Emergence of network communities driven by local rules

Natural systems are modeled by networks with nodes and links. Often the nodes are segregated into communities with different connectivity patterns. Node heterogeneity such as political affiliation in social networks or biological function in gene networks are highlighted as key factors driving the segregation of nodes into communities. Here I demonstrate that node heterogeneity is not a necessary requirement. To this end I introduce the Ramsey community number, $r_ \kappa$, the minimum graph size that warranties the emergence of network communities with almost certainty. Using the stochastic block model for community detection with correction for degree sequence, I show that networks generated by local rules have finite $r_ \kappa$ values while their randomized versions do not have emergent communities. I conclude that network communities are an emergent property of networks evolving with local rules.