18,206 research outputs found
Seeded Graph Matching: Efficient Algorithms and Theoretical Guarantees
In this paper, a new information theoretic framework for graph matching is
introduced. Using this framework, the graph isomorphism and seeded graph
matching problems are studied. The maximum degree algorithm for graph
isomorphism is analyzed and sufficient conditions for successful matching are
rederived using type analysis. Furthermore, a new seeded matching algorithm
with polynomial time complexity is introduced. The algorithm uses `typicality
matching' and techniques from point-to-point communications for reliable
matching. Assuming an Erdos-Renyi model on the correlated graph pair, it is
shown that successful matching is guaranteed when the number of seeds grows
logarithmically with the number of vertices in the graphs. The logarithmic
coefficient is shown to be inversely proportional to the mutual information
between the edge variables in the two graphs
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
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
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