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140
The structure and function of complex networks
 SIAM REVIEW
, 2003
"... Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, ..."
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Cited by 2591 (7 self)
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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the smallworld effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Closing the gap in the capacity of wireless networks via percolation theory
 IEEE Trans. Information Theory
, 2007
"... Abstract—An achievable bit rate per source–destination pair in a wireless network of � randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same Ia � � transmission rate of ..."
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Cited by 244 (7 self)
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Abstract—An achievable bit rate per source–destination pair in a wireless network of � randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same Ia � � transmission rate of arbitrarily located nodes. This contrasts with previous results suggesting that a Ia � � �� � � reduced rate is the price to pay for the randomness due to the location of the nodes. The network operation strategy to achieve the result corresponds to the transition region between order and disorder of an underlying percolation model. If nodes are allowed to transmit over large distances, then paths of connected nodes that cross the entire network area can be easily found, but these generate excessive interference. If nodes transmit over short distances, then such crossing paths do not exist. Percolation theory ensures that crossing paths form in the transition region between these two extreme scenarios. Nodes along these paths are used as a backbone, relaying data for other nodes, and can transport the total amount of information generated by all the sources. A lower bound on the achievable bit rate is then obtained by performing pairwise coding and decoding at each hop along the paths, and using a time division multiple access scheme. Index Terms—Adhoc networks, capacity, percolation theory, scaling laws, throughput, wireless networks.
Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks
"... Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communicationtheoretic results accoun ..."
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Cited by 231 (43 self)
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Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communicationtheoretic results accounting for the network’s geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs – including point process theory, percolation theory, and probabilistic combinatorics – have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.
Probabilistic Broadcast for Flooding in Wireless Mobile Ad hoc Networks
, 2002
"... Although far from optimal, flooding is an indispensable message dissemination technique for networkwide broadcast within mobile ad hoc networks (MANETs). As such, the plain flooding algorithm provokes a high number of unnecessary packet rebroadcasts, causing contention, packet collisions and ult ..."
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Cited by 171 (1 self)
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Although far from optimal, flooding is an indispensable message dissemination technique for networkwide broadcast within mobile ad hoc networks (MANETs). As such, the plain flooding algorithm provokes a high number of unnecessary packet rebroadcasts, causing contention, packet collisions and ultimately wasting precious limited bandwidth. Studies have been undertaken to optimize flooding using a deterministic approach. Because of the highly dynamic and mobile characteristics of MANETs, probabilistic algorithms may be better suited. We explore the phase transition phenomenon observed in percolation theory and random graphs as a basis for defining probabilistic flooding algorithms. We consider models with and without packet collisions to better understand when phase transition occurs. We show through simulation that in cases of no collision control, probabilistic flooding greatly enhances network performance while significantly reducing broadcast packets in dense networks, although phase transition is not observed.
The RandomCluster Model
, 2006
"... Abstract. The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, an ..."
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Cited by 70 (20 self)
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Abstract. The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the randomcluster representation. This systematic summary of randomcluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinitevolume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for twodimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the clusterweighting factor q, and the problem of proving that the critical randomcluster model in two
The critical probability for random Voronoi percolation in the plane is 1/2, Probability Theory and Related Fields 136
, 2006
"... We study percolation in the following random environment: let Z be a Poisson process of constant intensity on R 2, and form the Voronoi tessellation of R 2 with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is ..."
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Cited by 58 (7 self)
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We study percolation in the following random environment: let Z be a Poisson process of constant intensity on R 2, and form the Voronoi tessellation of R 2 with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is 1/2. More precisely, if p> 1/2 then the union of the black cells contains an infinite component with probability 1, while if p < 1/2 then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten’s results for bond percolation in Z 2. The result corresponding to Harris ’ Theorem for bond percolation in Z 2 is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten’s results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten’s Theorem for Z 2; we hope
Scale Invariance in Biology: Coincidence Or Footprint of a Universal Mechanism?
, 2001
"... In this article, we present a selfcontained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1}fnoise where f denotes the frequency of a signal (temporal scale i ..."
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Cited by 51 (1 self)
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In this article, we present a selfcontained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1}fnoise where f denotes the frequency of a signal (temporal scale invariance) and power laws (scale invariance in the size and duration of events in the dynamics of the system). A hypothesis recently put forward to explain these scalefree phenomomena is criticality, a notion introduced by physicists while studying phase transitions in materials, where systems spontaneously arrange themselves in an unstable manner similar, for instance, to a row of dominoes. Here, we review in a critical manner work which investigates to what extent this idea can be generalized to biology. More precisely, we start with a brief introduction to the concepts of absence of characteristic scale (powerlaw distributions, fractals and 1}fnoise) and of critical phenomena. We then review typical mathematical models exhibiting such properties : edge of chaos, cellular automata and selforganized critical models. These notions are then brought together to see to what extent they can account for the scale invariance observed in ecology, evolution of species, type III epidemics and some aspects of the central nervous system. This article also discusses how the notion of scale invariance can give important insights into the workings of biological systems.
The Wulff construction and asymptotics of the finite cluster distribution for twodimensional Bernoulli percolation
 COMMUN. MATH. PHYS
, 1990
"... We consider twodimensional Bernoulli percolation at density p> Pc and establish the following results: 1. The probability, PN(P), that the origin is in a finite cluster of size N obeys 1 ~(p)a(p) lim ~ log PN(P)where P~o (P) is the infinite cluster density, a(p) is the (zeroangle) surface ten ..."
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Cited by 46 (8 self)
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We consider twodimensional Bernoulli percolation at density p> Pc and establish the following results: 1. The probability, PN(P), that the origin is in a finite cluster of size N obeys 1 ~(p)a(p) lim ~ log PN(P)where P~o (P) is the infinite cluster density, a(p) is the (zeroangle) surface tension, and ~(p) is a quantity which remains positive and finite as P~Pc. Roughly speaking, ~(p)a(p) is the minimum surface energy of a &quot;percolation droplet&quot; of unit area. 2. For all supercritical densities p> Pc, the system obeys a microscopic Wulff construction: Namely, if the origin is conditioned to be in a finite cluster of size N, then with probability tending rapidly to 1 with N, the shape of this clustermeasured on the scale ~/Nwill be that predicted by the classical Wulffconstruction. Alternatively, ifa system of finite volume, N, is restricted to a &quot;microcanonical ensemble &quot; in which the infinite cluster density is below its usual value, then with probability tending rapidly to 1 with N, the excess sites in finite clusters will form a single large droplet, whichagain on the scale v/Nwill assume the classical Wulff shape.
The scaling limit of the incipient infinite cluster in highdimensional percolation. II. Integrated superBrownian excursion
, 1999
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Wextended fusion algebra of critical percolation
 J. Phys. A: Math. Theor
"... We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasirational in ..."
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Cited by 37 (13 self)
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We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasirational in the sense that the fusion of a finite number of these representations decomposes into a finite direct sum of these representations. The fusion rules are commutative, associative and exhibit an sℓ(2) structure. They involve representations which we call Kac representations of which some are reducible yet indecomposable representations of rank 1. In particular, the identity of the fusion algebra is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the recent results of EberleFlohr and ReadSaleur. Notably, in agreement with EberleFlohr, we find the appearance of indecomposable representations of rank 3. Our fusion rules are supported by extensive numerical studies of an integrable lattice model of critical percolation. Details of our lattice findings and numerical results will be presented elsewhere.