Dimensionality of Social Networks Using Motifs and Eigenvalues

<div><p>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 <i>m</i>-dimensional metric space with power-law distributed influence regions best fits samples from real-world networks when <i>m</i> 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.</p>