79 research outputs found
Computing symmetry groups of polyhedra
Knowing the symmetries of a polyhedron can be very useful for the analysis of
its structure as well as for practical polyhedral computations. In this note,
we study symmetry groups preserving the linear, projective and combinatorial
structure of a polyhedron. In each case we give algorithmic methods to compute
the corresponding group and discuss some practical experiences. For practical
purposes the linear symmetry group is the most important, as its computation
can be directly translated into a graph automorphism problem. We indicate how
to compute integral subgroups of the linear symmetry group that are used for
instance in integer linear programming.Comment: 20 pages, 1 figure; containing a corrected and improved revisio
Synthetic Generation of Social Network Data With Endorsements
In many simulation studies involving networks there is the need to rely on a
sample network to perform the simulation experiments. In many cases, real
network data is not available due to privacy concerns. In that case we can
recourse to synthetic data sets with similar properties to the real data. In
this paper we discuss the problem of generating synthetic data sets for a
certain kind of online social network, for simulation purposes. Some popular
online social networks, such as LinkedIn and ResearchGate, allow user
endorsements for specific skills. For each particular skill, the endorsements
give rise to a directed subgraph of the corresponding network, where the nodes
correspond to network members or users, and the arcs represent endorsement
relations. Modelling these endorsement digraphs can be done by formulating an
optimization problem, which is amenable to different heuristics. Our
construction method consists of two stages: The first one simulates the growth
of the network, and the second one solves the aforementioned optimization
problem to construct the endorsements.Comment: 5 figures, 2 algorithms, Journal of Simulation 201
The graph bottleneck identity
A matrix is said to determine a
\emph{transitional measure} for a digraph on vertices if for all
the \emph{transition inequality} holds and reduces to the equality (called the \emph{graph
bottleneck identity}) if and only if every path in from to contains
. We show that every positive transitional measure produces a distance by
means of a logarithmic transformation. Moreover, the resulting distance
is \emph{graph-geodetic}, that is,
holds if and only if every path in connecting and contains .
Five types of matrices that determine transitional measures for a digraph are
considered, namely, the matrices of path weights, connection reliabilities,
route weights, and the weights of in-forests and out-forests. The results
obtained have undirected counterparts. In [P. Chebotarev, A class of
graph-geodetic distances generalizing the shortest-path and the resistance
distances, Discrete Appl. Math., URL
http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to
fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic
The Walk Distances in Graphs
The walk distances in graphs are defined as the result of appropriate
transformations of the proximity measures, where
is the weighted adjacency matrix of a graph and is a sufficiently small
positive parameter. The walk distances are graph-geodetic; moreover, they
converge to the shortest path distance and to the so-called long walk distance
as the parameter approaches its limiting values. We also show that the
logarithmic forest distances which are known to generalize the resistance
distance and the shortest path distance are a subclass of walk distances. On
the other hand, the long walk distance is equal to the resistance distance in a
transformed graph.Comment: Accepted for publication in Discrete Applied Mathematics. 26 pages, 3
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