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

In studies of social dynamics, cohesion refers to a group's tendency to stay in unity, which -- as argued in sociometry -- arises from the network topology of interpersonal ties between members of the group. We follow this idea and propose a game-based model of cohesion that not only relies on the social network, but also reflects individuals' social needs. In particular, our model is a type of cooperative games where players may gain popularity by strategically forming groups. A group is socially cohesive if the grand coalition is core stable. We study social cohesion in some special types of graphs and draw a link between social cohesion and the classical notion of structural cohesion. We then focus on the problem of deciding whether a given social network is socially cohesive and show that this problem is CoNP-complete. Nevertheless, we give two efficient heuristics for coalition structures where players enjoy high popularity and experimentally evaluate their performances.

Authors

• 10 publications
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1 Introduction

Human has a natural desire to bind with others and needs to belong to groups. By understanding the basic instruments that generate coherent social groups, one can explain important phenomena such as the emergence of norms, group conformity, self-identity and social classes [12, 1, 19, 18]. For example, studies reveal that on arrival to Western countries, immigrants tend to form cohesive groups among relatives and acquaintances in their ethnic communities, which may hamper their acculturation into the new society [28]. Another study identifies cohesive groups of inhabitants in an Austrian village that correspond to stratified classes defined by succession to farmland ownership [4].

A social group arises when members of the groups are linked and develop bonds. Cohesion refers to a tendency for a group to stay in unity, which is considered from two traditional – and seemingly opposing – views: Firstly, group cohesion refers to a “pulling force” that draws members together [12]; Secondly, cohesion can also refer to a type of “resistance” of the group to disruption [17]. A common ground from both views is that cohesion amounts to a complex process characterised by both the micro-focus of psychology (fulfilment of personal objectives and needs), and the macro-focus of sociology (emergence of social classes) [5]. A challenge is therefore to build a general but rigorous model to bridge the micro- and the macro-foci.

Most theories of group dynamics rely on two fundamental drives: tasks, and social needs. Indeed, every group exists to accomplish a certain task; cooperation is desirable because combining skills and resources leads to a better collective outcome. Based solely on this drive, cohesion becomes an issue of economics: how collective gains can be distributed among members to satisfy each member’s goals. The theory of cooperative games tries to answer this question by assuming people as rational players who arrive at a stable outcome, i.e., a coalition formation where every coalition finds a stable division of the collective goods [3]. Social need is another important factor of group dynamics. A society embodies complex social relations such as friendship and trust. The theory of self categorization asserts that individuals mentally associate themselves into groups based on such traits [18]. Taking social relations into account, White and Harary describe cohesion as a network property and define structural cohesion [34]; through this notion they prove that the two seemingly opposing views of cohesion (pulling force versus resistance to disruption) are in fact equivalent. Their work is then followed by intensive effort on community detection in the last 10-15 years [10, 13].

We identify insufficiencies in the existing mathematical models for social cohesion: 1) Cooperative game is a general framework on the economic process of resource allocation. While cohesion may imply stability, cooperative games often do not capture cohesion, as they miss the crucial social network dimension. 2) Structural cohesion of a network refers to the minimum number of nodes whose removal results in network disintegration [34]; this is a property of the network on the whole, and does not embody individual needs. Hence it fails to link the macro- with the micro-focus of group dynamics.

In this paper, we define cooperative games on social networks whose nodes are rational players. Outcomes of the game not only rely on the network of social relations, but also reflect individuals’ social needs. Our model is consistent with the following theories: Firstly, we follow the network approach to study social phenomena, which is started by early pioneers such as Simmel and Dirkheim [11, 35]. Secondly, our game-theoretic formulation is in line with theories in group dynamics that focus on the interdependence of group members [23]. Thirdly, we rigorously verify that networks with high structural cohesion also tend to be socially cohesive according to our definition.

To define payoffs, we adopt the following intuition: People prefer to be in a group where they are seen as valuable and influential members. Thus, the payoff of players in a sub-network should reflect in some sense social positions. Social position is a multidimensional concept affected by a range of factors from behavioural and cognitive traits to structural and positional properties. Here we focus on the positional properties and follow the sociometric view that popularity – an important indicator of social position – arises from interpersonal ties such as liking or attraction among people [20]; a person is popular if she is liked by a large portion of other people. In particular, the authors of [9] adopt the degrees of nodes, i.e., the numbers of edges attached to nodes, as a measure of popularity and identify economical benefits for a person to become popular. Hence in our games, we define the payoffs of players based on their degrees.

