189,604 research outputs found
Relations Between Graphs
Given two graphs G and H, we ask under which conditions there is a relation R
that generates the edges of H given the structure of graph G. This construction
can be seen as a form of multihomomorphism. It generalizes surjective
homomorphisms of graphs and naturally leads to notions of R-retractions,
R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are
unique up to isomorphism and can be computed in polynomial time.Comment: accepted by Ars Mathematica Contemporane
On the max-algebraic core of a nonnegative matrix
The max-algebraic core of a nonnegative matrix is the intersection of column
spans of all max-algebraic matrix powers. Here we investigate the action of a
matrix on its core. Being closely related to ultimate periodicity of matrix
powers, this study leads us to new modifications and geometric
characterizations of robust, orbit periodic and weakly stable matrices.Comment: 27 page
Graph cluster randomization: network exposure to multiple universes
A/B testing is a standard approach for evaluating the effect of online
experiments; the goal is to estimate the `average treatment effect' of a new
feature or condition by exposing a sample of the overall population to it. A
drawback with A/B testing is that it is poorly suited for experiments involving
social interference, when the treatment of individuals spills over to
neighboring individuals along an underlying social network. In this work, we
propose a novel methodology using graph clustering to analyze average treatment
effects under social interference. To begin, we characterize graph-theoretic
conditions under which individuals can be considered to be `network exposed' to
an experiment. We then show how graph cluster randomization admits an efficient
exact algorithm to compute the probabilities for each vertex being network
exposed under several of these exposure conditions. Using these probabilities
as inverse weights, a Horvitz-Thompson estimator can then provide an effect
estimate that is unbiased, provided that the exposure model has been properly
specified.
Given an estimator that is unbiased, we focus on minimizing the variance.
First, we develop simple sufficient conditions for the variance of the
estimator to be asymptotically small in n, the size of the graph. However, for
general randomization schemes, this variance can be lower bounded by an
exponential function of the degrees of a graph. In contrast, we show that if a
graph satisfies a restricted-growth condition on the growth rate of
neighborhoods, then there exists a natural clustering algorithm, based on
vertex neighborhoods, for which the variance of the estimator can be upper
bounded by a linear function of the degrees. Thus we show that proper cluster
randomization can lead to exponentially lower estimator variance when
experimentally measuring average treatment effects under interference.Comment: 9 pages, 2 figure
The complexity of the list homomorphism problem for graphs
We completely classify the computational complexity of the list H-colouring
problem for graphs (with possible loops) in combinatorial and algebraic terms:
for every graph H the problem is either NP-complete, NL-complete, L-complete or
is first-order definable; descriptive complexity equivalents are given as well
via Datalog and its fragments. Our algebraic characterisations match important
conjectures in the study of constraint satisfaction problems.Comment: 12 pages, STACS 201
- …