Community detection in networks with unequal groups

TL;DR

This paper investigates how asymmetry in community sizes or degrees affects detectability in network community detection, showing that asymmetry can eliminate the phase transition in networks with four or fewer groups.

Contribution

It demonstrates that asymmetry in community sizes or degrees can remove the detectability transition in networks with up to four groups, using the cavity method.

Findings

Asymmetry removes the detectability transition for up to four groups.

For more than four groups, the transition persists up to a critical asymmetry.

Local neighborhood analysis improves community detection success.

Abstract

Recently, a phase transition has been discovered in the network community detection problem below which no algorithm can tell which nodes belong to which communities with success any better than a random guess. This result has, however, so far been limited to the case where the communities have the same size or the same average degree. Here we consider the case where the sizes or average degrees are different. This asymmetry allows us to assign nodes to communities with better-than- random success by examining their local neighborhoods. Using the cavity method, we show that this removes the detectability transition completely for networks with four groups or fewer, while for more than four groups the transition persists up to a critical amount of asymmetry but not beyond. The critical point in the latter case coincides with the point at which local information percolates, causing a global transition from a less-accurate solution to a more-accurate one.