218 research outputs found
Counting metastable states of Ising spin glasses on arbitrary graphs
Using a field-theoretical representation of the Tanaka-Edwards integral we
develop a method to systematically compute the number N_s of 1-spin-stable
states (local energy minima) of a glassy Ising system with nearest-neighbor
interactions and random Gaussian couplings on an arbitrary graph. In
particular, we use this method to determine N_s for K-regular random graphs and
d-dimensional regular lattices for d=2,3. The method works also for other
graphs. Excellent accuracy of the results allows us to observe that the number
of local energy minima depends mainly on local properties of the graph on which
the spin glass is defined.Comment: 8 pages, 4 figures (one in color), additional materials can be found
under http://www.physik.uni-leipzig.de/~waclaw/glasses-data.ht
Eigenvalue density of empirical covariance matrix for correlated samples
We describe a method to determine the eigenvalue density of empirical
covariance matrix in the presence of correlations between samples. This is a
straightforward generalization of the method developed earlier by the authors
for uncorrelated samples. The method allows for exact determination of the
experimental spectrum for a given covariance matrix and given correlations
between samples in the limit of large N and N/T=r=const with N being the number
of degrees of freedom and T being the number of samples. We discuss the effect
of correlations on several examples.Comment: 12 pages, 5 figures, to appear in Acta Phys. Pol. B (Proceedings of
the conference on `Applications of Random Matrix Theory to Economy and Other
Complex Systems', May 25-28, 2005, Cracow, Polan
Causal and homogeneous networks
Growing networks have a causal structure. We show that the causality strongly
influences the scaling and geometrical properties of the network. In particular
the average distance between nodes is smaller for causal networks than for
corresponding homogeneous networks. We explain the origin of this effect and
illustrate it using as an example a solvable model of random trees. We also
discuss the issue of stability of the scale-free node degree distribution. We
show that a surplus of links may lead to the emergence of a singular node with
the degree proportional to the total number of links. This effect is closely
related to the backgammon condensation known from the balls-in-boxes model.Comment: short review submitted to AIP proceedings, CNET2004 conference;
changes in the discussion of the distance distribution for growing trees,
Fig. 6-right change
Free zero-range processes on networks
A free zero-range process (FRZP) is a simple stochastic process describing
the dynamics of a gas of particles hopping between neighboring nodes of a
network. We discuss three different cases of increasing complexity: (a) FZRP on
a rigid geometry where the network is fixed during the process, (b) FZRP on a
random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical
network whose topology continuously changes during the process in a way which
depends on the current distribution of particles. The case (a) provides a very
simple realization of the phenomenon of condensation which manifests as the
appearance of a condensate of particles on the node with maximal degree. The
case (b) is very interesting since the averaging over typical ensembles of
graphs acts as a kind of homogenization of the system which makes all nodes
identical from the point of view of the FZRP. In the case (c), the distribution
of particles and the dynamics of network are coupled to each other. The
strength of this coupling depends on the ratio of two time scales: for changes
of the topology and of the FZRP. We will discuss a specific example of that
type of interaction and show that it leads to an interesting phase diagram.Comment: 11 pages, 4 figures, to appear in Proceedings of SPIE Symposium
"Fluctuations and Noise 2007", Florence, 20-24 May 200
Balls-in-boxes condensation on networks
We discuss two different regimes of condensate formation in zero-range
processes on networks: on a q-regular network, where the condensate is formed
as a result of a spontaneous symmetry breaking, and on an irregular network,
where the symmetry of the partition function is explicitly broken. In the
latter case we consider a minimal irregularity of the q-regular network
introduced by a single Q-node with degree Q>q. The statics and dynamics of the
condensation depends on the parameter log(Q/q), which controls the exponential
fall-off of the distribution of particles on regular nodes and the typical time
scale for melting of the condensate on the Q-node which increases exponentially
with the system size . This behavior is different than that on a q-regular
network where log(Q/q)=0 and where the condensation results from the
spontaneous symmetry breaking of the partition function, which is invariant
under a permutation of particle occupation numbers on the q-nodes of the
network. In this case the typical time scale for condensate melting is known to
increase typically as a power of the system size.Comment: 7 pages, 3 figures, submitted to the "Chaos" focus issue on
"Optimization in Networks" (scheduled to appear as Volume 17, No. 2, 2007
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