We study the link structure of online social networks (OSNs) and introduce a new model for such networks that may help in inferring their hidden underlying reality. In the geo-protean (GEO-P) model for OSNs, nodes are identified with points in Euclidean space, and edges are stochastically generated by a mixture of the relative distance of nodes and a ranking function. With high probability, the GEO-P model generates graphs satisfying many observed properties of OSNs, such as power-law degree distributions, the small-world property, densification power law, and bad spectral expansion. We introduce the dimension of an OSN based on our model and examine this new parameter using actual OSN data. We discuss how the geo-protean model may eventually be used as a tool to group users with similar attributes using only the link structure of the network.
"Geometric Protean Graphs." Internet Math. 8 (1-2) 2 - 28, 2012.