Exact recovery in Gaussian weighted stochastic block model and planted dense subgraphs: Statistical and algorithmic thresholds

In this paper, we study the exact recovery problem in the Gaussian weighted version of the Stochastic block model with two symmetric communities. We provide the information-theoretic threshold in terms of the signal-to-noise ratio (SNR) of the model and prove that when SNR $<1$, no statistical estimator can exactly recover the community structure with probability bounded away from zero. On the other hand, we show that when SNR $>1$, the Maximum likelihood estimator itself succeeds in exactly recovering the community structure with probability approaching one. Then, we provide two algorithms for achieving exact recovery. The Semi-definite relaxation as well as the spectral relaxation of the Maximum likelihood estimator can recover the community structure down to the threshold value of 1, establishing the absence of an information-computation gap for this model. Next, we compare the problem of community detection with the problem of recovering a planted densely weighted community within a graph and prove that the exact recovery of two symmetric communities is a strictly easier problem than recovering a planted dense subgraph of size half the total number of nodes, by establishing that when the same SNR$< 3/2$, no statistical estimator can exactly recover the planted community with probability bounded away from zero. More precisely, when $1 <$ SNR $< 3/2$ exact recovery of community detection is possible, both statistically and algorithmically, but it is impossible to exactly recover the planted community, even statistically, in the Gaussian weighted model. Finally, we show that when SNR $>2$, the Maximum likelihood estimator itself succeeds in exactly recovering the planted community with probability approaching one. We also prove that the Semi-definite relaxation of the Maximum likelihood estimator can recover the planted community structure down to the threshold value of 2.