Measuring network centrality using hypergraphs

Measuring Network Centrality Using Hypergraphs Sanjukta Roy Balaraman Ravindran Chennai Mathematical Institute Indian Institute of Technology Madras [email protected] ABSTRACT Networks abstracted as graph lose some information related to the super-dyadic relation among the nodes. We find natural occurrence of hyperedges in co-authorship, co-citation, social networks, e-mail networks, weblog networks etc. Treating these networks as hypergraph preserves the super-dyadic relations. But the primal graph or Gaifmann graph associated with hypergraphs converts each hyperedge to a clique losing again the n-ary relationship among nodes. We aim to measure Shapley Value based centrality on these networks without losing the super-dyadic information. For this purpose, we use co-operative games on single graph representation of a hypergraph such that Shapley value can be computed efficiently[1]. We propose several methods to generate simpler graphs from hypergraphs and study the efficacy of the centrality scores computed on these constructions. Categories and Subject Descriptors G.2.2 [Hypergraphs]: Network Centrality Keywords Hypergraph, Shapley value, Centrality 1. INTRODUCTION The study of networks represents an important area of multidisciplinary research involving Physics, Mathematics, Chemistry, Biology, Social sciences, and Information sciences. These systems are commonly represented using simple graphs in which the nodes represent the objects under investigation, e.g., people, proteins, molecules, computer systems etc., and the edges represent interactions between the nodes. But, there are occasions in which interactions involve more than two actors. For example, deliberations that take place in a committee are multi-way interactions [2]. In such circumstances, using simple graphs to represent complex networks does not provide complete description of the Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. CODS ’15, March 18 - 21, 2015, Bangalore, India Copyright 2015 ACM 978-1-4503-3436-5/15/03 $15.00. http://dx.doi.org/10.1145/2732587.2732595 [email protected] real- world systems. For example, if we model a collaboration network as a simple graph, we would have to reduce the interaction into a set of two way interactions, and if there are multiple actors collaborating on the same project, the representation will be a clique on all those actors. As we will show, we lose information about the set of actors collaborating on the same project in the clique construction. A natural way to model these interaction is through a hypergraph. In a simple graph, an edge is represented by a pair of vertices, whereas an hyperedge is a non-empty subset of vertices. Multi-way interactions, known as super-dyadic relations, characterize many real world applications and there has been much interest lately in modeling such relations, especially using hypergraphs [3, 4, 5, 6]. While there have been several approaches proposed for using various graph properties, such as connectivity, centrality, modularity, etc., for modeling interactions between actors, the extensions of these notions to hypergraphs is still an active area of research. The typical approach is to construct a simple graph from the hypergraph and compute the properties on the regular graph. The approaches differ in the construction of the simple graph. A hypergraph of tag co-occurrences has been used for detecting communities using betweenness centrality [3]. Puzis et al. [4] used betweenness as a centrality measure on hypergraphs and gave an efficient approach using Brandes algorithm. Zhou et al. [7] pointed out the information loss occurs when hypergraphs are treated as simple graphs and utilised hypergraphs to device clustering technique. Katherine Faust [5] used non-dyadic affiliation relationships which allowed to quantify the centrality of a subset of actors or a subset of events. She calculated an actor’s centrality as a function of the centrality of hyperedges of events to which the actor belongs and calculated the centrality of events as a function of centrality of the actors. Hypergraphs have been used to understand team assembly mechanisms to determine collaboration [8, 9, 10]. Jain et al. [11] studied Indian Rail network using hypergraph. Of particular interest to us in this work are centrality measures. A centrality measure identifies nodes that are in the ‘center’of a network. In practice, identifying exactly what we mean by ‘center’depends on the application of interest. In this work, we mainly look at this from the view point of information propagation. Independent Cascade and Linear Threshold are the two information diffusion models proposed in [12]. To find a seed set of k nodes that maximizes influence propagation is in general a hard problem and [12] gave an (1 − 1/e − ǫ) approximation under the assumption that the activation functions are submodular. We use the diffusion models of [12] to define measures of effectiveness of the different centrality scores that we investigate. There have been some earlier attempts at computing centrality scores on hypergraphs. Faust [5] looked at a hypergraph as a bipartite graph[6] between actors and events when computing closeness centrality. Puzis et al.