21,367 research outputs found
L\'evy-type diffusion on one-dimensional directed Cantor Graphs
L\'evy-type walks with correlated jumps, induced by the topology of the
medium, are studied on a class of one-dimensional deterministic graphs built
from generalized Cantor and Smith-Volterra-Cantor sets. The particle performs a
standard random walk on the sets but is also allowed to move ballistically
throughout the empty regions. Using scaling relations and the mapping onto the
electric network problem, we obtain the exact values of the scaling exponents
for the asymptotic return probability, the resistivity and the mean square
displacement as a function of the topological parameters of the sets.
Interestingly, the systems undergoes a transition from superdiffusive to
diffusive behavior as a function of the filling of the fractal. The
deterministic topology also allows us to discuss the importance of the choice
of the initial condition. In particular, we demonstrate that local and average
measurements can display different asymptotic behavior. The analytic results
are compared with the numerical solution of the master equation of the process.Comment: 9 pages, 9 figure
Classical dynamics on graphs
We consider the classical evolution of a particle on a graph by using a
time-continuous Frobenius-Perron operator which generalizes previous
propositions. In this way, the relaxation rates as well as the chaotic
properties can be defined for the time-continuous classical dynamics on graphs.
These properties are given as the zeros of some periodic-orbit zeta functions.
We consider in detail the case of infinite periodic graphs where the particle
undergoes a diffusion process. The infinite spatial extension is taken into
account by Fourier transforms which decompose the observables and probability
densities into sectors corresponding to different values of the wave number.
The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a
Frobenius-Perron operator corresponding to a given sector. The diffusion
coefficient is obtained from the hydrodynamic modes of diffusion and has the
Green-Kubo form. Moreover, we study finite but large open graphs which converge
to the infinite periodic graph when their size goes to infinity. The lifetime
of the particle on the open graph is shown to correspond to the lifetime of a
system which undergoes a diffusion process before it escapes.Comment: 42 pages and 8 figure
Transport and dynamics on open quantum graphs
We study the classical limit of quantum mechanics on graphs by introducing a
Wigner function for graphs. The classical dynamics is compared to the quantum
dynamics obtained from the propagator. In particular we consider extended open
graphs whose classical dynamics generate a diffusion process. The transport
properties of the classical system are revealed in the scattering resonances
and in the time evolution of the quantum system.Comment: 42 pages, 13 figures, submitted to PR
Lock-in Problem for Parallel Rotor-router Walks
The rotor-router model, also called the Propp machine, was introduced as a
deterministic alternative to the random walk. In this model, a group of
identical tokens are initially placed at nodes of the graph. Each node
maintains a cyclic ordering of the outgoing arcs, and during consecutive turns
the tokens are propagated along arcs chosen according to this ordering in
round-robin fashion. The behavior of the model is fully deterministic. Yanovski
et al.(2003) proved that a single rotor-router walk on any graph with m edges
and diameter stabilizes to a traversal of an Eulerian circuit on the set of
all 2m directed arcs on the edge set of the graph, and that such periodic
behaviour of the system is achieved after an initial transient phase of at most
2mD steps. The case of multiple parallel rotor-routers was studied
experimentally, leading Yanovski et al. to the conjecture that a system of k
\textgreater{} 1 parallel walks also stabilizes with a period of length at
most steps. In this work we disprove this conjecture, showing that the
period of parallel rotor-router walks can in fact, be superpolynomial in the
size of graph. On the positive side, we provide a characterization of the
periodic behavior of parallel router walks, in terms of a structural property
of stable states called a subcycle decomposition. This property provides us the
tools to efficiently detect whether a given system configuration corresponds to
the transient or to the limit behavior of the system. Moreover, we provide
polynomial upper bounds of and on the
number of steps it takes for the system to stabilize. Thus, we are able to
predict any future behavior of the system using an algorithm that takes
polynomial time and space. In addition, we show that there exists a separation
between the stabilization time of the single-walk and multiple-walk
rotor-router systems, and that for some graphs the latter can be asymptotically
larger even for the case of walks
Sigma models for quantum chaotic dynamics
We review the construction of the supersymmetric sigma model for unitary
maps, using the color- flavor transformation. We then illustrate applications
by three case studies in quantum chaos. In two of these cases, general Floquet
maps and quantum graphs, we show that universal spectral fluctuations arise
provided the pertinent classical dynamics are fully chaotic (ergodic and with
decay rates sufficiently gapped away from zero). In the third case, the kicked
rotor, we show how the existence of arbitrarily long-lived modes of excitation
(diffusion) precludes universal fluctuations and entails quantum localization
A Unifying Model for Representing Time-Varying Graphs
Graph-based models form a fundamental aspect of data representation in Data
Sciences and play a key role in modeling complex networked systems. In
particular, recently there is an ever-increasing interest in modeling dynamic
complex networks, i.e. networks in which the topological structure (nodes and
edges) may vary over time. In this context, we propose a novel model for
representing finite discrete Time-Varying Graphs (TVGs), which are typically
used to model dynamic complex networked systems. We analyze the data structures
built from our proposed model and demonstrate that, for most practical cases,
the asymptotic memory complexity of our model is in the order of the
cardinality of the set of edges. Further, we show that our proposal is an
unifying model that can represent several previous (classes of) models for
dynamic networks found in the recent literature, which in general are unable to
represent each other. In contrast to previous models, our proposal is also able
to intrinsically model cyclic (i.e. periodic) behavior in dynamic networks.
These representation capabilities attest the expressive power of our proposed
unifying model for TVGs. We thus believe our unifying model for TVGs is a step
forward in the theoretical foundations for data analysis of complex networked
systems.Comment: Also appears in the Proc. of the IEEE International Conference on
Data Science and Advanced Analytics (IEEE DSAA'2015
Analytical computation of the epidemic threshold on temporal networks
The time variation of contacts in a networked system may fundamentally alter
the properties of spreading processes and affect the condition for large-scale
propagation, as encoded in the epidemic threshold. Despite the great interest
in the problem for the physics, applied mathematics, computer science and
epidemiology communities, a full theoretical understanding is still missing and
currently limited to the cases where the time-scale separation holds between
spreading and network dynamics or to specific temporal network models. We
consider a Markov chain description of the Susceptible-Infectious-Susceptible
process on an arbitrary temporal network. By adopting a multilayer perspective,
we develop a general analytical derivation of the epidemic threshold in terms
of the spectral radius of a matrix that encodes both network structure and
disease dynamics. The accuracy of the approach is confirmed on a set of
temporal models and empirical networks and against numerical results. In
addition, we explore how the threshold changes when varying the overall time of
observation of the temporal network, so as to provide insights on the optimal
time window for data collection of empirical temporal networked systems. Our
framework is both of fundamental and practical interest, as it offers novel
understanding of the interplay between temporal networks and spreading
dynamics.Comment: 22 pages, 6 figure
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