97 research outputs found
Localization transition, Lifschitz tails and rare-region effects in network models
Effects of heterogeneity in the suspected-infected-susceptible model on
networks are investigated using quenched mean-field theory. The emergence of
localization is described by the distributions of the inverse participation
ratio and compared with the rare-region effects appearing in simulations and in
the Lifschitz tails. The latter, in the linear approximation, is related to the
spectral density of the Laplacian matrix and to the time dependent order
parameter. I show that these approximations indicate correctly Griffiths Phases
both on regular one-dimensional lattices and on small world networks exhibiting
purely topological disorder. I discuss the localization transition that occurs
on scale-free networks at degree exponent.Comment: 9 pages, 9 figures, accepted version in PR
Critical dynamics on a large human Open Connectome network
Extended numerical simulations of threshold models have been performed on a
human brain network with N=836733 connected nodes available from the Open
Connectome project. While in case of simple threshold models a sharp
discontinuous phase transition without any critical dynamics arises, variable
thresholds models exhibit extended power-law scaling regions. This is
attributed to fact that Griffiths effects, stemming from the
topological/interaction heterogeneity of the network, can become relevant if
the input sensitivity of nodes is equalized. I have studied the effects effects
of link directness, as well as the consequence of inhibitory connections.
Non-universal power-law avalanche size and time distributions have been found
with exponents agreeing with the values obtained in electrode experiments of
the human brain. The dynamical critical region occurs in an extended control
parameter space without the assumption of self organized criticality.Comment: 7 pages, 6 figures, accepted version to appear in PR
Griffiths phases and localization in hierarchical modular networks
We study variants of hierarchical modular network models suggested by Kaiser
and Hilgetag [Frontiers in Neuroinformatics, 4 (2010) 8] to model functional
brain connectivity, using extensive simulations and quenched mean-field theory
(QMF), focusing on structures with a connection probability that decays
exponentially with the level index. Such networks can be embedded in
two-dimensional Euclidean space. We explore the dynamic behavior of the contact
process (CP) and threshold models on networks of this kind, including
hierarchical trees. While in the small-world networks originally proposed to
model brain connectivity, the topological heterogeneities are not strong enough
to induce deviations from mean-field behavior, we show that a Griffiths phase
can emerge under reduced connection probabilities, approaching the percolation
threshold. In this case the topological dimension of the networks is finite,
and extended regions of bursty, power-law dynamics are observed. Localization
in the steady state is also shown via QMF. We investigate the effects of link
asymmetry and coupling disorder, and show that localization can occur even in
small-world networks with high connectivity in case of link disorder.Comment: 18 pages, 20 figures, accepted version in Scientific Report
The role of diffusion in branching and annihilation random walk models
Different branching and annihilating random walk models are investigated by
cluster mean-field method and simulations in one and two dimensions. In case of
the A -> 2A, 2A -> 0 model the cluster mean-field approximations show diffusion
dependence in the phase diagram as was found recently by non-perturbative
renormalization group method (L. Canet et al., cond-mat/0403423). The same type
of survey for the A -> 2A, 4A -> 0 model results in a reentrant phase diagram,
similar to that of 2A -> 3A, 4A -> 0 model (G. \'Odor, PRE {\bf 69}, 036112
(2004)). Simulations of the A -> 2A, 4A -> 0 model in one and two dimensions
confirm the presence of both the directed percolation transitions at finite
branching rates and the mean-field transition at zero branching rate. In two
dimensions the directed percolation transition disappears for strong diffusion
rates. These results disagree with the predictions of the perturbative
renormalization group method.Comment: 4 pages, 4 figures, 1 table include
The phase transition of triplet reaction-diffusion models
The phase transitions classes of reaction-diffusion systems with
multi-particle reactions is an open challenging problem. Large scale
simulations are applied for the 3A -> 4A, 3A -> 2A and the 3A -> 4A, 3A->0
triplet reaction models with site occupation restriction in one dimension.
