Spectral Functional-Digraph Theory, Stability, and Entropy for Gene Regulatory Networks

Abstract

Spectral graph theory is an indispensable tool in the rich interdisciplinary field of network science, which includes as objects ordinary abstract graphs as well as directed graphs such as the Internet, semantic networks, electrical circuits, and gene regulatory networks (GRN). However, its contributions sometimes get lost in the code, and network theory occasionally becomes overwhelmed with problems specific to undirected graphs. In this paper, we will study functional digraphs, calculate the eigenvalues and eigenvectors of their adjacency matrices, describe how to compute their automorphism groups, and define a notion of entropy in terms of their symmetries. We will then introduce gene regulatory networks (GRNs) from scratch, and consider their phase spaces, which are functional digraphs describing the deterministic progression of the overall state of a GRN. Finally, we will redefine the stability of a GRN and assert that it is closely related to the entropy of its phase space.