IMPORTANCE OF AGENTS IN NETWORKS: CLIQUE BASED GAME-THEORETIC APPROACH Текст научной статьи по специальности «Математика»

Contributions to Game Theory and Management, XV, 189—199

Importance of Agents in Networks: Clique Based Game-Theoretic Approach*

Juping Li1, Anna Tur2 and Maksim Zavrajnov2

1 National Pipeline Network Group Southwest Pipeline Co, Ltd. Nanning Oil and Gas Branch, Liangqing District, Guangxi Zhuang Autonomous Region, China 2 St. Petersburg State University, Faculty of Applied Mathematics and Control Processes, 7/9, Universitetskaya nab., St. Petersburg, 199034, Russia E-mail: [email protected]

Abstract Centrality measures are commonly used to detect important nodes. There are some metrics that measure a node's connectivity to different communities. This paper extends the standard network centrality measures and proposes to estimate the importance of nodes in network as a solution of a cooperative game. Three ways of defining such cooperative game are introduced. Each of them uses the concept of a clique in graph. Examples are considered

Keywords: cooperative game, game on graph, Shapley value, clique.

1. Introduction

There are many ways to determine the importance (centrality) of network nodes, each based on some selected fairness property (Jackson, 2010; Klein, 2010). If we consider, for example, a social network, an important property of a community is the ability of its members to communicate with each other. Therefore, we can say that a community is the better the more participants it has who are able to interact with each other. Thus, if a community is defined by a graph, then we need to consider the size of the cliques in it when assessing the value of the community and use a metric for nodes that measures their connectivity to different cliques. This principle of measuring the centrality of nodes was considered in (Faghani, 2013). A metric has been proposed that measures the connectivity of a node to different communities or cliques. The cross-clique connectivity of a node is the number of cliques to which this node belongs.

On the other hand game-theoretic methods are successfully used to identification of key nodes (Mazalov et al., 2016; Mazalov and Khitraya, 2021; Skibski et al., 2017; del Pozo et al., 2011). According to this approach, the worth of each coalition is estimated based on a characteristic function introduced in a special way. And then importance of each node can be measured as its payoff in such a cooperative game.

In this paper, we propose to apply a game-theoretic approach to determine the centrality of network nodes based on the concept of clique and compare results with the notion of the cross-clique connectivity.

2. Basic Definitions

This section will briefly review some concepts from graph theory and game theory.

*The work of the second author was supported by the Russian Science Foundation grant No. 22-11-00051, https://rscf.ru/en/project/22-11-00051/

https://doi.org/10.21638/11701/spbu31.2022.14

2.1. Graph

A graph G = (N, A) is a set of vertices N = {1, 2,..., n}, and a set of edges A joining all or some of these vertices from N. If two vertices i G N and j G N are connected by an edge (i, j) G A, we say the vertices are adjacent.

A subgraph GS is the graph (NS, AS) that contains only a subset NS of the set of vertices N of the original graph but contains all the edges whose initial and final vertices are both within this subset.

Definition 1. A clique is a subset of vertices of an undirected graph G such that every two distinct vertices in the clique are adjacent.

Definition 2. A maximal clique is a clique that can not be extended by including one more adjacent vertex, that is, a clique which does not exist exclusively within the vertex set of a larger clique.

Definition 3. A maximum clique of a graph G is a clique, such that there is no clique with more vertices.

Definition 4. The clique number w (G) of a graph G is the number of vertices in a maximum clique in G.

2.2. Game on graph

Let r = (G, V) be a cooperative game on graph G = (N, A), where N is the set of agents (vertices represent agents) and V : 2N ^ R is the characteristic function, where V (0) = 0. A subset S of N is called a coalition and N is called the grand coalition.

There are some properties that are usually checked for characteristic functions.

1. Characteristic function is monotonic, if for every S c N and T c N such that S c T, we have V(S) < V(T).

2. Characteristic function is superadditive if for every S c N and T c N such that S n T = 0, we have V (S U T) > V (S) + V (T).

