79 research outputs found

    Computing symmetry groups of polyhedra

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    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

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    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

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    A matrix S=(sij)∈Rn×nS=(s_{ij})\in{\mathbb R}^{n\times n} is said to determine a \emph{transitional measure} for a digraph GG on nn vertices if for all i,j,k∈{1,.˙.,n},i,j,k\in\{1,\...,n\}, the \emph{transition inequality} sijsjk≤siksjjs_{ij} s_{jk}\le s_{ik} s_{jj} holds and reduces to the equality (called the \emph{graph bottleneck identity}) if and only if every path in GG from ii to kk contains jj. We show that every positive transitional measure produces a distance by means of a logarithmic transformation. Moreover, the resulting distance d(⋅,⋅)d(\cdot,\cdot) is \emph{graph-geodetic}, that is, d(i,j)+d(j,k)=d(i,k)d(i,j)+d(j,k)=d(i,k) holds if and only if every path in GG connecting ii and kk contains jj. 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

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    The walk distances in graphs are defined as the result of appropriate transformations of the ∑k=0∞(tA)k\sum_{k=0}^\infty(tA)^k proximity measures, where AA is the weighted adjacency matrix of a graph and tt 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 tt 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 figure
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