Matrix Energy as a Measure of Topological Complexity of a Graph

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’s 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. AMS classification: 05C50 Keywords: Matrix Energy, Topological Complexity, Weyuker’s Criteria, Architectural Regime, P point. 1 INTRODUCTION In the context of complex interconnected systems, the quantification of complexity has increasingly gained importance over the last few years. While working with large, complex systems, the challenge of quantifying complexity is central and rigorous, formalized framework to compute and compare their respective complexities and aid decision-­‐making. In particular, the consideration of connectivity structure attracts attention because they affect system behavior. The term “structure” is directly linked to the definition of a system. In general, the term “structure” is understood as the network formed by dependencies between components of any system [9]. One emerging area of application is in the characterization and impact of complex system architectures that are fast becoming highly networked and distributed in nature [17, 19, 21]. The concept of network dimension can be used to determine the underlying network structure and its function. A network with higher dimension is said to be more complex than one with a lower dimension. Here, we focus on the spectral dimension of the binary adjacency matrix that represents the connectivity structure of the system. Recently, the idea of spectral dimension has been used to estimate the reconstructability of networks [13, 19]. In this paper, we propose matrix energy of the underlying binary adjacency