Concentration and Stability of Community-Detecting Functions on Random Networks

We propose a general form of community-detecting functions for finding communities—an optimal partition of a random network—and examine the concentration and stability of the function values using the bounded difference martingale method. We derive LDP inequalities for both the general case and several specific community-detecting functions: modularity, graph bipartitioning, and q-Potts community structure. We also discuss the concentration and stability of community-detecting functions on different types of random networks: sparse and nonsparse networks and some examples such as ER and CL networks.