Spectral gap in random bipartite biregular graphs and applications

TL;DR

This paper proves a spectral gap property for random bipartite biregular graphs, with implications for community detection, coding, and matrix completion, using advanced spectral analysis techniques.

Contribution

It establishes a spectral gap analogue for bipartite biregular graphs and links non-backtracking spectra to adjacency matrices via the Ihara-Bass formula.

Findings

Spectral gap exists for the non-backtracking matrix in random bipartite biregular graphs.

Random rectangular zero-one matrices with fixed margins are full-rank with high probability.

Applications demonstrated in community detection, coding theory, and matrix completion.

Abstract

We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular graphs. We use the Ihara-Bass formula to connect the non-backtracking spectrum to that of the adjacency matrix, employing the moment method to show there exists a spectral gap for the non-backtracking matrix. A byproduct of our main theorem is that random rectangular zero-one matrices with fixed row and column sums are full-rank with high probability. Finally, we illustrate applications to community detection, coding theory, and deterministic matrix completion.