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.

Random Intersection Graph Process

“…From the inequality EA 2 (T t ) = i∈T t i −1 a 2 ≤ c a 2 we conclude that A 2 (T t ) is stochastically bounded. Hence t −1 Y 2 t A 2 (T t ) = o P (1). This bound and (3.23) combined with (3.10) implyd…”

“…In a recent article [4], a related intersection graph process is considered, where a newly arrived user is linked preferentially to already highly popular items, thus, further increasing their popularity/attractiveness measured by the number of users that have chosen this item. We note that the spatial preferred attachment model [1] can also be viewed as a geometric intersection graph process.…”

“…Let us prove the second bound of (3.21). In view of (3.4) it suffices to show that (1). For this purpose we write, by the triangle inequality,…”

Percolation and Random Graphs

“…The model is directed, but it can be easily adapted to an undirected setting. The idea behind the model in (Aiello, Bonato, Cooper, Janssen and Pralat, 2007) is that for normal preferential attachment models, new vertices should be aware of the degrees of the already present vertices. In reality, it is quite hard to observe the degrees of vertices, and, therefore, in (Aiello, Bonato, Cooper, Janssen and Pralat, 2007), vertices instead have a region of influence in some metric space, for example the torus [0,1] m for some dimension m, for which the metric equals d(x, y) = min{x − y + u ∞ : u ∈ {0, 1, −1} m }.…”

“…In (Aiello, Bonato, Cooper, Janssen and Pralat, 2007), a spatial preferential attachment model with local influence regions is studied, as a model for the Web graph. The model is directed, but it can be easily adapted to an undirected setting.…”