Phase transitions in semisupervised clustering of sparse networks

TL;DR

This paper investigates how partial label information influences the phase transitions in semisupervised community detection within sparse networks, revealing critical points where accuracy improves abruptly or continuously.

Contribution

It characterizes the phase diagram of semisupervised clustering in stochastic block models, identifying discontinuous and continuous transitions in detection accuracy as a function of known labels.

Findings

Detectability transition disappears for two groups with any label knowledge.

For larger groups, a line of easy/hard transition points emerges, with accuracy jumping at a critical label fraction.

Similar phase transition behaviors are observed in real-world network data.

Abstract

Predicting labels of nodes in a network, such as community memberships or demographic variables, is an important problem with applications in social and biological networks. A recently-discovered phase transition puts fundamental limits on the accuracy of these predictions if we have access only to the network topology. However, if we know the correct labels of some fraction $\alpha$ of the nodes, we can do better. We study the phase diagram of this "semisupervised" learning problem for networks generated by the stochastic block model. We use the cavity method and the associated belief propagation algorithm to study what accuracy can be achieved as a function of $\alpha$. For $k = 2$ groups, we find that the detectability transition disappears for any $\alpha > 0$, in agreement with previous work. For larger $k$ where a hard but detectable regime exists, we find that the easy/hard transition (the point at which efficient algorithms can do better than chance) becomes a line of transitions where the accuracy jumps discontinuously at a critical value of $\alpha$. This line ends in a critical point with a second-order transition, beyond which the accuracy is a continuous function of $\alpha$. We demonstrate qualitatively similar transitions in two real-world networks.