Computers, Materials & Continua DOI:10.32604/cmc.2022.021804 | |

Article |

Extremal Coalitions for Influence Games Through Swarm Intelligence-Based Methods

1Escuela de Ingeniería Informática, Universidad de Valparaíso, Valparaíso, Chile

2Computer Science Department, Universitat Politècnica de Catalunya, Barcelona, Spain

3Mathematics Department, Universitat Politècnica de Catalunya, Terrassa, Spain

*Corresponding Author: Fabián Riquelme. Email: fabian.riquelme@uv.cl

Received: 14 July 2021; Accepted: 19 August 2021

Abstract: An influence game is a simple game represented over an influence graph (i.e., a labeled, weighted graph) on which the influence spread phenomenon is exerted. Influence games allow applying different properties and parameters coming from cooperative game theory to the contexts of social network analysis, decision-systems, voting systems, and collective behavior. The exact calculation of several of these properties and parameters is computationally hard, even for a small number of players. Two examples of these parameters are the length and the width of a game. The length of a game is the size of its smaller winning coalition, while the width of a game is the size of its larger losing coalition. Both parameters are relevant to know the levels of difficulty in reaching agreements in collective decision-making systems. Despite the above, new bio-inspired metaheuristic algorithms have recently been developed to solve the NP-hard influence maximization problem in an efficient and approximate way, being able to find small winning coalitions that maximize the influence spread within an influence graph. In this article, we apply some variations of this solution to find extreme winning and losing coalitions, and thus efficient approximate solutions for the length and the width of influence games. As a case study, we consider two real social networks, one formed by the 58 members of the European Union Council under nice voting rules, and the other formed by the 705 members of the European Parliament, connected by political affinity. Results are promising and show that it is feasible to generate approximate solutions for the length and width parameters of influence games, in reduced solving time.

Keywords: Influence game; influence spread; collective behavior; swarm intelligence; bio-inspired computing

Cooperative game theory deals with the study of players forming coalitions to achieve a common benefit, enforcing cooperative behavior [1]. Simple games are cooperative games where the benefit obtained by the coalitions is always binary, i.e., coalitions can either win or lose [2]. Simple games are one of the most fundamental models for decision-making [3], and have been used to solve problems arising voting systems, social choice theory, logic and threshold logic, circuit complexity, artificial intelligence, geometry, linear programming, Sperner theory, order theory, among other disciplines [3–5]. Besides the traditional forms of representation of simple games [6], since the 2010s, different formulations based on graphs have emerged, with the aim of applying the knowledge acquired in cooperative game theory in network science [7]. In this context, influence games arise as simple games defined by influence graphs (i.e., weighted, labeled graphs) on which an influence spread phenomenon is exerted. In an influence game, an initial activation of nodes (i.e., a coalition of players) is winning if it is able to influence at least a minimum number of nodes, called the quota of the game [8]. It is known that any simple game can be represented by an influence game, and vice versa [8]. Furthermore, influence games and influence graphs have been already used to solve problems of multi-agent systems [8], social network analysis [9,10], and collective decision-making models [11,12].

Despite the great expressiveness of influence games, it is a compact form of representation of simple games. Therefore, the computation of several parameters and properties coming from cooperative game theory on influence games is computationally hard (NP-hard, coNP-complete, or #P-complete) [8,11,12]. In the same vein, there are problems inherent to the influence spread phenomenon that are computationally hard. The best known is the influence maximization problem (IMP), one of the main problems of social network analysis [13]. This problem refers to finding k seed nodes in a network such that, under some influence spread model, the expected number of nodes influenced by the k seeds is the largest possible [14]. As a minimization problem, another possibility is to find the minimum seed set that allows influencing a given number of nodes [15]. This variation is also known as the least cost influence problem (LCI) [16].

Although the influence spread phenomenon was already studied in the 1970s to analyze collective behavior [17,18], its computational applications began to develop widely in the area of marketing in the early 2000s [19,20]. Two of the best-known general influence spread models are the linear threshold model (LT-model) and the independent cascade model (IC-model). For both, IMP and LCI turn out to be NP-hard [14,15]. Although the IC-model is often used more, due to its ease of implementation, as it is a predictive model, it produces less replicable results to experiment in case studies [8,21]. The high complexity of IMP and LCI makes it impractical to obtain optimal solutions through linear programming, integer programming, dynamic optimization, or other exact algorithms on networks with a large number of nodes. Therefore, since the 2010s different proposals have emerged to solve these problems in an approximate and efficient way. The solutions include greedy, approximation and community-based algorithms, heuristics, bio-inspired metaheuristics, among others [9]. Recently, a multi-objective optimization model was proposed to find the smallest set of nodes that can influence the largest number of nodes in the network [9].

