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

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 Barabási 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.

Geographical influences of an emerging network of gang rivalries

“…One method for incorporating geographical information into the random graph construction is by using a Geographical Threshold Graph [34,7,8]. This is a random graph on a set of randomly weighted nodes, where the nodes are located in a metric space and the connections are determined by thresholding a function of the distance and the weights.…”

“…to assist in determining whether or not two nodes are connected with an edge [34,7,8]. Geographical Threshold Graphs randomly assign weights η i to the N nodes.…”

Spatial Models for Virtual Networks

“…In [29], nodes are linked independently with probability proportional to the dot product of the vectors reprenting the nodes. In [21], and later also [7], nodes are assigned random weights w i , and two nodes v i and v j are linked precisely when a function of the weights and the distance, for example (w i + w j )/d(v i , v j ), exceeds a given threshold θ. In [8], each new node v t links to the node v i that minimizes a function which is the convex combination of the graph distance of v i to the center of the graph, and the metric distance between v i and v t .…”

Planarity as a driver of Spatial Network structure

“…The random geometric graph just mentioned is the fundamental example of these. Subsequent adaptions that demonstrate the power law property create connections based on a product of node weight and a power of the distance (Masuda et al, 2005;Manna and Sen, 2002).…”