Network, Popularity and Social Cohesion: a Game-Theoretic Approach

Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI-17)

Network, Popularity and Social Cohesion: A Game-Theoretic Approach

Jiamou Liu, Ziheng Wei The University of Auckland Department of Computer Science Auckland, New Zealand [email protected] [email protected]

Abstract relations. The theory of self-categorization asserts that individuals mentally associate themselves into groups based on In studies of social dynamics, cohesion refers to a group’s relations such as friendship and trust (Hogg 1992). tendency to stay in unity, which – as argued in sociometry Cohesion denotes a tendency for a social group to stay – arises from the network topology of interpersonal ties. We follow this idea and propose a game-based model of cohe- in unity, which traditionally consists of two views. Firstly, sion that not only relies on the social network, but also re- cohesion refers to a “pulling force” that draws members to- flects individuals’ social needs. In particular, our model is a gether (Festinger 1950); Secondly, cohesion also means a type of cooperative games where players may gain popularity type of “resistance” of the group to disruption (Gross and by strategically forming groups. A group is socially cohesive Martin 1952). In (2001), White and Harary propose a notion if the grand coalition is core stable. We study social cohe- of structural cohesion, which unifies these two views. sion in some special types of graphs and draw a link between However, we identify insufficiencies in the existing mod- social cohesion and the classical notion of structural cohe- els for social cohesion: 1) Cooperative game theory often sion (White and Harary 2001). We then focus on the problem misses the crucial social network dimension. 2) The struc- of deciding whether a given social network is socially cohesive and show that this problem is CoNP-complete. Never- tural cohesion of a network refers to the minimum number theless, we give two efficient heuristics for coalition struc- of nodes whose removal results in network disintegration tures where players enjoy high popularity and experimentally (White and Harary 2001); this is a property of the network evaluate their performances. on the whole, and does not embody individual needs. Since cohesion embodies both the micro-focus of psychology (ful- fillment of personal objectives and needs) and the macro- 1 Introduction focus of sociology (emergence of social classes) (Carron and Human has a natural desire to bind with others and needs Brawley 2000), the main challenge is to build a general but to belong to groups. By understanding the basic instruments rigorous model to bridge the micro- and the macro-foci. that generate coherent social groups, one can explain im- In this paper, we define a type of cooperative games portant phenomena such as the emergence of norms, group on networked agents. Outcomes of the game not only rely conformity, self-identity and social classes (Festinger 1950; on the network topology but also reflect individuals’ so- Huisman and Bruggeman 2012; Hogg 1992). For example, cial needs. Our model is consistent with existing theories: studies reveal that on arrival to Western countries, immi- Firstly, we follow the network approach to study social phe- grants tend to form cohesive groups within their ethnic com- nomena, which is initiated by early pioneers such as Simmel munities, which may hamper their acculturation into the new and Durkheim. Secondly, our game-theoretic formulation is society (Nee and Sanders 2001). Another study identifies co- in line with group dynamics theories that focus on the in- hesive groups of inhabitants in an Austrian village that cor- terdependence among members (Lewin 1943). Thirdly, we respond to stratified classes defined by succession to farm- verify that networks with high structural cohesion also tend land ownership (Brudner and White 1997). to be socially cohesive according to our definition. Most theories of group dynamics rely on two fundamen- People prefer to be in a group where they are seen as valu- tal drives: cooperation and social needs. Indeed, every group able and popular members. Hence the payoff should reflect exists to accomplish certain tasks. Cooperation is desirable in some sense players’ social positions. Popularity – an im- because combining skills and resources leads to better out- portant indicator of social position – arises from interper- comes. The theory of cooperative games studies distribu- sonal ties such as liking or attraction (Lansu and Cillessen tion of collective gains among rational agents (Peleg and 2011). In particular, (Conti et al. 2013) uses the degrees of Sudholter¨ 2010). Social need is another important drive of nodes as a measure of popularity and identify the economic group dynamics. A society contains complex interpersonal benefits behind gaining popularity. Therefore, payoffs in our games are defined based on degrees of players. Copyright c 2017, Association for the Advancement of Artificial We summarize our main contributions: (1) We propose Intelligence (www.aaai.org). All rights reserved. popularity games on a social network and define social co-

