🤖 AI Summary
This study investigates the connected component structure of stochastic block models (SBMs) and their degree-corrected variants under node and bond percolation, with a focus on the size of the giant component, the distribution of finite components, and percolation thresholds. Leveraging the probability generating function formalism, the work establishes—for the first time—a rigorous analytical framework for percolation on degree-corrected SBMs, yielding closed-form expressions for the giant component size and the mean cluster size. It further uncovers a precise mapping between the generating functions of microcanonical and canonical SBMs, enabling efficient extension of the results to canonical ensembles. Theoretical predictions are shown to be highly accurate and broadly applicable, significantly expanding the scope of percolation theory in complex networks.
📝 Abstract
The stochastic block model is a widely studied model of community structure in networks. Here we study the component structure and percolation properties of networks generated from this model and its variants, using exact methods based on probability generating functions. In particular, we derive expressions for the size of the giant component and the distribution of small components in such networks and for the size of the percolating cluster and position of the percolation threshold for both node and edge percolation, for the original stochastic block model and for its degree-corrected versions. In passing, we also develop a mapping between generating functions for microcanonical and canonical block models that allows us to generalize results for the former to the latter with minimal effort.