · KPvi (G) nodes corresponding to node vi. Since the with rational proportion of weights of every two edges. sum of Katz prestige of all nodes in G is equal to the sum of node weights, i.e., 1, we get that there are N nodes in graph • Then, we consider arbitrary strongly connected out- ∗ ∗ regular graphs. G . In our example, there are indeed 13 nodes in G and 4 nodes correspond to node v4. These proofs are based on the following observation: for ev- Now, from CY, we know that according to F every node ery strongly connected out-regular graph G with rational pro- in G∗ has the same centrality equal to 1/N. It is easy to ver- portion of weights of its edges it is possible to construct a ify that if we merge using proportional combining for every cycle graph G∗, from which G can be obtained using pro- node vi all nodes corresponding to vi we will obtain graph portional combining. As a result, if some centrality measure G. Based on NC this implies that node vi has centrality in G satisfies CY and NC, then it is uniquely defined on graph G. equal to N · KPvi (G) · 1/N = KPvi (G) which concludes We will present this construction on the graph G from Fig- the proof. ure 4. Node weights are arbitrary, but we assume they sum So far, we considered out-regular graphs and four axioms: up to 1. This graph is x-out-regular. In such graphs, it can be LOC, ED, NC and CY. Here is where the proof splits: shown that Katz prestige of any node is rational; in this graph 2 3 • If F satisfies EM, then it is equal to Katz prestige for we have: KPv (G) = , KPv (G) = KPv (G) = , 1 13 2 3 13 every graph; it is easy to see that every graph can be KP (G) = 4 and KP (G) = 1 . v4 13 v5 13 made out-regular if we divide weights of outgoing edges Now, let us define an impact of an edge (u, v), de- of every node by its outdegree. noted by I(u, v), as Katz prestige of node u multiplied by + 0 • If F satisfies EC, then it is equal to Eigenvector cen- c(u, v)/ degu (G). Impacts of edges are depicted in graph G trality for every graph; here, more detail analysis is re- in Figure 4. For example, I(v4, v1) = Kv4 (G) · 1/2 = 2/13. quired, as the EC operation changes weights of both out- 2https://arxiv.org/pdf/2105.03146.pdf going and incoming edges.

447 Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21)

v v v a a a v0 a v0 a v0 u b u u0 b u u0 b u0 b u0 a+b+c a+b+c b+c c G c G0 c G00 c G∗ w w

+ Figure 5: A sequence of graphs that shows the profit from edge (u, v) for node v is equal to pF (Fu(G), c(u, v), degu (G)).

Now, let us turn our attention to PageRank and Katz cen- Let us illustrate the scheme of this proof on graph G from trality. We have the following results: Figure 5. Consider node v and pick one incoming edge, say 0 Theorem 3. A centrality measure defined on G satisfies LOC, (u, v). First, we create graph G in which we extract from 0 ED, NC, EM and BL if and only if it is PageRank for some node u all its outgoing edges and attach them to a new node u decay factor a (Equation (5)). with weight Fu(G). To keep the out-degree of u unchanged, K(a) we add a new node w with an edge from u. Using NC, it can Theorem 4. A centrality measure defined on G satisfies 0 0 be shown that F (G ) = F (G) and F 0 (G ) = F (G). LOC, ED, NC, EC and BL if and only if it is Katz centrality v v u u In the second step, we split v into two nodes with the same for some decay factor a (Equation (7)). outgoing edges, but disjoint incoming edges: v0 has one edge It is easy to check that both centralities satisfy their corre- (u0, v0), and v has the remaining edges. From NC we know 00 00 sponding axioms. Let us focus on the proof of uniqueness. that Fv(G) = Fv(G ) + Fv0 (G ). Since v has less incoming The key role in our proof will be played by semi-out- edges in G00 than in G, we can use the inductive assumption. regular graphs which is a class of graphs in which all nodes Hence, it remains to determine the centrality of v0 in G00. except for sinks (nodes with no outgoing edges) have equal To this end, observe that only u0 is the predecessor of v0. out-degrees. Most lemmas described below applies only to Hence, from ED, removing all outgoing edges of other nodes semi-out-regular graphs. does not affect the centrality of v. Also, from LOC, remov- K(a) Let F be a centrality measure defined on G or G that ing nodes other than u0 and its direct predecessors does not satisfies LOC, ED, NC, BL and EM or EC. First, we show change the centrality of v. Hence, we can focus on graph G∗. several technical properties of F that we use later in the Using NC we can merge all of the nodes in G∗ except for u0 proof: Node Combination generalized to all semi-out-regular and v0. In this way, we get a graph with three nodes and two graphs, minimal centrality equal to the node weight, linearity 00 + edges, so we have Fv0 (G ) = pF (Fu(G), c(u, v), degu (G)) with respect to node weights and existence of a constant aF what we needed to prove. such that Fv({u, v}, {(u, v)}, b, c) = aF · b(u) if b(v) = 0 Next, we relax the assumption that the node does not have and c(u, v) = 1. a self-. Finally, using EM or EC we get uniqueness for Next, we show that the profit from edge (u, v) for node v PageRank and Katz centrality. depends only on its weight and the centrality and out-degree of node u; moreover, if F satisfies EM or EC, then this profit is equal to the profit in PageRank or Katz centrality. 5 Conclusions To this end, for x, y, z > 0, y ≤ z, we define a profit We proposed the first joint axiomatization of four classic function pF (x, y, z) as follows: feedback centralities: Eigenvector centrality, Katz central- x y,z ity, Katz prestige and PageRank. We used seven axioms and pF (x, y, z) = Fv({u, v, w}, {(u, v), (u, w)}, b , c ), proved that each centrality measure is uniquely characterized where bx(u) = x, bx(v) = bx(w) = 0, cy,z(u, v) = y and y,z by a set of five axioms. Our axiomatization highlights the c (u, w) = z − y (if z = y, then we remove edge (u, w) similarities and differences between these measures which from the graph). To put in words, value pF (x, y, z) is the help in making an informed choice of a centrality measure profit from an incoming edge with weight y that starts in node for a specific application at hand. with centrality x and out-degree z in the smallest such graph There are many possible directions of further research. It possible: with three nodes and two edges. It is easy to check would be interesting to extend our axiomatization to Degree that if F satisfies EM, then pF (x, y, z) = aF · x · y/z as centrality and Beta measure which constitute the borderline PageRank and if F satisfies EC, then pF (x, y, z) = aF · x · y cases of feedback centralities. Another direction is consider- as Katz centrality. ing centrality measures based on random walks, such as Ran- Now, it remains to prove that the profit from any edge + dom Walk Closeness or Decay. Also, several axioms consid- (u, v) equals pF (Fu, c(u, v), degu (G)). More precisely, we ered in our paper can form a basis for the first axiomatization prove that: of -based centrality measures for directed graphs. X + Fv(G) = b(v) + pF (Fu(G), c(u, v), degu (G)). + (u,v)∈Γu (G) Acknowledgments First, we consider nodes without a self-loop only. We prove This work was supported by the Polish National Science Cen- the thesis by induction over the number of incoming edges. tre grant 2018/31/B/ST6/03201.

448 Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence (IJCAI-21)

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