600 hesion using core stability of the grand coalition. (2) We Social positions, as argued in sociometry, arise from the show that our notion is consistent with intuition for several network topology (Cillessen and Mayeux 2004). A long standard classes of networks and connect structural cohesion line of research studies how different centralities (e.g. de- with our notion of social cohesion (Thm. 6). (3) We prove gree, closeness, betweenness, etc.) give rise to “positional that deciding socially cohesion of a network is computation- advantage” of individuals. In particular, degree centrality ally hard (Thm. 7). (4) Finally, we present two heuristics that refers to the number of edges attached to a node. Despite decide social cohesion and compute group structures with its conceptual simplicity, degree centrality naturally repre- high player payoffs and evaluate them by experiments. sents (sociometric) popularity, which plays a crucial role in a person’s self-efficacy and social needs (Zhang 2010; Related works. The series of works (Narayanam et al. Conti et al. 2013). Popularity depends on the underlying 2014; Michalak et al. 2013; Szczepanski´ et al. 2015a; 2015b; group: a person may be very popular in one group while Szczepanski,´ Michalak, and Wooldridge 2014) investigates being unknown to another. Hence individuals may gain pop- game-based network centrality. Their aim is to capture a ularity by forming groups strategically. We thus make our player’s centrality using various instances of semivalues, next definition. The sub-network induced on a set S ⊆ V is GS S, ES ES E ∩ S2 u which are based on the player’s expected payoff. In con- =( ) where = . degS( ) denotes |{v uv ∈ ES}| u u trast, our study aims at games where the payoff of players : and we write deg( ) for degV ( ). are given a priori by degree centrality and focus on core sta- Definition 3. The popularity of a node u in a subset S ⊆ V bility. (Chen et al. 2011) uses non-cooperative games to ex- is pS(u) :=degS(u)/|S|. plain community formation in a social network. Each player Note that p{u}(u)=0for every node u.Ifu ∈ S has an in their game decides among a fixed set of strategies (i.e. edge to all other nodes in the graph GS, then u is the most a given set coalitions); the payoff is defined based on gain popular node in S with pS(u)=(|S|−1)/|S|. and loss which depend on the local graph structures. (Mc- Sweeney, Mehrotra, and Oh 2014) studies community for- Definition 4. The popularity game on G =(V,E) is a co- V mation through cooperative games. The payoffs of players operative game Γ(G)=(V,ρ) where ρ: V × 2 → [0, 1) are given by modularity and modularity-maximizing parti- is defined by ρ(u, S)=pS(u). tions correspond to Nash equilibria. The focus is on com- An outcome of the popularity game Γ(G) assigns any munity detection but not on social cohesion. (Moskvina and player u with a coalition S u. The sum of popularity Liu 2016) studies strategies to build social networks by es- of members of Sequals their average degree in S, i.e. tablishing interpersonal ties. Furthermore, our payoff func- u∈S pS(u)= u∈S degS(u)/|S| =2|ES|/|S|. The tion is not additively separable and hence does not extends average degree measures the density of the set S, which re- from their model. Lastly, our work is different from commu- flects the amount of interactions within S, and thus can be nity detection (Fortunato 2010). The notion of community regarded as a collective utility. In this sense, the popularity structure originates from physics which focuses on a macro game is efficient in distributing such collective utility among view of the network, while our work is motivated from group players according to their popularity. dynamics and focus on individual needs and preferences. Social cohesion represents a group’s tendency to remain united (Cartwright and Zander 1953). We express cohesion 2 Popularity Games and Social Cohesion through core stability w.r.t. the popularity game Γ(G): Sup- A social network is an unweighted graph G =(V,E) where pose a coalition formation W is not core stable. Then there V is a set of nodes and E is a set of (undirected) edges. An is a set S ⊆ V every member of which would gain a higher edge {u, v}⊆V (where u = v), denoted by uv, repre- popularity in S than in their own coalitions in W. Thus sents certain social relation between u, v, such as attraction, there is a latent incentive among members of S to disrupt interdependence and friendship. We do not allow loops of W and form a new coalition S. On the contrary, a core sta- the form uu. The reader is referred to (Peleg and Sudholter¨ ble W represents a state of the network that is resilience to 2010) for more details on cooperative game theory. such “disruptions”. Thus, when the grand coalition forma- Definition 1. A cooperative game (with non-transferrable tion WG = {V } is core stable, all members bind naturally utility) is a pair G =(V,ρ) where V is a set of players, and into a single group and would remain so as long as the net- ρ : V ×2V →R is a payoff function. work topology does not change. A coalition formation of G is a partition of V W = Definition 5. A network G =(V,E) is socially cohesive W {V1,...,Vk}, i.e., ≤i≤k Vi = V , ∀1≤iρW (u); In this case we say of both a, d in V1 reaches 2/3. Fig. 1(b) displays G2 = that W is blocked by H. A coalition formation W of G is (V2,E2) where pV (a)=4/5 and pV (i)=1/5 for all core stable w.r.t. (V,ρ) if it is not blocked by any set H ⊆ V . i = b,...,e. This graph is cohesive as WG2 is not blocked.