We now summarise the main contributions of the paper: (1) We propose popularity games on a social network and present a game-based notion of social cohesion, which refers to the situation when the grand coalition is core stable, a well-known stability concept.  (2) To justify our model, we show consistency between popularity games and intuition over several special classes of networks. We also build a natural connection between structural cohesion and our notion of social cohesion (Theorem 4.1).  (3) We prove that deciding whether a network is socially cohesive is computationally hard (Theorem 7)  (4) Finally, we present two heuristics that decide social cohesion and compute group structures with high player payoffs and evaluate them by experiments.

Related works. The series of works [27, 26, 30, 32, 31] investigates game-based network centrality. Their aim is to capture a player’s centrality using various instances of semivalues, which are based on the player’s expected payoff. In contrast, our study aims at games where the payoff of players are given a priori by degree centrality and focus on core stability. [7] uses non-cooperative games to explain community formation in a social network. Each player in their game decides among a fixed set of strategies (i.e. a given set coalitions); the payoff is defined based on gain and loss which depend on the local graph structures. [25] studies community formation through cooperative games. The payoffs of players are given by modularity and modularity-maximising partitions correspond to Nash equilibria. The focus is on community detection but not on social cohesion. Furthermore, our payoff function is not additively separable and hence does not extends from their model. Lastly, our work is different from community detection [13]. The notion of community structure originates from physics which focuses on a macro view of the network, while our work is motivated from group dynamics and focus on individual needs and preferences.

Paper organisation.

Section 2 presents the game model and discusses notions of popularity and social cohesion. Section 3 looks at several standard graph classes and characterizes core stability in each class. Section 4 links structural cohesion with our game-based social cohesion. Section 5 proves that deciding social cohesion of a given network is -complete. Section 6 relaxes social cohesion to a notion of social rationality and through experiments, this section connects this notion with community structures. Finally, Section 7 concludes and discusses future work.

2 Popularity Games and Social Cohesion

Network and games.

A social network is an unweighted graph where is a set of nodes and is a set of (undirected) edges. An edge (where ), denoted by , represents certain social relation between , such as attraction, interdependence and friendship. We do not allow loops of the form . If , we say that are adjacent. A path from a node to a node is a finite sequence of nodes where for all . The network is connected if a path exists between any pairs of nodes. Let be a social network. We define a cooperative game on where each node in is a rational player. The reader is referred to [29]

for more details on cooperative game theory.

Definition 1

A cooperative game (with non-transferrable utility) is a pair where is a set of players, and is a payoff function.

A coalition formation of is a partition of , i.e., , ; each set is called a coalition. The grand coalition formation is where is called the grand coalition. Cooperative games describe situations where players strategically build coalitions based on individual payoffs. A predicted outcome of the game is a stable coalition formation in the sense that no set of players have the incentive to “disrupt” the formation by binding into a new coalition. More precisely, let be a cooperative game. Take a coalition formation and set where and .

Definition 2

A non-empty set of players is blocking for if ; In this case we say that is blocked by .

In other words, if blocks , then every will get a higher payoff if they join .

Definition 3

A coalition formation of is core stable w.r.t. if it is not blocked by any set .

Popularity.

Social positions, as argued in sociometric studies, arise from the network topology [8]. A long line of research studies how different centralities (e.g. degree, closeness, betweenness, etc.) give rise to “positional advantage” of individuals. In particular, degree centrality refers to the number of edges attached to a node. Despite its conceptual simplicity, degree centrality naturally represents (sociometric) popularity, which plays a crucial role in a person’s self-efficacy and social needs [37, 9]. Popularity depends on the underlying group: a person may be very popular in one group while being unknown to another. Hence individuals may gain popularity by forming groups strategically. We thus make our next definition. The sub-network induced on a set is where . denotes and we write for .

Definition 4

The popularity of a node in a subset is .

Note that for every node . If has an edge to all other nodes in the graph , then is the most popular node in with . The popularity of any player is in the range .

Definition 5

The popularity game on is a cooperative game where is defined by .

An outcome of the popularity game assigns any player with a coalition . The sum of popularity of members of equals their average degree in , i.e. . The average degree measures the density of the set , which reflects the amount of interactions within , and thus can be regarded as a collective utility. In this sense, the popularity game is efficient in distributing such collective utility among players according to their popularity.

Social cohesion.

Social cohesion represents a group’s tendency to remain united in satisfying members’ social needs [6]. We express cohesion through core stability w.r.t. the popularity game : Suppose a coalition formation is not core stable. Then there is a set every member of which would gain a higher popularity in than in their own coalitions in . Thus there is a latent incentive among members of to disrupt and form a new coalition . On the contrary, a core stable represents a state of the network that is resilience to such “disruptions”. Thus, when the grand coalition formation is core stable, all members bind naturally and harmoniously into a single group and would remain so as long as the network topology does not change.

