Geographical threshold graphs with small-world and scale-free properties

TL;DR

This paper introduces a geographical non-growing network model with vertex weights that can generate small-world, scale-free networks, providing a more realistic explanation for many real-world networks' properties.

Contribution

It proposes a new model combining geography and vertex weights to produce small-world, scale-free networks without requiring network growth.

Findings

Model reproduces small-world and scale-free properties

Generalizes existing models like unit disk and gravity models

Links network topology with vertex weight distributions

Abstract

Many real networks are equipped with short diameters, high clustering, and power-law degree distributions. With preferential attachment and network growth, the model by Barabasi and Albert simultaneously reproduces these properties, and geographical versions of growing networks have also been analyzed. However, nongrowing networks with intrinsic vertex weights often explain these features more plausibly, since not all networks are really growing. We propose a geographical nongrowing network model with vertex weights. Edges are assumed to form when a pair of vertices are spatially close and/or have large summed weights. Our model generalizes a variety of models as well as the original nongeographical counterpart, such as the unit disk graph, the Boolean model, and the gravity model, which appear in the contexts of percolation, wire communication, mechanical and solid physics, sociology, economy, and marketing. In appropriate configurations, our model produces small-world networks with power-law degree distributions. We also discuss the relation between geography, power laws in networks, and power laws in general quantities serving as vertex weights.