Phase Transitions in Community Detection: A Solvable Toy Model
This work provides insights into community detection transitions, but it is incremental as it uses a toy model that reproduces only qualitative features without correct thresholds.
The paper tackles the problem of understanding phase transitions in community detection by analyzing a simplified 'zero-temperature inference' model, which predicts a first-order detectability transition for q>2 and a discontinuous accuracy jump in a semi-supervised setting, though it does not match exact threshold values from more complex methods.
Recently, it was shown that there is a phase transition in the community detection problem. This transition was first computed using the cavity method, and has been proved rigorously in the case of $q=2$ groups. However, analytic calculations using the cavity method are challenging since they require us to understand probability distributions of messages. We study analogous transitions in so-called "zero-temperature inference" model, where this distribution is supported only on the most-likely messages. Furthermore, whenever several messages are equally likely, we break the tie by choosing among them with equal probability. While the resulting analysis does not give the correct values of the thresholds, it does reproduce some of the qualitative features of the system. It predicts a first-order detectability transition whenever $q > 2$, while the finite-temperature cavity method shows that this is the case only when $q > 4$. It also has a regime analogous to the "hard but detectable" phase, where the community structure can be partially recovered, but only when the initial messages are sufficiently accurate. Finally, we study a semisupervised setting where we are given the correct labels for a fraction $ρ$ of the nodes. For $q > 2$, we find a regime where the accuracy jumps discontinuously at a critical value of $ρ$.