194,456 research outputs found
Random graphs with clustering
We offer a solution to a long-standing problem in the physics of networks,
the creation of a plausible, solvable model of a network that displays
clustering or transitivity -- the propensity for two neighbors of a network
node also to be neighbors of one another. We show how standard random graph
models can be generalized to incorporate clustering and give exact solutions
for various properties of the resulting networks, including sizes of network
components, size of the giant component if there is one, position of the phase
transition at which the giant component forms, and position of the phase
transition for percolation on the network.Comment: 5 pages, 2 figure
Vortices Clustering: The Origin of the Second Peak in the Magnetisation Loops of High Temperature Superconductors
We study vortex clustering in type II Superconductors. We demonstrate that
the ``second peak'' observed in magnetisation loops may be a dynamical effect
associated with a density driven instability of the vortex system. At the
microscopic level the instability shows up as the clustering of individual
vortices at (rare) preferential regions of the pinning potential. In the limit
of quasi-static ramping the instability is related to a phase transition in the
equilibrium vortex system.Comment: 11 pages + 3 figure
Phase transitions in a network with range dependent connection probability
We consider a one-dimensional network in which the nodes at Euclidean
distance can have long range connections with a probabilty in addition to nearest neighbour connections. This system has been
shown to exhibit small world behaviour for above which its
behaviour is like a regular lattice. From the study of the clustering
coefficients, we show that there is a transition to a random network at . The finite size scaling analysis of the clustering coefficients obtained
from numerical simulations indicate that a continuous phase transition occurs
at this point. Using these results, we find that the two transitions occurring
in this network can be detected in any dimension by the behaviour of a single
quantity, the average bond length. The phase transitions in all dimensions are
non-trivial in nature.Comment: 4 pages, revtex4, submitted to Physical Review
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure
Spectral density of the non-backtracking operator
The non-backtracking operator was recently shown to provide a significant
improvement when used for spectral clustering of sparse networks. In this paper
we analyze its spectral density on large random sparse graphs using a mapping
to the correlation functions of a certain interacting quantum disordered system
on the graph. On sparse, tree-like graphs, this can be solved efficiently by
the cavity method and a belief propagation algorithm. We show that there exists
a paramagnetic phase, leading to zero spectral density, that is stable outside
a circle of radius , where is the leading eigenvalue of the
non-backtracking operator. We observe a second-order phase transition at the
edge of this circle, between a zero and a non-zero spectral density. That fact
that this phase transition is absent in the spectral density of other matrices
commonly used for spectral clustering provides a physical justification of the
performances of the non-backtracking operator in spectral clustering.Comment: 6 pages, 6 figures, submitted to EP
A kinetic model and scaling properties for non-equilibrium clustering of self-propelled particles
We demonstrate that the clustering statistics and the corresponding phase
transition to non-equilibrium clustering found in many experiments and
simulation studies with self-propelled particles (SPPs) with alignment can be
obtained from a simple kinetic model. The key elements of this approach are the
scaling of the cluster cross-section with the cluster mass -- characterized by
an exponent -- and the scaling of the cluster perimeter with the
cluster mass -- described by an exponent . The analysis of the kinetic
approach reveals that the SPPs exhibit two phases: i) an individual phase,
where the cluster size distribution (CSD) is dominated by an exponential tail
that defines a characteristic cluster size, and ii) a collective phase
characterized by the presence of non-monotonic CSD with a local maximum at
large cluster sizes. At the transition between these two phases the CSD is well
described by a power-law with a critical exponent , which is a function
of and only. The critical exponent is found to be in the range
in line with observations in experiments and simulations
Dynamic instability transitions in 1D driven diffusive flow with nonlocal hopping
One-dimensional directed driven stochastic flow with competing nonlocal and
local hopping events has an instability threshold from a populated phase into
an empty-road (ER) phase. We implement this in the context of the asymmetric
exclusion process. The nonlocal skids promote strong clustering in the
stationary populated phase. Such clusters drive the dynamic phase transition
and determine its scaling properties. We numerically establish that the
instability transition into the ER phase is second order in the regime where
the entry point reservoir controls the current and first order in the regime
where the bulk is in control. The first order transition originates from a
turn-about of the cluster drift velocity. At the critical line, the current
remains analytic, the road density vanishes linearly, and fluctuations scale as
uncorrelated noise. A self-consistent cluster dynamics analysis explains why
these scaling properties remain that simple.Comment: 11 pages, 14 figures (25 eps files); revised as the publised versio
Clustering and Synchronization of Oscillator Networks
Using a recently described technique for manipulating the clustering
coefficient of a network without changing its degree distribution, we examine
the effect of clustering on the synchronization of phase oscillators on
networks with Poisson and scale-free degree distributions. For both types of
network, increased clustering hinders global synchronization as the network
splits into dynamical clusters that oscillate at different frequencies.
Surprisingly, in scale-free networks, clustering promotes the synchronization
of the most connected nodes (hubs) even though it inhibits global
synchronization. As a result, scale-free networks show an additional, advanced
transition instead of a single synchronization threshold. This cluster-enhanced
synchronization of hubs may be relevant to the brain with its scale-free and
highly clustered structure.Comment: Submitted to Phys. Rev.
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