183,101 research outputs found
The effect of network structure on phase transitions in queuing networks
Recently, De Martino et al have presented a general framework for the study
of transportation phenomena on complex networks. One of their most significant
achievements was a deeper understanding of the phase transition from the
uncongested to the congested phase at a critical traffic load. In this paper,
we also study phase transition in transportation networks using a discrete time
random walk model. Our aim is to establish a direct connection between the
structure of the graph and the value of the critical traffic load. Applying
spectral graph theory, we show that the original results of De Martino et al
showing that the critical loading depends only on the degree sequence of the
graph -- suggesting that different graphs with the same degree sequence have
the same critical loading if all other circumstances are fixed -- is valid only
if the graph is dense enough. For sparse graphs, higher order corrections,
related to the local structure of the network, appear.Comment: 12 pages, 7 figure
Condensation of degrees emerging through a first-order phase transition in classical random graphs
Due to their conceptual and mathematical simplicity, Erd\"os-R\'enyi or
classical random graphs remain as a fundamental paradigm to model complex
interacting systems in several areas. Although condensation phenomena have been
widely considered in complex network theory, the condensation of degrees has
hitherto eluded a careful study. Here we show that the degree statistics of the
classical random graph model undergoes a first-order phase transition between a
Poisson-like distribution and a condensed phase, the latter characterized by a
large fraction of nodes having degrees in a limited sector of their
configuration space. The mechanism underlying the first-order transition is
discussed in light of standard concepts in statistical physics. We uncover the
phase diagram characterizing the ensemble space of the model and we evaluate
the rate function governing the probability to observe a condensed state, which
shows that condensation of degrees is a rare statistical event akin to similar
condensation phenomena recently observed in several other systems. Monte Carlo
simulations confirm the exactness of our theoretical results.Comment: 8 pages, 6 figure
Width of percolation transition in complex networks
It is known that the critical probability for the percolation transition is
not a sharp threshold, actually it is a region of non-zero width
for systems of finite size. Here we present evidence that for complex networks
, where is the average
length of the percolation cluster, and is the number of nodes in the
network. For Erd\H{o}s-R\'enyi (ER) graphs , while for
scale-free (SF) networks with a degree distribution
and , . We show analytically
and numerically that the \textit{survivability} , which is the
probability of a cluster to survive chemical shells at probability ,
behaves near criticality as . Thus
for probabilities inside the region the behavior of the
system is indistinguishable from that of the critical point
Localization Transition of Biased Random Walks on Random Networks
We study random walks on large random graphs that are biased towards a
randomly chosen but fixed target node. We show that a critical bias strength
b_c exists such that most walks find the target within a finite time when
b>b_c. For b<b_c, a finite fraction of walks drifts off to infinity before
hitting the target. The phase transition at b=b_c is second order, but finite
size behavior is complex and does not obey the usual finite size scaling
ansatz. By extending rigorous results for biased walks on Galton-Watson trees,
we give the exact analytical value for b_c and verify it by large scale
simulations.Comment: 4 pages, includes 4 figure
Random graph ensembles with many short loops
Networks observed in the real world often have many short loops. This
violates the tree-like assumption that underpins the majority of random graph
models and most of the methods used for their analysis. In this paper we sketch
possible research routes to be explored in order to make progress on networks
with many short loops, involving old and new random graph models and ideas for
novel mathematical methods. We do not present conclusive solutions of problems,
but aim to encourage and stimulate new activity and in what we believe to be an
important but under-exposed area of research. We discuss in more detail the
Strauss model, which can be seen as the `harmonic oscillator' of `loopy' random
graphs, and a recent exactly solvable immunological model that involves random
graphs with extensively many cliques and short loops.Comment: 18 pages, 10 figures,Mathematical Modelling of Complex Systems (Paris
2013) conferenc
A Bag-of-Paths Node Criticality Measure
This work compares several node (and network) criticality measures
quantifying to which extend each node is critical with respect to the
communication flow between nodes of the network, and introduces a new measure
based on the Bag-of-Paths (BoP) framework. Network disconnection simulation
experiments show that the new BoP measure outperforms all the other measures on
a sample of Erdos-Renyi and Albert-Barabasi graphs. Furthermore, a faster
(still O(n^3)), approximate, BoP criticality relying on the Sherman-Morrison
rank-one update of a matrix is introduced for tackling larger networks. This
approximate measure shows similar performances as the original, exact, one
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