3. Characteristic function is convex if for every S c N and T c N, we have V (S U T) > V (S) + V (T) - V(S n T).

As a solution of cooperative game we consider the Shapley value. The Shapley value was proposed by Shapley (1953) in order to solve the problem of conflicts arising from the distribution of benefits among multiple players in the process of cooperation.

The Shapley value in the game r = (G, V) is the vector defined by

Shi = E (S - 1)n(n - S)! [V (S) - V (S\{i})], i G N. (1)

scn ies !

We have Shi = V (N) and, for superadditive games, Shi > V ({i}) for all

ieN

i G N.

3. Construction of Characteristic Function

As mentioned earlier, in this paper we will assume that the more agents in the coalition that can directly interact with each other, the higher this coalition should be rated. Next, three methods for constructing a characteristic function based on the concept of maximum cliques in a graph will be proposed.

3.1. Characteristic function Vi

In cooperative game theory, one of the main problems is the method of specifying the characteristic function. The value of the characteristic function of some coalition should reflect the worth of this coalition. But how to measure the strength of a coalition in a game on a graph? If we assume that the interaction of players in a coalition is possible only if the players can directly interact with each other, then it turns out to be logical to use the size of the maximum clique of this coalition as the value of the characteristic function.

Definition 5. ri = (G, Vi) is a cooperative game on a graph G = (N, A) with the characteristic function Vi : 2N ^ R defined by the rule

Vi (S) = w (Gs), Vi (0) = 0.

Here w (GS) - the clique number of subgraph GS.

Example 1. Consider an example of game A = (G, V1) on the graph G with five vertices presented by Figure 1.

Fig. 1. Graph G

Here Vl(N) = w(G) = 3, V1({2, 3,4}) = V1({3, 5, 4}) = V1 ({1, 2, 3,4}) = V1({5, 2, 3,4}) = V1({1, 3,4, 5}) = 3, Vi({1, 2, 5, 4}) = V1({1, 2}) = V1({2, 3}) = V1({2, 4}) = V1({3, 5}) = V1({3, 4}) = Vi({4, 5}) = V1({1, 2, 3}) = V1({1, 2, 4}) = V1({1, 2, 5}) = Vi({2, 3, 5}) = V1({1,4, 5}) = 2, Vi({i}) = 1 V e N.

3.2. Properties of characteristic function Vi

Consider some properties of the characteristic function Vl.

Note that if Q is a clique in Si c N and Si c S2, then Q is a clique in S2 too. So, w(GSl) < w(GS2) and Vi(Si) < Vi(S2). We can conclude, that Vi(S) is monotonic.

But the characteristic function Vi is not superadditive in common case. In Example 1, let S = {1,2}, T = {3,4,5}, then Vl(S) = 2, Vl(T) = 3, Vl(S U T) = 3 and we see, that Vl(S) + Vl(T) > Vl(S U T). This means the characteristic function Vl is not superadditive.

4. Characteristic function V2

Let's consider one more way of assessing the worth of a coalition, based on the concept of a clique.

Definition 6. r2 = (G, V2) is a cooperative game on a graph G = (N, A) with the characteristic function V2 : 2N ^ R defined by the rule

n

V2 (S) = E Skkpk(S), V2 (0) = 0.

k = 1

Here y#k(S) - the number of maximal cliques with cardinality k belonging to GS, and S > 1.

Note that multiplier Sk allows to increase the value of large cliques.

Example 2. Consider again the graph G presented by Figure 1. There are three maximal cliques in G: {1, 2}, {2, 3,4}, {3, 5,4}. Let S = 3. So

V2(N) = 33 • 3 • 2 + 32 • 2 = 180.

Calculating V2(S) for S c N we need to find maximal cliques in subgraphs GS.