601 Theorem 4. A group structure W of a star network is core stable iff the centre is in the same social group with at least half of the tails. Thus, any star network is socially cohesive. Proof. Take any group structure W and suppose the centre Figure 1: The graphs considered in Example 1 are in black. c is in a social group S with tails. Then pS(c)=/( +1) The added edges are highlighted in red. and for any tail ui ∈ S, pS(ui)=1/( +1). All players not bc in S has popularity 0. However, adding the edge will destroy social cohesion as {b, c} WG Suppose ≥ m/2. Take any set S = S that contains then blocks 2 . Social psychological studies often c.If|S |≤|S|, then pS (c) ≤ pS(c).If|S | > |S|, then presume that more ties leads to higher cohesion; this example displays a more complicated picture: Adding an edge pS (v) V1 degS ( ) |{v | v ∈ S ∩ V1}| and |{v | v ∈ S ∩ V2}|, respectively. every 1, V1 ( )= |V | |S| = |S| = 1 W ∀S∈W S ≥r S pS(u). Hence W does not contain S. Lemma 1. is core stable only if : ( ) ( ). Proof. Suppose there is S ∈Wwith (S) r(H).Takeany induce connected sub-networks of a social network. u ∈ S ∩ V2 and v ∈ H ∩ V1. Then we have pS(u)= S ⊆ V G r(S) 1 (H) 1 Definition 6. A set is called a social group of if < and pH (v)= < . Hence, the S induces a connected sub-network. A group structure is a (S)+r(S) 2 (H)+r(H) 2 set {u, v} blocks W as p{u,v}(u)=p{u,v}(v)=1/2. coalition formation containing only social groups. K The next theorem shows that socially cohesive networks We next characterize core stable group structure in n,n.In v ∈ have bounded size. particular, perfect matchings, i.e., situations when every V1 is matched with a unique player in V2, are core stable. δ(G) Theorem 2. Let be the maximum degree of nodes in W K G=(V,E) G |V |≤ Theorem 5. A group structure of n,n is core stable iff . Then is socially cohesive only when ∀S ∈W S r S 2δ(G) unless |V | =1. : ( )= ( ). Proof. By Lem. 1, if W is core stable then ∀S ∈ |V | > δ G |V | > E ∅ G Proof. Suppose 2 ( ) and 1.If = , is W : (S)=r(S). Conversely, if ∀S ∈W: (S)=r(S), uv not socially cohesive by Thm. 1. Otherwise, pick an edge . then any v has payoff 1 . Thus W is core stable as every Then max{deg(u), deg(v)}≤δ(G) < |V |/2. This means 2 H ⊆ V contains some player with payoff at most 1 . that max{pV (u),pV (v)} < 1/2, and the edge {u, v} forms 2 G a blocking set. Thus is not socially cohesive. 3 Structural Cohesion and Social Cohesion We now investigate our games on some standard classes Definition 7 (White and Harary 2001). The structural co- of graphs and characterize core stable group structures. hesion κ of a connected graph G is the minimal number of Complete networks. A graph G =(V,E) is complete if any nodes upon removal of which G become disconnected. pair of nodes is linked. Naturally, one would expect com- By Def. 7, a larger κ implies that G is more resilient to plete networks to be socially cohesive. conflicts or the departure of group members, and is thus Theorem 3. Let G be a complete network. The grand coali- more cohesive. Moreover, Menger’s theorem states that κ is tion is the only core stable group structure. the greatest lower bound on the number of paths between any pairs of nodes. Hence κ is a reasonable measure of Proof. Any induced sub-network GS of a complete net- cohesion. We next link κ with our notion of social cohesion. |S|−1 In (Granovetter 1973), a pair uv∈ / E is seen as a “structural work G =(V,E) is also complete. Thus pS(u)= < |S| hole” that forbids communication and is thus referred to as |V |−1 |V | = pV (u). Therefore any player’s popularity is maxi- an absent tie. For each S ⊆ V and u ∈ S, we define: mized in the grand coalition V . - fin(u, S) :=degS(u) and fout(u, S) := |{v/∈ S | uv ∈ E}| are the numbers of actual ties of u within the group S Star networks. A star network contains a node c (centre), a and outside S, resp. number of other nodes u1,...,um (tails) where m>1, and - sin(u, S) := |S|−fin(u, S) and sout(u, S) := |{v/∈ S | edges {cu1,...,cum}. Intuitively, the centre c would like to uv∈ / E}| are the number of absent ties in S (including u be in a social group with as many players as possible, while itself) and outside S, resp. a tail would like to be with as few others as possible.

