27,430 research outputs found
Spectral redemption: clustering sparse networks
Spectral algorithms are classic approaches to clustering and community
detection in networks. However, for sparse networks the standard versions of
these algorithms are suboptimal, in some cases completely failing to detect
communities even when other algorithms such as belief propagation can do so.
Here we introduce a new class of spectral algorithms based on a
non-backtracking walk on the directed edges of the graph. The spectrum of this
operator is much better-behaved than that of the adjacency matrix or other
commonly used matrices, maintaining a strong separation between the bulk
eigenvalues and the eigenvalues relevant to community structure even in the
sparse case. We show that our algorithm is optimal for graphs generated by the
stochastic block model, detecting communities all the way down to the
theoretical limit. We also show the spectrum of the non-backtracking operator
for some real-world networks, illustrating its advantages over traditional
spectral clustering.Comment: 11 pages, 6 figures. Clarified to what extent our claims are
rigorous, and to what extent they are conjectures; also added an
interpretation of the eigenvectors of the 2n-dimensional version of the
non-backtracking matri
Properties of dense partially random graphs
We study the properties of random graphs where for each vertex a {\it
neighbourhood} has been previously defined. The probability of an edge joining
two vertices depends on whether the vertices are neighbours or not, as happens
in Small World Graphs (SWGs). But we consider the case where the average degree
of each node is of order of the size of the graph (unlike SWGs, which are
sparse). This allows us to calculate the mean distance and clustering, that are
qualitatively similar (although not in such a dramatic scale range) to the case
of SWGs. We also obtain analytically the distribution of eigenvalues of the
corresponding adjacency matrices. This distribution is discrete for large
eigenvalues and continuous for small eigenvalues. The continuous part of the
distribution follows a semicircle law, whose width is proportional to the
"disorder" of the graph, whereas the discrete part is simply a rescaling of the
spectrum of the substrate. We apply our results to the calculation of the
mixing rate and the synchronizability threshold.Comment: 14 pages. To be published in Physical Review
Index statistical properties of sparse random graphs
Using the replica method, we develop an analytical approach to compute the
characteristic function for the probability that a
large adjacency matrix of sparse random graphs has eigenvalues
below a threshold . The method allows to determine, in principle, all
moments of , from which the typical sample to sample
fluctuations can be fully characterized. For random graph models with localized
eigenvectors, we show that the index variance scales linearly with
for , with a model-dependent prefactor that can be exactly
calculated. Explicit results are discussed for Erd\"os-R\'enyi and regular
random graphs, both exhibiting a prefactor with a non-monotonic behavior as a
function of . These results contrast with rotationally invariant
random matrices, where the index variance scales only as , with an
universal prefactor that is independent of . Numerical diagonalization
results confirm the exactness of our approach and, in addition, strongly
support the Gaussian nature of the index fluctuations.Comment: 10 pages, 5 figure
Dynamical Scaling Behavior of Percolation Clusters in Scale-free Networks
In this work we investigate the spectra of Laplacian matrices that determine
many dynamic properties of scale-free networks below and at the percolation
threshold. We use a replica formalism to develop analytically, based on an
integral equation, a systematic way to determine the ensemble averaged
eigenvalue spectrum for a general type of tree-like networks. Close to the
percolation threshold we find characteristic scaling functions for the density
of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic
power laws rho(lambda) ~ lambda^alpha_1 or rho(lambda) ~ lambda^alpha_2 for
small lambda, where alpha_1 holds below and alpha_2 at the percolation
threshold. In the range where the spectra are accessible from a numerical
diagonalization procedure the two methods lead to very similar results.Comment: 9 pages, 6 figure
Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
We consider the problem of coloring Erdos-Renyi and regular random graphs of
finite connectivity using q colors. It has been studied so far using the cavity
approach within the so-called one-step replica symmetry breaking (1RSB) ansatz.
We derive a general criterion for the validity of this ansatz and, applying it
to the ground state, we provide evidence that the 1RSB solution gives exact
threshold values c_q for the q-COL/UNCOL phase transition. We also study the
asymptotic thresholds for q >> 1 finding c_q = 2qlog(q)-log(q)-1+o(1) in
perfect agreement with rigorous mathematical bounds, as well as the nature of
excited states, and give a global phase diagram of the problem.Comment: 23 pages, 10 figures. Replaced with accepted versio
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