19 research outputs found
Network alignment and similarity reveal atlas-based topological differences in structural connectomes
The interactions between different brain regions can be modeled as a graph, called connectome, whose nodes correspond to parcels from a predefined brain atlas. The edges of the graph encode the strength of the axonal connectivity between regions of the atlas which can be estimated via diffusion Magnetic Resonance Imaging (MRI) tractography. Herein, we aim at providing a novel perspective on the problem of choosing a suitable atlas for structural connectivity studies by assessing how robustly an atlas captures the network topology across different subjects in a homogeneous cohort. We measure this robustness by assessing the alignability of the connectomes, namely the possibility to retrieve graph matchings that provide highly similar graphs. We introduce two novel concepts. First, the graph Jaccard index (GJI), a graph similarity measure based on the well-established Jaccard index between sets; the GJI exhibits natural mathematical properties that are not satisfied by previous approaches. Second, we devise WL-align, a new technique for aligning connectomes obtained by adapting the Weisfeiler-Lehman (WL) graph-isomorphism test.We validated the GJI and WL-align on data from the Human Connectome Project database, inferring a strategy for choosing a suitable parcellation for structural connectivity studies. Code and data are publicly available
Seeded Graph Matching via Large Neighborhood Statistics
We study a well known noisy model of the graph isomorphism problem. In this
model, the goal is to perfectly recover the vertex correspondence between two
edge-correlated Erd\H{o}s-R\'{e}nyi random graphs, with an initial seed set of
correctly matched vertex pairs revealed as side information. For seeded
problems, our result provides a significant improvement over previously known
results. We show that it is possible to achieve the information-theoretic limit
of graph sparsity in time polynomial in the number of vertices . Moreover,
we show the number of seeds needed for exact recovery in polynomial-time can be
as low as in the sparse graph regime (with the average degree
smaller than ) and in the dense graph regime.
Our results also shed light on the unseeded problem. In particular, we give
sub-exponential time algorithms for sparse models and an
algorithm for dense models for some parameters, including some that are not
covered by recent results of Barak et al
Partial Recovery in the Graph Alignment Problem
In this paper, we consider the graph alignment problem, which is the problem
of recovering, given two graphs, a one-to-one mapping between nodes that
maximizes edge overlap. This problem can be viewed as a noisy version of the
well-known graph isomorphism problem and appears in many applications,
including social network deanonymization and cellular biology. Our focus here
is on partial recovery, i.e., we look for a one-to-one mapping which is correct
on a fraction of the nodes of the graph rather than on all of them, and we
assume that the two input graphs to the problem are correlated
Erd\H{o}s-R\'enyi graphs of parameters . Our main contribution is then
to give necessary and sufficient conditions on under which partial
recovery is possible with high probability as the number of nodes goes to
infinity. In particular, we show that it is possible to achieve partial
recovery in the regime under certain additional assumptions. An
interesting byproduct of the analysis techniques we develop to obtain the
sufficiency result in the partial recovery setting is a tighter analysis of the
maximum likelihood estimator for the graph alignment problem, which leads to
improved sufficient conditions for exact recovery