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Match Graphs given a seeding of vertex correspondences

Description

Given two adjacency matrices A and B of the same size, match the two graphs with the help of m seed vertex pairs which correspond to the first m rows (and columns) of the adjacency matrices.

Usage

match_vertices(A, B, m, start, iteration)

Arguments

A

a numeric matrix, the adjacency matrix of the first graph

B

a numeric matrix, the adjacency matrix of the second graph

m

The number of seeds. The first m vertices of both graphs are matched.

start

a numeric matrix, the permutation matrix estimate is initialized with start

iteration

The number of iterations for the Frank-Wolfe algorithm

Details

The approximate graph matching problem is to find a bijection between the vertices of two graphs , such that the number of edge disagreements between the corresponding vertex pairs is minimized. For seeded graph matching, part of the bijection that consist of known correspondences (the seeds) is known and the problem task is to complete the bijection by estimating the permutation matrix that permutes the rows and columns of the adjacency matrix of the second graph.

It is assumed that for the two supplied adjacency matrices A and B, both of size n*n, the first m rows(and columns) of A and B correspond to the same vertices in both graphs. That is, the n*n permutation matrix that defines the bijection is I_{m} \bigoplus P for a (n-m)*(n-m) permutation matrix P and m times m identity matrix I_{m}. The function match_vertices estimates the permutation matrix P via an optimization algorithm based on the Frank-Wolfe algorithm.

See references for further details.

Value

A numeric matrix which is the permutation matrix that determines the bijection between the graphs of A and B

Author(s)

Vince Lyzinski http://www.ams.jhu.edu/~lyzinski/

References

Vogelstein, J. T., Conroy, J. M., Podrazik, L. J., Kratzer, S. G., Harley, E. T., Fishkind, D. E.,Vogelstein, R. J., Priebe, C. E. (2011). Fast Approximate Quadratic Programming for Large (Brain) Graph Matching. Online: http://arxiv.org/abs/1112.5507

Fishkind, D. E., Adali, S., Priebe, C. E. (2012). Seeded Graph Matching Online: http://arxiv.org/abs/1209.0367

See Also

sample_correlated_gnp,sample_correlated_gnp_pair

Examples


 #require(Matrix)
 g1 <- erdos.renyi.game(10, .1)
 randperm <- c(1:3, 3+sample(7))
 g2 <- sample_correlated_gnp(g1, corr=1, p=g1$p, perm=randperm)
 A  <- as.matrix(get.adjacency(g1))
 B  <- as.matrix(get.adjacency(g2))
 P  <-match_vertices (A, B, m=3, start=diag(rep(1, nrow(A)-3)), 20)
 P

[Package igraph version 1.2.4.1 Index]