Rethinking balance theory: a probabilistic approach to the study of signed networks

Networks provide a powerful language to model interacting sys- tems, by representing their units as nodes and the interactions be- tween them as links. Interactions can be connotated in several ways, such as binary/weighted or undirected/directed. In this thesis, we focus on the positive/negative connotation - modelling trust and distrust, alliance and enmity, etc. - by considering the so-called signed networks. Rooted in the formulation of Balance Theory, a psy- chological paradigm that has been proposed to model human re- lationships, the study of signed networks has found application in fields as different as biology, ecology and economics. Here, we ap- proach it from the perspective of statistical physics by extending the framework of Exponential Random Graph Models (ERGMs) to the classes of binary, (un)directed, monopartite, and bipartite networks. First, we employ signed ERGMs to assess the statistical significance of frustrated patterns in biological, economic, and social, undirected, signed networks. As our results reveal, the level and nature of struc- tural balance embodied in these networks critically depends on i) the considered system and ii) the employed benchmark: while so- cial networks align with balance theory, biological networks often exhibit a pronounced level of frustration. We further investigate undirected, signed structures at the mesoscopic scale, by evaluating the tendency of a configuration to be balanced either in the ‘tradi- tional’ or in a ‘relaxed’ sense. In the case of binary, directed net- works, we employ ERGMs to provide a first analysis of the level of signed reciprocity and its relationship with frustration, suggesting an alternative interpretation of balance in the light of directionality. Finally, we propose an unsupervised algorithm to obtain statisti- cally validated projections of bipartite, signed networks, according to which any two nodes sharing a statistically significant number of concordant (discordant) relationships are connected by a positive (negative) edge. We test our method on several real-world config- urations: in all cases, non-trivial mesoscopic structures, induced by relationships that cannot be traced back to the constraints defining the employed benchmarks, are detected.