5,139 research outputs found
Trajectories entropy in dynamical graphs with memory
In this paper we investigate the application of non-local graph entropy to
evolving and dynamical graphs. The measure is based upon the notion of Markov
diffusion on a graph, and relies on the entropy applied to trajectories
originating at a specific node. In particular, we study the model of
reinforcement-decay graph dynamics, which leads to scale free graphs. We find
that the node entropy characterizes the structure of the network in the two
parameter phase-space describing the dynamical evolution of the weighted graph.
We then apply an adapted version of the entropy measure to purely memristive
circuits. We provide evidence that meanwhile in the case of DC voltage the
entropy based on the forward probability is enough to characterize the graph
properties, in the case of AC voltage generators one needs to consider both
forward and backward based transition probabilities. We provide also evidence
that the entropy highlights the self-organizing properties of memristive
circuits, which re-organizes itself to satisfy the symmetries of the underlying
graph.Comment: 15 pages one column, 10 figures; new analysis and memristor models
added. Text improve
Entrograms and coarse graining of dynamics on complex networks
Using an information theoretic point of view, we investigate how a dynamics
acting on a network can be coarse grained through the use of graph partitions.
Specifically, we are interested in how aggregating the state space of a Markov
process according to a partition impacts on the thus obtained lower-dimensional
dynamics. We highlight that for a dynamics on a particular graph there may be
multiple coarse grained descriptions that capture different, incomparable
features of the original process. For instance, a coarse graining induced by
one partition may be commensurate with a time-scale separation in the dynamics,
while another coarse graining may correspond to a different lower-dimensional
dynamics that preserves the Markov property of the original process. Taking
inspiration from the literature of Computational Mechanics, we find that a
convenient tool to summarise and visualise such dynamical properties of a
coarse grained model (partition) is the entrogram. The entrogram gathers
certain information-theoretic measures, which quantify how information flows
across time steps. These information theoretic quantities include the entropy
rate, as well as a measure for the memory contained in the process, i.e., how
well the dynamics can be approximated by a first order Markov process. We use
the entrogram to investigate how specific macro-scale connection patterns in
the state-space transition graph of the original dynamics result in desirable
properties of coarse grained descriptions. We thereby provide a fresh
perspective on the interplay between structure and dynamics in networks, and
the process of partitioning from an information theoretic perspective. We focus
on networks that may be approximated by both a core-periphery or a clustered
organization, and highlight that each of these coarse grained descriptions can
capture different aspects of a Markov process acting on the network.Comment: 17 pages, 6 figue
Order Parameter Flow in the SK Spin-Glass I: Replica Symmetry
We present a theory to describe the dynamics of the Sherrington- Kirkpatrick
spin-glass with (sequential) Glauber dynamics in terms of deterministic flow
equations for macroscopic parameters. Two transparent assumptions allow us to
close the macroscopic laws. Replica theory enters as a tool in the calculation
of the time- dependent local field distribution. The theory produces in a
natural way dynamical generalisations of the AT- and zero-entropy lines and of
Parisi's order parameter function . In equilibrium we recover the
standard results from equilibrium statistical mechanics. In this paper we make
the replica-symmetric ansatz, as a first step towards calculating the order
parameter flow. Numerical simulations support our assumptions and suggest that
our equations describe the shape of the local field distribution and the
macroscopic dynamics reasonably well in the region where replica symmetry is
stable.Comment: 41 pages, Latex, OUTP-94-29S, 14 figures available in hardcop
"0-1" test chaosu
The goal of this thesis is to research the 0-1 test for chaos, its application in Matlab, and testing on suitable models. Elementary tools of the dynamical systems analysis are introduced, that are later used in the main results part of the thesis. The 0-1 test for chaos is introduced in detail, defined, and implemented in Matlab. The application is then performed on two one-dimensional discrete models where the first one is in explicit and the second one in implicit form. In both cases, simulations in dependence of the state parameter were done and main results are given - the 0-1 test for chaos, phase, and bifurcation diagrams.Hlavním cílem bakalářské práce je studium 0-1 testu chaosu, jeho implementace v Matlabu a následné testování na vhodných modelech. V práci jsou zavedeny základní nástroje analýzy dynamických systémů, které jsou později použity v části hlavních výsledků. 0-1 test chaosu je podrobně uveden, řádně definován a implementován v Matlabu. Aplikace je provedena na dvou jednodimenzionálních diskrétních modelech z nichž jeden je v explicitním a druhý v implicitním tvaru. V obou případech byly provedeny simulace v závislosti na stavovém parametru a hlavní výsledky byly demonstrovány formou 0-1 testu chaosu, fázových a bifurkačních diagramů.470 - Katedra aplikované matematikyvýborn
Asymptotic behavior of memristive circuits
The interest in memristors has risen due to their possible application both
as memory units and as computational devices in combination with CMOS. This is
in part due to their nonlinear dynamics, and a strong dependence on the circuit
topology. We provide evidence that also purely memristive circuits can be
employed for computational purposes. In the present paper we show that a
polynomial Lyapunov function in the memory parameters exists for the case of DC
controlled memristors. Such Lyapunov function can be asymptotically
approximated with binary variables, and mapped to quadratic combinatorial
optimization problems. This also shows a direct parallel between memristive
circuits and the Hopfield-Little model. In the case of Erdos-Renyi random
circuits, we show numerically that the distribution of the matrix elements of
the projectors can be roughly approximated with a Gaussian distribution, and
that it scales with the inverse square root of the number of elements. This
provides an approximated but direct connection with the physics of disordered
system and, in particular, of mean field spin glasses. Using this and the fact
that the interaction is controlled by a projector operator on the loop space of
the circuit. We estimate the number of stationary points of the approximate
Lyapunov function and provide a scaling formula as an upper bound in terms of
the circuit topology only.Comment: 20 pages, 8 figures; proofs corrected, figures changed; results
substantially unchanged; to appear in Entrop
Complex network classification using partially self-avoiding deterministic walks
Complex networks have attracted increasing interest from various fields of
science. It has been demonstrated that each complex network model presents
specific topological structures which characterize its connectivity and
dynamics. Complex network classification rely on the use of representative
measurements that model topological structures. Although there are a large
number of measurements, most of them are correlated. To overcome this
limitation, this paper presents a new measurement for complex network
classification based on partially self-avoiding walks. We validate the
measurement on a data set composed by 40.000 complex networks of four
well-known models. Our results indicate that the proposed measurement improves
correct classification of networks compared to the traditional ones
- …