Definition 6

A network is socially cohesive (or simply cohesive) if the grand coalition formation is core stable w.r.t. the popularity game .

Example 1. Fig. 1(a) displays a network . The popularity is if , and the popularity is if . The set blocks as each member has popularity . The only core stable formation is . Adding the edge (shown in red) would make socially cohesive as the popularity of both in reaches . Fig. 1(b) displays another network where and for all . Note that this graph is socially cohesive as the grand coalition structure is not blocked. However, adding the edge (shown in red) will destroy social cohesion as then blocks . Social psychological studies often presume that more ties leads to higher cohesion; this example displays a more complicated picture: Adding an edge may establish cohesion, but may also sabotage cohesion.

Theorem 2.1 (Connectivity)

If a coalition formation of is core stable then any either consists of a set of isolated nodes, or induces a connected subgraph.

Proof

Suppose that is not a set of isolated nodes, and that where are non-empty and no edge exists between any pair in . Take a node with non-zero degree, and say, without loss of generality, that . Then

 pV1(u)=degV1(u)|V1|>degV1(u)|S|=degS(u)|S|=pS(u).

Hence does not contain .∎

Theorem 2.1 states that any two nodes (that are not isolated nodes themselves) not connected by a path have no incentive to be in the same coalition. Hence it is sufficient to only consider coalitions that induce connected sub-networks of a social network.

Definition 7

A set is called a social group of if induces a connected sub-network. A group structure is a coalition formation containing only social groups.

The next theorem shows that social cohesion is inherently a small group property, i.e., socially cohesive networks have bounded size.

Theorem 2.2

Let be the maximum degree of nodes in . Then is socially cohesive only when unless .

Proof

Suppose and . If , is not socially cohesive by Theorem 2.1. Otherwise, pick an edge . Then . This means that , and the edge forms a blocking set. Thus is not socially cohesive. ∎

3 Social Cohesion in Special Classes of Graphs

We now investigate our games on some standard classes of graphs and characterize core stable group structures.

Complete networks.

A graph is complete if an edge exists between any pair of nodes. It corresponds to the tightest social structure where all members mutually interact. Naturally, one would expect a complete network to be socially cohesive.

Theorem 3.1

Let be a complete network. The grand coalition is the only core stable group structure. Hence is socially cohesive.

Proof

Any induced sub-network of a complete network (where ) is also complete. Thus

 pS(u)=|S|−1|S|<|V|−1|V|=pV(u).

Therefore any player’s popularity is maximised in the grand coalition . ∎

Star networks.

A star network contains a node (centre), a number of other nodes (tails) where , and edges . Intuitively, the centre would like to be in a social group with as many players as possible, while a tail would like to be with as few others as possible.

Theorem 3.2

A group structure of a star network is core stable if and only if 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 and suppose the centre is in a social group with tails. Then and for any tail , . All players not in has popularity 0 as their social groups contain only one node.

• Suppose . Take any set that contains . If , then . If , then for some tail . In either case does not block . Hence is core stable.

• Suppose . Then let be the set of all tails not in . Then . Hence the set is a blocking set for as .

Thus is core stable if and only if . ∎

Complete Bipartite Graph.

A complete bipartite graph consists of disjoint sets of nodes with nodes each and (where ). Let be a group structure. For every , we use and to denote and , respectively.

Lemma 1

is core stable only if .

Proof

Suppose there is with . Since , there is with . Take any and . Then we have

 pS(u)=r(S)ℓ(S)+r(S)<12

,   and

 pH(v)=ℓ(H)ℓ(H)+r(H)<12.

Hence, the set blocks as . ∎

We next characterize core stable group structure in . In particular, perfect matchings, i.e., situations when every is matched with a unique player in , are core stable.

Theorem 3.3

A group structure of is core stable if and only if .

Proof

By Lemma 1, if is core stable then . Conversely, if , then any has payoff . Thus is core stable as every contains some player with payoff at most . ∎

We now turn our attention to with arbitrary and focus on a special type of group structures: A clan structure is a group structure that contains at most one non-singleton social group, called the clan; all other social groups contain only single players, called the exiles. The number is the number of exiles, i.e., . It is clear that any group structure of a star network is a clan structure. Theorem 3.2 then becomes a special case (when ) of the next theorem, which characterizes core stable clans structures of .

