long route to consensus two stage coarsening in a binary choice voting model formation of consensus in binary yes no type of voting is a well defined process however even in presence of clear incentives the dynamics involved can be incredibly complex specifically formations of large groups of similarly opinionated individuals could create a condition of support bubbles or spontaneous polarization that renders consensus virtually unattainable e g the question of the uk exiting the eu there have been earlier attempts in capturing the dynamics of consensus formation in societies through simple z 2 symmetric models hoping to capture the essential dynamics of average behavior of a large number of individuals in a statistical sense however in absence of external noise they tend to reach a frozen state with fragmented and polarized states i e two or more groups of similarly opinionated groups with frozen dynamics here we show in a kinetic exchange opinion model considered on l l square lattices that while such frozen states could be avoided an exponentially slow approach to consensus is manifested specifically the system could either reach consensus in a time that scales as l 2 or a long lived metastable state termed a domain wall state for which formation of consensus takes a time scaling as l 3 6 the latter behavior is comparable to some voterlike models with intermediate states studied previously the late time anomaly in the timescale is reflected in the persistence probability of the model finally the interval of zero crossing of the average opinion i e the time interval over which the average opinion does not change sign is shown to follow a scale free distribution which is compared with that seen in the opinion surveys regarding brexit and associated issues since the late 1970s the issue of minority spreading is also addressed by calculating the exit probability

rare region effects in the contact process on networks networks and dynamical processes occurring on them have become a paradigmatic representation of complex systems studying the role of quenched disorder both intrinsic to nodes and topological is a key challenge with this in mind here we analyze the contact process i e the simplest model for propagation phenomena with node dependent infection rates i e intrinsic quenched disorder on complex networks we find griffiths phases and other rare region effects leading rather generically to anomalously slow algebraic logarithmic etc relaxation on erd ifmmode mbox h o else h o fi s r enyi networks we predict similar effects to exist for other topologies as long as a nonvanishing percolation threshold exists more strikingly we find that griffiths phases can also emerge char22 even with constant epidemic rates char22 as a consequence of mere topological heterogeneity in particular we find griffiths phases in finite dimensional networks as for instance a family of generalized small world networks these results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks and are relevant for the analysis of both models and empirical data

complex networks effect of subtle changes in nature of randomness abstract in two different classes of network models namely the watts strogatz type and the euclidean type subtle changes have been introduced in the randomness in the watts strogatz type network rewiring has been done in different ways and although the qualitative results remain the same finite differences in the exponents are observed in the euclidean type networks where at least one finite phase transition occurs two models differing in a similar way have been considered the results show a possible shift in one of the phase transition points but no change in the values of the exponents the ws and euclidean type models are equivalent for extreme values of the parameters we compare their behaviour for intermediate values

computing a shape s moments from its boundary abstract moments have been among the most commonly used tools in the analysis of 2d shapes important information about a shape such as its size center location and orientation are all moment based attributes there is also a number of very useful shape features such as maximum and minimum moments of inertia moment invariants shape spreadness and shape elongation which are also derived from moments traditionally moments of a shape are computed from the occupancy array representation of the shape that is all the shape s pixels are involved in the computation in this paper we propose a method which computes the same set of moments using only the corner pixels along a shape s boundary the basic approach is to construct a set of triangles using the shape s corners and the origin of the coordinate system the moments of these triangles are computed first the moments of the shape are then derived from the triangles moments we have found the suggested method to be much more efficient than the traditional occupancy array based methods

optimized 3d watermarking for minimal surface distortion this paper proposes a new approach to 3d watermarking by ensuring the optimal preservation of mesh surfaces a new 3d surface preservation function metric is defined consisting of the distance of a vertex displaced by watermarking to the original surface to the watermarked object surface as well as the actual vertex displacement the proposed method is statistical blind and robust minimal surface distortion according to the proposed function metric is enforced during the statistical watermark embedding stage using levenberg marquardt optimization method a study of the watermark code crypto security is provided for the proposed methodology according to the experimental results the proposed methodology has high robustness against the common mesh attacks while preserving the original object surface during watermarking

3d symmetry detection using the extended gaussian image symmetry detection is important in the area of computer vision a 3d symmetry detection algorithm is presented in this paper the symmetry detection problem is converted to the correlation of the gaussian image once the gaussian image of the object has been obtained the algorithm is independent of the input format the algorithm can handle different kinds of images or objects simulated and real images have been tested in a variety of formats and the results show that the symmetry can be determined using the gaussian image

epidemics and percolation in small world networks we study some simple models of disease transmission on small world networks in which either the probability of infection by a disease or the probability of its transmission is varied or both the resulting models display epidemic behavior when the infection or transmission probability rises above the threshold for site or bond percolation on the network and we give exact solutions for the position of this threshold in a variety of cases we confirm our analytic results by numerical simulation

