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.

We propose a novel graph matching algorithm that uses ideas from graph signal processing to match vertices of graphs using alternative graph representations. Specilically, we consider a multi-scale heat diffusion on the graphs to create multiple weighted graph representations that incorporate both direct adjacencies as well as local structures induced from the heat diffusion. Then a multi-objective optimization method is used to match vertices across all pairs of graph representations simultaneously. We show that our proposed algorithm performs significantly better than the algorithm that only uses the adjacency matrices, especially when the number of known latent alignments between vertices (seeds) is small. We test the algorithm on a set of graphs and show that at the low seed level, the proposed algorithm performs at least 15-35% better than the traditional graph matching algorithm.

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 [14] 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.

In brain imaging and connectomics, the study of brain networks, estimating the mean of a population of graphs based on a sample is a core problem. Often, this problem is especially difficult because the sample or cohort size is relatively small, sometimes even a single subject, while the number of nodes can be very large with noisy estimates of connectivity. While the element-wise sample mean of the adjacency matrices is a common approach, this method does not exploit the underlying structural properties of the graphs. We propose using a low-rank method that incorporates dimension selection and diagonal augmentation to smooth the estimates and improve performance over the naïve methodology for small sample sizes. Theoretical results for the stochastic block model show that this method offers major improvements when there are many vertices. Similarly, we demonstrate that the low-rank methods outperform the standard sample mean for a variety of independent edge distributions as well as human connectome data derived from the magnetic resonance imaging, especially when the sample sizes are small. Moreover, the low-rank methods yield "eigen-connectomes," which correlate with the lobe-structure of the human brain and superstructures of the mouse brain. These results indicate that the low-rank methods are the important parts of the toolbox for researchers studying populations of graphs in general and statistical connectomics in particular.

In this work, we show that using the eigen-decomposition of the adjacency matrix, we can consistently estimate latent positions for random dot product graphs provided the latent positions are i.i.d. from some distribution. If class labels are observed for a number of vertices tending to infinity, then we show that the remaining vertices can be classified with error converging to Bayes optimal using the $(k)$-nearest-neighbors classification rule. We evaluate the proposed methods on simulated data and a graph derived from Wikipedia.