The Limiting Eigenvalue Distribution of Iterated k-Regular Graph Cylinders

TL;DR

This paper investigates the eigenvalue distributions of matrices derived from iterated k-regular graph cylinders, revealing symmetric binomial distributions for bipartite graphs and asymmetric ones for non-bipartite graphs, with implications for neural network design.

Contribution

The study characterizes the limiting eigenvalue distributions of iterated k-regular graph matrices, distinguishing between bipartite and non-bipartite cases, and explores their relevance to neural network structures.

Findings

Bipartite graph cylinders produce symmetric binomial eigenvalue distributions.

Non-bipartite graph cylinders produce asymmetric eigenvalue distributions.

The graph cylinder construction does not yield suitable expander graphs for neural networks.

Abstract

We explore the limiting empirical eigenvalue distributions arising from matrices of the form \[A_{n+1} = \begin{bmatrix} A_n & I\\ I & A_n \end{bmatrix} , \]where $A_0$ is the adjacency matrix of a $k$-regular graph. We find that for bipartite graphs, the distributions are centered symmetric binomial distributions, and for non-bipartite graphs, the distributions are asymmetric. This research grew out of our work on neural networks in $k$-regular graphs. Our original question was whether or not the graph cylinder construction would produce an expander graph that is a suitable candidate for the neural networks being developed at Nousot. This question is answered in the negative for our computational purposes. However, the limiting distribution is still of theoretical interest to us; thus, we present our results here.