11,581 research outputs found
Boolean Dynamics with Random Couplings
This paper reviews a class of generic dissipative dynamical systems called
N-K models. In these models, the dynamics of N elements, defined as Boolean
variables, develop step by step, clocked by a discrete time variable. Each of
the N Boolean elements at a given time is given a value which depends upon K
elements in the previous time step.
We review the work of many authors on the behavior of the models, looking
particularly at the structure and lengths of their cycles, the sizes of their
basins of attraction, and the flow of information through the systems. In the
limit of infinite N, there is a phase transition between a chaotic and an
ordered phase, with a critical phase in between.
We argue that the behavior of this system depends significantly on the
topology of the network connections. If the elements are placed upon a lattice
with dimension d, the system shows correlations related to the standard
percolation or directed percolation phase transition on such a lattice. On the
other hand, a very different behavior is seen in the Kauffman net in which all
spins are equally likely to be coupled to a given spin. In this situation,
coupling loops are mostly suppressed, and the behavior of the system is much
more like that of a mean field theory.
We also describe possible applications of the models to, for example, genetic
networks, cell differentiation, evolution, democracy in social systems and
neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical
Sciences Serie
Automatic Network Fingerprinting through Single-Node Motifs
Complex networks have been characterised by their specific connectivity
patterns (network motifs), but their building blocks can also be identified and
described by node-motifs---a combination of local network features. One
technique to identify single node-motifs has been presented by Costa et al. (L.
D. F. Costa, F. A. Rodrigues, C. C. Hilgetag, and M. Kaiser, Europhys. Lett.,
87, 1, 2009). Here, we first suggest improvements to the method including how
its parameters can be determined automatically. Such automatic routines make
high-throughput studies of many networks feasible. Second, the new routines are
validated in different network-series. Third, we provide an example of how the
method can be used to analyse network time-series. In conclusion, we provide a
robust method for systematically discovering and classifying characteristic
nodes of a network. In contrast to classical motif analysis, our approach can
identify individual components (here: nodes) that are specific to a network.
Such special nodes, as hubs before, might be found to play critical roles in
real-world networks.Comment: 16 pages (4 figures) plus supporting information 8 pages (5 figures
Self-avoiding walks and connective constants in small-world networks
Long-distance characteristics of small-world networks have been studied by
means of self-avoiding walks (SAW's). We consider networks generated by
rewiring links in one- and two-dimensional regular lattices. The number of
SAW's was obtained from numerical simulations as a function of the number
of steps on the considered networks. The so-called connective constant,
, which characterizes the long-distance
behavior of the walks, increases continuously with disorder strength (or
rewiring probability, ). For small , one has a linear relation , and being constants dependent on the underlying
lattice. Close to one finds the behavior expected for random graphs. An
analytical approach is given to account for the results derived from numerical
simulations. Both methods yield results agreeing with each other for small ,
and differ for close to 1, because of the different connectivity
distributions resulting in both cases.Comment: 7 pages, 5 figure
Long range action in networks of chaotic elements
We show that under certain simple assumptions on the topology (structure) of
networks of strongly interacting chaotic elements a phenomenon of long range
action takes place, namely that the asymptotic (as time goes to infinity)
dynamics of an arbitrary large network is completely determined by its boundary
conditions. This phenomenon takes place under very mild and robust assumptions
on local dynamics with short range interactions. However, we show that it is
unstable with respect to arbitrarily weak local random perturbations.Comment: 15 page
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 Physics of Living Neural Networks
Improvements in technique in conjunction with an evolution of the theoretical
and conceptual approach to neuronal networks provide a new perspective on
living neurons in culture. Organization and connectivity are being measured
quantitatively along with other physical quantities such as information, and
are being related to function. In this review we first discuss some of these
advances, which enable elucidation of structural aspects. We then discuss two
recent experimental models that yield some conceptual simplicity. A
one-dimensional network enables precise quantitative comparison to analytic
models, for example of propagation and information transport. A two-dimensional
percolating network gives quantitative information on connectivity of cultured
neurons. The physical quantities that emerge as essential characteristics of
the network in vitro are propagation speeds, synaptic transmission, information
creation and capacity. Potential application to neuronal devices is discussed.Comment: PACS: 87.18.Sn, 87.19.La, 87.80.-y, 87.80.Xa, 64.60.Ak Keywords:
complex systems, neuroscience, neural networks, transport of information,
neural connectivity, percolation
http://www.weizmann.ac.il/complex/tlusty/papers/PhysRep2007.pdf
http://www.weizmann.ac.il/complex/EMoses/pdf/PhysRep-448-56.pd
Recent advances and open challenges in percolation
Percolation is the paradigm for random connectivity and has been one of the
most applied statistical models. With simple geometrical rules a transition is
obtained which is related to magnetic models. This transition is, in all
dimensions, one of the most robust continuous transitions known. We present a
very brief overview of more than 60 years of work in this area and discuss
several open questions for a variety of models, including classical, explosive,
invasion, bootstrap, and correlated percolation
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