Multiplex graph matching matched filters

Abstract

We consider the problem of detecting a noisy induced multiplex template network in a larger multiplex background network. Our approach, which extends the framework of [36] to the multiplex setting, leverages a multiplex analogue of the classical graph matching problem to use the template as a matched filter for efficiently searching the background for candidate template matches. The effectiveness of our approach is demonstrated both theoretically and empirically, with particular attention paid to the potential benefits of considering multiple channels.for a definition of multiplex isomorphism), accounting for the reality that relatively large, complex subgraph templates may only errorfully occur in the larger background network. These errors may be due to missing edges/vertices in the template or background, and arise in a variety of real data settings [34]. The subgraph isomorphism problem-given a template A, determine if an isomorphic copy of A exists in a larger network B and find the isomorphic copy (or copies) if it exists-has been the subject of voluminous research in the monoplex (i.e., single layer) setting, with approaches based on efficient tree search [38], color coding [3, 2], graph homomorphisms [18], rule-based/filter-based matchings [10,29], among others; for a survey of the literature circa 2012, see [24]. In contrast, the problem of multilayer homomorphic/isomorphic subgraph detection is still in its relative infancy, with comparatively fewer existing methods in the literature; see, for example, [41,37,29]. Notation: The following notation will be used throughout. For an integer n > 0, we will define [n] := {1, 2, . . . , n}, J n to be the n × n hollow matrix with all off-diagonal entries identically set to 1, 0 n to be the n × n matrix with all entries identically set to 0,

Bisected graph matching improves automated pairing of bilaterally homologous neurons from connectomes

“…Pantazis et al [30] introduced a version of FAQ for multilayer networks, which may be helpful when considering different kinds of synaptic connections such as gap junction vs. chemical [38] or axo-axonic vs. axo-dendritic. Further, other approaches to matching neurons in neuroscience do not use connectivity at all: Costa et al [39] introduced an algorithm for matching neurons on the basis of morphology which has been widely used on connectomic reconstructions.…”

“…With this formulation, the graph matching problem can be seen as a special case of the bisected graph matching problem, since the objective function in Equation 2.2 reduces to that of Equation 2.1 in the special case where A LR and A RL are both the zero matrix. Note that this problem is also distinct from the multiplex graph matching problem described in Pantazis et al [30], as the contralateral subgraphs require only a permutation of their rows or their columns (not both) to maintain the correct structure of the adjacency matrix. Given this notion of what it means to find a good matching between the hemispheres, a state-ofthe-art graph matching technique can be adapted to solve this new bisected graph matching problem.…”