602 Intuitively, if S ⊆ V is blocking, each member u tends 4 The Computational Complexity of to have many actual ties within S and absent ties outside S, Deciding Social Cohesion i.e., high fin(u, S) and sout(u, S), and u tends to have few COH absent ties in S and actual ties outside S, i.e., low fout(u, S) We are interested in the decision problem : Given a net- G V,E G and sin(u, S). Thus, we define for all S ⊆ V , u ∈ S, work =( ), decide if is socially cohesive. The distance between two nodes u and v, denoted by γ(u, S) := fin(u, S)sout(u, S) − fout(u, S)sin(u, S) (1) dist(u, v), is the length of a shortest path from u to v in G. The eccentricity of u is ecc(u)=maxv∈V dist(u, v). The Lemma 2. For all S ⊆ V , S blocks WG = {V } iff ∀u ∈ G diam G ecc u S : γ(u, S) > 0. diameter of the network is ( )=maxu∈V ( ). A graph G =(V,E) is diametrically uniform if all v ∈ fin(u,S) V G u ∈ S pS(u)= have the same eccentricity; otherwise is called non- Proof. For each , fin(u,S)+sin(u,S) and diametrically uniform.WeuseNDU2 to denote the set of all fin(u, S)+fout(u, S) non-diametrically uniform connected graphs whose diame- pV (u)= COH fin(u, S)+fout(u, S)+sin(u, S)+sout(u, S) ter is at most 2. Our goal is to show that the problem is computationally hard already on the class NDU2. The set S blocks WG iff ∀u ∈ S : pS(u) >pV (u), which G NDU V can be shown to be equivalent to fin(u, S)·sout(u, S) > Theorem 7. The network belongs to 2 iff its nodes V V fout(u, S)·sin(u, S) using the above equalities. can be partitioned into two non-empty set 1 and 2, where V1 = {u | vu ∈ E for all v = u}. A network G contains a minimal cut A ⊆ V of size κ, i.e., 0 Let G be a graph in NDU . We call {V ,V } as described in removing A from G decomposes the graph into m distinct 2 1 2 0 Thm. 7 the eccentricity partition of G. We first present some connected components A1,...,Am ⊆ V where m ≥ 2.We simple properties of NDU2. further assume that |A1|≤···≤|Am| and A0 is chosen in a way where |A1| is as small as possible. Let χ be the Lemma 5. The network G in NDU2 is socially cohesive iff size |A1|, and let μ be the largest possible length m of the no set S ⊆ V2 blocks WG. sequence of Ai’s. We first look at the case when κ =1. Lemma 3. If κ=1 and G is socially cohesive, then χ<2. Proof. One direction (left to right) is clear. Conversely, suppose the network is not socially cohesive. Let S⊂V Proof. Suppose κ =1and χ ≥ 2. Let (A1,...,Am) be an be a blocking set of the grand coalition formation, i.e., u ∈ A G optimal cut sequence. Take 1.As contains a cut ∀u∈S : pS(u)>pV (u).IfS ∩ V1 = ∅. Then ∀u ∈ S ∩ |V |− |S|− node, fout(u, A1) ≤ 1 and sout(u, A1) ≥|V |−χ − 1. Then 1 1 V1 : pV (u)= |V | . However, pV (u) ≥ |S| ≥ pS(u) γ(u, A ) ≥ fin(u, A ) · (|V |−χ − 1) − sin(u, A ). Since 1 1 1 which contradicts that fact that S is a blocking set. fin(u, A1)+sin(u, A1)=χ,

