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.
Community detectability has been studied under the stochastic block model with restricted assumptions such as @math , @math and fixed average degree as the network size @math increases @cite_25 @cite_22 @cite_12 @cite_3 @cite_33 . The planted clique detection problem in @cite_21 is a special case of the stochastic block model when @math and @math . A less restricted stochastic block model is studied in @cite_10 where a universal phase transition in community detectability is established for which the critical value does not depend on the community sizes. A similar model to our network model is studied in @cite_23 for interconnected networks. However, in @cite_23 the subnetworks are of equal size and the external edges are known (i.e., non-random). Phase transitions in spectral community detection under noiseless network setting is studied in @cite_8 .
@cite_22 @cite_33 @cite_8 @cite_21 @cite_3 @cite_23 @cite_10 @cite_25 @cite_12
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This paper studies the problem of detecting the presence of a small dense community planted in a large Erd o s-R 'enyi random graph @math , where the edge probability within the community exceeds @math by a constant factor. Assuming the hardness of the planted clique detection problem, we show that the computational complexity of detecting the community exhibits the following phase transition phenomenon: As the graph size @math grows and the graph becomes sparser according to @math , there exists a critical value of @math , below which there exists a computationally intensive procedure that can detect far smaller communities than any computationally efficient procedure, and above which a linear-time procedure is statistically optimal. The results also lead to the average-case hardness results for recovering the dense community and approximating the densest @math -subgraph. We prove the existence of an asymptotic phase transition threshold on community detectability for the spectral modularity method [M. E. J. Newman, Phys. Rev. E 74, 036104 (2006) and Proc. National Academy of Sciences. 103, 8577 (2006)] under a stochastic block model. The phase transition on community detectability occurs as the inter-community edge connection probability @math grows. This phase transition separates a sub-critical regime of small @math , where modularity-based community detection successfully identifies the communities, from a super-critical regime of large @math where successful community detection is impossible. We show that, as the community sizes become large, the asymptotic phase transition threshold @math is equal to @math , where @math is the within-community edge connection probability. Thus the phase transition threshold is universal in the sense that it does not depend on the ratio of community sizes. The universal phase transition phenomenon is validated by simulations for moderately sized communities. Using the derived expression for the phase transition threshold we propose an empirical method for estimating this threshold from real-world data. We consider the problem of community detection in the Stochastic Block Model with a finite number @math of communities of sizes linearly growing with the network size @math . This model consists in a random graph such that each pair of vertices is connected independently with probability @math within communities and @math across communities. One observes a realization of this random graph, and the objective is to reconstruct the communities from this observation. We show that under spectral algorithms, the number of misclassified vertices does not exceed @math with high probability as @math grows large, whenever @math , @math and where @math and @math denote the (fixed) proportions of vertices in the two smallest communities. In view of recent work by and , this establishes that the proposed spectral algorithms are able to exactly recover communities whenever this is at all possible in the case of networks with two communities with equal sizes. We conjecture that condition (1) is actually necessary to obtain less than @math misclassified vertices asymptotically, which would establish the optimality of spectral method in more general scenarios. Detecting hidden communities from observed interactions is a classical problem. Theoretical analysis of community detection has so far been mostly limited to models with non-overlapping communities such as the stochastic block model. In this paper, we provide guaranteed community detection for a family of probabilistic network models with overlapping communities, termed as the mixed membership Dirichlet model, first introduced in (2008). This model allows for nodes to have fractional memberships in multiple communities and assumes that the community memberships are drawn from a Dirichlet distribution. Moreover, it contains the stochastic block model as a special case. We propose a unified approach to learning communities in these models via a tensor spectral decomposition approach. Our estimator uses low-order moment tensor of the observed network, consisting of 3-star counts. Our learning method is based on simple linear algebraic operations such as singular value decomposition and tensor power iterations. We provide guaranteed recovery of community memberships and model parameters, and present a careful finite sample analysis of our learning method. Additionally, our results match the best known scaling requirements for the special case of the (homogeneous) stochastic block model.