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

Abstract

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 JensenShannon 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.

Discrete Quantum Walks on Graphs and Digraphs

“…it is maximized when the entire average mixing matrix is flat. In [10], Bai, Rossi, Cui, and Hancock proposed a graph signature based on the total entropy of continuous quantum walks. According to their experimental results, this entropic measure provides significant information on the properties of graphs.…”

A Quantum-inspired Entropic Kernel for Multiple Financial Time Series Analysis

“…Remarks. The CTQW has been successfully employed to develop novel approaches in machine learning and data mining Bai et al, 2016], because of the richer structure than their classical counterparts. The reason of utilizing the CTQW in this work is that the state vector of the CTQW is complex-valued and its evolution is governed by a time-varying unitary matrix.…”

“…is the time-averaged probability of the CTQW visiting v j from v i . The quantum based Shannon entropy [Bai et al, 2016] of vertex v i can be defined as…”

“…The main advantage of employing graph kernels is that they can offer us an effective way of mapping the network structures into a high dimensional space so that the standard kernel machinery for * Corresponding Author vectorial data is applicable to the network analysis. Most existing graph kernels are based on the idea of decomposing graphs or networks into substructures and then measuring pairs of isomorphic substructures [Haussler, 1999], e.g., graph kernels based on counting pairs of isomorphic a) paths [Borgwardt and Kriegel, 2005], b) walks [Kashima et al, 2003], and c) subgraphs [Bai et al, 2015b] or subtrees [Shervashidze et al, 2009]. Unfortunately, directly adopting these graph kernels to analyze the time-varying financial networks inferred from original vectorial time series tends to be elusive.…”

Discrete-time quantum walks and graph structures

“…This invariant has been applied in structural pattern recognition. For example, Bai et al [4] proposed a graph signature based on the entropy of the average mixing matrix of a graph. According to the experimental results, this entropic measure allows us to distinguish different structures.…”