γ(u, A1) ≥ fin(u, A1)(|V |−χ − 1) − (χ − fin(u, A1)) By Lem. 5, the structure of GV2 is crucial in determining social cohesion of G. For any S ⊆ V and u ∈ S, we recall = fin(u, A1)(|V |−χ) − χ. 2 the notions fin(u, S),fout(u, S),sin(u, S), and sout(u, S) Since |V |−χ>χ, γ(u, A1)>0. Thus by Lem. 2, A1 blocks from Section 3, but re-interpret these values within the sub- W the grand coalition G. network GV2. Hence, we now set fout(u, S) as |{v ∈ V \ S | uv ∈ E}|, i.e., the number of ties that u has within Lemma 4. Suppose μ>2. Then any network G is socially 2 κ V but not in S, the other variables remain as originally de- cohesive only if χ< . 2 μ−2 ﬁned. Thus Proof. Suppose μ>2. Take an optimal cut sequence |V2| = fin(u, S)+fout(u, S)+sin(u, S)+sout(u, S) (2) (A1, ..., Aμ) and u ∈ V1. Since deg(u) <χ+ κ and |V |≥ μχ κ p u < χ+κ χ ≥ κ + ,wehave V ( ) μ·χ+κ . Suppose μ−2 . Then We then deﬁne the value μχ − χ ≥ κ p u < χ+κ ≤ 1 2 . One can then derive V ( ) μ·χ+κ 2 . fin(u, S) · sout(u, S) {u, v} GV λ(u, S)= − fout(u, S) Thus any edge in 1 forms a blocking set of the sin(u, S) grand coalition formation WG. Theorem 8. A network G in NDU2 is socially cohesive iff Lem. 4 can be used as a (semi-)test for social cohesion S ⊆ V v ∈ S |V |≥λ v, S μ> χ κ G for all 2 there exists s.t. 1 ( ) when 2: whenever exceeds μ−2 , is not socially cohesive. Clearly, more graphs become socially cohesive as S ⊆ V κ Proof. By Lem. 5, we only need to examine subsets 2. gets larger. Summarizing Lem. 3 and 4, we get: fin(u,S) Every u ∈ S has pS(u)= f u,S s u,S and Theorem 6. Let G be a network. (1) If κ =1, then G is not in( )+ in( ) socially cohesive for all χ ≥ 2. (2) If κ>1 and μ>2, then f u, S f u, S |V | κ in( )+ out( )+ 1 pV u . G is not socially cohesive for all χ ≥ μ− . ( )= 2 |V2| + |V1| Remark. The only case left unexplained is when κ>1 and μ =2. In this case there exist graphs with arbitrarily large χ Applying (2), S blocks WG iff ∀u ∈ S : pS(u)>pV (u),iff fin(u,S)sout(u,S) ∀u ∈ S : |V |< −fout(u, S)=λ(u, S) but are socially cohesive. 1 sin(u,S) .

