12 research outputs found
Dimension Reduction via Colour Refinement
Colour refinement is a basic algorithmic routine for graph isomorphism
testing, appearing as a subroutine in almost all practical isomorphism solvers.
It partitions the vertices of a graph into "colour classes" in such a way that
all vertices in the same colour class have the same number of neighbours in
every colour class. Tinhofer (Disc. App. Math., 1991), Ramana, Scheinerman, and
Ullman (Disc. Math., 1994) and Godsil (Lin. Alg. and its App., 1997)
established a tight correspondence between colour refinement and fractional
isomorphisms of graphs, which are solutions to the LP relaxation of a natural
ILP formulation of graph isomorphism.
We introduce a version of colour refinement for matrices and extend existing
quasilinear algorithms for computing the colour classes. Then we generalise the
correspondence between colour refinement and fractional automorphisms and
develop a theory of fractional automorphisms and isomorphisms of matrices.
We apply our results to reduce the dimensions of systems of linear equations
and linear programs. Specifically, we show that any given LP L can efficiently
be transformed into a (potentially) smaller LP L' whose number of variables and
constraints is the number of colour classes of the colour refinement algorithm,
applied to a matrix associated with the LP. The transformation is such that we
can easily (by a linear mapping) map both feasible and optimal solutions back
and forth between the two LPs. We demonstrate empirically that colour
refinement can indeed greatly reduce the cost of solving linear programs
Unsupervised Learning of Invariance Transformations
The need for large amounts of training data in modern machine learning is one
of the biggest challenges of the field. Compared to the brain, current
artificial algorithms are much less capable of learning invariance
transformations and employing them to extrapolate knowledge from small sample
sets. It has recently been proposed that the brain might encode perceptual
invariances as approximate graph symmetries in the network of synaptic
connections. Such symmetries may arise naturally through a biologically
plausible process of unsupervised Hebbian learning. In the present paper, we
illustrate this proposal on numerical examples, showing that invariance
transformations can indeed be recovered from the structure of recurrent
synaptic connections which form within a layer of feature detector neurons via
a simple Hebbian learning rule. In order to numerically recover the invariance
transformations from the resulting recurrent network, we develop a general
algorithmic framework for finding approximate graph automorphisms. We discuss
how this framework can be used to find approximate automorphisms in weighted
graphs in general
An Optimization-based Approach To Node Role Discovery in Networks: Approximating Equitable Partitions
Similar to community detection, partitioning the nodes of a network according
to their structural roles aims to identify fundamental building blocks of a
network. The found partitions can be used, e.g., to simplify descriptions of
the network connectivity, to derive reduced order models for dynamical
processes unfolding on processes, or as ingredients for various graph mining
tasks. In this work, we offer a fresh look on the problem of role extraction
and its differences to community detection and present a definition of node
roles related to graph-isomorphism tests, the Weisfeiler-Leman algorithm and
equitable partitions. We study two associated optimization problems (cost
functions) grounded in ideas from graph isomorphism testing, and present
theoretical guarantees associated to the solutions of these problems. Finally,
we validate our approach via a novel "role-infused partition benchmark", a
network model from which we can sample networks in which nodes are endowed with
different roles in a stochastic way
Neighborhood Structure Configuration Models
We develop a new method to efficiently sample synthetic networks that
preserve the d-hop neighborhood structure of a given network for any given d.
The proposed algorithm trades off the diversity in network samples against the
depth of the neighborhood structure that is preserved. Our key innovation is to
employ a colored Configuration Model with colors derived from iterations of the
so-called Color Refinement algorithm. We prove that with increasing iterations
the preserved structural information increases: the generated synthetic
networks and the original network become more and more similar, and are
eventually indistinguishable in terms of centrality measures such as PageRank,
HITS, Katz centrality and eigenvector centrality. Our work enables to
efficiently generate samples with a precisely controlled similarity to the
original network, especially for large networks
Subjectively interesting connecting trees and forests
Consider a large graph or network, and a user-provided set of query vertices between which the user wishes to explore relations. For example, a researcher may want to connect research papers in a citation network, an analyst may wish to connect organized crime suspects in a communication network, or an internet user may want to organize their bookmarks given their location in the world wide web. A natural way to do this is to connect the vertices in the form of a tree structure that is present in the graph. However, in sufficiently dense graphs, most such trees will be large or somehow trivial (e.g. involving high degree vertices) and thus not insightful. Extending previous research, we define and investigate the new problem of mining subjectively interesting trees connecting a set of query vertices in a graph, i.e., trees that are highly surprising to the specific user at hand. Using information theoretic principles, we formalize the notion of interestingness of such trees mathematically, taking in account certain prior beliefs the user has specified about the graph. A remaining problem is efficiently fitting a prior belief model. We show how this can be done for a large class of prior beliefs. Given a specified prior belief model, we then propose heuristic algorithms to find the best trees efficiently. An empirical validation of our methods on a large real graphs evaluates the different heuristics and validates the interestingness of the given trees