  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.