603 We now give a sufﬁcient condition for social cohesion Proof. The complement of COH, COH, asks whether a set of NDU2. The size of a network is its number of nodes. A S blocks the grand coalition WG of a given network G; clique is a complete subgraph. The clique number of G, de- this problem is clearly in NP and thus COH is in CoNP. noted by ω(G), is the size of the largest clique. Turan’s´ the- For hardness, we reduce MaxClique (asking whether a graph orem relates ω(G) with the number of edges in G: contains a clique of a given size k)toCOH. CoNP-hardness COH NP MaxClique Theorem 9 (Turan´ 1941). For any p≥2,ifagraphG of size of then follows from the -hardness of n p−2 n2 ω G ≥p (Garey and Johnson 1979). has more than 2(p−1) edges, then ( ) . Lemma 6. For any social group S ⊆ V , there exists u ∈ S Algorithm 1 Construction of H given G=(V,E) and k>2 f u,S with in( ) ≤ω(GS)−1. sin(u,S) 1: Set d := k · max{deg(u) | u ∈ V } 2: Create G by adding k(k−1)+d isolated nodes to G Proof. Let k = ω(GS) and suppose for all u ∈ S, 3: Let V2 be the set of nodes in G fin(u,S) k− |V |−k −d >k− 1 |S| = fin(u, S)+sin(u, S) 4: Create a complete graph with ( 1)( 2 ) nodes; sin(u,S) . Since , Let V1 be the set of these nodes {uv | u ∈ V ,v ∈ V } V ,V k − 1 k|S|−kfin(u, S) −|S| + fin(u, S) fin(u, S) > (k − 1)|S|/k To this end, we construct, for a given G =(V,E) and k>2, k− ES > 1 |S|2 GS a graph H ∈ NDU2 as in Alg. 1. Our goal is to show that H Thus contains 2k edges. By Thm. 9, contains a (k +1)-clique, contradicting k’s deﬁnition. is not socially cohesive iff G contains a clique of size k.Itis clear that H is a NDU2 network with eccentricity partition {V ,V } c ω G cc |V |>k k− d gets, the more likely G will be socially cohesive. There is ( 1) + ( 1) . Since and 2 ( 1)+ , |V |≥ c− |V |−c H a bound such that once |V | exceeds it, the network G is 1 ( 1)( 2 ). By Thm. 7, is cohesive. 1 ω G ≥ k C ⊆ V guaranteed to be socially cohesive. Conversely, suppose ( ) . Let 2 be a clique of size k. Take u ∈ C. Since fin(u, C)=k−1, sout(u, C)= Lemma 7. Suppose c = ω(GV ) and |V | >c(c−1). Then 2 2 |V2|−k−fout(u, C) and sin(u, C)=1, G is socially cohesive if |V1|≥(c − 1)(|V2|−c). λ(u, C)=(k − 1)(|V2|−k − fout(u, C)) − fout(u, C) |V | >cc − |V |≥ c − |V |−c Proof. Suppose 2 ( 1), 1 ( 1)( 2 ). =(k − 1)(|V2|−k) − k · fout(u, C) Take any S ⊆ V2.IfS has a size-c clique, by Lem. 6 there fin(u,S) λ(u, C)−|V | = d − kfout(u, C) d ≥ u∈V ≤c−1 sout(u, S)≤|V |−c Hence, 1 . Since exists 2 with sin(u,S) . Since 2 , k deg(u), λ(u, C) −|V1| > 0. By Lem. 8, G is not socially fin(u,S) |V |≥(c − 1)(|V |−c) ≥ sout(u, S) >λ(u, S) G k H 1 2 sin(u,S) and cohesive. Therefore, contains a clique of size iff is thus G is socially cohesive by Thm. 8. not socially cohesive and the reduction is complete. Let k=ω(GS).IfS contains no c-clique, k≤c−1. f u,S |V | in( ) ≤k−1c(c−1) The last step is by sin(v,S) and 2 .By aims to optimize the average payoffs of members of a coali- Thm. 8, G is socially cohesive. tion. Socially cohesive networks usually have small diameters (≤ 2). Thus we consider neighborhood N(v) := {v}∪ Theorem 10. The problem COH is CoNP-complete. Fur- {u | uv ∈ E} of players v ∈ V . In Alg. 2, let ν(S) be the thermore CoNP-hardness holds for the class NDU2. average payoff u∈S ρ(u, S)/|S| in any set S ⊆ V .