[4] essentially converted each hyperedge to a clique and computed betweenness centrality on the modified graph efficiently. Representing each hyperedge as a clique converts the hypergraph to a general graph. We call this conversion as clique construction. Clique construction is the method that is widely used for computations on hypergraph. But in this work we argue that the clique construction loses the non-dyadic information represented by the hyperedges. Another, slightly different way of looking at the hypergraph is using the edge multiplicity between two nodes as the weight on the edge between those nodes in the clique construction. This conversion we call as wt-clique construction. The following is a toy example where we look at a small network so that we can easily measure influence of each node. This example shows that the traditional computations on hypergraphs indeed give away some information. Example 1. Let H1 = (V, E1 ) be the hypergraph with V = {i, j, k, l} and E1 = {{i, j}, {i, k}, {j, k}, {i, k, l}, {i, j, k}, {j, l}} and H2 = (V, E2 ) be another hypergraph with E2 = {{i, j, k}, {i, j, k, l}, {i, k}}. These two hypergraphs can be thought of as two collaboration networks where i, j, k, l are authors and each edge represents a document the authors have written. For example, in the first network (given by H1 ), the first edge i, j represents a document written by author i and j together. Notice here, in both the networks each author has written a document with every one else but the number of times two authors have written a document together is different. For example, in the second network (given by H2 ) i has written one document with l and three documents with k. Let us assume we want to measure the centrality of nodes in these two networks to calculate fluency of information propagation between two authors. Our philosophy is that it must be easy to propagate information from k to i than from l to i. If we convert the hyperedges of H1 and H2 to clique (clique construction) for both H1 and H2 , we get the same graph, a 4-clique. Apparently, we lose the edge multiplicities. If we put the multiplicities as edge weights in the clique construction (wt-clique construction), both H1 and H2 again translate into same graph as in figure 1, though the above two hypergraphs are not the same. Here we propose three different reduced representations of hypergraph which use the non-dyadic information of the hyperedges efficiently and show how ranking of nodes changes according to their influence on the network. As stated before, finding maximum spread is a hard problem. [13] uses a game theoretic approach to find the ranking of the nodes which is comparable to that given by the natural greedy algorithm[12] but is much more time efficient that the greedy algorithm [1]. It captures the marginal contribution that each player makes to the dynamics of the game. Con- i 2 j 1 1 2 3 l 1 k Figure 1: wt-clique construction for H1 and H2 : W (H1 ) and W (H2 ) are the same sider a cooperative game with transferable utilities, (N, ν), where N = 1, 2, . . . , n is the set of players and ν : 2N → R is a characteristic function, that assigns a value to each coalition (subset of N ) , with ν(φ) = 0. The Shapley value (SV) of player i ∈ N , is 1 X ν(Si (π) ∪ {i}) − ν(Si (π)) (1) SV (i) = . n! π∈Ω where Ω is the set of all permutations over N , and Si (π) is the set of players appearing before the ith player in permutation π. We use the same game (Topk Game) that is described in [13]. Then we extend the game for multi-hop on weighted graphs as done in Topk Game with weights in [1]. We measure influence of a node using Shapley Value (SV) of the node as in [13, 1]. This computation is fast on graphs. As shown in [1], it is linear in number of edges and vertices. We give an equally efficient algorithm for computing ranking on hypergraphs. We compute these ranking using clique construction; wt-clique construction and three of our proposed methods and compare the results using diffusion models. The main contributions of this work are: (1) Different methods of reducing hypergraphs to simpler forms for the purposes of centrality computations. These reductions could possibly yield more efficient mechanisms for computing other network properties; (2) Computation of game-theoretic centrality measures on super-dyadic relations. These have been shown to be efficient to compute on simple graphs. And, this extension enables us to compute centrality scores on large super-dyadic networks; and (3) Empirical study of how centrality scores are distributed over the nodes of the network under different graph constructions. We establish that traditional constructions of simple graphs from hypergraphs lead to inefficient centrality distributions. We also introduce the notion of dominance for comparing various centrality measures that allows us to identify which measures are truly different in their ranking of the nodes. 