Theorem 3.4

For any , a clan structure of is core stable if and only if the clan contains all nodes in and . Thus, is socially cohesive.

Proof

Suppose is a core stable clan structure. Lemma 1 implies that the clan contains all nodes in and . If , and thus . Then the set blocks where and is the set of exiles in . Hence we must have .

Conversely, suppose and . Assume for a contradiction that blocks . Then for any , . This means that must contain a player that belongs to , and we must have . However,

 1 =nℓ(S)+n+ℓ(S)ℓ(S)+n =pS(u)+pS(v)

Contradiction. Thus is core stable. ∎

4 Structural Cohesion and Social Cohesion

White and Harary in [34] describe group cohesion using graph connectivity.

Definition 8 (White and Harary [34])

Let be a connected graph. The structural cohesion of a connected graph is the minimal number of nodes upon removal of which become disconnected.

As stated in [34], a larger implies that is more resilient to conflicts or the departure of group members, and is thus more cohesive. Moreover, Menger’s theorem states that is the greatest lower bound on the number of paths between any pairs of nodes. Hence is a reasonable measure of cohesion. We next link with our notion of social cohesion. In [16], a pair is seen as a type of “structural hole” that forbids communication and is thus referred to as an absent tie. For each and we define the following:

1. is the number of actual ties of within the group ,

2. is the number of actual ties of outside ,

3. is the number of absent ties (including itself) in , and

4. is the number of absent ties outside .

These variables give rise to a characterization of social cohesion. Intuitively, if is a blocking set, each member tends to have many actual ties within and absent ties outside , i.e., high and , and tends to have few absent ties in and actual ties outside , i.e., low and . Thus, we define for all , ,

 γ(u,S)\coloneqqfin(u,S)sout(u,S)−fout(u,S)sin(u,S) (1)
Lemma 2

For all , blocks if and only if .

Proof

For each , and

 pV(u)=deg(u)|V|=fin(u,S)+fout(u,S)fin(u,S)+fout(u,S)+sin(u,S)+sout(u,S)

The set blocks if and only if , which can be shown to be equivalent to using the above equalities. ∎

A network contains a minimal cut of size , i.e., removing from decomposes the graph into distinct connected components where , i.e., contains disjoint sets such that each is connected, and for any all paths between and go through . We further assume that and is chosen in a way where is as small as possible. Let be the size , and let be the largest possible length of the sequence of ’s. We first look at the case when .

Lemma 3

If and is socially cohesive, then .

Proof

Suppose and . Let be an optimal cut sequence. Take . As contains a cut node, and . Then . Since ,

 γ(u,A1) ≥fin(u,A1)(|V|−χ−1)−(χ−fin(u,A1)) =fin(u,A1)(|V|−χ)−χ.

Since , . Thus by Lemma 2, blocks the grand coalition . ∎

We now generalize Lemma 3 to graphs with higher structural cohesion.

Lemma 4

Suppose . Then any network is socially cohesive only if .

Proof

Suppose . Take an optimal cut sequence and . Since and , we have . Suppose . Then . One can then derive that . Thus any edge in forms a blocking set of the grand coalition formation . ∎

Lemma 4 can be used as a (semi-)test for social cohesion when : whenever exceeds , is not socially cohesive. Clearly, more graphs become socially cohesive as gets larger. Summarizing Lemma 3 and Lemma 4, we obtain the following result.

Theorem 4.1

Let be a network.

• If , then is not socially cohesive as long as .

• If and , then is not socially cohesive as long as

Remark 1

The only case left unexplained is when and . In this case there exist graphs with arbitrarily large but are socially cohesive.

5 The Computational Complexity of Deciding Social Cohesion

We focus on the computational problem of deciding if a network is socially cohesive. More precisely, We are interested in the decision problem :

INPUT

A network

OUTPUT

Decide if is socially cohesive.

Instead of considering networks in general, we restrict our attention to a special type of networks. The distance between two nodes and , denoted by , is the length of a shortest path from to in . The eccentricity of is . The diameter of the network is .

Definition 9

A graph is diametrically uniform if all have the same eccentricity; otherwise is called non-diametrically uniform.

We use to denote the set of all non-diametrically uniform connected graphs whose diameter is at most . Our goal is to show that the problem is computationally hard already on the class . The following is a characterization theorem for graphs.

Theorem 5.1

The network belongs to if and only if its nodes can be partitioned into two non-empty set and , where .

Proof

Since has diameter 2 and is not diametrically uniform, there is a non-empty set of nodes with eccentricity 1, and the other nodes (call them ) have eccentricity 2. The sets satisfy the condition in the theorem. ∎

Let be a graph in . We call as described in Theorem 5.1 the eccentricity partition of . We first present some simple properties of .

