30 research outputs found
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 -dimensional metric space with power-law
distributed influence regions best fits samples from real-world networks when
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
Complex network view of evolving manifolds
We study complex networks formed by triangulations and higher-dimensional
simplicial complexes representing closed evolving manifolds. In particular, for
triangulations, the set of possible transformations of these networks is
restricted by the condition that at each step, all the faces must be triangles.
Stochastic application of these operations leads to random networks with
different architectures. We perform extensive numerical simulations and explore
the geometries of growing and equilibrium complex networks generated by these
transformations and their local structural properties. This characterization
includes the Hausdorff and spectral dimensions of the resulting networks, their
degree distributions, and various structural correlations. Our results reveal a
rich zoo of architectures and geometries of these networks, some of which
appear to be small worlds while others are finite-dimensional with Hausdorff
dimension equal or higher than the original dimensionality of their simplices.
The range of spectral dimensions of the evolving triangulations turns out to be
from about 1.4 to infinity. Our models include simplicial complexes
representing manifolds with evolving topologies, for example, an h-holed torus
with a progressively growing number of holes. This evolving graph demonstrates
features of a small-world network and has a particularly heavy-tailed degree
distribution.Comment: 14 pages, 15 figure