2. REDUCED REPRESENTATION OF HYPERGRAPH We already talked about two types of reduced graph formation from hypergraphs viz, clique construction and wtclique construction. We give three other constructions which, we will show, do better in terms of centrality measurement based on SV. For completeness we briefly describe clique construction and wt-clique construction as well. 2.1 Clique Construction This is also known as primal of the hypergraph. Let H = (V (H), E(H)) be a hypergraph with the vertex set V (H) and the edge set E(H). A hyperedge e ∈ E(H) is a subset of V (H). The primal graph, also called the Gaifmann graph is defined as follows, P (H) = (V ′ , E ′ ) such that V ′ = V (H) and E ′ = {{u, v} | e ∈ E(H) and u, v ∈ e} This is also known as 2-section of hypergraph. Hypergraphs are represented as its primal graph in many problems like partition connectedness, betweenness computation etc. 2.2 Wt-Clique Construction Let H = (V (H), E(H)) be a hypergraph with the vertex set V (H) and the edge set E(H). we say, u, v ∈ V (H) share x hyperedges if there are x hyperedges, e1 , e2 , . . . , ex such that ei ∈ E(H) and u, v ∈ ei ∀i ∈ {1, 2, . . . , x}. Let W (H) be the reduced representation of H using wtclique construction. W (H) = (V ′ , E ′ , w) such that V ′ = V (H) and E ′ = {{u, v} | e ∈ E(H) and u, v ∈ e} and w : E ′ → R with w(u, v) = number of hyperedges u, v share. A similar kind of approach is used in [4] for computing betweenness centrality on hypergraphs where they call it one mode projection of hypergraph. 2.3 Line Graphs This is also known as the dual of the hypergraph. Let L(H) be the line graph of the hypergraph, H. L(H) = (V ′ , E ′ ), where V ′ = E(H) and E ′ = {{e1 , e2 } | e1 , e2 ∈ E(H), e1 ∩ e2 6= φ} 2.3.1 Why Line Graph We expect the line graph will give a better centrality measure than the clique construction on hypergraph. This is because, in clique construction we lose some information like edge multiplicities of the vertices. Observe, even in wt-clique construction where edge multiplicities do appear as weights on the edges, we do not consider whether two edges are contribution of two different hyperedges or the same hyperedge. For instance in Example 1, consider the edges (j, l) and (i, j) in the weighted clique construction of graphs H1 and H2 that is W (H1 ) and W (H2 ) respectively. The hyperedge {i, j, k, l} ∈ E2 is responsible for the edge between j and l in W (H2 ). The same hyperedge adds to the weight of the edge between i and j in W (H2 ). But in W (H1 ), (j, l) is present because of the hyperedge {j, l} ∈ E1 . And the same hyperedge is not responsible for the edge (i, j) ∈ W (H1 ). To summarise, it can not be said from W (H1 ) and W (H2 ) whether the same hyperedge is responsible for two different edges in the wt-clique. The above observation is crucial in centrality measurement because from W (H1 ) and W (H2 ) it can be said that i, j and k are equally central for both H1 and H2 . Though this can be true for H1 but for H2 , it can be clearly seen that nodes i and k appear more often than j.Later our results on line graph shows that this is correctly predicted by the computation using L(H). Let us now look at the line graph construction of the two hypergraphs of Example 1. This gives us some insight why Shapley Value measure computed on hypergraph using line graph can be more accurate compared to the one computed using clique or wt-clique construction. Figure 2 shows the line graphs for H1 and H2 . 2.4 Weighted Line Graph We also look at weighted line graph where weight on each edge is the size of the intersection between the two hyperedges. We compare the centrality measure with other reduced representations of hypergraph and show which gives a better measure for centrality. Let Lw (H) be the weighted line graph for H. Lw (H) = (V ′ , E ′ , w), where V ′ = E(H) , E ′ = {{e1 , e2 } | e1 , e2 ∈ E(H), e1 ∩ e2 6= φ} and w : E ′ → R, w is defined as follows. 1 Weight of an edge {e, e′ } ∈ Lw (H) is w({e, e′ }) = |e∩e ′| , ′ where e, e ∈ E. This representation keeps the hyperedges intact as in the line graph representation. Also it uses the information that how many nodes are common between two hyperedges. Two hyperedges which has more common nodes are closer. That is, it is easier to pass some information since more nodes (rational agents) can be involved in passing the information. 2.5 Multi-Graph This is one variant of clique construction for hypergraph. Here, instead of putting the edge multiplicities as edge weights, multiple edges are kept between two nodes. Define multi-graph of hypergraph, H, M (H) = (V ′ , E ′ ) such that V ′ = V (H) and E ′ is a multiset, E ′ = {{u, v} | e ∈ E(H) and u, v ∈ e}. If two vertices u, v ∈ V (H) share x hyperedges in H then the multi-graph of H, M (H) has x edges between u and v. Though this construction is similar to wt-clique construction, for example, both H1 , H2 of example 1 have same multi-graph construction, we will show it fares better in SV computation using the definition of fringe[1] (value of a coalition) given in the next section. Proof of the next claim given in the following section gives an intuitive idea why it performs better than W (H) or sometimes even L(H). Claim 1. If u, v belongs to many hyperedges then they marginally affect each other more in case of M (H) compared to W (H). 