A Spatial Web Graph Model with Local Influence Regions

Abstract

Abstract. We present a new stochastic model for complex networks, based on a spatial embedding of the nodes, called the Spatial Preferred Attachment (SPA) model. In the SPA model, nodes have influence regions of varying size, and new nodes may only link to a node if they fall within its influence region. The spatial embedding of the nodes models the background knowledge or identity of the node, which will influence its link environment. In our model, nodes can determine their link environment based only on local knowledge of the network. We prove that our model gives a power law in-degree distribution, with exponent in [2, ∞) depending on the parameters, and with concentration for a wide range of in-degree values. We show that the model allows for edges that span a large distance in the underlying space, modelling a feature often observed in real-world complex networks.

Spatial Models for Virtual Networks

“…A new node can link to an existing node only when it falls within its associated sphere. The SPA model, proposed in [1], is based on the same principle, except that the radii of the spheres are not uniform, but depend on the in-degree of the node (the SPA model generates directed graphs). Precisely, each node v i is the center of a sphere whose radius is chosen so that its volume equals…”

“…First, we make a simplifying assumption. In [1], it was shown that the expected degree, at time t, of node v i born at time i is proportional to (t/i) p . In the following, assuming that the degree of each node is close to its expected degree, we will set the the volume of the sphere of a node equal to its expected value:…”

“…For each node v i so that v t falls inside the sphere of v i , independently, a link from v t to v i is created with probability p. Thus, a node with high degree is "visible" over a larger area of S than a node with small degree. It was shown in [1] that the SPA model produces graphs with a power law degree distribution, with exponent 1 + 1/pA 1 . Moreover, because of the unequal size of the spheres, long links can occur, but only to nodes of high degree.…”

Some Typical Properties of the Spatial Preferred Attachment Model

“…In [3] (see also [4] for a preceding version of this paper), the model is defined for a variety of metric spaces S. In this paper, we let S be the unit hypercube in R m , equipped with the torus metric derived from any of the L p norms. This means that for any two points x and y in S, d(x, y) = min ||x − y + u|| p : u ∈ {−1, 0, 1} m .…”

“…The original model as presented in [3] has a third parameter, A 3 , which is assumed to be zero here. This causes no loss of generality, since all asymptotic results presented here are unaffected by A 3 .…”