Spectral gap in random bipartite biregular graphs and applications

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. Finally, we give some applications in machine learning and coding theory.

Existence and polynomial time construction of biregular, bipartite Ramanujan graphs of all degrees

“…goes to 1 as n → ∞. This was extended to (c, d)-biregular bipartite graphs in [3]. However, we do note that although one might expect almost-Ramanujan and Ramanujan graphs to have almost-identical properties, there are certain proof techniques (related to the Ihara Zeta function) for which Ramanujan graphs are distinctly better, for reasons that are beyond the scope of this article (see [2]).…”

Lower bounds for testing graphical models: colorings and antiferromagnetic Ising models

“…Finally, we mention that the main technical challenge in our approach is to establish that in every gadget, with high probability, either every vertex of L is assigned "+" and every vertex of R is assigned "−" or vice versa; see Theorems 4.1 and 4.2. To show this, we require very precise bounds on the edge expansion of the random bipartite graph G. When d → ∞, these bounds can be derived in a fairly straightforward manner from the results in [10]. However, the case of d = O(1) is more difficult, and it requires for us to define the notion of edge expansion with respect to the ports of the gadget and extending some of the ideas in [39] (see Theorem 5.3).…”

Balanced Nonadaptive Redundancy Scheduling

“…When G is d-regular and bipartite, the eigenvalues of the adjacency matrix of G are symmetric around 0. In other words, the eigenvalues of the adjacency matrix are either 0 or pairs of λ and −λ [34]. Thus, max{λ 2 , λ n } has the trivial value of d. In such cases, the spectral gap is defined as follows.…”

“…[34] Let G = (V, E) be an d-regular bipartite graph, where|V| = n. Let λ 1 λ 2 • • • −λ 2 −λ 1 bethe eigenvalues of the adjacency matrix of G. The spectral gap of the graph G is defined as ∆(G) = d − λ The associated graphs with round-robin and BIBD scheduling policies are d-regular bipartite, with spectral gap d − λ 2 . In the following, we derive the spectral gap of these two policies.…”