V2({2, 3,4, 5}) = 33 • 3 • 2 = 162,

V2({1, 2, 3, 4}) = 33 • 3 + 32 • 2 = 99, V2({1, 2,4, 5}) = V2({1, 2, 3, 5}) = 32 • 2 • 3 = 54, V2({1, 3, 4, 5}) = 33 • 3 + 31 = 84, Vi({1, 2, 3}) = V2({1, 2, 4}) = V2({2, 3, 5}) = V>({2, 4, 5}) = 32 • 2 • 2 = 36,

V>({2, 3,4}) = V2({3, 4, 5}) = 33 • 3 = 81, V2({1, 2, 5}) = V2({1, 3, 5}) = V2({1, 3,4}) = V2({1,4, 5}) = 32 • 2 + 31 = 21, V2({1, 2}) = V>({2, 3}) = V>({3, 5}) = V>({2, 4}) = = V2({3, 4}) = V2({4, 5}) = 32 • 2 = 18, V2({1, 5}) = V2({1,4}) = V2({1, 3}) = V2({2, 5}) = 31 • 2 = 6, V2({i}) =3 Vi G N.

4.1. Properties of characteristic function V2

Consider some properties of the characteristic function V2.

Monotonicity Note that if add some vertices to coalition S c N, then this can increase existing maximal cliques or create new ones. So if S c T, then V2(S) < V2(T), and characteristic function V2 is monotonic.

Superadditivity

Proposition 1. Characteristic function V2 is superadditive.

Proof. Consider S, T c N, S n T = 0. We need to prove that

V2 (S u T) > V2 (S) + V2 (T).

Let S1, S2, S3,..., Sk be the maximal cliques in GS, and T1, T2, T3,..., T; be the maximal cliques in Gt .

Then S1, S2, S3,..., Sk, T1, T2, T3,..., T; are cliques in GSUT. But perhaps they are no longer the maximal cliques.

Let R1,..., Rt be the maximal cliques in GSUT. For any Sj the following situations are possible:

— Sj = Rj for some j and |Rj | = |Sj|;

— Sj c Rj, for some j, then |Rj| > |Sj|;

It is not possible that Sj c Rj and Sz c Rj for i = z. The same is for every Tj. But it is possible, that Sj c Rj and Tz c Rj for some i, z, j. In this case, |Rj | = |Sj| + |Tz |.

Then

S 1 Rl1 |R1| + ... + S 1 1 |Rt| > S 1 Sl1 |S11 + ... + S 1 Sk 1 |Sk | + S 1 Tl1 |T1| + ... + S 1 Ti| |T; |. We can conclude, that

V2 (S u T) > V2 (S) + V2 (T).

Another important property of characteristic functions is convexity. Characteristic function V2 (S) is not convex in the common case. We can demonstrate it on the example presented by Figure 2.

Fig. 2. Graph G

Let S = {1, 2, 3,4}, T = {1, 2, 3, 5}, S =1.2. Then

V2(S) = V2(T) = S3 • 3 • 2 = 10.368, V2(S U T) = S4 • 4 + S3 • 3 = 13.4784, V2(S n T) = S3 • 3 = 5.184.

So

V2(S U T) + V2(S n T) < V2(S) + V2(T). And we can conclude, that V2(S) is not convex.

5. Characteristic function V3

We have seen that the previous two methods of constructing a characteristic function are not universal, since in the general case such functions may not be superadditive or convex. Now introduce another way of solving this problem.

Definition 7. r3 = (G, V3) is a cooperative game on a graph G = (N, A) with the characteristic function V3 : 2N ^ R defined by the rule

n

V3 (S) = ^ Sfckafc(S), S> 1, V3 (0) = 0.

fc=i

Here ak (S) - the number of cliques of cardinality k belonging to GS and S > 1.

This function, unlike the previous one, takes into account all cliques of the subgraph formed by the coalition, not just the maximal cliques.

Example 3. As an example consider the graph G presented by Figure 1. Let S = 3. Find all cliques in the graph.

There are 5 cliques of cardinality 1: {1}, {2}, {3}, {4}, {5}; 6 cliques of cardinality 2: {(1, 2), (2, 3), (2,4), (3,4), (3, 5), (4, 5)}; 2 cliques of cardinality 3: {(2, 3,4), (3,4, 5)}. Note that V3 ({i}) = S = 3, Vi G N .

