A novel entropy-based graph signature from the average mixing matrix

Abstract

In this paper, we propose a novel entropic signature for graphs, where we probe the graphs by means of continuous-time quantum walks. More precisely, we characterise the structure of a graph through its average mixing matrix. The average mixing matrix is a doubly-stochastic matrix that encapsulates the time-averaged behaviour of a continuous-time quantum walk on the graph, i.e., the ij-th element of the average mixing matrix represents the time-averaged transition probability of a continuous-time quantum walk from the vertex vi to the vertex vj. With this matrix to hand, we can associate a probability distribution with each vertex of the graph. We define a novel entropic signature by concatenating the average Shannon entropy of these probability distributions with their Jensen-Shannon divergence. We show that this new entropic measure can encaspulate the rich structural information of the graphs, thus allowing to discriminate between different structures. We explore the proposed entropic measure on several graph datasets abstracted from bioinformatics databases and we compare it with alternative entropic signatures in the literature. The experimental results demonstrate the effectiveness and efficiency of our method.