Navigability, Walkability, and Perspicacity Associated with Canonical Ensembles of Walks in Finite Connected Undirected Graphs—Toward Information Graph Theory

Abstract

Canonical ensembles of walks in a finite connected graph assign the properly normalized probability distributions to all nodes, subgraphs, and nodal subsets of the graph at all time and connectivity scales of the diffusion process. The probabilistic description of graphs allows for introducing the quantitative measures of navigability through the graph, walkability of individual paths, and mutual perspicacity of the different modes of the (diffusion) processes. The application of information theory methods to problems about graphs, in contrast to geometric, combinatoric, algorithmic, and algebraic approaches, can be called information graph theory. As it involves evaluating communication efficiency between individual systems’ units at different time and connectivity scales, information graph theory is in demand for a wide range of applications, such as designing network-on-chip architecture and engineering urban morphology within the concept of the smart city.