2,974 research outputs found
Symmetric motifs in random geometric graphs
We study symmetric motifs in random geometric graphs. Symmetric motifs are
subsets of nodes which have the same adjacencies. These subgraphs are
particularly prevalent in random geometric graphs and appear in the Laplacian
and adjacency spectrum as sharp, distinct peaks, a feature often found in
real-world networks. We look at the probabilities of their appearance and
compare these across parameter space and dimension. We then use the Chen-Stein
method to derive the minimum separation distance in random geometric graphs
which we apply to study symmetric motifs in both the intensive and
thermodynamic limits. In the thermodynamic limit the probability that the
closest nodes are symmetric approaches one, whilst in the intensive limit this
probability depends upon the dimension.Comment: 11 page
Emergence of Symmetry in Complex Networks
Many real networks have been found to have a rich degree of symmetry, which
is a very important structural property of complex network, yet has been rarely
studied so far. And where does symmetry comes from has not been explained. To
explore the mechanism underlying symmetry of the networks, we studied
statistics of certain local symmetric motifs, such as symmetric bicliques and
generalized symmetric bicliques, which contribute to local symmetry of
networks. We found that symmetry of complex networks is a consequence of
similar linkage pattern, which means that nodes with similar degree tend to
share similar linkage targets. A improved version of BA model integrating
similar linkage pattern successfully reproduces the symmetry of real networks,
indicating that similar linkage pattern is the underlying ingredient that
responsible for the emergence of the symmetry in complex networks.Comment: 7 pages, 7 figure
Spectral statistics of random geometric graphs
We use random matrix theory to study the spectrum of random geometric graphs,
a fundamental model of spatial networks. Considering ensembles of random
geometric graphs we look at short range correlations in the level spacings of
the spectrum via the nearest neighbour and next nearest neighbour spacing
distribution and long range correlations via the spectral rigidity Delta_3
statistic. These correlations in the level spacings give information about
localisation of eigenvectors, level of community structure and the level of
randomness within the networks. We find a parameter dependent transition
between Poisson and Gaussian orthogonal ensemble statistics. That is the
spectral statistics of spatial random geometric graphs fits the universality of
random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert
and Watts-Strogatz random graph.Comment: 19 pages, 6 figures. Substantially updated from previous versio
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
CONTEST : a Controllable Test Matrix Toolbox for MATLAB
Large, sparse networks that describe complex interactions are a common feature across a number of disciplines, giving rise to many challenging matrix computational tasks. Several random graph models have been proposed that capture key properties of real-life networks. These models provide realistic, parametrized matrices for testing linear system and eigenvalue solvers. CONTEST (CONtrollable TEST matrices) is a random network toolbox for MATLAB that implements nine models. The models produce unweighted directed or undirected graphs; that is, symmetric or unsymmetric matrices with elements equal to zero or one. They have one or more parameters that affect features such as sparsity and characteristic pathlength and all can be of arbitrary dimension. Utility functions are supplied for rewiring, adding extra shortcuts and subsampling in order to create further classes of networks. Other utilities convert the adjacency matrices into real-valued coefficient matrices for naturally arising computational tasks that reduce to sparse linear system and eigenvalue problems
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