Seeded graph matching

Abstract

Given two graphs, the graph matching problem is to align the two vertex sets so as to minimize the number of adjacency disagreements between the two graphs. The seeded graph matching problem is the graph matching problem when we are first given a partial alignment that we are tasked with completing. In this article, we modify the state-of-the-art approximate graph matching algorithm "FAQ" of to make it a fast approximate seeded graph matching algorithm, adapt its applicability to include graphs with differently sized vertex sets, and extend the algorithm so as to provide, for each individual vertex, a nomination list of likely matches. We demonstrate the effectiveness of our algorithm via simulation and real data experiments; indeed, knowledge of even a few seeds can be extremely effective when our seeded graph matching algorithm is used to recover a naturally existing alignment that is only partially observed.

Alignment strength and correlation for graphs

“…φ ∈Πn d(G, H, φ ) in the definition of alignment strength (Equation 2) involves an exponentially sized summation, nonetheless it can be computed efficiently using Equation 5 from Section 3. Also, although the computation of the graph matching problem solution φ GM is intractable [2], nonetheless there are effective, efficient approximate graph matching algorithms that can be used [11], [5], one of which we discuss and use later in this paper.…”

“…The subsequent sections, Section 4 and Section 5, follow up with empirical illustrations that total correlation T is a meaningful measure. As we vary the edge correlation e together with the heterogeneity correlation h for correlated Bernoulli graphs G and H in broad families of parameter settings, it turns out that the value of T dictates (in Section 4) how successful the approximate seeded graph matching algorithm called SGM [5], [9] is in recovering the identity bijection (which is the natural alignment here) and (in Section 5) T dictates how much time it takes to perform seeded graph matching exactly via binary integer linear programming. The seeded graph matching problem is the graph matching problem wherein we seek to compute φ GM ∈ arg min φ ∈Πn d(G, H, φ ), except that part of the natural alignment is known; having these "seeds" can substantially help recover the rest of the natural alignment correctly.…”

“…The seeded graph matching problem is the graph matching problem wherein we seek to compute φ GM ∈ arg min φ ∈Πn d(G, H, φ ), except that part of the natural alignment is known; having these "seeds" can substantially help recover the rest of the natural alignment correctly. In Section 4, we utilize the SGM Algorithm [5], [9] for approximate seeded graph matching on moderately sized graphs, on the order of 1000 vertices, since, unfortunately, exact seeded graph matching can only be done on very small, toysize graphs (a few tens of non-seed vertices). In Section 5, where we are interested in comparing runtime, the approximate seeded graph matching algorithms are not appropriate to use, since their run times tend to be monolithic (given the number of vertices) and less sensitive to the parameters of the random graph distribution.…”

Multiplex graph matching matched filters

“…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.…”

“…The formulation in Eq. (2) effectively seeks to maximize the number of common edges between the multiplex template and multiplex background, where all edges across all channels are weighted equally (see [16,6] for the monoplex analogue). The Centered Multiplex Graph Matching Problem is defined as finding an element P ∈ Π n in arg min…”