Dimensionality of social networks using motifs and eigenvalues

TL;DR

This paper investigates the intrinsic dimensionality of social networks, proposing that their structure can be modeled with a logarithmic dimension relative to the number of nodes, supported by motif and eigenvalue analyses on real networks.

Contribution

It introduces a model linking social network dimensionality to a logarithmic scale and validates this with empirical data from Facebook and LinkedIn using motif and eigenvalue methods.

Findings

Social network dimension scales logarithmically with network size.

Motif count distributions support the logarithmic dimension hypothesis.

Eigenvalue distributions corroborate the model's predictions.

Abstract

We consider the dimensionality of social networks, and develop experiments aimed at predicting that dimension. We find that a social network model with nodes and links sampled from an $m$-dimensional metric space with power-law distributed influence regions best fits samples from real-world networks when $m$ scales logarithmically with the number of nodes of the network. This supports a logarithmic dimension hypothesis, and we provide evidence with two different social networks, Facebook and LinkedIn. Further, we employ two different methods for confirming the hypothesis: the first uses the distribution of motif counts, and the second exploits the eigenvalue distribution.