Geometric Graph Properties of the Spatial Preferred Attachment model

The spatial preferred attachment (SPA) model is a model for networked information spaces such as domains of the World Wide Web, citation graphs, and on-line social networks. It uses a metric space to model the hidden attributes of the vertices. Thus, vertices are elements of a metric space, and link formation depends on the metric distance between vertices. We show, through theoretical analysis and simulation, that for graphs formed according to the SPA model it is possible to infer the metric distance between vertices from the link structure of the graph. Precisely, the estimate is based on the number of common neighbours of a pair of vertices, a measure known as {\sl co-citation}. To be able to calculate this estimate, we derive a precise relation between the number of common neighbours and metric distance. We also analyze the distribution of {\sl edge lengths}, where the length of an edge is the metric distance between its end points. We show that this distribution has three different regimes, and that the tail of this distribution follows a power law.

💡 Research Summary

The paper provides a rigorous investigation of the Spatial Preferred Attachment (SPA) model, a stochastic graph generation process that embeds vertices in a metric space and makes link formation dependent on geometric distance as well as preferential attachment. In the SPA model each vertex v, born at time i, is assigned a sphere of influence S(v,t) whose volume at time t is |S(v,t)| = A₁·deg⁻(v,t) + A₂·t (capped at the unit hyper‑cube). A new vertex u, uniformly sampled from the space, creates a directed edge to any existing vertex v for which u lies inside S(v,t) with probability p. The condition p·A₁ < 1 guarantees that the graph does not become overly dense.

Degree Growth.
The authors first derive the expected in‑degree of a vertex born at time i:
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