In this paper, we study the sensitivity of the spectral clustering based community detection algorithm subject to a Erdos-Renyi type random noise model. We prove phase transitions in community detectability as a function of the external edge connection probability and the noisy edge presence probability under a general network model where two arbitrarily connected communities are interconnected by random external edges. Specifically, the community detection performance transitions from almost perfect detectability to low detectability as the intercommunity edge connection probability exceeds some critical value.We derive upper and lower bounds on the critical value and show that the bounds are identical when the two communities have the same size. The phase transition results are validated using network simulations. Using the derived expressions for the phase transition threshold we propose a method for estimating this threshold from observed data.
@cite_31: Abstract A stochastic model is proposed for social networks in which the actors in a network are partitioned into subgroups called blocks. The model provides a stochastic generalization of the blockmodel. Estimation techniques are developed for the special case of a single relation social network, with blocks specified a priori. An extension of the model allows for tendencies toward reciprocation of ties beyond those explained by the partition. The extended model provides a one degree-of-freedom test of the model. A numerical example from the social network literature is used to illustrate the methods.
Consider two arbitrarily connected communities with internal adjacency matrices @math and @math and network sizes @math and @math , respectively. The external connections between these two communities are characterized by an @math adjacency matrix @math , where each entry in @math is a Bernoulli( @math ) random variable. Let @math . The overall @math adjacency matrix of the community structure can be represented as . The widely used stochastic block model @cite_31 is a special case of ) when the two community structures are generated by connected Erdos-Renyi random graphs parameterized by the within-community connection probability @math ( @math ). Our network model is more general since we only assume random connection probability @math on the external edges and we allow the within-community adjacency matrices @math to be arbitrary. In this paper we consider the noisy setting in which the adjacency matrix @math is corrupted by a random adjacency matrix @math such that the observed adjacency matrix is @math . The adjacency matrix @math is generated by a Erdos-Renyi random graph with edge connection probability @math . Note that this model only allows random insertions and not deletions of edges.