Phase transitions in semisupervised clustering of sparse networks
This addresses fundamental limits in network analysis for applications like social and biological networks, but is incremental as it builds on prior work on phase transitions.
The paper tackles the problem of predicting node labels in sparse networks using semisupervised learning, finding that for networks with more than two groups, accuracy jumps discontinuously at a critical fraction of labeled nodes, with a line of transitions ending in a critical point.
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 $α$ 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 $α$. For $k = 2$ groups, we find that the detectability transition disappears for any $α> 0$, 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 $α$. This line ends in a critical point with a second-order transition, beyond which the accuracy is a continuous function of $α$. We demonstrate qualitatively similar transitions in two real-world networks.