Matrix Energy as a Measure of Topological Complexity of a Graph

Abstract: The complexity of highly interconnected systems is rooted in the interwoven architecture defined by its connectivity structure. In this paper, we develop matrix energy of the underlying connectivity structure as a measure of topological complexity and highlight interpretations about certain global features of underlying system connectivity patterns. The proposed complexity metric is shown to satisfy the Weyuker criteria as a measure of its validity as a formal complexity metric. We also introduce the notion of P point in the graph density space. The P point acts as a boundary between multiple connectivity regimes for finite-size graphs.