280,357 research outputs found
Average nearest neighbor degrees in scale-free networks
The average nearest neighbor degree (ANND) of a node of degree is widely
used to measure dependencies between degrees of neighbor nodes in a network. We
formally analyze ANND in undirected random graphs when the graph size tends to
infinity. The limiting behavior of ANND depends on the variance of the degree
distribution. When the variance is finite, the ANND has a deterministic limit.
When the variance is infinite, the ANND scales with the size of the graph, and
we prove a corresponding central limit theorem in the configuration model (CM,
a network with random connections). As ANND proved uninformative in the
infinite variance scenario, we propose an alternative measure, the average
nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic
function whenever the degree distribution has finite mean. We then consider the
erased configuration model (ECM), where self-loops and multiple edges are
removed, and investigate the well-known `structural negative correlations', or
`finite-size effects', that arise in simple graphs, such as ECM, because large
nodes can only have a limited number of large neighbors. Interestingly, we
prove that for any fixed , ANNR in ECM converges to the same limit as in CM.
However, numerical experiments show that finite-size effects occur when
scales with
Limit Theorems for Network Dependent Random Variables
This paper is concerned with cross-sectional dependence arising because
observations are interconnected through an observed network. Following Doukhan
and Louhichi (1999), we measure the strength of dependence by covariances of
nonlinearly transformed variables. We provide a law of large numbers and
central limit theorem for network dependent variables. We also provide a method
of calculating standard errors robust to general forms of network dependence.
For that purpose, we rely on a network heteroskedasticity and autocorrelation
consistent (HAC) variance estimator, and show its consistency. The results rely
on conditions characterized by tradeoffs between the rate of decay of
dependence across a network and network's denseness. Our approach can
accommodate data generated by network formation models, random fields on
graphs, conditional dependency graphs, and large functional-causal systems of
equations
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
Cut Size Statistics of Graph Bisection Heuristics
We investigate the statistical properties of cut sizes generated by heuristic
algorithms which solve approximately the graph bisection problem. On an
ensemble of sparse random graphs, we find empirically that the distribution of
the cut sizes found by ``local'' algorithms becomes peaked as the number of
vertices in the graphs becomes large. Evidence is given that this distribution
tends towards a Gaussian whose mean and variance scales linearly with the
number of vertices of the graphs. Given the distribution of cut sizes
associated with each heuristic, we provide a ranking procedure which takes into
account both the quality of the solutions and the speed of the algorithms. This
procedure is demonstrated for a selection of local graph bisection heuristics.Comment: 17 pages, 5 figures, submitted to SIAM Journal on Optimization also
available at http://ipnweb.in2p3.fr/~martin
Cluster tails for critical power-law inhomogeneous random graphs
Recently, the scaling limit of cluster sizes for critical inhomogeneous
random graphs of rank-1 type having finite variance but infinite third moment
degrees was obtained (see previous work by Bhamidi, van der Hofstad and van
Leeuwaarden). It was proved that when the degrees obey a power law with
exponent in the interval (3,4), the sequence of clusters ordered in decreasing
size and scaled appropriately converges as n goes to infinity to a sequence of
decreasing non-degenerate random variables.
Here, we study the tails of the limit of the rescaled largest cluster, i.e.,
the probability that the scaling limit of the largest cluster takes a large
value u, as a function of u. This extends a related result of Pittel for the
Erd\H{o}s-R\'enyi random graph to the setting of rank-1 inhomogeneous random
graphs with infinite third moment degrees. We make use of delicate large
deviations and weak convergence arguments.Comment: corrected and updated first referenc
Analytic solution of the resolvent equations for heterogeneous random graphs : spectral and localization properties
The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states (LDOSs). For random graphs with a negative binomial degree distribution, we show that all eigenvectors are extended and that the spectral density exhibits a logarithmic or a power-law divergence when the variance of the degree distribution is large enough. We elucidate this singular behaviour by showing that the distribution of the LDOSs at the centre of the spectrum displays a power-law tail controlled by the variance of the degree distribution. In the regime of weak degree fluctuations the spectral density has a finite support, which promotes the stability of large complex systems on random graphs
- …