Detecting hierarchical organization of pervasive communities by modular decomposition of Markov chain

Connecting nodes that contingently co-appear, which is a common process of networking in social and biological systems, normally leads to modular structure characterized by the absence of definite boundaries. This study seeks to find and evaluate methods to detect such modules, which will be called 'pervasive' communities. We propose a mathematical formulation to decompose a random walk spreading over the entire network into localized random walks as a proxy for pervasive communities. We applied this formulation to biological and social as well as synthetic networks to demonstrate that it can properly detect communities as pervasively structured objects. We further addressed a question that is fundamental but has been little discussed so far: What is the hierarchical organization of pervasive communities and how can it be extracted? Here we show that hierarchical organization of pervasive communities is unveiled from finer to coarser layers through discrete phase transitions that intermittently occur as the value for a resolution-controlling parameter is quasi-statically increased. To our knowledge, this is the first elucidation of how the pervasiveness and hierarchy, both hallmarks of community structure of real-world networks, are unified.