3. SHAPLEY VALUE BASED CENTRALITY MEASURE ON HYPERGRAPH We try to find the centrality measure on hypergraph using the concept of Shapley Value. But we define the game on L(H) instead of H and find a ranking for the nodes of L(H). That is, we first measure the centrality of each of the hyperedges than the vertices then translate it for the vertices of H. We have to maintain collective rationality and individual rationality when converting the Shapley values of the hyperedges to vertices of hypergraph. Let xi be the value player i gets. Then, |N | X xi = ν(N ) and ∀ player j, xj ≥ v({j}) (2) i=1 must hold for the nodes of hypergraph. The value of the grand coalition, ν(N ) is 1. To achieve this, we normalise the Shapley values we get for the hyperedges and produce normalised values for the vertices. We model the games presented in [1] for hypergraphs. Then we compute the Shapley value for these games when they are played on line graph and weighted line graph of the {i, k, l} {i, j} {i, k} {i, k, } {i, j, k} {j, k} {j, l} {i, j, k, l} (b) {i, j, k} (a) Figure 2: (a)L(H1 ) (b)L(H2 ) hypergraph. Let H = (V,E) be a hypergraph. Define the neighbourhood of a hyperedge, e ∈ E in the line graph of H, L(H) as follows, N g(e) = {e′ | e′ ∈ E, e ∩ e′ 6= φ} (3) We use the following equation to translate the values to the nodes of H from the Shapley value svL (.) of the hyperedges. For node vi ∈ V X svL (e) sv(vi ) = (4) |e| e∈E,v ∈e i Next, we briefly introduced two games, one on unweighted graphs and another on weighted graphs (Game 1 and Game 3 from [1]). These games are defined over undirected network graphs. Then, we describe how we use these two games in case of hypergraphs. 3.1 TopK Game: ν1 (C) = #agents at-most 1 hop away Given an unweighted, undirected network G(V, E). Fringe of a subset C ⊆ V (G) is defined as the set {v ∈ V (G) : v ∈ C(or)∃u ∈ C such that (u, v) ∈ E(G)}. Based on the fringe, the cooperative game g1(V (G), ν1 ) is defined with respect to the network G(V, E) by the characteristic function ν1 : 2V (G) → R given by ( 0 if C = φ (5) ν1 (C) = size(f ringe(C)) otherwise Using g1 on line graph, L(H) Shapley value i ∈ E is given as follows, X 1 svL (i) = (6) 1 + |N g(i)| j∈E,i∩j6=φ This is computed using algorithm 1 of [1] which runs in O(V + E) and finds SV for each of the hyperedges. 3.2 TopK Game with Weights: ν3 (C) = #agents at-most dcutoff away This is an extension of TopK game for weighted undirected networks G(V, E, W ), where W : E → R+ is the weight function. Here the fringe of a set, s ⊆ C is defined as the vertices which are atmost dcutoff away from some node in s. Let g3(V (G), ν3 ) be the game defined on the network, G(V, E, W ). The definition of fringe gives the following definition for ν3 ,   if C = φ 0 ν3 (C) = size({vi :  ∃v ∈ C(or)distance(v , v ) ≤ d otherwise j i j cutoff }) (7) for each coalition C ⊆ V (G). [1] gives an exact formula for SVs using ν3 . However, in this case the algorithm for implementing the formula has complexity O(V E + V 2 log(V )). For weighted line graph, Lw (H), g3 is used to find the Shapley Value of the hyperedges. For TopK Game with Weights as in [1], we define the extended neighbourhood as follows, extN g(e) = {e′ | dee′ < dcutof f } (8) ′ where, dee′ = shortest path distance between e and e in Lw (H). Shapley value of i ∈ E in Lw (H) is X 1 (9) svL (i) = 1 + |extN g(i)| j∈{i}∪extN g(i) As we see in the final equations, the marginal contribution of each node depends mainly on its neighbourhood. When looking at clique construction, any two vertices which share a common edge are neighbours. Line graph gives a dual of this relation. So it is important to find the actual neighbourhood for each node in the line graph that can influence the marginal contribution of a node of the line graph. Here what we mean by actual neighbourhood is the neighbours who can influence or take part in the computation of marginal contribution of this node, in other the words, neighbours who can change the node’s influence over the network. The physical inference of Equation 7 tells the same notion mathematically. Next we analyse the neighbourhood for line graph and give an analogous equation which is more appropriate. 3.2.1 Neighbourhood Analysis: Line Graph Instead of considering the degree as the size of the neighborhood we define volume that is analogous to degree for the nodes of line graph L(H). Volume of a node v, vol(v) is defined as the sum of weights of its neighbours. And, we do not consider a node in a line graph has a contribution of one, instead we take the cardinality of the corresponding hyperedge. Let us explain why a different definition of size is required. Let us assume e1 , e2 , e3 be three different edges of the hypergraph, H such that i, j ∈ e1 , k, j ∈ e2 and j, l ∈ e3 . Let vl1 , vl2 , vl3 be the vertices of L(H) corresponding to e1 , e2 and e3 respectively. Using Equation 6, each node in neighbourhood of vl2 reduces the probability that vl1 brings vl2 in the coalition. Hence reduces the marginal contribution of vl1 . So, vl3 reduces MC of vl1 . Observe, a vertex in L(H), corresponding to a hyperedge