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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
IOP Publishing
2015-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/17/2/023013 |
| _version_ | 1797751268337254400 |
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| author | B Krüger E M Schmidt K Mecke |
| author_facet | B Krüger E M Schmidt K Mecke |
| author_sort | B Krüger |
| collection | DOAJ |
| description | 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 $k\geqslant 5$ . 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. |
| first_indexed | 2024-03-12T16:45:00Z |
| format | Article |
| id | doaj.art-e876f96cd39146a795cf9e561cc799be |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:45:00Z |
| publishDate | 2015-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-e876f96cd39146a795cf9e561cc799be2023-08-08T14:17:19ZengIOP PublishingNew Journal of Physics1367-26302015-01-0117202301310.1088/1367-2630/17/2/023013Unimodular lattice triangulations as small-world and scale-free random graphsB Krüger0E M Schmidt1K Mecke2Institute for Theoretical Physics, Universität Erlangen-Nürnberg , Staudtstr. 7, 91058 Erlangen, GermanyInstitute for Theoretical Physics, Universität Erlangen-Nürnberg , Staudtstr. 7, 91058 Erlangen, GermanyInstitute for Theoretical Physics, Universität Erlangen-Nürnberg , Staudtstr. 7, 91058 Erlangen, GermanyReal-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 $k\geqslant 5$ . 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.https://doi.org/10.1088/1367-2630/17/2/023013unimodular lattice triangulationsnetworksmaximal planar graphs64.60.aq02.10.Ox05.10.Ln |
| spellingShingle | B Krüger E M Schmidt K Mecke Unimodular lattice triangulations as small-world and scale-free random graphs New Journal of Physics unimodular lattice triangulations networks maximal planar graphs 64.60.aq 02.10.Ox 05.10.Ln |
| title | Unimodular lattice triangulations as small-world and scale-free random graphs |
| title_full | Unimodular lattice triangulations as small-world and scale-free random graphs |
| title_fullStr | Unimodular lattice triangulations as small-world and scale-free random graphs |
| title_full_unstemmed | Unimodular lattice triangulations as small-world and scale-free random graphs |
| title_short | Unimodular lattice triangulations as small-world and scale-free random graphs |
| title_sort | unimodular lattice triangulations as small world and scale free random graphs |
| topic | unimodular lattice triangulations networks maximal planar graphs 64.60.aq 02.10.Ox 05.10.Ln |
| url | https://doi.org/10.1088/1367-2630/17/2/023013 |
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