37,543 research outputs found
Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees
In this paper, we analyze the limiting spectral distribution of the adjacency
matrix of a random graph ensemble, proposed by Chung and Lu, in which a given
expected degree sequence
is prescribed on the ensemble. Let if there is an edge
between the nodes and zero otherwise, and consider the normalized
random adjacency matrix of the graph ensemble: . The empirical spectral distribution
of denoted by is the empirical
measure putting a mass at each of the real eigenvalues of the
symmetric matrix . Under some technical conditions on the
expected degree sequence, we show that with probability one,
converges weakly to a deterministic
distribution . Furthermore, we fully characterize this
distribution by providing explicit expressions for the moments of
. We apply our results to well-known degree distributions,
such as power-law and exponential. The asymptotic expressions of the spectral
moments in each case provide significant insights about the bulk behavior of
the eigenvalue spectrum
Concentration of the Stationary Distribution on General Random Directed Graphs
We consider a random model for directed graphs whereby an arc is placed from
one vertex to another with a prescribed probability which may vary from arc to
arc. Using perturbation bounds as well as Chernoff inequalities, we show that
the stationary distribution of a Markov process on a random graph is
concentrated near that of the "expected" process under mild conditions. These
conditions involve the ratio between the minimum and maximum in- and
out-degrees, the ratio of the minimum and maximum entry in the stationary
distribution, and the smallest singu- lar value of the transition matrix.
Lastly, we give examples of applications of our results to well-known models
such as PageRank and G(n, p).Comment: 14 pages, 0 figure
Testing goodness-of-fit of random graph models
Random graphs are matrices with independent 0, 1 elements with probabilities
determined by a small number of parameters. One of the oldest model is the
Rasch model where the odds are ratios of positive numbers scaling the rows and
columns. Later Persi Diaconis with his coworkers rediscovered the model for
symmetric matrices and called the model beta. Here we give goodnes-of-fit tests
for the model and extend the model to a version of the block model introduced
by Holland, Laskey, and Leinhard
Spectra of complex networks
We propose a general approach to the description of spectra of complex
networks. For the spectra of networks with uncorrelated vertices (and a local
tree-like structure), exact equations are derived. These equations are
generalized to the case of networks with correlations between neighboring
vertices. The tail of the density of eigenvalues at large
is related to the behavior of the vertex degree distribution
at large . In particular, as , . We propose a simple approximation, which enables us to
calculate spectra of various graphs analytically. We analyse spectra of various
complex networks and discuss the role of vertices of low degree. We show that
spectra of locally tree-like random graphs may serve as a starting point in the
analysis of spectral properties of real-world networks, e.g., of the Internet.Comment: 10 pages, 4 figure
- …