with large cardinality or having influential nodes of H, gets high Shapley value and the other vertices which are connected to it gets comparatively low Shapley value. If l is an influential node that means vl3 becomes more influential lowering the influence of vl1 . Therefore, though l is not directly connected to i it can reduce SV of i. To eliminate this, we define voli (.) as the size of intersection of a vertex of L(H) with i. To maintain consistency, instead of counting i as 1 as in Equation 6, we take the contribution of i to the coalition as the size of i. What did we say intuitively in the above argument? Did we say that the clique construction gives the correct neighbourhood? No, We showed that the neighbourhood must depend not only on how the vertices share an edge, if they do share, but also how many edges they share and most importantly which vertices are connecting two edges. This gives us the basic idea why line graph construction works better. Therefore vol : 2V (G) → R and is defined as, vol(v) = P u∈N g(v) | u ∩ v |. We propose the following equation based on the above argument. X 1 sv(i) = (10) vol(v) + size(v) v∈N g(i) 3.3 Games with Multi-Graph 3.3.1 Proof of Claim 1 Proof. If two agents have been together in k hyperedges then they contribute k times to each others degree. Let us assume there are k edges between u and v. Further assume, w is connected to v by an edge. Since marginal contribution (MC) depends inversely on the degree of the neighbors, probability that w brings v into a coalition is less in this scenario compared to the case when u and v are connected by a single edge. Thus, adding more to the degree of a neighbor decreases the MC of any other node which are less strongly connected to it. Hence it increases the expected MC of node u. 3.3.2 TopK Game on Multi-Graph As stated in the beginning, multi-graph construction is similar to wt-clique construction. So the Shapley value of vi ∈ V (H) using TopK Game from [1, 13] on M(H) is given in the following equation. X w(vi , vj ) (11) SVtopk (vi ) = 1 + degM (H) (vj ) vj ∈{vi }∪NM (H) (vi ) where w(vi , vj ) is the edge multiplicity between vi and vj , NM (H) (vi ) is the neighbourhood of vi in M (H) which is same as neighbourhood of vi in H and degM (H) =| NM (H) (vi ) |. 4. DIFFUSION Let f (S) be the expected number of nodes active (influenced) at the end if set S is targeted for initial activation. For example S can be the top k nodes,for some constant k. We pick this top k from the ranking we computed, which are not directly connected by an edge as done in the spin algorithm[13]. We want to maximise f (S). This optimization problem is NP-hard as proved in [12]. Hence we look for an approximate solution for this problem. Kempe et al.[12] generalised two diffusion models viz, Independent Cascade and Linear Threshold. They showed that greedy hill climbing algorithm using generalisation of both the models gives (1 − 1/e − ǫ) approximation. In fact, as shown by them, both the generalised models are equivalent. To compare the rankings we get for the nodes of H using P (H), W (H), L(H), Lw (H) and M (H), we use these two methods. These models only look at the rankings and not how the scores are distributed among the nodes. That is useful when the scores do not have any other inference. We come up with a heuristic using the concept of dominance from cooperative games to compare two set of SVs we get by different reduced representations. Dominance is a fundamental concept in cooperative game theory and is used for deciding whether a vector,x from the imputation space is preferred over another vector, y from the same imputation space. 4.1 Measure of Diffusion using Game Theory The solution concepts in cooperative games give a measure of power (centrality score) of each player. Imputation P Space is the set Iv = {x ∈ Rn | x = (x1 , x2 , . . . , xn ), n i=1 xi = 1, xi ≥ 0}. It is the set of all possible assignment of values to each of the players. Since this set can contain uncountably many vectors we try to find a smaller set. We use the rules of the game to get a smaller set such that each vector from this subset assigns some meaningful values to the players. Shapley Value is the solution concept that we are using in our work to find the centrality scores (power) of the players. Marginal contribution network enables us to find the Shapley Value scores in polynomial time[1]. Now, we want to use the idea of dominant imputation to show which method is better in assigning the scores. 4.1.1 Motivation Idea: Core is another solution concept like Shapley value to measure power of each of the agents. Finding core is a computationally hard problem too. It is defined as follows. Definition 1. Core. Core is the set of all undominated imputation. Let x, y ∈ Iv , Iv = set of all imputations. Definition 2. Dominance. We say x ≻ y ( x dominates y or x is preferred to y), if there existsPa non-empty coalition S ⊆ N such that ∀i ∈ S, xi > yi and i∈S xi ≤ v(S) Where N is the set of all players and v(S) denotes the value of the coalition S. Checking if an imputation (centrality score) is in the core is NP-hard[14](reduction from vertex cover) even with MCNet representation. So instead of checking if an imputation is undominated, the domination of the imputation has been checked. Domination is easy to compute when we know two imputations (say x and y as in the definition). If x dominates y and the agents who are getting more score in x are amongst the top agents according to both the imputations then x correctly assigns high score to the important nodes and low score to less important nodes since value of grand coaltion is 1. 