300 research outputs found
Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs
We consider the adjacency matrix of a large random graph and study
fluctuations of the function
with .
We prove that the moments of fluctuations normalized by in the limit
satisfy the Wick relations for the Gaussian random variables. This
allows us to prove central limit theorem for and then extend
the result on the linear eigenvalue statistics of any
function which increases, together with its
first two derivatives, at infinity not faster than an exponential.Comment: 22 page
Robust Emergent Activity in Dynamical Networks
We study the evolution of a random weighted network with complex nonlinear
dynamics at each node, whose activity may cease as a result of interactions
with other nodes. Starting from a knowledge of the micro-level behaviour at
each node, we develop a macroscopic description of the system in terms of the
statistical features of the subnetwork of active nodes. We find the asymptotic
characteristics of this subnetwork to be remarkably robust: the size of the
active set is independent of the total number of nodes in the network, and the
average degree of the active nodes is independent of both the network size and
its connectivity. These results suggest that very different networks evolve to
active subnetworks with the same characteristic features. This has strong
implications for dynamical networks observed in the natural world, notably the
existence of a characteristic range of links per species across ecological
systems.Comment: 4 pages, 5 figure
On the large N expansion in hyperbolic sigma-models
Invariant correlation functions for hyperbolic sigma-models
are investigated. The existence of a large asymptotic expansion is proven
on finite lattices of dimension . The unique saddle point
configuration is characterized by a negative gap vanishing at least like 1/V
with the volume. Technical difficulties compared to the compact case are
bypassed using horospherical coordinates and the matrix-tree theorem.Comment: 15 pages. Some changes in introduction and discussion; to appear in
J. Math. Phy
Numerical evaluation of the upper critical dimension of percolation in scale-free networks
We propose a numerical method to evaluate the upper critical dimension
of random percolation clusters in Erd\H{o}s-R\'{e}nyi networks and in
scale-free networks with degree distribution ,
where is the degree of a node and is the broadness of the degree
distribution. Our results report the theoretical prediction, for scale-free networks with and
for Erd\H{o}s-R\'{e}nyi networks and scale-free networks with .
When the removal of nodes is not random but targeted on removing the highest
degree nodes we obtain for all . Our method also yields
a better numerical evaluation of the critical percolation threshold, , for
scale-free networks. Our results suggest that the finite size effects increases
when approaches 3 from above.Comment: 10 pages, 5 figure
Modular networks emerge from multiconstraint optimization
Modular structure is ubiquitous among complex networks. We note that most
such systems are subject to multiple structural and functional constraints,
e.g., minimizing the average path length and the total number of links, while
maximizing robustness against perturbations in node activity. We show that the
optimal networks satisfying these three constraints are characterized by the
existence of multiple subnetworks (modules) sparsely connected to each other.
In addition, these modules have distinct hubs, resulting in an overall
heterogeneous degree distribution.Comment: 5 pages, 4 figures; Published versio
Dynamic Computation of Network Statistics via Updating Schema
In this paper we derive an updating scheme for calculating some important
network statistics such as degree, clustering coefficient, etc., aiming at
reduce the amount of computation needed to track the evolving behavior of large
networks; and more importantly, to provide efficient methods for potential use
of modeling the evolution of networks. Using the updating scheme, the network
statistics can be computed and updated easily and much faster than
re-calculating each time for large evolving networks. The update formula can
also be used to determine which edge/node will lead to the extremal change of
network statistics, providing a way of predicting or designing evolution rule
of networks.Comment: 17 pages, 6 figure
Cavity method for quantum spin glasses on the Bethe lattice
We propose a generalization of the cavity method to quantum spin glasses on
fixed connectivity lattices. Our work is motivated by the recent refinements of
the classical technique and its potential application to quantum computational
problems. We numerically solve for the phase structure of a connectivity
transverse field Ising model on a Bethe lattice with couplings, and
investigate the distribution of various classical and quantum observables.Comment: 27 pages, 9 figure
Approximating Spectral Impact of Structural Perturbations in Large Networks
Determining the effect of structural perturbations on the eigenvalue spectra
of networks is an important problem because the spectra characterize not only
their topological structures, but also their dynamical behavior, such as
synchronization and cascading processes on networks. Here we develop a theory
for estimating the change of the largest eigenvalue of the adjacency matrix or
the extreme eigenvalues of the graph Laplacian when small but arbitrary set of
links are added or removed from the network. We demonstrate the effectiveness
of our approximation schemes using both real and artificial networks, showing
in particular that we can accurately obtain the spectral ranking of small
subgraphs. We also propose a local iterative scheme which computes the relative
ranking of a subgraph using only the connectivity information of its neighbors
within a few links. Our results may not only contribute to our theoretical
understanding of dynamical processes on networks, but also lead to practical
applications in ranking subgraphs of real complex networks.Comment: 9 pages, 3 figures, 2 table
The Bak-Sneppen Model on Scale-Free Networks
We investigate by numerical simulations and analytical calculations the
Bak-Sneppen model for biological evolution in scale-free networks. By using
large scale numerical simulations, we study the avalanche size distribution and
the activity time behavior at nodes with different connectivities. We argue the
absence of a critical barrier and its associated critical behavior for infinite
size systems. These findings are supported by a single site mean-field analytic
treatment of the model.Comment: 5 pages and 3 eps figures. Final version appeared in Europhys. Let
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