Title: Strong, weak or no balance? Testing structural hypotheses against real networks

Abstract: The abundance of data about social, economic and political relationships has opened an era in which social theories can be tested against empirical evidence, allowing human behaviour to be analyzed just as any other natural phenomenon. The present contribution focuses on balance theory, stating that social agents tend to avoid the formation of `unbalanced', or `frustrated', cycles, i.e. cycles with an odd number of negative links. Such a statement can be made statistically rigorous only after a comparison with a null model. Since the existing ones cannot account for the heterogeneity of individual actors, we, first, extend the Exponential Random Graphs framework to binary, undirected, signed networks with local constraints and, then, employ both homogeneous and heterogeneous benchmarks to compare the empirical abundance of short cycles with its expected value, on several, real-world systems. What emerges is that the level of balance in real-world networks crucially depends on (at least) three factors, i.e. the measure adopted to quantify it, the nature of the data, the null model employed for the analysis. As an example, the study of triangles reveals that homogeneous null models with global constraints tend to favour the weak version of balance theory, according to which only the triangle with one negative link should be under-represented in real, social and political networks; on the other hand, heterogeneous null models with local constraints tend to favour the strong version of balance theory, according to which also the triangle with all negative links should be under-represented in real, social networks. Biological networks, instead, are found to be significantly frustrated under any benchmark considered here.