A graph theoretic approach to ultrafast information distribution: Borel Cayley graph resizing algorithm

Abstract

A graph theoretic approach is proposed to formulate communication graphs that enable ultrafast information distribution. In our earlier work, we reported that Borel Cayley graph (BCG) is a promising candidate as a logical topology for fast information distribution. However, the practical applications of BCG have been challenging because of its inflexible sizes. In this paper, we propose a simple but effective graph resizing algorithm that removes nodes from an oversized BCG to achieve a desired network size. The proposed resizing algorithm consists of two parts: a random pruning algorithm that identifies nodes to be removed uniformly at random; and a novel cut-through rewiring (CTR) algorithm that rewires the remaining nodes. The proposed resizing algorithm preserves the superior properties of the original BCGs, including a small diameter, a short average path length, a large algebraic connectivity, and ultrafast information distribution performance. Furthermore, analytical formulae were derived to compute the graph disconnection probability of the BCGs after resizing. Analytical results showed that the resized graphs are almost surely connected even after 80 ∼ 90% size reduction, depending on the original BCG size.