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.Comment: 26 page

Geometric protean graphs

We study the link structure of on-line social networks (OSNs), and introduce a new model for such networks which may help infer 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 evolution of complex networks with degree correlations

We present a general class of geometric network growth mechanisms by homogeneous attachment in which the links created at a given time t are distributed homogeneously between a new node and the existing nodes selected uniformly. This is achieved by creating links between nodes uniformly distributed in a homogeneous metric space according to a Fermi-Dirac connection probability with inverse temperature β and general time-dependent chemical potential μ(t). The chemical potential limits the spatial extent of newly created links. Using a hidden variable framework, we obtain an analytical expression for the degree sequence and show that μ(t) can be fixed to yield any given degree distributions, including a scale-free degree distribution. Additionally, we find that depending on the order in which nodes appear in the network—its history—the degree-degree correlations can be tuned to be assortative or disassortative. The effect of the geometry on the structure is investigated through the average clustering coefficient ⟨c⟩. In the thermodynamic limit, we identify a phase transition between a random regime where ⟨c⟩→ 0 when β 0 when β>βc

Mathematical Models of Abstract Systems: Knowing abstract geometric forms

Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models