In this work, we propose two optimization models inspired on [9] to find approximations to the length and width of an influence game. The length of a game is the size of the smallest winning coalition, while the width is the size of the largest losing coalition. Both parameters were firstly defined in 1990 as indicators of efficiency for decision-making [22]. The problems of computing the length and the width in an influence game are NP-hard [8]. As far as we know, this is the first time that both problems have been studied on influence games with a large number of players. As a case study, we compute the length and width of an influence game representing the simple majority voting system of the 705 members of the European Parliament, connected according to their political affinity. To face these case studies, we employ particle swarm optimization (PSO), which is a widely known swarm intelligence algorithm. The have three reasons to choose this technique over others: (a) any bio-inspired methods designed under the swarm collaborative behavior can be applied to this proposal; (b) PSO is one of the most popular optimizer algorithms, so there is extensive literature that reports its excellent efficiency [23]; and (c) it is a metaheuristic algorithm that has shown excellent performance to solve the similar IMP problem on which the formulation of the length and width approximation problems are based [21]. For the parameter adjustment of this optimizer, we will previously use a small case study of only 27 players, corresponding to the European Union Council under nice rules [24], in which we can compare the results with the exact solutions.

The article continues as follows. Section 2 presents the fundamental concepts related to influence games, along with the length and width of a game, and the influence maximization problem. Section 3 is devoted to model the optimization problems to approximate the length and the width of an influence game, and to present as well the bio-inspired metaheuristic algorithms used to solve those problems. In Section 4 we present the case studies, related with real voting systems. In Section 5, we show and discuss the results obtained by applying the models and methods defined in Section 3 to find approximations of the length and the width in influence games with a large number of nodes. Finally, Section 6 presents our main conclusions and future work.

Here we use notation from [3,8]. A simple game is a type of cooperative game defined by a pair

An influence graph is a tuple

Influence graphs are structures on which different diffusion dynamics can be applied, determined by influence spread models. In an influence graph, the vertices can adopt two states, active or inactive, depending on whether or not they have been influenced through a certain influence spread model. In general, the idea is that from an initial set of vertex activation (the so-called seed), these vertices can be iteratively activating others in the graph or network until the active vertices can no longer continue to spread their influence. This iterative process stops when either all the network has been influenced, or there is not enough influence power to activate the remaining vertices. Note that the latter can be the case when the remaining vertices are not reachable from the active ones.

Formally, given an influence graph

This process stops when no additional activation occurs. The final set of activated nodes is

An influence game is a tuple

In an influence game, minimal winning coalitions represent non-redundant alliances between players, such that each player is essential to “win” (by “win,” we mean passing a vote, adopting a product or service, making a collective decision, etc.). On the other hand, maximum losing coalitions are alliances that only need one more player to win. Despite the above, two minimal winning coalitions (or maximal losing coalitions) may have different cardinalities, so the coordination efforts required to be formed are different. Indeed, in a context of limited resources, we might be interested in forming an initial seed of players as small as possible, but with the ability to spread their influence to the other nodes in the network, in order to reach the necessary quota to win. Similarly, we might also be interested in knowing in advance what is the greatest wear or coordination effort to invest so as not to win the game. The latter would help us to know the inherent risk within the game. All the above leads us to the concepts of length and width of a game, as efficiency indicators for decision-making processes [22].

Let

• The length of the game is the minimum size between the winning coalition of the game, i.e.,

• The width of the game is the maximum size between the losing coalition of the game, i.e.,

Sometimes, the width of a game is also defined as the minimum size of a blocking coalition [25], or as

2.3 Influence Maximization Problem

Let

Although we have worked with set notation so far, optimization problems often work with vector notation as well. In fact, a set

A similar problem is the least cost influence problem (LCI), also known as target set selection problem (TSS) [27], which aims to minimize the seed

Although both IMP and LCI are NP-hard [14,15], there exist several efforts to find approximate solutions to these problems [21]. Both problems are closely related. As we already mentioned in Section 2.2, sometimes it is not enough to maximize the spread of influence, but it is also necessary to find a satisfactory seed. In this sense, by “satisfactory” we mean that it has a low cardinality of elements, and that it is minimal, i.e., if any actor

3.1 Optimization Models to Compute Extremal Coalitions

First, we define the length and width problems for an influence game using the integer programming paradigm. Given an influence game

while the expression (5) determines how maximum losing coalitions must be:

Note that both optimization problems allow us to approximate extreme coalitions of a given game. The length corresponds to a global optimal solution for (4) which is also an optimal solution for (3). The width is a global optimal solution for (5). These metrics are not usually studied as combinatorial optimization problems. We remark again that computing the global optimal solutions for both problems on influence games is computationally intractable [8]. Note also that, unlike in [21], these problems are not multi-objective.

