Unimodular lattice triangulations as small-world and scale-free random graphs

Real-world networks, e.g., the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulation...

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Bibliographic Details
Published in:New journal of physics 2015-02, Vol.17 (2), p.23013-11
Main Authors: Krüger, B, Schmidt, E M, Mecke, K
Format: Article
Language:English
Subjects:
02.10.Ox
05.10.Ln
64.60.aq
Apexes
Clustering
Computer simulation
Graph theory
Graphs
Inverse
Lattices
Mathematical models
maximal planar graphs
Networks
Nodes
Physics
Shortest-path problems
Triangulation
unimodular lattice triangulations
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Summary:Real-world networks, e.g., the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages, using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real-world graphs; for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees . Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.
ISSN:1367-2630
1367-2630
DOI:10.1088/1367-2630/17/2/023013