The Average Mixing Matrix Signature

Abstract

Abstract. Laplacian-based descriptors, such as the Heat Kernel Signature and the Wave Kernel Signature, allow one to embed the vertices of a graph onto a vectorial space, and have been successfully used to find the optimal matching between a pair of input graphs. While the HKS uses a heat di↵usion process to probe the local structure of a graph, the WKS attempts to do the same through wave propagation. In this paper, we propose an alternative structural descriptor that is based on continuoustime quantum walks. More specifically, we characterise the structure of a graph using its average mixing matrix. The average mixing matrix is a doubly-stochastic matrix that encodes the time-averaged behaviour of a continuous-time quantum walk on the graph. We propose to use the rows of the average mixing matrix for increasing stopping times to develop a novel signature, the Average Mixing Matrix Signature (AMMS). We perform an extensive range of experiments and we show that the proposed signature is robust under structural perturbations of the original graphs and it outperforms both the HKS and WKS when used as a node descriptor in a graph matching task.

The Average Mixing Kernel Signature

“…In this section we present quantitative and qualitative analysis testing the performance of our descriptor with respect to two alternative state-of-the-art spectral signatures, the WKS [2] and the scaled HKS [33], as these represent two of the most successful and used non-learned descriptors for deformable shapes. We omit the comparison with the AMMS [26] doe to its performance in this domain. Indeed, the AMMS was introduced to work on graphs of limited size and with long-range connections and, as we observed, its poor performance in the shape domain is further exacerbated by the need to truncate the spectrum.…”

“…In [26] Rossi et al introduced another structural signature based on the average mixing matrix (AMMS) for the analysis of graphs. However, we stress that our work significantly differs from [26] in several key aspects:…”

“…Directly applying the AMMS to our problem would lead to a computationally infeasible, noise-dominated signature where we cannot focus on the energy levels of interest. One way of reducing the computational cost of the AMMS would have been truncating the spectrum, however this would not have led to the quantum-mechanically feasible process one which our framework is based; in the AMMS the sampling is done over the time domain, as opposed to the energy domain in the AMKS; while a limited empirical analysis suggests that the AMMS is robust to structural noise, we perform the first theoretical analysis of the noise-robustness resulting from the average mixing matrix use, revealing that the mixing factors naturally account for the noise-induced mixing of eigenspaces; while there is no analysis of the computational complexity of the AMMS in [26], here we perform a thorough complexity as well as runtime analysis of the AMKS in addition to showing that its computation can be fully expressed in terms of matrix operations; finally, as the application domains and communities of interest of the AMMS and AMKS are different, in this paper we perform an entirely separate and more extensive set of experiments.…”