3,221 research outputs found
A Spectral Assignment Approach for the Graph Isomorphism Problem
In this paper, we propose algorithms for the graph isomorphism (GI) problem
that are based on the eigendecompositions of the adjacency matrices. The
eigenvalues of isomorphic graphs are identical. However, two graphs and
can be isospectral but non-isomorphic. We first construct a graph
isomorphism testing algorithm for friendly graphs and then extend it to
unambiguous graphs. We show that isomorphisms can be detected by solving a
linear assignment problem. If the graphs possess repeated eigenvalues, which
typically correspond to graph symmetries, finding isomorphisms is much harder.
By repeatedly perturbing the adjacency matrices and by using properties of
eigenpolytopes, it is possible to break symmetries of the graphs and
iteratively assign vertices of to vertices of , provided that an
admissible assignment exists. This heuristic approach can be used to construct
a permutation which transforms into if the graphs are
isomorphic. The methods will be illustrated with several guiding examples
Projected Power Iteration for Network Alignment
The network alignment problem asks for the best correspondence between two
given graphs, so that the largest possible number of edges are matched. This
problem appears in many scientific problems (like the study of protein-protein
interactions) and it is very closely related to the quadratic assignment
problem which has graph isomorphism, traveling salesman and minimum bisection
problems as particular cases. The graph matching problem is NP-hard in general.
However, under some restrictive models for the graphs, algorithms can
approximate the alignment efficiently. In that spirit the recent work by Feizi
and collaborators introduce EigenAlign, a fast spectral method with convergence
guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm
Projected Power Alignment, which is a projected power iteration version of
EigenAlign. We numerically show it improves the recovery rates of EigenAlign
and we describe the theory that may be used to provide performance guarantees
for Projected Power Alignment.Comment: 8 page
A quantum-walk-inspired adiabatic algorithm for graph isomorphism
We present a 2-local quantum algorithm for graph isomorphism GI based on an
adiabatic protocol. By exploiting continuous-time quantum-walks, we are able to
avoid a mere diffusion over all possible configurations and to significantly
reduce the dimensionality of the visited space. Within this restricted space,
the graph isomorphism problem can be translated into the search of a satisfying
assignment to a 2-SAT formula without resorting to perturbation gadgets or
projective techniques. We present an analysis of the execution time of the
algorithm on small instances of the graph isomorphism problem and discuss the
issue of an implementation of the proposed adiabatic scheme on current quantum
computing hardware.Comment: 10 pages, 5 figure
Graph matching: relax or not?
We consider the problem of exact and inexact matching of weighted undirected
graphs, in which a bijective correspondence is sought to minimize a quadratic
weight disagreement. This computationally challenging problem is often relaxed
as a convex quadratic program, in which the space of permutations is replaced
by the space of doubly-stochastic matrices. However, the applicability of such
a relaxation is poorly understood. We define a broad class of friendly graphs
characterized by an easily verifiable spectral property. We prove that for
friendly graphs, the convex relaxation is guaranteed to find the exact
isomorphism or certify its inexistence. This result is further extended to
approximately isomorphic graphs, for which we develop an explicit bound on the
amount of weight disagreement under which the relaxation is guaranteed to find
the globally optimal approximate isomorphism. We also show that in many cases,
the graph matching problem can be further harmlessly relaxed to a convex
quadratic program with only n separable linear equality constraints, which is
substantially more efficient than the standard relaxation involving 2n equality
and n^2 inequality constraints. Finally, we show that our results are still
valid for unfriendly graphs if additional information in the form of seeds or
attributes is allowed, with the latter satisfying an easy to verify spectral
characteristic
On the equivalence between graph isomorphism testing and function approximation with GNNs
Graph neural networks (GNNs) have achieved lots of success on
graph-structured data. In the light of this, there has been increasing interest
in studying their representation power. One line of work focuses on the
universal approximation of permutation-invariant functions by certain classes
of GNNs, and another demonstrates the limitation of GNNs via graph isomorphism
tests.
Our work connects these two perspectives and proves their equivalence. We
further develop a framework of the representation power of GNNs with the
language of sigma-algebra, which incorporates both viewpoints. Using this
framework, we compare the expressive power of different classes of GNNs as well
as other methods on graphs. In particular, we prove that order-2 Graph
G-invariant networks fail to distinguish non-isomorphic regular graphs with the
same degree. We then extend them to a new architecture, Ring-GNNs, which
succeeds on distinguishing these graphs and provides improvements on real-world
social network datasets
- …