## Computational Lower Bounds for Community Detection on Random Graphs

author: Bruce Hajek, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign
published: Aug. 20, 2015,   recorded: July 2015,   views: 1659
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# Description

This paper studies the problem of detecting the presence of a small dense community planted in a large Erd\H{o}s-R\'enyi random graph $\calG(N,q)$, where the edge probability within the community exceeds $q$ 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 $N$ grows and the graph becomes sparser according to $q=N^{-\alpha}$,there exists a critical value of $\alpha = \frac{2}{3}$, 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 $K$-subgraph.

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