3.2 Swarm Intelligence Methods

Swarm intelligence methods are a type of naturally bio-inspired metaheuristic algorithms. Like evolutionary algorithms, they are based on an initial population of possible solutions that improve over time. The swarm intelligence methods look for a common collective behavior between the particles that are part of this population. In this work, we use the particle swarm optimization method (PSO), which has already shown a correct performance for related problems [21].

The particles in PSO represent birds or fishes, each of which has two components: position and velocity. Each of these particles is a candidate solution. A set of particles forms a swarm, which evolves over time, where time is represented by discrete iterations. For each particle, the algorithm changes its velocity through the search space and then updates its position, depending on its own experience and the position of neighboring particles [28,29,23]. PSO has two well identifiable phases. In the initial phase, the particles increase their velocity so that current solutions lead to greater diversification. Once a high diversification is reached, the velocities progressively decrease to zero, and the second phase of intensification begins to search for more suitable solutions. This phase is executed in positions memorized as pBest. This memory capacity is precisely what allows PSO to continue improving its solutions over time. Therefore, the objective of the initial phase is to find pBest positions within the fittest attraction basins.

Velocity and position are represented by the vectors

where

This algorithm allows solving both the length and the width problems. Section 4.3 explains how the functions of lines 4, 7, 28 and 30, were set. In lines 7 and 30, each time the domain of the particles is altered, its feasibility must be evaluated. These feasibility restrictions must be adapted to our context, so that they work for any influence game. Instead, the population generation (line 4) and the binarization process (line 28) must be adjusted on a case-by-case basis for each specific network or case study.

4 Voting Systems Experimentation

The main objective of this work is to apply the solution developed in Section 3 to find approximations to the length and width of influence games with a large number of players. As case studies, we consider real voting systems. In this context, players represent voters, and winning coalitions are sets of voters who can approve a law or amendment. Therefore, the quota of an influence game defines the minimum number of votes necessary to approve said law or amendment. As in [21], we first use a small case study on which the length and width can be computed by brute force. In this way, we fix the parameters of the metaheuristic algorithms to reach the ideal solutions. Then, with these parameter settings, we apply the algorithms to approximate the length and width of a much larger voting system, for which finding the exact solution is not feasible.

4.1 European Union Council Under the Nice Rules

The first case study corresponds to the qualified majority voting system of the Council of the European Union, modified by the Treaty of Nice in 2005 [24]. This simple game can be represented as an influence game

•

•

• Let be

• The node labels are

•

To see the formal details on the construction of this case study, see Appendix A. The resulting network is illustrated in Fig. 1.

The second case study corresponds to the majority voting system of the European Parliament (EP) in 2021. In this case, the EP is directly represented as an influence graph, corresponding to the social network of the different parliamentarians related by political affinity. To model this voting as an influence game, we must quantify the political affinity leading to the influence game

•

•

•

• both voters represent the same country;

• both voters represent the same country and the same local political party;

• both voters represent the same EP political group;

• some of the two voters are “Non-Inscrits” but the political wing of both coincides (if some voter has no political wing, then it is not comparable with any other).

For each condition that is met above, add +1 to the edge between both players.

•

•

The whole dataset can be found in [30]. The resultant network is illustrated in Fig. 2. Note that this is a new social network defined for this specific work.

All experiments were executed on a computer with an Intel Core i7-8700 processor, 32GB RAM 2666, 256GB SSD NVMe storage, on a 64-bit Windows 10 Pro operating system.

First, a brute force algorithm was implemented to calculate the length and width of an influence game given as input. The algorithm searches for all possible combinations of initial activations and determines its influence spread within the network, saving the smallest activations capable of reaching the quota (length) and the largest ones that do not reach the quota (width). This algorithm was run for the case study of the Council of the European Union presented in Section 4.1. The execution took 44 min. As a result, a width = 24 and a length = 14 was obtained. For the width, the following three optimal combinations were found (players are ordered according to Appendix A):

Solution 1: {2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}

Solution 2: {3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}

Solution 3: {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}

For the length, 2317 optimal combinations were found. This big difference in the number of solutions should not be surprising because of the second weighted voting game considered in the case study (see Appendix A). All solutions were saved in a human readable file to identify the efficient solutions. Note that using this same algorithm for the European Union Parliament case study (Section 4.2) is intractable due to the large number of players. Therefore, we do not know the optimal solutions for this second influence game.

