25 research outputs found
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
The Umeyama algorithm for matching correlated Gaussian geometric models in the low-dimensional regime
Motivated by the problem of matching two correlated random geometric graphs,
we study the problem of matching two Gaussian geometric models correlated
through a latent node permutation. Specifically, given an unknown permutation
on and given i.i.d. pairs of correlated Gaussian
vectors in with noise parameter ,
we consider two types of (correlated) weighted complete graphs with edge
weights given by , . The goal is to recover the hidden vertex correspondence based
on the observed matrices and . For the low-dimensional regime where
, Wang, Wu, Xu, and Yolou [WWXY22+] established the information
thresholds for exact and almost exact recovery in matching correlated Gaussian
geometric models. They also conducted numerical experiments for the classical
Umeyama algorithm. In our work, we prove that this algorithm achieves exact
recovery of when the noise parameter , and
almost exact recovery when . Our results approach the
information thresholds up to a factor in the
low-dimensional regime.Comment: 31 page
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
Seeded graph matching for the correlated Wigner model via the projected power method
In the graph matching problem we observe two graphs and the goal is to
find an assignment (or matching) between their vertices such that some measure
of edge agreement is maximized. We assume in this work that the observed pair
has been drawn from the correlated Wigner model -- a popular model for
correlated weighted graphs -- where the entries of the adjacency matrices of
and are independent Gaussians and each edge of is correlated with
one edge of (determined by the unknown matching) with the edge correlation
described by a parameter . In this paper, we analyse the
performance of the projected power method (PPM) as a seeded graph matching
algorithm where we are given an initial partially correct matching (called the
seed) as side information. We prove that if the seed is close enough to the
ground-truth matching, then with high probability, PPM iteratively improves the
seed and recovers the ground-truth matching (either partially or exactly) in
iterations. Our results prove that PPM works even in
regimes of constant , thus extending the analysis in (Mao et al.,2021)
for the sparse Erd\"os-Renyi model to the (dense) Wigner model. As a byproduct
of our analysis, we see that the PPM framework generalizes some of the
state-of-art algorithms for seeded graph matching. We support and complement
our theoretical findings with numerical experiments on synthetic data