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Abstract Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G , with o ( K ) misclassified vertices on average, in the sublinear regime n 1- o (1) ≤ K ≤ o ( n ). A critical parameter is the effective signal-to-noise ratio λ = K 2 ( p - q ) 2 / (( n - K ) q ), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log * n + O (1) iterations, with the total time complexity O (| E |log * n ), where log * n is the iterated logarithm of n . Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying a power iteration to the nonbacktracking matrix of the graph is shown to attain weak recovery if and only if λ > 1. In addition, the BP algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all K ≥ ( n / log n ) (ρ BP + o (1)), where ρ BP is a function of p / q .
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Transfer Learning for Latent Variable Network Models
We study transfer learning for estimation in latent variable network models. In our setting, the conditional edge probability matrices given the latent variables are represented by P for the source and Q for the target. We wish to estimate Q given two kinds of data: (1) edge data from a subgraph induced by an o(1) fraction of the nodes of Q, and (2) edge data from all of P. If the source P has no relation to the target Q, the estimation error must be Ω(1). However, we show that if the latent variables are shared, then vanishing error is possible. We give an efficient algorithm that utilizes the ordering of a suitably defined graph distance. Our algorithm achieves o(1) error and does not assume a parametric form on the source or target networks. Next, for the specific case of Stochastic Block Models we prove a minimax lower bound and show that a simple algorithm achieves this rate. Finally, we empirically demonstrate our algorithm's use on real-world and simulated graph transfer problems.
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- Award ID(s):
- 2505865
- PAR ID:
- 10631817
- Publisher / Repository:
- https://doi.org/10.48550/arXiv.2406.03437
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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