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This paper studies the problem of detecting the presence of a small densecommunity planted in a large Erd\H{o}s-R\'enyi random graph $\mathcal{G}(N,q)$,where the edge probability within the community exceeds $q$ by a constantfactor. Assuming the hardness of the planted clique detection problem, we showthat the computational complexity of detecting the community exhibits thefollowing phase transition phenomenon: As the graph size $N$ grows and thegraph becomes sparser according to $q=N^{-\alpha}$, there exists a criticalvalue of $\alpha = \frac{2}{3}$, below which there exists a computationallyintensive procedure that can detect far smaller communities than anycomputationally efficient procedure, and above which a linear-time procedure isstatistically optimal. The results also lead to the average-case hardnessresults for recovering the dense community and approximating the densest$K$-subgraph.

Computational Lower Bounds for Community Detection on Random Graphs