A Logic of Allies and Enemies Wiebe van der Hoek Louwe B. Kuijer Y`ı N. Wang´ [email protected] [email protected] [email protected] Dept. of Computer Science Dept. of Computer Science Dept. of Philosophy University of Liverpool University of Liverpool Zhejiang University Abstract We introduce the Logic of Allies and Enemies (LAE). This logic describes the dynamics of relations in a social network, under the assumption that agents have an tendency to estab- lish more stable relationships with others in the network, which was studied extensively in structural balance theory. Stability scores are introduced to evaluate the stability of a social network. We demonstrate a number of validities of this logic, apply it to concrete cases, and show that its model checking problem is PSPACE-complete. 1 Introduction Structural balance theory, also called “social balance theory” or simply “balance theory”, was initiated in Heider’s work on social psychology [15, 16, 17], and later reinvented by Harary et al. using graph theory in [12, 3, 14], where they use signed graphs to represent social networks of agents, with positive signs for allies or friends and negative signs for enemies or antagonists. This has become a basic framework for studying positive and negative relationships, and has since then become an active area in the field of social network analysis. A social network is (structurally) balanced if it meets certain structural conditions on its pos- itive or negative relationships between agents. For example, a “triad” of three agents that are all enemies of one another is considered unbalanced. The concept of structural balance has a dynamic nature in the sense that one may expect an unbalanced network to have a tendency to become bal- anced over time by switching the relationships between agents. There has been a growing interest in a dynamic context of social networks, carrying out empirical studies or building theories of how a social network evolve over time in regard to structural balance theory [8, 18, 32, 2, 20, 1, 33]. In this paper we study the dynamics of social networks from a new perspective, namely that of temporal logic. Following the ideas from structural balance theory, we define stability scores to evaluate the degree of balance of a network, and use it to determine the transition relation between social networks, so that a network evolves to a more stable one by switching a link between two agents. This generates a time tree with its nodes representing social networks, in which more stable ones are closer to the leaves. We develop a Logic of Allies and Enemies (LAE) based on Computation Tree Logic (CTL) [5] to characterize the evolution process. We discuss a few validities of LAE, and show that its model checking problem is PSPACE-complete. 1 To the best of our knowledge, this is the first investigation of social network dynamics using logic. There is existing work in which logic is used to reason about social networks, most notably a number of works [27, 29, 21, 4, 31] use logic of knowledge and belief to describe the way information spreads through a social network. The difference between those works and the current paper is that they describe how a network influences the exchange of information, whereas we describe how the relations in a network influence each other. The structure of the paper is as follows. First, in Section 2, we give a brief introduction to the structural balance theory, from the viewpoint of our work. In Section 3, we give an informal description of the dynamics of social networks. Then, in Section 4, we formalize the dynamics and define the logic LAE. Following that, in Section 5, we consider an extended example. Finally, in Section 6 we show that the model checking problem is PSPACE-complete. We conclude and discuss future work in the last section. 2 Background Heider [15, 16, 17], furthered by Newcomb [23, 24, 25], argues that attitudes of a person (p) towards an other person (o) and a third object (x) can influence each other. For example, p’s attitude towards o is affected by p’s attitude towards x if x is positively associated with o (say, o owns or likes x), and in a similar way p’s attitude towards o is affected by p’s attitude towards x. The concept balance arises naturally in a p-o-x model. For example, p and o like each other and they both like (or both dislike) x is in a balanced state, while it is unbalanced if one likes x while the other does not. Harary et al. [12, 13, 3] formalized Heider’s model using graph theory, proved some theorems on balance, and provided a basic method for calculating degrees of balance. Many further devel- opments and investigations were made in Harary et al.’s framework. This paper lies also in this strand. In this section, we briefly introduce the concepts and results of Harary et al.’s framework that are relevant to this paper. A signed graph or simply graph consists of a finite set of vertices, together with edges (a set of pairs of vertices1) which are divided into positive and negative ones. The sign of a positive edge is +, and that of a negative edge is −. An undirected graph is a graph whose edges are all unordered pairs. If a graph is not undirected, it is a directed graph, or a digraph. A cycle is a sequence of the form ha1a2; : : : ; an−1an; ana1i, such that each pair aiai+1 in it is an edge and a1; : : : ; an are all distinct. In the case with a digraph, a cycle by definition may be −−! −− −−! a sequence like ha1a2; a2a3; a3a1i (the direction of the edges do not matter) and is therefore also called a semicycle to avoid possible confusion. The sign of a cycle is the product of the signs of all its edges (where + is read as +1 and − is read as −1). When the sign of a cycle is positive, i.e., there is an even number of negative edges in the cycle, we say the cycle is positive, or otherwise it is negative. For example, a cycle of the form +−−+ is positive, because +1 · −1 · −1 · +1 = +1 is positive; and −+−−+ is negative. A graph is balanced if all of its cycles are positive, and unbalanced otherwise. 1Self-loops and parallel edges (i.e., edges that join the same pair of vertices) are usually not considered. 2 The length of a cycle is the number of edges in it. An n-cycle is a cycle of length n. For many real-world situations, cycles of different lengths play different roles regarding structural balance. For example, 3-cycles and 4-cycles are usually more important than, say, an 11-cycle, for determining balance. In general, the influence of a cycle decreases as the distance increases between vertices. A graph is n-balanced if all of its cycles of length not exceeding n are positive. An important special case is that of 3-balance, which only considers “triads” of three actors. For a complete undirected graph, i.e., an undirected graph where every pair of distinct vertices is connected by a (positive or negative) edge, there are four possible cases (up to equivalence of the vertices) for a triad of three agents. These cases are depicted in Figure 1, where a solid line represents a positive edge and a dashed line represents a negative edge. For the graphs illustrated by 1(a) and 1(c), the product of the edges is positive so they are balanced, while for 1(b) and 1(d) the product is negative so they are unbalanced. (a) +++ (b) ++− (c) +−− (d) −−− Figure 1: Illustration of four triads, where a solid line stands for a positive edge, and a dashed line stands for a negative edge. Instead of considering the products of the edge signs, the (un)balance of the triads in Figure 1 can also be seen as deriving from the following four sayings: • My friend’s friend is my friend. • My enemy’s enemy is my friend. • My friend’s enemy is my enemy. • My enemy’s friend is my enemy. In the balanced triads (a) and (c) the sayings hold, whereas in the unbalanced triads (b) and (d) one or more of the sayings are violated. Few networks are balanced in real life, yet it is conjectured and argued that a network has a tendency towards a balanced state over time. Some research, e.g., [25, 32] and the Sampson monastery dataset [28], supports this, and some studies describe the difficulty of a network, espe- cially a big network, reaching a balanced state [9, 7, 18]. 3 Pressures on Friendship and Enmity In some, but not all, variants of balance theory neutral edges are allowed, in addition to positive and negative ones. Here, we allow such neutral edges, so two agents a and b are either friends (+), enemies (−) or neither (0).2 When such neutral edges are present, it is no longer the case that the four sayings are satisfied if and only if the edge product is positive. For example, the triads +00 and ++0 both have edge product 0, but in +00 all four sayings are satisfied while in ++0 2We do not consider so-called “frenemies”, that are both friends and enemies of one another. 3 the saying “my friend’s friend is my friend” is violated. We consider ++0 to be at least partially unbalanced, so in our semantics we follow the four sayings.