Aligning Spatially Constrained Graphs

Abstract

We focus on the problem of aligning graphs that have a spatial basis. In such graphs, which we refer to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rigid graphs</i> , nodes have preferred positions relative to their graph neighbors. Rigid graphs can be used to abstract objects in diverse applications such as large biomolecules, where edges corresponding to chemical bonds have preferred lengths, functional connectomes of the human brain, where edges corresponding to co-firing regions of the brain have preferred anatomical distances, and mobile device/ sensor communication logs, where edges corresponding to point-to-point communications across devices have distance constraints. Effective analysis of such graphs must account for edge lengths in addition to topological features. For instance, when identifying conserved patterns through graph alignment, it is important for matched edges to have correlated lengths, in addition to topological similarity. In this paper, we formulate the problem of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rigid graph alignment</i> and present a method for solving it. Our formulation of rigid graph alignment simultaneously aligns the topology of the input graphs, as well as the geometric structure represented by the edge lengths, which is solved using a block coordinate descent technique. Using detailed experiments on real and synthetic datasets, we demonstrate a number of important desirable features of our method: (i) it significantly outperforms topological and structural aligners on a wide range of problems; (ii) it scales to problems in important real-world applications; and (iii) it has excellent stability properties, in view of noise and missing data in typical applications.