dorsal/arxiv
View SchemaGeographical networks evolving with optimal policy
| Authors | Yan-Bo Xie, Tao Zhou, Wen-Jie Bai, Guanrong Chen, Wei-Ke Xiao, Bing-Hong Wang |
|---|---|
| Categories | |
| ArXiv ID | physics/0605054 |
| URL | https://arxiv.org/abs/physics/0605054 |
| DOI | 10.1103/PhysRevE.75.036106 |
| Journal | Physical Review E, 75, 036106 (2007) |
Abstract
In this article, we propose a growing network model based on an optimal policy involving both topological and geographical measures. In this model, at each time step, a new node, having randomly assigned coordinates in a $1 \times 1$ square, is added and connected to a previously existing node $i$, which minimizes the quantity $r_i^2/k_i^\alpha$, where $r_i$ is the geographical distance, $k_i$ the degree, and $\alpha$ a free parameter. The degree distribution obeys a power-law form when $\alpha=1$, and an exponential form when $\alpha=0$. When $\alpha$ is in the interval $(0,1)$, the network exhibits a stretched exponential distribution. We prove that the average topological distance increases in a logarithmic scale of the network size, indicating the existence of the small-world property. Furthermore, we obtain the geographical edge-length distribution, the total geographical length of all edges, and the average geographical distance of the whole network. Interestingly, we found that the total edge-length will sharply increase when $\alpha$ exceeds the critical value $\alpha_c=1$, and the average geographical distance has an upper bound independent of the network size. All the results are obtained analytically with some reasonable approximations, which are well verified by simulations.
{
"annotation_id": "3986012f-edba-43a7-b12f-0d2c649bb874",
"date_created": "2026-03-02T18:01:07.548000Z",
"date_modified": "2026-03-02T18:01:07.548000Z",
"file_hash": "a575781728113239961f84c0badbd4018553133d9236ed406a79ee10a6cb0819",
"private": false,
"record": {
"abstract": "In this article, we propose a growing network model based on an optimal\npolicy involving both topological and geographical measures. In this model, at\neach time step, a new node, having randomly assigned coordinates in a $1 \\times\n1$ square, is added and connected to a previously existing node $i$, which\nminimizes the quantity $r_i^2/k_i^\\alpha$, where $r_i$ is the geographical\ndistance, $k_i$ the degree, and $\\alpha$ a free parameter. The degree\ndistribution obeys a power-law form when $\\alpha=1$, and an exponential form\nwhen $\\alpha=0$. When $\\alpha$ is in the interval $(0,1)$, the network exhibits\na stretched exponential distribution. We prove that the average topological\ndistance increases in a logarithmic scale of the network size, indicating the\nexistence of the small-world property. Furthermore, we obtain the geographical\nedge-length distribution, the total geographical length of all edges, and the\naverage geographical distance of the whole network. Interestingly, we found\nthat the total edge-length will sharply increase when $\\alpha$ exceeds the\ncritical value $\\alpha_c=1$, and the average geographical distance has an upper\nbound independent of the network size. All the results are obtained\nanalytically with some reasonable approximations, which are well verified by\nsimulations.",
"arxiv_id": "physics/0605054",
"authors": [
"Yan-Bo Xie",
"Tao Zhou",
"Wen-Jie Bai",
"Guanrong Chen",
"Wei-Ke Xiao",
"Bing-Hong Wang"
],
"categories": [
"physics.soc-ph"
],
"doi": "10.1103/PhysRevE.75.036106",
"journal_ref": "Physical Review E, 75, 036106 (2007)",
"title": "Geographical networks evolving with optimal policy",
"url": "https://arxiv.org/abs/physics/0605054"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "52768219-3488-4e25-a256-4f25c199df53",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}