Computing global structural balance in large-scale signed social networks

Abstract

Structural balance theory affirms that signed social networks (i.e., graphs whose signed edges represent friendly/hostile interactions among individuals) tend to be organized so as to avoid conflictual situations, corresponding to cycles of negative parity. Using an algorithm for ground-state calculation in large-scale Ising spin glasses, in this paper we compute the global level of balance of very large online social networks and verify that currently available networks are indeed extremely balanced. This property is explainable in terms of the high degree of skewness of the sign distributions on the nodes of the graph. In particular, individuals linked by a large majority of negative edges create mostly "apparent disorder," rather than true "frustration."combinatorial optimization | social network theory O nline social networks are examples of large-scale communities of interacting individuals in which local ties between users (friend, fan, colleague, but also friend/foe, trust/distrust, etc.) give rise to a complex, multidimensional web of aggregated social behavior (1-4). For such complex networks, the emergence of global properties from local interactions is an intriguing subject, so far investigated mostly at structural and topological level (2,(5)(6)(7)(8). In social network theory (9-11), however, the content of the relationships is often even more important than their topology, and this calls for the development of appropriate analytical and computational tools, able to extrapolate content-related features out of the set of interactions of a social community. Obtaining efficient tools is particularly challenging when, as in social networks retrieved from online media, the size of the community is very big, of the order of 10 5 individuals or higher.A global property that has recently attracted some attention (1,(12)(13)(14) is determining the structural balance of a signed social network. Structural (or social) balance theory was first formulated by Heider (15) in order to understand the structure and origin of tensions and conflicts in a network of individuals whose mutual relationships are characterizable in terms of friendship and hostility. It was modeled in terms of signed graphs by Cartwright and Harary (16); see refs. 10 and 11 for an overview of the theory. The nodes of the graph represent users and the positive/ negative edges their friendly/hostile relationships. It has been known for some time how to interpret structural balance on such networks (16): The potential source of tensions are the cycles of the graph (i.e., the closed paths beginning and ending on the same node), notably those of negative sign (i.e., having an odd number of negative edges). It follows that the concept of balance is not related to the actual number of negative edges on the cycles but only to their parity; see Fig. 1 for an illustration on basic graphs. In particular, a signed graph is exactly balanced (i.e., tensions are completely absent) if and only if all its cycles are positive (16). As such, structural balance ...

Deep Belief Network-Based Approaches for Link Prediction in Signed Social Networks

“…Finally, we could use the evaluations introduced in Section 5.3 to show the FriendTNS + − 's ability at predicting the sign value of links. We select three k values (5,10,20) in this experiment, and k is set to M IN (numberof positivelinkof u, numberof negativelinkof u) when node u does not have at least k positive links or negative links. The experiment result is shown in Section 6.1 for this algorithm; it does not need to know the link value when calculating the linked nodes' similarities, which point looks like the unsupervised learning.…”

“…However, there are not many structure features that can be extracted from the structure of the SSN itself. Recently, researchers started to study the "meaning" of SSN structure features [8][9][10] and tried to find methods by some social psychology theories, such as structural balance theory [11]. However, the theory-based method is so strict that it cannot model the principles that drive the behaviors of social members effectively, especially the hidden principles.…”

Interaction sign patterns in biological networks: From qualitative to quantitative criteria

“…. , σ n ), with σ i ∈ {±1}, and the associated gauge matrix Σ = diag(σ) (as defined in [11]), system (1) is monotone w. r. t. σ if, for all x 1 (0), [25], [26]. The ordering is strict if, in addition, strict inequality holds for at least one of the coordinates of…”

“…Section III deals with criteria to determine when a given stable Jacobian J can admit a SSIM M = sgn(−J −1 ) that is fully nonnegative, or is the gauge transformation [11] of a nonnegative matrix (i.e., is similar to a nonnegative matrix through a diagonal signature matrix). This happens when the system is respectively cooperative, or monotone: in these cases, the SSIM can be computed in a purely qualitative way.…”

Dynamics of social balance under temporal interaction

“…Heider's social balance for triads was generalized to a concept of balance for networks, i.e., the so-called structural balance [16]. Heider's social balance and its variants are empirically found in human [17][18][19][20][21][22][23][24][25] and other animal [26] societies (also see [27,28] for related measurements).…”