For coalition N we have ai(N) = 5, a2(N) = 6, a3(N) = 2, a4(N) = a5(N) = 0 V3 (N) = S • 1 • 5 + S2 • 2 • 6 + S3 • 3 • 2 = 285, Vs({1, 2}) = V2({2, 3}) = V3({2, 4}) = V3({3, 5}) = V3({3, 4}) =

= V3({4, 5}) = S • 1 • 2 + S2 • 2 • 1 = 24, V3({1, 3}) = V3({1, 4}) = V3({1, 5}) = V3({2, 5}) = S • 1 • 2 = 6, V3({1, 2, 3}) = V3({1, 2,4}) = V3({2, 3, 5}) = S • 1 • 3 + S2 • 2 • 2 = 45, V3({1, 2, 5}) = V2({1, 4, 5}) = S • 1 • 3 + S2 • 2 • 1 = 27, V3({2, 3,4}) = Vi({3, 5,4}) = S • 1 • 3 + S2 • 2 • 3 + S3 • 3 • 1 = 144, V3({1, 2, 3,4}) = S • 1 • 4 + S2 • 2 • 4 + S3 • 3 • 1 = 165, V3({5, 2, 3,4}) = S • 1 • 4 + S2 • 2 • 5 + S3 • 3 • 2 = 264, V3({1, 3,4, 5}) = S • 1 • 4 + S2 • 2 • 3 + S3 • 3 • 1 = 147, V3({1, 2, 5,4}) = S • 1 • 4 + S2 • 2 • 3 = 66.

5.1. Properties of characteristic function V3

Consider some properties of the characteristic function V3. Note that if Si c S2, then V3(Si) < V3(S2), so the characteristic function V3 is monotonic.

Another important property of characteristic functions is convexity. Proposition 2. Characteristic function V3 (S) is convex.

Proof. It was shown in (Shapley, 1971) that the definition of convexity of characteristic function is equivalent to fulfilling the following condition: for each S1 c N, S2 c S1, and each i G N \ S1

V3(Si U {i}) - V3(Si) > V3(S2 u {i}) - V3(S2). (2)

Consider arbitrary player i G N, S1, S2 c N such that i G S1, S2 c S1. Check if the condition (2) is satisfied.

Denote the set of all cliques of cardinality k with player i in GS as Pj (i). Then

|Si| + 1

V3 (Si U{i}) - V3 (Si)= E ¿kIPklU{i}(i)|k,

V3 (S2 U{i}) - V3 (S2)= E |P|2u{i}(i)|k.

k = 1 | S2I + I

xk\ Ttk

k=1

Since S2 c S1, then every clique in S2 is also a clique in S1.

So' PJ2u{i}(i) c Pl1u{i}(i) for each and

IS1I + 1 IS2I + 1

E ^k|PkiU{i}(i)|k > E ^k|Ps2U{i}(i)|k.

k=1 k=1

Condition (2) is satisfied, so the characteristic function is convex.

5.2. The Shapley value

Consider the Shapley value as the cooperative optimality principle in the game A = (G, V3) .

Proposition 3. In the game I3 = (G, V3) the Shapley value has the form:

n

Shi(G3) = E ,

k=1

where A], is the number of cliques in G with k elements containing the node i.

Proof. Consider a clique in G with k elements. This clique contributes k units to the total payoff V3(N). If to delete the link between any i and j from this clique, then N will lost this gain. So, each player from this clique hopes to receive at least equally from these k units.

We get that from each clique in G with k elements player from this clique hopes to receive at least = ¿k.

If A|. is the number of all clique with k elements containing the node i, then

S2 A S3 A"i Sk A" »n

Shi = S + ^-A2 • 2+ • 3+ ... + ^-An • n = £SkA-.

k = i

Note that if S =1, then the Shapley value Sh(G3) coincides with the cross-clique connectivity proposed by Faghani (2013).

5.3. The Importance of Agents

As mentioned earlier, the cooperative solution in a game on a graph can be used to evaluate the importance of agents (vertices). Assume that the Shapley value is chosen as cooperative solution. Denote as Sh(r) the Shapley value in game rj. Let

Sh"(rj) a(r)= Vj(N) .

And it becomes clear that the value «¿(rj) G [0,1] can be considered as the importance of vertex i according to game rj.

6. Comparing of Results. Examples

To compare three characteristic functions, we consider some special types of graphs for which we can obtain formulas for calculating the Shapley value and importance a in explicit form.

