Matched Filters for Noisy Induced Subgraph Detection

Abstract

The problem of finding the vertex correspondence between two noisy graphs with different number of vertices where the smaller graph is still large has many applications in social networks, neuroscience, and computer vision. We propose a solution to this problem via a graph matching matched filter: centering and padding the smaller adjacency matrix and applying graph matching methods to align it to the larger network. The centering and padding schemes can be incorporated into any algorithm that matches using adjacency matrices. Under a statistical model for correlated pairs of graphs, which yields a noisy copy of the small graph within the larger graph, the resulting optimization problem can be guaranteed to recover the true vertex correspondence between the networks. However, there are currently no efficient algorithms for solving this problem. To illustrate the possibilities and challenges of such problems, we use an algorithm that can exploit a partially known correspondence and show via varied simulations and applications to Drosophila and human connectomes that this approach can achieve good performance.

Multiplex graph matching matched filters

“…In this section, we will explore the benefit of considering multiplex versus monoplex networks when considering template matchability in the MS model. We note here that while the formal theory underlying the matchability results in the multiplex setting differs only slightly from the monoplex setting of [36], we stress that the end results demonstrate the utility of considering multiple channels. In the MS model, let T ∈ M c m and W ∈ M c n be the respective template and background source graphs, with respective centered, padded adjacency matrices given respectively by C and D satisfying…”

“…In this paper, we presented a framework for finding noisy copies of a multiplex template in a large multiplex background network. Our strategy, which extends [36] to the multiplex setting, uses graph matching combined with multiple random restarts to search the background for locally optimal matches to the template. To formalize this strategy, we provided a very natural extension of the classical graph matching problem to the multiplex that is easily amended to matching graphs of different orders (both across networks and channels).…”

“…To lift the monoplex GMP to the general multiplex definition presented above, we consider the following padded formulations of our general multiplex networks (adapted here from [16,36]). Letting H ∈ M c m and G ∈ M c n with m ≤ n, we consider the following two schemes for ameliorating the differing graph orders.…”

“…Given A, a multiplex graph matching algorithm designed to approximately solve Eq. (2-3), our multiplex graph matching matched filter (M-GMMF), generalizing the monoplex filtering setting of [36], proceeds as in Algorithm 1. Note that in our experiments (and in the pseudocode below), we make use of A = MFAQ (see Algorithm 2 in Appendix A.1), but we stress that our approach can utilize any suitable A equally well.…”

“…The above noisy induced subgraph detection problem depends greatly on the definition of "best fitting" employed. Our approach, generalizing [36] to the multiplex setting, will employ the multiplex template H to search the multiplex background graph G for possible matches, with goodness of fit measured via a multiplex formulation of the classical graph matching problem (see [9,17,15,40] for excellent reviews of the voluminous graph matching literature).…”