4.1.2 Why We Need When we know some vectors from the core or we know what is the best way to divide the value of the grand coalition, we have an optimal possible imputation. We can compute which centrality score is better between two of them by comparing them with the optimal one (undominated imputation). Precisely, we look at how much each deviates from the optimal one, co-ordinate wise. But in lack of that standard or optimal measure we might want to compare two rankings(imputations) to check which one is giving a better estimate of ranking. Here is where the idea of dominance can be used. 4.1.3 Dominance to Diffusion Let X and Y are two imputations. S be a set of agents such that ∀i ∈ S xi > yi . We can check the marginal contribution of xi s in general over the sum of the values given to xi s in the said imputation. If marginal contribution is these xi s are high and X assigns high score to these xi s, that implies X is more appropriate measure than Y . Computing diffusion with general diffusion model is hard. This gives us an efficiently computable method to compare two centrality measures. But since domination is not transitive to compare n different measures dominance need to be checked for O(n2 ) times. 5. RESULTS We use the five different reduced representations of a hypergraph described earlier and compute the centrality of the nodes in each reduced representation. We performed our experiments on three datasets, viz., JMLR, Citeseer, and Cora. The hypergraph on the JMLR dataset is a co-authorship network where each hyperedge corresponds to a paper published in JMLR and links the authors of that paper. The other two are co-citation network where all the papers cited in a particular paper form a hyperedge. As mentioned before we compare the five methods using (i) two diffusion models[12], Linear Threshold and Independent Cascade and (ii) the concept of dominance from co-operative game theory. We performed three different experiments on the different datasets: (1) As done in [13], we use top k nodes with spin algorithm to find diffusion using Linear Threshold and Independent Cascade, where, k is our budget of how many nodes we activate initially. We compared the different centrality scores on the extent of the diffusion at the end of the process. (2) Let us assume, we do not want to care about the budget but want to activate atleast p% of the nodes at the end of diffusion. Say, p is 90%. To achieve this we fix a SV, val such that all nodes having SV more than val is activated initially. We call val, the limiting SV. We observe as we decrease the limiting SV p increases rapidly in the beginning but after a certain value of limiting SV change in p in negligible. We also observe that for networks with thousand nodes the limiting value is 10−4 for p = 85. Method H1 Unweighted i, j, k, l Clique score Weighted Clique score Line Graph score Weighted Line Graph score MultiGraph score H2 i, j, k, l Summary Gives equal importance to all the vertices where as l is much less central than i or k .25, .25, .25, .25 .25, .25, .25, .25 k, j, i, l k, j, i, l .27, .27, .27, .18 .27, .27, .27, .18 j, k, i, l k, i, j, l correctly assigns lower fraction to l .33, .29, .29, .08 .4, .4, .15, .03 k, i, j, l k, i, j, l correctly assigns lower fraction to l .31, .31, .3, .08 .4, .4, .15, .03 k, i, j, l k, i, j, l note the difference of sore from wt-clique construction .28, .28, .24, .17 .28, .28, .24, .17 Table 1: No of nodes influenced for example 1 Dataset No of nodes No of edges Size of largest component 1159 2240 No of connected components 11 438 JMLR Citeseer [15] Cora [15] 3217 3312 2708 2223 78 2485 2275 2110 Table 2: Description of the Datasets Used (3) We examined how the SV is distributed as function of the ranking of the nodes. If a larger fraction of the SV is concentrated on the higher ranked nodes, then the measure discriminates better between the truly important nodes and the rest. Most influential nodes can be prominently identified by any of the methods.But as we try to extend the reach of diffusion, we increase k that is we activate more nodes in the beginning. We observed that the top 10 nodes we get employing different methods have at least 70% intersection. But when we look at how much power has been assigned to each of these nodes the methods differ largely. 5.1 Example 1 Table 1 shows the empirical results that we get for the five mentioned reduced representation of hypergraph for the hypergraphs of Example 1. It shows the distribution of score, for the four nodes, produced by the five methods. It clearly shows that the ranking using P (H) (unweighted clique) wrongly assigns same score to all the nodes. Also, observe the difference in scoring pattern of the methods W (H) (weighted clique) and M (H) (multi-graph). Figure 3: Compare SV distribution on JMLR Data Table 2 describes the three datasets on which we have performed the experiments. But before looking at the results on the datasets, let us look at the results on Example 1 that we introduced in introduction. 