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.
Highlights
Empirical studies of on-line social networks as undirected graphs suggest these graphs have several intrinsic properties: highly skewed or even power-law degree distributions [1,2], large local clustering [3], constant [3] or even shrinking diameter with network size [4], densification [4], and localized information flow bottlenecks [5,6]
One model that captures these properties asymptotically is the geometric protean model (GEO-P) [10]. It differs from other network models [1,4,11,12] because all links in geometric protean networks arise based on an underlying metric space
Many NP-hard optimization problems related to graph properties and community detection are polynomial time solvable in a low dimensional metric space, and our findings suggest new techniques to explore for understanding why we may expect to solve these problems in social networks
Summary
Empirical studies of on-line social networks as undirected graphs suggest these graphs have several intrinsic properties: highly skewed or even power-law degree distributions [1,2], large local clustering [3], constant [3] or even shrinking diameter with network size [4], densification [4], and localized information flow bottlenecks [5,6]. One model that captures these properties asymptotically is the geometric protean model (GEO-P) [10] It differs from other network models [1,4,11,12] because all links in geometric protean networks arise based on an underlying metric space. This metric space mirrors a construction in the social sciences called Blau space [13]. In Blau space, agents in the social network correspond to points in a metric space, and the relative position of nodes follows the principle of homophily [14]: nodes with similar sociodemographics are closer together in the space
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