  We propose a general form of community detecting functions for finding the
communities or the 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: the sparse and non-sparse networks and some examples such
as ER and CL networks.
