13 research outputs found
A fast Monte Carlo algorithm for site or bond percolation
We describe in detail a new and highly efficient algorithm for studying site
or bond percolation on any lattice. The algorithm can measure an observable
quantity in a percolation system for all values of the site or bond occupation
probability from zero to one in an amount of time which scales linearly with
the size of the system. We demonstrate our algorithm by using it to investigate
a number of issues in percolation theory, including the position of the
percolation transition for site percolation on the square lattice, the
stretched exponential behavior of spanning probabilities away from the critical
point, and the size of the giant component for site percolation on random
graphs.Comment: 17 pages, 13 figures. Corrections and some additional material in
this version. Accompanying material can be found on the web at
http://www.santafe.edu/~mark/percolation
Percolation of randomly distributed growing clusters: Finite Size Scaling and Critical Exponents
We study the percolation properties of the growing clusters model. In this
model, a number of seeds placed on random locations on a lattice are allowed to
grow with a constant velocity to form clusters. When two or more clusters
eventually touch each other they immediately stop their growth. The model
exhibits a discontinuous transition for very low values of the seed
concentration and a second, non-trivial continuous phase transition for
intermediate values. Here we study in detail this continuous transition
that separates a phase of finite clusters from a phase characterized by the
presence of a giant component. Using finite size scaling and large scale Monte
Carlo simulations we determine the value of the percolation threshold where the
giant component first appears, and the critical exponents that characterize the
transition. We find that the transition belongs to a different universality
class from the standard percolation transition.Comment: 5 two-column pages, 6 figure
Fibrillar organization in tendons: a pattern revealed by percolation characteristics of the respective geometric network
Since the tendon is composed by collagen fibrils of various sizes connected between them through molecular cross-links, it sounds logical to model it via a heterogeneous network of fibrils. Using cross sectional images, that network is operatively inferred from the respective Gabriel graph of the fibril mass centers. We focus on network percolation characteristics under an ordered activation of fibrils (progressive recruitment going from the smallest to the largest fibril). Analyses of percolation were carried out on a repository of images of digital flexor tendons obtained from samples of lizards and frogs. Observed percolation thresholds were compared against values derived from hypothetical scenarios of random activation of nodes. Strikingly, we found a significant delay for the occurrence of percolation in actual data. We interpret this finding as the consequence of some non-random packing of fibrillar units into a size-constrained geometric pattern. We erect an ideal geometric model of balanced interspersion of polymorphic units that accounts for the delayed percolating instance. We also address the circumstance of being percolation curves mirrored by the empirical curves of stress-strain obtained from the same studied tendons. By virtue of this isomorphism, we hypothesize that the inflection points of both curves are different quantitative manifestations of a common transitional process during mechanical load transference.Fil: Dos Santos, Daniel Andrés. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Tucuman. Instituto de Biodiversidad Neotropical; Argentina. Universidad Nacional de Tucumán. Facultad de Ciencias Naturales e Instituto Miguel Lillo; ArgentinaFil: Ponssa, MarÃa Laura. Fundación Miguel Lillo. Dirección de ZoologÃa. Instituto de HerpetologÃa; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Tulli, MarÃa José. Fundación Miguel Lillo. Dirección de ZoologÃa. Instituto de HerpetologÃa; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Abdala, Virginia Sara Luz. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Tucuman. Instituto de Biodiversidad Neotropical; Argentina. Universidad Nacional de Tucumán. Facultad de Ciencias Naturales e Instituto Miguel Lillo; Argentina. Fundación Miguel Lillo. Dirección de ZoologÃa. Instituto de HerpetologÃa; Argentin
Inhomogeneous percolation models for spreading phenomena in random graphs
Percolation theory has been largely used in the study of structural
properties of complex networks such as the robustness, with remarkable results.
Nevertheless, a purely topological description is not sufficient for a correct
characterization of networks behaviour in relation with physical flows and
spreading phenomena taking place on them. The functionality of real networks
also depends on the ability of the nodes and the edges in bearing and handling
loads of flows, energy, information and other physical quantities. We propose
to study these properties introducing a process of inhomogeneous percolation,
in which both the nodes and the edges spread out the flows with a given
probability.
Generating functions approach is exploited in order to get a generalization
of the Molloy-Reed Criterion for inhomogeneous joint site bond percolation in
correlated random graphs. A series of simple assumptions allows the analysis of
more realistic situations, for which a number of new results are presented. In
particular, for the site percolation with inhomogeneous edge transmission, we
obtain the explicit expressions of the percolation threshold for many
interesting cases, that are analyzed by means of simple examples and numerical
simulations. Some possible applications are debated.Comment: 28 pages, 11 figure