5.2 JMLR Data From JMLR data we extract the hypergraph of the coauthorship network that is, in this network each node represents an author and each paper they published is a hyperedge consisting of the authors of that paper. Statistics of the network is given in table 2 in row 1. We compare the distribution of SV among the nodes, computed from the JMLR hypergraph using the five mentioned representations, in figure 3. It plots the number of nodes, having SV more than a particular value v, vs v. The trend that we observe in figure 3 is that, the Shapley value computation through line graph always assigns very low (almost zero) value to powerless nodes where as the value, for important nodes go as high as 0.01. On the other hand, the highest value assigned to the most important node using P(H) or even M(H) is 0.0016 and 0.0018 respectively. The idea of dominance clearly shows this difference. Let us look at the top nodes given by P (H), L(H) and M (H). We apply Linear Threshold model to measure diffusion. In Linear Threshold model, threshold for each node is kept as 1/3 of degree of that node and we find the diffusion when top 10 nodes are activated initially. The number of nodes that are activated in the beginning is written in brackets in the heading of the columns in the table. The result is shown in figure 4. Notice, when top 15 or top 20 nodes are activated in the beginning M (H) outperforms L(H) but as we take more nodes for initial activation the green bar for L(H) stands taller. 5.3 Citeseer Data It is a co-citation network that is, each publication is a node and the authors who co-authored a publication form an hyperedge. Statistics of the network is given in table 2 in row 2. Table 3 compares three of our proposed single graph representations of hypergraph with primal graph of hypergraph, P (H) and W (H).We followed spin algorithm[13] for picking Figure 4: Diffusion using P(H), M(H) and L(H) on JMLR Data Graph Representation P (H) W (H) L(H) Lw (H) M (H) P (H) (Betweenness) L(H) (Betweenness) Linear Threshold (top 10) Independent Independent time Cascade Cascade (ms) (top5) (top10) 698 627 709 691 698 832 800.86 837.68 833.34 837.68 1002.13 1039.12 914.15 933.73 966.04 754 2583 1573 3042 751 625 901.55 951.81 18520 664 791.35 1009.29 15731 Table 3: Diffusion using Linear Threshold and Independent Cascade model on Citeseer up the top k nodes, for initial activation, from the rankings produced by the five methods. As stated earlier, it is easier to find out nodes which are important (e.g., the top node or top 5 nodes). Here, both L(H) and M (H) gives the same set of top 5 nodes. Table 3 shows the results of between-ness centrality too, when found using primal graph and our proposed line graph method. For between-ness too the performance of line graph is better both in terms of time requirement and ranking. Computations on line graph takes the advantage of the fact that in general these hypergraphs are sparse compared to general graphs. Figure 5 compares the SV assigned to nodes using L(H) and M (H). It plots number of nodes, having Shapley Value above v, vs v. The most important node gets SV 0.2 and 0.0089 using L(H) and M (H) respectively. In the figure the purple line for M (H) is almost grounded to x − axis after 2 × 10−3 where as many nodes have been assigned SV more than 5 × 10−3 by L(H). Figure 6 compares the distribution of SV assigned to the nodes by all the five methods. Table 4 gives an analogous analysis in tabular format. It also puts together the number of nodes activated at the end when all Method L(H) LW (H) Figure 5: Compare SV distribution on Citeseer Data: L(H) vs M(H) Figure 6: Compare SV distribution on Citeseer data nodes, having SV above the limiting value v, are activated initially. For example, when the sum of Shapley value of the nodes picked is fixed at 0.07273128, we get only 3 nodes by L(H) compared to 68 nodes by M (H) and the limiting value is only 0.01 for L(H) compared to 6.14 × 10−4 for M (H). The sum of SV of top 1022 nodes, chosen by spin algorithm, manages to reach 0.3421102 only for M (H). Dominates M (H) via 660 nodes of total score 0.685, P (H) via 702 nodes of total score 0.705 M (H) via 717 nodes of total score 0.709, P (H) via 748 nodes of total score 0.728 Dominated By None L(H) via 1695 nodes of total score 0.609 Table 5: Comparison of Diffusion Using Dominance on Citeseer Figure 7: Diffusion using linear threshold model on Cora data citation network from Cora Data. It gives the diffusion for between-ness too, when computed using P (H) and L(H). Performance of later is better when compared with respect to diffusion. Notice, since this hypergraph is not as sparse as in the case of Citeseer, the computation time when using L(H) is more compared to P (H) in case of between-ness measurement. If a method M1 dominates another method M2 via x nodes, more value of x implies stronger M2 , less value of x implies M2 has not assigned more power to more important nodes when compared to M1 . Observe table 5, it takes only 26% nodes in favour of Lw (H) when dominating M (H) compared to 62% when L(H) dominates Lw (H). This is because, important nodes are assigned less score using M (H). So, they go in favour of Lw (H). But, when comparing Lw (H) to L(H) all the important nodes are assigned high value so it takes more nodes to sum up to a score of 0.609. 