We consider the algorithmic problem of community detection in networks. Given an undirected friendship graph G, a subset

S of vertices is an (a,b)-community if: * Every member of the community is friends with an (a)-fraction of the community; and

* every non-member is friends with at most a (b)-fraction of the

community.

[Arora, Ge, Sachdeva, Schoenebeck 2012] gave a quasi-polynomial

time algorithm for enumerating all the (a,b)-communities

for any constants a>b.

Here, we prove that, assuming the Exponential Time Hypothesis (ETH),

quasi-polynomial time is in fact necessary - and even for a much weaker

approximation desideratum. Namely, distinguishing between:

* G contains an (1,o(1))-community; and

* G does not contain a (b,b+o(1))-community

for any b.

We also prove that counting the number of (1,o(1))-communities

requires quasi-polynomial time assuming the weaker #ETH.