110,354 research outputs found
Random Sequential Renormalization of Networks I: Application to Critical Trees
We introduce the concept of Random Sequential Renormalization (RSR) for
arbitrary networks. RSR is a graph renormalization procedure that locally
aggregates nodes to produce a coarse grained network. It is analogous to the
(quasi-)parallel renormalization schemes introduced by C. Song {\it et al.}
(Nature {\bf 433}, 392 (2005)) and studied more recently by F. Radicchi {\it et
al.} (Phys. Rev. Lett. {\bf 101}, 148701 (2008)), but much simpler and easier
to implement. In this first paper we apply RSR to critical trees and derive
analytical results consistent with numerical simulations. Critical trees
exhibit three regimes in their evolution under RSR: (i) An initial regime
, where is the number of nodes at some step in the
renormalization and is the initial size. RSR in this regime is described
by a mean field theory and fluctuations from one realization to another are
small. The exponent is derived using random walk arguments. The
degree distribution becomes broader under successive renormalization --
reaching a power law, with and a variance
that diverges as at the end of this regime. Both of these results
are derived based on a scaling theory. (ii) An intermediate regime for
, in which hubs develop, and
fluctuations between different realizations of the RSR are large. Crossover
functions exhibiting finite size scaling, in the critical region , connect the behaviors in the first two regimes. (iii)
The last regime, for , is characterized by the
appearance of star configurations with a central hub surrounded by many leaves.
The distribution of sizes where stars first form is found numerically to be a
power law up to a cutoff that scales as with
The Harris-Luck criterion for random lattices
The Harris-Luck criterion judges the relevance of (potentially) spatially
correlated, quenched disorder induced by, e.g., random bonds, randomly diluted
sites or a quasi-periodicity of the lattice, for altering the critical behavior
of a coupled matter system. We investigate the applicability of this type of
criterion to the case of spin variables coupled to random lattices. Their
aptitude to alter critical behavior depends on the degree of spatial
correlations present, which is quantified by a wandering exponent. We consider
the cases of Poissonian random graphs resulting from the Voronoi-Delaunay
construction and of planar, ``fat'' Feynman diagrams and precisely
determine their wandering exponents. The resulting predictions are compared to
various exact and numerical results for the Potts model coupled to these
quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one
figure added for clarification, minor re-wordings and typo cleanu
Supersaturation Problem for Color-Critical Graphs
The \emph{Tur\'an function} \ex(n,F) of a graph is the maximum number
of edges in an -free graph with vertices. The classical results of
Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs
where the key question is to determine , the minimum number of copies
of that a graph with vertices and \ex(n,F)+q edges can have.
We determine asymptotically when is \emph{color-critical}
(that is, contains an edge whose deletion reduces its chromatic number) and
.
Determining the exact value of seems rather difficult. For
example, let be the limit superior of for which the extremal
structures are obtained by adding some edges to a maximum -free graph.
The problem of determining for cliques was a well-known question of Erd\H
os that was solved only decades later by Lov\'asz and Simonovits. Here we prove
that for every {color-critical}~. Our approach also allows us to
determine for a number of graphs, including odd cycles, cliques with one
edge removed, and complete bipartite graphs plus an edge.Comment: 27 pages, 2 figure
Building Damage-Resilient Dominating Sets in Complex Networks against Random and Targeted Attacks
We study the vulnerability of dominating sets against random and targeted
node removals in complex networks. While small, cost-efficient dominating sets
play a significant role in controllability and observability of these networks,
a fixed and intact network structure is always implicitly assumed. We find that
cost-efficiency of dominating sets optimized for small size alone comes at a
price of being vulnerable to damage; domination in the remaining network can be
severely disrupted, even if a small fraction of dominator nodes are lost. We
develop two new methods for finding flexible dominating sets, allowing either
adjustable overall resilience, or dominating set size, while maximizing the
dominated fraction of the remaining network after the attack. We analyze the
efficiency of each method on synthetic scale-free networks, as well as real
complex networks
Dynamics of Unperturbed and Noisy Generalized Boolean Networks
For years, we have been building models of gene regulatory networks, where
recent advances in molecular biology shed some light on new structural and
dynamical properties of such highly complex systems. In this work, we propose a
novel timing of updates in Random and Scale-Free Boolean Networks, inspired by
recent findings in molecular biology. This update sequence is neither fully
synchronous nor asynchronous, but rather takes into account the sequence in
which genes affect each other. We have used both Kauffman's original model and
Aldana's extension, which takes into account the structural properties about
known parts of actual GRNs, where the degree distribution is right-skewed and
long-tailed. The computer simulations of the dynamics of the new model compare
favorably to the original ones and show biologically plausible results both in
terms of attractors number and length. We have complemented this study with a
complete analysis of our systems' stability under transient perturbations,
which is one of biological networks defining attribute. Results are
encouraging, as our model shows comparable and usually even better behavior
than preceding ones without loosing Boolean networks attractive simplicity.Comment: 29 pages, publishe
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