5.4 Cora Data It is again a co-citation network. Statistics of the network is given in table 2 in row 3. Total number of nodes activated at the end when top 10 nodes are activated at the start is shown in figure 7 for the five methods discussed. The diffusion model employed here is linear threshold. Table 7 gives comparison of three of our proposed representations with P (H) and W (H) for the co- Figure 8: Compare SV distribution on Cora data 0.4164433 0.41616583 0.4039153 Using Line Graph limiting Nodes Diffusion Diffusion value Started (Linear (Indeof SV with Threshold) pendent Cascade) −6 10 990 3067 2788.81 10−5 945 3060 2775.0 10−4 644 2864 2629.12 0.22036676 0.07273128 0.04144367 0.001 0.01 0.05 Sum of SV 84 3 1 1216 608 541 1787.81 1596.57 1573.94 limiting value of SV 7.47112E5 (sum 0.3421102) 3.02 × 10−4 6.14 × 10−4 8.87 × 10−4 Using Multi-Graph Nodes Diffusion Started (Linear with Threshold) Diffusion (Independent Cascade) 1022 3107 2831.49 472 68 24 2518 1212 769 2438.02 1757.99 1622.98 Table 4: Diffusion: Linear Threshold and Independent Cascade on Citeseer Data 0.37252015 0.371984 0.35889158 Using Line Graph limiting Nodes Diffusion Diffusion value Started (Linear (Indeof SV with Threshold) pendent Cascade) −6 10 752 2686 2489.74 10−5 678 2678 2476.13 10−4 309 2639 2374.86 0.27076006 0.14932592 0.09491563 0.001 0.01 0.05 Sum of SV 49 4 1 2255 994 416 2186.8 2127.4 2128.83 limiting value of SV 1.087832E4 sum 0.2821837 1.5 × 10−4 3.96 × 10−4 7.59 × 10−4 Using Multi-Graph Nodes Diffusion Started (Linear with Threshold) Diffusion (Independent Cascade) 722 2686 2489.13 636 153 51 2686 2466 2227 2478.1 2254.25 2168.42 Table 6: Diffusion: Linear Threshold and Independent Cascade on Cora data Figure 8 gives the SV distribution for different methods. The y-axis gives the number of nodes having Shapley Value more than the value of x-axis. The line corresponding to the SV distribution produced using W (H) ends even before 0.005 which means no nodes has been assigned value above 0.005. Where as the lines for L(H) and Lw (H) goes beyond 0.05 and has many nodes with values more than that. This depicts the important nodes are also assigned low values by W (H), P (H) and even M (H). Therefore it is hard to conclude which are the more powerful node from the scores in case of W (H), P (H) and M (H). A similar analysis between L(H) and M (H) is shown in table 6 where the corresponding diffusion values are also given. As we analysed for Citeseer Data, observe from Table 8, it takes only 14% nodes in favour of Lw (H) when dominating M (H) compared to 61% when dominating L(H). This is because, important nodes are assigned less score using M (H) so they go in favour of Lw (H). But when comparing Lw (H) to L(H), all the important nodes are assigned high value so it takes more nodes to sum upto a score of 0.877 that can give preference to Lw (H). Since, we do not care about these less important nodes, we can safely employ any one of L(H) or Lw (H) to get the ranking as well as the Shapley value based scores. 6. CONCLUSION We established that computation of game theoretic cen- Graph Representation P (H) W (H) L(H) LW (H) M (H) P (H) Betweenness L(H) Betweenness Linear Threshold Threshold (top 10) 1725 1721 1864 1934 1821 Independent Independent time Cascade Cascade (top 10) (top 5) 1654.73 1670.13 1656.82 1685.98 1659.83 1588.34 1596.37 1625.5 1625.5 1598.67 840 7998 7981 16438 2523 1831 1663.82 1606.36 12226 1934 1680.68 1623.73 22932 Table 7: No of nodes influenced on Cora data trality on hypergraphs using the clique construction indeed loses some information which is preserved by line graph construction. The results also demonstrate that the clique construction does not perform well compared to Weighted Line Graph and Multi-Graph construction. Looking at the results on all the three datasets we conclude that Weighted Line Graph and Multi-graph reductions of a hypergraph lead Method Dominates L(H) M (H) via 403 nodes of total score 0.845, P (H) via 442 nodes of total score 0.873, W (H) via 457 nodes of total score 0.891 LW (H) M (H) via 399 nodes of total score 0.853, P (H) via 443 nodes of total score 0.887, W (H) via 493 nodes of total score 0.909, L(H) via 1669 nodes of total score 0.877 Dominated By Lw (H) via 1669 nodes of total score 0.877 None Table 8: Comparison of Diffusion Using Dominance on Cora to more meaningful centrality scores. It would be an interesting line of work to investigate if these two reductions of a hypergraph yield better algorithms/ computations in other domains, like, community detection. 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