3,055 research outputs found
On the Complexity of Role Colouring Planar Graphs, Trees and Cographs
We prove several results about the complexity of the role colouring problem.
A role colouring of a graph is an assignment of colours to the vertices of
such that two vertices of the same colour have identical sets of colours in
their neighbourhoods. We show that the problem of finding a role colouring with
colours is NP-hard for planar graphs. We show that restricting the
problem to trees yields a polynomially solvable case, as long as is either
constant or has a constant difference with , the number of vertices in the
tree. Finally, we prove that cographs are always -role-colourable for
and construct such a colouring in polynomial time
Guessing Numbers of Odd Cycles
For a given number of colours, , the guessing number of a graph is the
base logarithm of the size of the largest family of colourings of the
vertex set of the graph such that the colour of each vertex can be determined
from the colours of the vertices in its neighbourhood. An upper bound for the
guessing number of the -vertex cycle graph is . It is known that
the guessing number equals whenever is even or is a perfect
square \cite{Christofides2011guessing}. We show that, for any given integer
, if is the largest factor of less than or equal to
, for sufficiently large odd , the guessing number of with
colours is . This answers a question posed by
Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also
present an explicit protocol which achieves this bound for every . Linking
this to index coding with side information, we deduce that the information
defect of with colours is for sufficiently
large odd . Our results are a generalisation of the case which was
proven in \cite{bar2011index}.Comment: 16 page
Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths
We introduce several novel and computationally efficient methods for
detecting "core--periphery structure" in networks. Core--periphery structure is
a type of mesoscale structure that includes densely-connected core vertices and
sparsely-connected peripheral vertices. Core vertices tend to be well-connected
both among themselves and to peripheral vertices, which tend not to be
well-connected to other vertices. Our first method, which is based on
transportation in networks, aggregates information from many geodesic paths in
a network and yields a score for each vertex that reflects the likelihood that
a vertex is a core vertex. Our second method is based on a low-rank
approximation of a network's adjacency matrix, which can often be expressed as
a tensor-product matrix. Our third approach uses the bottom eigenvector of the
random-walk Laplacian to infer a coreness score and a classification into core
and peripheral vertices. We also design an objective function to (1) help
classify vertices into core or peripheral vertices and (2) provide a
goodness-of-fit criterion for classifications into core versus peripheral
vertices. To examine the performance of our methods, we apply our algorithms to
both synthetically-generated networks and a variety of networks constructed
from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on
"Network Analysis and Modelling" (available at
https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB
A Failure To Impose Control: Private Security And The Mexican State
The private security industry is growing rapidly across the globe and governments are implementing numerous strategies to attempt to control it. Scholars tend to agree that drafting and imposing comprehensive regulations is one of the most effective strategies states can use to control and direct private security companies to reduce insecurity and operate effectively, safely, and accountably. A number of studies on private security tend to highlight – and rightfully so – the weaknesses or absence of states’ private security regulations (Argueta, 2012; Born; Buzatu, 2008; Lopes, 2018; McCrie, 2017; Nagaraj, 2012; Simelane, 2008). However, strengthening a regulatory regime is not sufficient for solving many problems plaguing the private security industry and can even be counterproductive if state institutions are weak
In Appreciation of Buffoonery, Egotism, and the Shōmon School: Koikawa Harumachi's Kachō kakurenbō (1776)
Early Modern Japan Networ
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