Lemma 5

The network in is socially cohesive if and only if no set blocks .

Proof

One direction (left to right) is clear. Conversely, suppose the network is not socially cohesive. Let be a blocking set of the grand coalition formation, i.e., . If . Then . However, which contradicts that fact that is a blocking set. ∎

By the lemma above, the structure of is crucial in determining social cohesion of . For any and , we recall the notions , and from Section 4, but re-interpret these values within the sub-network . Hence, we now set as , i.e., the number of ties that has within but not in , the other variables remain as originally defined. Thus

 |V2|=fin(u,S)+fout(u,S)+sin(u,S)+sout(u,S) (2)

We then define the value

Theorem 5.2

A network in is socially cohesive if and only if for all there exists such that

Proof

By Lemma 5, we only need to examine subsets . Every has

 pS(u)=fin(u,S)fin(u,S)+sin(u,S) and pV(u)=fin(u,S)+fout(u,S)+|V1||V2|+|V1|.

A set blocks if and only if . Note that

 fin(u,S)fin(u,S)+sin(u,S)>fin(u,S)+fout(u,S)+|V1||V2|+|V1| if % and only if

.

Applying (2), blocks if and only if , if and only if , as required. ∎

We now give a sufficient condition for social cohesion of an network. The size of a network is its number of nodes. A clique is a complete subgraph. The clique number of , denoted by , is the size of the largest clique. Turán’s theorem relates with the number of edges in :

Theorem 5.3 (Turán [33])

For any , if a graph with nodes has more than edges, then .

Lemma 6

For any social group , there exists with .

Proof

Let and suppose for all , . Since ,

 k−1 k−1 fin(v,S) >(c−1)(|S|−fin(v,S)) fin(u,S) >k|S|−kfin(u,S)−|S|+fin(u,S) kfin(u,S) >|S|⋅(k−1) fin(u,S) >(k−1)|S|/k

Thus contains edges. By Theorem 5.3, contains a size- clique, contradicting ’s definition. ∎

The following is a sufficient condition for social cohesion of an network . Intuitively, the set contains the most socially active members – those who interact with everyone else. Hence they serve as “socializers” who hold the group together. The larger gets, the more likely will be socially cohesive. There is a bound such that once exceeds it, the network is guaranteed to be socially cohesive.

Lemma 7

Suppose and . Then is socially cohesive whenever .

Proof

Suppose , . Take any . If has a size- clique, by Lemma 6 there exists with . Since , we have

 |V1|≥(c−1)(|V2|−c)≥fin(u,S)sin(u,S)sout(u,S)>λ(u,S)

and thus is socially cohesive by Theorem 5.2.

If contains no clique of size , then . Let . By Lemma 6, . Thus . Since ,

 λ(u,S) =fin(u,S)sin(u,S)(|V2|−fin(u,V2)−sin(u,S))−fout(u,S) =fin(u,S)sin(u,S)(|V2|−fin(u,V2))−fin(u,S)−fout(u,S) =fin(u,S)sin(u,S)(|V2|−fin(u,V2))−fin(u,V2).

Furthermore, since ,

 λ(u,S) =(|S|−sin(u,S))(|V2|−fin(u,V2))sin(u,S)−fin(u,V2) =|S|⋅|V2|−|S|fin(u,V2)−|V2|sin(u,S)sin(u,S)

Hence, is at least

 (c−1)(|V2|−c)−(|S|⋅|V2|sin(u,S)−|S|fin(u,V2)sin(u,S)−|V2|) =|V2|(c−|S|sin(u,S))−c(c−1)+|S|fin(u,V2)sin(u,S)≥0

The last step is by and . By Theorem 5.2, is socially cohesive. ∎

Theorem 5.4

The problem is -complete. Furthermore -hardness holds already for the class .

Proof

The complement of , , asks whether a set blocks the grand coalition of a given network ; this problem is clearly in and thus is in . For hardness, we reduce (asking whether a graph contains a clique of a given size ) to . -hardness of then follows from the -hardness of [14].

To this end, we construct, for a given and , a graph as in Alg. 1. Our goal is to show that is not socially cohesive if and only if contains a clique of size . It is clear that is a network with eccentricity partition . Let . Suppose . By definition of and , we have

 |V1|−(c−1)(|V2|−c)=|V2|(k−c)−k(k−1)+c(c−1)−d.

Since and , . By Theorem 7, is cohesive.

Conversely, suppose . Let be a clique of size . Take . Since