Example 4. First consider the example shown in the Figure 3. Here we can see a graph star.

Fig. 3. Star graph

Let N = {1,...,n}.

Here in the game ri we have:

Vi (N) = 2, Vi (0) = 0,

Vi ({1}) = 1, Vi(S) = 1 VS : 1 GS, |S|> 1 Vi(S) = 2 VS : 1 G S, |S| > 1.

Then we can get the formula for the Sapley value and importance of agents:

Shi(ri) = 1, ai(ri) = 1,

In the game r2

And

Shj (A) =

n - 1' j 2(n - 1)

, (A) =

j = 1.

V2 (N) = 2(2(n - 1), V2 (0) = 0, V2({i}) = (, Vi = 1,...,n, V2(S) = 2(|S|- 1)(2 VS : 1 G S, |S| > 1.

„, , , ((2 + n - n2) 12 + n - n2

Shi(A) = (n - 1)(2 + -i---, ai (A) = - +

2n

2 4(n(n - 1)'

Sh (r ) (2 , ((n2 - n - 2) (r ) 1 ((n2 - n - 2) . = 1 Shj(r 2) = ( +--—-——, aj(A) = ^-TT +—77Z7Z-TvT, j = 1

2n(n - 1)

2(n - 1) 4(n(n - 1)2

In the game A

And

V3 (N) = 2(2(n - 1) + n (, V3 (0) = 0, V3({i}) = (, Vi = 1, ••• ,n,

V3(S) = 2(|S| - 1)(2 + (|S| VS : 1 G S, |S| > 1.

Shi(A) = (n - 1)(2 + (, ai(A)

1 + (n - 1)( 2 ((n - 1) + n,

Shj (A) = (2 + (, aj (r2) =

1+

2 ((n - 1) + -

j = 1.

Note, that ai(ri) > ai(r2) and ai(ri) > ai(r3) for n > 2. Thus, the importance of player 1 is rated higher when using the first characteristic function.

Consider another example with a more complex graph structure.

Example 5. Find the importance of every vertex of graph G presented by Figure 4.

Calculate the Shapley value and importance for each vertex using characteristic functions of different types. Table 1 demonstrates the results obtained.

It can be seen that in all three cases vertex 3 has the highest weight. Moreover, the characteristic function V2 gives the highest value a3 (with S = 2).

When increasing S the importance of vertices 3, 4, 5, 6 has increased because they are in 4-element clique, while the importance of vertices 1, 2 has decreased.

1

1

Fig. 4. Graph G Table 1. Comparing of Results

S = = 1 S= 2

i Shi (A) at(ri) Shi(r2) ai(r2) Shi (A) o.i(r3) Shi(r2) ai (/2) Shi (A) ai(r3)

1 0.35 0.088 1.133 0.16 4 0.093 8.27 0.094 18 0.068

2 0.35 0.088 1.133 0.16 4 0.093 8.27 0.094 18 0.068

3 1 0.25 1.583 0.23 11 0.256 23.16 0.263 70 0.261

4 0.767 0.192 1.05 0.15 8 0.186 16.1 0.183 54 0.201

5 0.767 0.192 1.05 0.15 8 0.186 16.1 0.183 54 0.201

6 0.767 0.192 1.05 0.15 8 0.186 16.1 0.183 54 0.201

Vi(N = 4 V (N =7 V3(N) = 43 V2 (N) = 88 V3(N) = 268

7. Conclusion

The problem of finding influential agents in networks is considered. The game-theoretic approach is applied. Three ways of constructing a characteristic function are proposed. The first characteristic function assigns to each coalition a worth equal to the clique number of the subgraph built on this coalition. It is shown that such characteristic function is not superadditive in common case. The second way of characteristic function constructing assigns to each coalition a worth equal to the sum of cardinalities of all maximal cliques formed by the participants of that coalition. It is shown that such a characteristic function is not convex in the common case, but it is superadditive. The third method assigns each coalition a value equal to the sum of cardinalities of all cliques formed by the players of this coalition. It is proved that such characteristic function is always convex. A formula for calculating the Shapley value is obtained. A method for measuring the importance of vertices based on the Shapley value in these games is proposed.

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