1,000 research outputs found
Renormalization group theory for percolation in time-varying networks
Motivated by multi-hop communication in unreliable wireless networks, we
present a percolation theory for time-varying networks. We develop a
renormalization group theory for a prototypical network on a regular grid,
where individual links switch stochastically between active and inactive
states. The question whether a given source node can communicate with a
destination node along paths of active links is equivalent to a percolation
problem. Our theory maps the temporal existence of multi-hop paths on an
effective two-state Markov process. We show analytically how this Markov
process converges towards a memory-less Bernoulli process as the hop distance
between source and destination node increases. Our work extends classical
percolation theory to the dynamic case and elucidates temporal correlations of
message losses. Quantification of temporal correlations has implications for
the design of wireless communication and control protocols, e.g. in
cyber-physical systems such as self-organized swarms of drones or smart traffic
networks.Comment: 8 pages, 3 figure
Critical dynamics of ballistic and Brownian particles in a heterogeneous environment
The dynamic properties of a classical tracer particle in a random, disordered
medium are investigated close to the localization transition. For Lorentz
models obeying Newtonian and diffusive motion at the microscale, we have
performed large-scale computer simulations, demonstrating that universality
holds at long times in the immediate vicinity of the transition. The scaling
function describing the crossover from anomalous transport to diffusive motion
is found to vary extremely slowly and spans at least 5 decades in time. To
extract the scaling function, one has to allow for the leading universal
corrections to scaling. Our findings suggest that apparent power laws with
varying exponents generically occur and dominate experimentally accessible time
windows as soon as the heterogeneities cover a decade in length scale. We
extract the divergent length scales, quantify the spatial heterogeneities in
terms of the non-Gaussian parameter, and corroborate our results by a thorough
finite-size analysis.Comment: 14 page
Fracture strength: Stress concentration, extreme value statistics and the fate of the Weibull distribution
The fracture strength distribution of materials is often described in terms
of the Weibull law which can be derived by using extreme value statistics if
elastic interactions are ignored. Here, we consider explicitly the interplay
between elasticity and disorder and test the asymptotic validity of the Weibull
distribution through numerical simulations of the two-dimensional random fuse
model. Even when the local fracture strength follows the Weibull distribution,
the global failure distribution is dictated by stress enhancement at the tip of
the cracks and sometimes deviates from the Weibull law. Only in the case of a
pre-existing power law distribution of crack widths do we find that the failure
strength is Weibull distributed. Contrary to conventional assumptions, even in
this case, the Weibull exponent can not be simply inferred from the exponent of
the initial crack width distribution. Our results thus raise some concerns on
the applicability of the Weibull distribution in most practical cases
Causal evolution of spin networks
A new approach to quantum gravity is described which joins the loop
representation formulation of the canonical theory to the causal set
formulation of the path integral. The theory assigns quantum amplitudes to
special classes of causal sets, which consist of spin networks representing
quantum states of the gravitational field joined together by labeled null
edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also
be interepreted as a theory of labeled timelike surfaces. The dynamics is
specified by a choice of functions of the labelings of d+1 dimensional
simplices,which represent elementary future light cones of events in these
discrete spacetimes. The quantum dynamics thus respects the discrete causal
structure of the causal sets. In the 1+1 dimensional case the theory is closely
related to directed percolation models. In this case, at least, the theory may
have critical behavior associated with percolation, leading to the existence of
a classical limit.Comment: latex, 32 pages, 17 figure
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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