Static and dynamic mean-field scaling is observed with signs of logarithmic
corrections suggesting d_c=1 upper critical dimension for this family of
models.Comment: 4 pages, 4 figures, updated version prior publication in PR
The topology of large Open Connectome networks for the human brain
The structural human connectome (i.e.\ the network of fiber connections in
the brain) can be analyzed at ever finer spatial resolution thanks to advances
in neuroimaging. Here we analyze several large data sets for the human brain
network made available by the Open Connectome Project. We apply statistical
model selection to characterize the degree distributions of graphs containing
up to nodes and edges. A three-parameter
generalized Weibull (also known as a stretched exponential) distribution is a
good fit to most of the observed degree distributions. For almost all networks,
simple power laws cannot fit the data, but in some cases there is statistical
support for power laws with an exponential cutoff. We also calculate the
topological (graph) dimension and the small-world coefficient of
these networks. While suggests a small-world topology, we found that
showing that long-distance connections provide only a small correction
to the topology of the embedding three-dimensional space.Comment: 14 pages, 6 figures, accepted version in Scientific Report
Critical behavior of the two dimensional 2A->3A, 4A->0 binary system
The phase transitions of the recently introduced 2A -> 3A, 4A -> 0
reaction-diffusion model (G.Odor, PRE 69 036112 (2004)) are explored in two
dimensions. This model exhibits site occupation restriction and explicit
diffusion of isolated particles. A reentrant phase diagram in the diffusion -
creation rate space is confirmed in agreement with cluster mean-field and
one-dimensional results. For strong diffusion a mean-field transition can be
observed at zero branching rate characterized by density decay
exponent. In contrast with this for weak diffusion the effective 2A ->3A->4A->0
reaction becomes relevant and the mean-field transition of the 2A -> 3A, 2A ->
0 model characterized by also appears for non-zero branching
rates.Comment: 5 pages, 5 figures included, small correction
Universality of (2+1)-dimensional restricted solid-on-solid models
Extensive dynamical simulations of Restricted Solid on Solid models in
dimensions have been done using parallel multisurface algorithms
implemented on graphics cards. Numerical evidence is presented that these
models exhibit KPZ surface growth scaling, irrespective of the step heights
. We show that by increasing the corrections to scaling increase, thus
smaller step-sized models describe better the asymptotic, long-wave-scaling
behavior
Bit-Vectorized GPU Implementation of a Stochastic Cellular Automaton Model for Surface Growth
Stochastic surface growth models aid in studying properties of universality
classes like the Kardar--Paris--Zhang class. High precision results obtained
from large scale computational studies can be transferred to many physical
systems. Many properties, such as roughening and some two-time functions can be
studied using stochastic cellular automaton (SCA) variants of stochastic
models. Here we present a highly efficient SCA implementation of a surface
growth model capable of simulating billions of lattice sites on a single GPU.
We also provide insight into cases requiring arbitrary random probabilities
which are not accessible through bit-vectorization.Comment: INES 2016, Budapest http://www.ines-conf.org/ines-conf/2016index.htm
Griffiths phases in infinite-dimensional, non-hierarchical modular networks
Griffiths phases (GPs), generated by the heterogeneities on modular networks,
have recently been suggested to provide a mechanism, rid of fine parameter
tuning, to explain the critical behavior of complex systems. One conjectured
requirement for systems with modular structures was that the network of modules
must be hierarchically organized and possess finite dimension. We investigate
the dynamical behavior of an activity spreading model, evolving on
heterogeneous random networks with highly modular structure and organized
non-hierarchically. We observe that loosely coupled modules act as effective
rare-regions, slowing down the extinction of activation. As a consequence, we
find extended control parameter regions with continuously changing dynamical
exponents for single network realizations, preserved after finite size
analyses, as in a real GP. The avalanche size distributions of spreading events
exhibit robust power-law tails. Our findings relax the requirement of
hierarchical organization of the modular structure, which can help to
rationalize the criticality of modular systems in the framework of GPs.Comment: 14 pages, 8 figure
- …