Dimensionality of social networks using motifs and eigenvalues
  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. 
Anthony Bonato, David F. Gleich, Myunghwan Kim, Dieter Mitsche,   Pawe{\l} Pra{\l}at, Amanda Tian, Stephen J. Young
cs.SI physics.soc-ph