Second, PSO was implemented from the coding of [31], by following the structure presented in Algorithm 1. The new code is available in [30]. We performed previous executions to test different parameter configurations on a smaller case study in which the length and width can be computed by brute force. These preliminary experiments allowed us to study the parameter values of PSO, and which were later used for the toughest instances. Consistently, the best configuration was the already defined in [32]:

Third, the bio-inspired algorithms were executed on the second case study with the same parameter settings obtained from the first case study. The results are discussed in the next section.

For the feasibility constraint (lines 8 and 31 of Alg. 1) two cases are considered. For the length calculation, it was considered that the particle is feasible if

Note that our proposal only works with feasible solutions, i.e., the solution vectors that do not satisfy the constraint, are not included in the swarm of solutions. To meet this requirement, a simple but powerful criterion of rejection is implemented. This method operates in a cycle way, each time a new solution is found. While a feasible solution is not reached, evolutionary changes depending on the algorithms are performed. This method seems to require an additional computational effort but in practice it is not. These computational experiments show it.

The results obtained for computing the length and the width in the case study of the network of European parliamentarians related by political affinity are presented and discussed below. To compute the length and the width, 60 executions were carried out in parallel. For the weight we considered 5000 iterations for each execution. For the length, we consider just 100 iterations, since the results were improving faster than for the weight.

5.1 Length of the European Parliament

Parallel executions of length all took around 900 s, that is, 15 min. The general results are shown in Tab. 1. Note that the values obtained for each row correspond to different executions, as they represent the “best” solutions of each type. In this context, the best solution found for the length is

Fig. 3 presents a convergence plot for the best solution as the number of iterations progresses. The size of the coalition (initial activation

5.2 Width of the European Parliament

Parallel executions of width all took around 11400 s, that is, 3.15 h. Although this is still a fairly low execution time (remember that computing the exact solution is intractable), it is much longer than for the length computation. This difference is because the solution space for width is much smaller than for length, or in other words, this game seems to have many more winning coalitions than losing coalitions. Therefore, PSO takes much longer between one iteration and the next to achieve a new improvement. This same reason was what led us to modify the randomization of the initial particles, described in Section 3.2; because the algorithm took a long time to find a particle that corresponded to a losing coalition (with a uniform random distribution, the initial particles turned out to be all winning coalitions).

The general results are shown in Tab. 1. In this context, the best solution found for the width is

However, in this case, something different happens than in the length computation. Due to the monotony of the linear threshold model, note that the 14 or 15 players influenced by the best solutions could have been included as part of the initial activation. Thus, we would have an initial particle with

Fig. 4a presents the convergence plot for the solution with

Many NP-hard problems, computationally intractable from a certain input size, can be solved by approximation algorithms, such as metaheuristics based on bio-inspired algorithms. These types of solutions, very common in graph theory, have been little explored to solve game theory problems. In this article, we use the PSO swarm intelligence algorithm to find efficient approximations for the length and width of an influence game. The modeling of these problems was based on other problems already known as the influence maximization problem. In cooperative game theory, influence games are a way to represent simple games as graphs or social networks.

As a case study, a social network formed by the 705 members of the European Parliament, related by political affinity, was considered. The best approximation for the length (i.e., the minimum winning coalition) was 213 players. Therefore, a coalition made up of only 30% of the parliamentarians was found, being capable of influencing another 145 players and thus exceeding the simple majority quota of 353 votes. For the width (i.e., the maximum losing coalition), two solutions of 330 members (44% of the parliamentarians) were found. These initial coalitions can influence a few additional players but still not capable of exceeding the game quota. Remarkably, in the case of the width, implicit solutions can be seen, given by the influence spread of the losing coalitions. As the higher influence spread found is 352, there is a coalition of 352 members, which is a loser and is not capable of influencing anyone else. Therefore, the best approximation of the width found for this case study is 352, although we do not know which players form this coalition.

Note that the solutions found include European MPs from different countries and political ideologies. As future work, ideological or country restrictions could be considered for the optimization problems (4) and (5). Thus, we could find the extremal coalitions made up of members of a certain political spectrum. This issue is beyond the original formulation of the length or width problems, but it would be useful from the point of view of voting theory and cooperative game theory.

Finally, we hope this approach could inspire new ways to solve game theory problems for larger instances than usual. In this sense, the problems representable as optimization problems are not few.

Funding Statement: F. Riquelme has been partially supported by Fondecyt de Iniciación 11200113, Chile, and by the SEGIB scholarship of Fundación Carolina, Spain; X. Molinero under grants PID2019-104987GB-I00 (JUVOCO); M. Serna under grants PID2020-112581GB-C21 (MOTION) and 2017-SGR-786 (ALBCOM).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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Appendix A. European Union Council voting system as influence game

Let

Let

• The set of players

• The vertex set

• The edges set

• The weight function

•

•

Under this construction, note that a coalition (of players) wins iff it is capable of activates the auxiliary nodes

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