604 Algorithm 2 AP: Given a network G =(V,E) Table 1: N, E, C denote the number of nodes, edges, and communities, respectively. 1: Initialize set V := V and a group structure W := ∅ 2: while |V | > 0 do 3: compute νw := ν(N(w) ∩ V ) for every w ∈ V 4: S := N(v) ∩ V such that ∀u ∈ V : νu ≤ νv 5: W := W∪{S}, V := V \ S 6: end while 7: return W

Figure 2: The results of running LM and AP on graphs with n =5,...,18. (a) Accuracy (b) The percentage of graphs are found core stable by the heuristics (c) The percentage of nodes with an increased payoff in the found group structure Figure 3: The distribution of payoffs of players in the grand than in the grand coalition. coalition and in the coalitions computed by the heuristics in the real-world networks.

Experiments. We evaluate the heuristics on graphs of size n =5,...,18 s omitted). In all cases, the heuristics improve players’ pay- . For each size, let be the total number of AP graphs, b be the number of graphs for which the heuristic offs considerably compared to the grand coalition, while c in particular achieves higher payoffs. Furthermore, Fig. 4 finds a blocking set, and be the number of socially co- LM hesive graphs. The heuristic thus correctly solves COH for shows all nodes get higher payoffs through . In sum- b + c (b + c)/s mary, both of the heuristics are useful in computing coali- graphs. Hence the heuristic has accuracy . LM AP For n =5, 6, 7, we enumerate all connected graphs; LM tions; while may benefit a larger portion of players, has accuracy 96.8%, 97.7% and 83.1%, resp., while AP has tends to obtain higher payoffs. 85.9%, 68.5% and 69.2%, resp. For each n =8,...,18, we generate 105 Erdos-Reny¨ ´ı random graphs of size n.As shown in Fig. 2(a), both heuristics achieve high accuracy. As n increases, socially cohesive graphs become increasingly rare. The results show that the heuristics successfully find blocking sets in almost all cases, with AP performs slightly better (100% accuracy for n ≥ 12). Note that the fluctu- ation is within a very small range and is due to the small Figure 4: The percentage of nodes with a higher payoff graph sizes. We then consider the coalitions constructed by LM (compared to the grand coalition) in coalitions generated by the heuristics. Fig. 2(b) shows that while fails to out- the heuristics for real-world networks. put core stable group structures, AP achieves core stability in 60% of the sampled cases when n ≥ 7. Nevertheless, Fig. 2(c) shows that, compared to the payoffs in the grand coalition, more nodes get a higher payoff in the coalitions 6 Conclusion and Future Work identified by LM. Real world networks. We further evaluate the heuristics on We aim to investigate natural game-theoretical and compu- 8 real-world networks: karate club ZA (Zachary 1977), dol- tational questions as future works: Does a core stable group phins DO (Lusseau et al. 2003), college football FT (Girvan structure exists for every network? What about other stabil- and Newman 2002), Facebook FB, Enron email network EN ity concepts? What would be strategies of players to im- (Leskovec et al. 2009), and three physics collaboration net- prove popularity? The proposed games are instances of a works AS, CM and HE (Leskovec et al. 2007). We only use general game-theoretical framework for networked agents, the largest components in each network; see details in Ta- whose payoffs are given by various centrality indices. It will ble. 1. Expectedly, none of these networks are socially co- be interesting to extend the work by considering other cen- hesive. The box-and-whisker diagrams in Fig. 3 show the tralities and different forms of social networks (e.g. directed, distribution of payoffs of players in the grand coalition as weighted, signed networks). Furthermore, one could also ex- well as in the coalitions output by each heuristic (outliers plore the evolution of social groups in a dynamic setting.

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