25,482 research outputs found
Critical phenomena in complex networks
The combination of the compactness of networks, featuring small diameters,
and their complex architectures results in a variety of critical effects
dramatically different from those in cooperative systems on lattices. In the
last few years, researchers have made important steps toward understanding the
qualitatively new critical phenomena in complex networks. We review the
results, concepts, and methods of this rapidly developing field. Here we mostly
consider two closely related classes of these critical phenomena, namely
structural phase transitions in the network architectures and transitions in
cooperative models on networks as substrates. We also discuss systems where a
network and interacting agents on it influence each other. We overview a wide
range of critical phenomena in equilibrium and growing networks including the
birth of the giant connected component, percolation, k-core percolation,
phenomena near epidemic thresholds, condensation transitions, critical
phenomena in spin models placed on networks, synchronization, and
self-organized criticality effects in interacting systems on networks. We also
discuss strong finite size effects in these systems and highlight open problems
and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references,
extende
Cortical free association dynamics: distinct phases of a latching network
A Potts associative memory network has been proposed as a simplified model of
macroscopic cortical dynamics, in which each Potts unit stands for a patch of
cortex, which can be activated in one of S local attractor states. The internal
neuronal dynamics of the patch is not described by the model, rather it is
subsumed into an effective description in terms of graded Potts units, with
adaptation effects both specific to each attractor state and generic to the
patch. If each unit, or patch, receives effective (tensor) connections from C
other units, the network has been shown to be able to store a large number p of
global patterns, or network attractors, each with a fraction a of the units
active, where the critical load p_c scales roughly like p_c ~ (C S^2)/(a
ln(1/a)) (if the patterns are randomly correlated). Interestingly, after
retrieving an externally cued attractor, the network can continue jumping, or
latching, from attractor to attractor, driven by adaptation effects. The
occurrence and duration of latching dynamics is found through simulations to
depend critically on the strength of local attractor states, expressed in the
Potts model by a parameter w. Here we describe with simulations and then
analytically the boundaries between distinct phases of no latching, of
transient and sustained latching, deriving a phase diagram in the plane w-T,
where T parametrizes thermal noise effects. Implications for real cortical
dynamics are briefly reviewed in the conclusions
Traffic Analysis in Random Delaunay Tessellations and Other Graphs
In this work we study the degree distribution, the maximum vertex and edge
flow in non-uniform random Delaunay triangulations when geodesic routing is
used. We also investigate the vertex and edge flow in Erd\"os-Renyi random
graphs, geometric random graphs, expanders and random -regular graphs.
Moreover we show that adding a random matching to the original graph can
considerably reduced the maximum vertex flow.Comment: Submitted to the Journal of Discrete Computational Geometr
Global and Local Information in Clustering Labeled Block Models
The stochastic block model is a classical cluster-exhibiting random graph
model that has been widely studied in statistics, physics and computer science.
In its simplest form, the model is a random graph with two equal-sized
clusters, with intra-cluster edge probability p, and inter-cluster edge
probability q. We focus on the sparse case, i.e., p, q = O(1/n), which is
practically more relevant and also mathematically more challenging. A
conjecture of Decelle, Krzakala, Moore and Zdeborova, based on ideas from
statistical physics, predicted a specific threshold for clustering. The
negative direction of the conjecture was proved by Mossel, Neeman and Sly
(2012), and more recently the positive direction was proven independently by
Massoulie and Mossel, Neeman, and Sly.
In many real network clustering problems, nodes contain information as well.
We study the interplay between node and network information in clustering by
studying a labeled block model, where in addition to the edge information, the
true cluster labels of a small fraction of the nodes are revealed. In the case
of two clusters, we show that below the threshold, a small amount of node
information does not affect recovery. On the other hand, we show that for any
small amount of information efficient local clustering is achievable as long as
the number of clusters is sufficiently large (as a function of the amount of
revealed information).Comment: 24 pages, 2 figures. A short abstract describing these results will
appear in proceedings of RANDOM 201
Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model
We compute the expected number of cars that have at least one two-hop path to
a fixed roadside unit in a one-dimensional vehicular ad hoc network in which
other cars can be used as relays to reach a roadside unit when they do not have
a reliable direct link. The pairwise channels between cars experience Rayleigh
fading in the random connection model, and so exist, with probability function
of the mutual distance between the cars, or between the cars and the roadside
unit. We derive exact equivalents for this expected number of cars when the car
density tends to zero and to infinity, and determine its behaviour using
an infinite oscillating power series in , which is accurate for all
regimes. We also corroborate those findings to a realistic situation, using
snapshots of actual traffic data. Finally, a normal approximation is discussed
for the probability mass function of the number of cars with a two-hop
connection to the origin. The probability mass function appears to be well
fitted by a Gaussian approximation with mean equal to the expected number of
cars with two hops to the origin.Comment: 21 pages, 7 figure
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