Phase transitions and overlapping modules in complex networks

Complex systems can be described in terms of networks capturing the intricate web of connections among the units they are made of. Here we review two aspects of the possible organization of such networks. First, we provide a phenomenological theory for topological transitions in restructuring networks. In this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges. In our studies we find a rich variety of topological phase transitions when the temperature is varied. These transitions signal singular changes in the essential features of the global structure. Next, we address a question of great current interest which is about the modular structure of networks. We describe, how to interpret the global organization as the coexistence of structural sub-units (modules or communities) associated with more highly interconnected parts. The existing deterministic methods used for large networks find separated communities, while most of the actual networks are made of highly overlapping cohesive groups of nodes. We describe a recently introduced an approach to analyze the main statistical features of the interwoven sets of overlapping communities making a step towards the uncovering of the modular structure of complex systems. Our approach is based on defining communities as clusters of percolating complete subgraphs called k-cliques. We present the basic features of the associated percolation transition of overlapping k-cliques. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique to explore overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks.