408 research outputs found
The sum of degrees in cliques
We investigate lower bounds on the average degree in r-cliques in graphs of
order n and size greater than t(r,n), where t(r,n) is the size of the Turan
graph on n vertices and r color classes. Continuing earlier research of Edwards
and Faudree, we completely prove a conjecture of Bollobas and Erdoes from 1975.Comment: 10 page
Metric dimension for random graphs
The metric dimension of a graph is the minimum number of vertices in a
subset of the vertex set of such that all other vertices are uniquely
determined by their distances to the vertices in . In this paper we
investigate the metric dimension of the random graph for a wide range
of probabilities
Knot Graphs
We consider the equivalence classes of graphs induced by the unsigned
versions of the Reidemeister moves on knot diagrams.
Any graph which is
reducible by some finite sequence of these moves, to a graph with no
edges is called a knot graph. We show that the class of knot graphs
strictly contains the set of delta-wye graphs. We prove that the
dimension of the intersection of the cycle and cocycle spaces is an
effective numerical invariant of these classes
Modularity from Fluctuations in Random Graphs and Complex Networks
The mechanisms by which modularity emerges in complex networks are not well
understood but recent reports have suggested that modularity may arise from
evolutionary selection. We show that finding the modularity of a network is
analogous to finding the ground-state energy of a spin system. Moreover, we
demonstrate that, due to fluctuations, stochastic network models give rise to
modular networks. Specifically, we show both numerically and analytically that
random graphs and scale-free networks have modularity. We argue that this fact
must be taken into consideration to define statistically-significant modularity
in complex networks.Comment: 4 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
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
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
Ising Model on Edge-Dual of Random Networks
We consider Ising model on edge-dual of uncorrelated random networks with
arbitrary degree distribution. These networks have a finite clustering in the
thermodynamic limit. High and low temperature expansions of Ising model on the
edge-dual of random networks are derived. A detailed comparison of the critical
behavior of Ising model on scale free random networks and their edge-dual is
presented.Comment: 23 pages, 4 figures, 1 tabl
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
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