Multiscale Localized Inference for Networks with a Measured Vertex Coordinate

Abstract

In many networks each vertex has a position measured from outside the network: a neuron's location along the body axis, a residue's index along a protein sequence, a genomic bin's position in base pairs. The chance that two vertices connect is then a surface over pairs of positions, and questions about the network become questions about small regions of that surface. A region on the diagonal covers pairs inside one stretch of the axis, and a departure there is a community with a boundary. A region off the diagonal covers pairs spanning two separated stretches, and a departure there is a bridge. Existing methods address one part of this at a time: community detection returns groups without placing them on the axis, block models fix the width in advance, and scan statistics test one window at one scale. We expand the surface in a wavelet dictionary whose elements are exactly such regions, at every position and width. We derive in closed form the modularity, edge length, transitivity and degree spread each element produces, and the generating element is recovered from those summaries. A scan estimates every coefficient against a background fitted on separate edges and controls the error rate across all positions and widths at once. Applied to a connectome, a protein contact map and a chromatin contact map, it recovers known anatomy in the first and is calibrated in all three. When positions are estimated with error near the width sought, the location of a departure is not identified at any signal strength, and a check decides this before any analysis.