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

shortest paths on systems with power law distributed long range connections we discuss shortest path lengths l r on periodic rings of size l supplemented with an average of pl randomly located long range links whose lengths are distributed according to p l approximately l mu using rescaling arguments and numerical simulation on systems of up to 10 7 sites we show that a characteristic length xi exists such that l r approximately r for r xi for small p we find that the shortest path length satisfies the scaling relation l r mu p xi f mu r xi three regions with different asymptotic behaviors are found respectively a mu 2 where theta s 1 b 1 mu 2 where 0 theta s mu 1 2 and c mu 1 where l r behaves logarithmically i e theta s 0 the characteristic length xi is of the form xi approximately p nu with nu 1 2 mu in region b but depends on l as well in region c a directed model of shortest paths is solved and compared with numerical results

critical behavior of propagation on small world networks we report numerical evidence that an epidemiclike model which can be interpreted as the propagation of a rumor exhibits critical behavior at a finite randomness of the underlying small world network the transition occurs between a regime where the rumor dies in a small neighborhood of its origin and a regime where it spreads over a finite fraction of the whole population critical exponents are evaluated through finite size scaling analysis and the dependence of the critical randomness with the network connectivity is studied the behavior of this system as a function of the small network randomness bears noticeable similarities with an epidemiological model reported recently m kuperman and g abramson phys rev lett 86 2909 2001 in spite of substantial differences in the respective dynamical rules

a general approach to moment calculation for polygons and line segments abstract the calculation of moments from the boundary of a solid digital area is a quite useful technique used in pattern recognition a straightforward way of doing the calculation that does not have any numerical instabilities is presented all the symmetries available in moment calculations are preserved and it is shown how different moment formulas are related to each other this includes relations between different moments and the relationships among different methods for calculating moments finally a direct relation between the moment expansion of a polygonal area and a stick figure made from one dimensional lines embedded in a two dimensional plane is derived

existence results for brezis nirenberg problems with hardy potential and singular coefficients abstract in this paper we consider the existence of non trivial solutions to semi linear brezis nirenberg type problems with hardy potential and singular coefficients first we shall study the corresponding eigenvalue problem and obtain some basic properties of eigenvalues and asymptotic estimates of the eigenfunctions and approximating eigenfunctions secondly we consider the extremal functions of the best embedding constant and get some crucial estimates for the cut off function of the extremal functions thirdly applying different variational theorems for distinct cases of those parameters appearing in the equation we obtain two existence results for non trivial solutions to semi linear brezis nirenberg type problems our existence results are divided into non resonant and resonant cases

simple and fast computation of moments abstract in this paper we address the problem of efficient computation of moments from the boundary of a digital area boundary based computation is superior to usual region based approaches as the data dimension of boundary representations is substantially smaller than that of region representations we investigate the inter order relationship of moments one of our results is that moments of higher order can be computed from moments of lower order based on this relationship a simple iterative algorithm is proposed for the computation of moments from a polygonal approximation of the boundary in comparison with a direct computation method our algorithm is simpler to program the memory requirement is minimum simulation results show that a speed up of factor 8 can be achieved using our algorithm a special version of the algorithm can be utilized to compute moments from the run length chain code of the boundary our algorithm can be applied to compute the most popular geometric moments as well as other types of moments like legendre zernike rotational and complex moments

zero temperature dynamics for the ferromagnetic ising model on random graphs abstract we consider glauber dynamics at zero temperature for the ferromagnetic ising model on the usual random graph model on n vertices with on average edges incident to each vertex in the limit as n based on numerical simulations svenson phys rev e 64 2001 036122 reported that the dynamics fails to reach a global energy minimum for a range of values of the present paper provides a mathematically rigorous proof that this failure to find the global minimum in fact happens for all 0 a lower bound on the residual energy is also given

navigation in a small world with local information it is commonly known that there exist short paths between vertices in a network showing the small world effect yet vertices for example the individuals living in society usually are not able to find the shortest paths due to the very serious limit of information to study this issue theoretically here the navigation process of launching messages toward designated targets is investigated on a variant of the one dimensional small world network swn in the network structure considered the probability of a shortcut falling between a pair of nodes is proportional to r alpha where r is the lattice distance between the nodes when alpha 0 it reduces to the swn model with random shortcuts the system shows the dynamic small world effect which is different from the well studied static sw effect we study the effective network diameter the path length as a function of the lattice distance and the dynamics they are controlled by multiple parameters and we use data collapse to show that the parameters are correlated the central finding is that in the one dimensional network studied the dynamic sw effect exists for 0 alpha 2 for each given value of alpha in this region the point where the dynamic sw effect arises is m l approximately 1 where m is the number of useful shortcuts and l is their average reduced effective length

effect of the nature of randomness on quenching dynamics of the ising model on complex networks randomness is known to affect the dynamical behavior of many systems to a large extent in this paper we investigate how the nature of randomness affects the dynamics in a zero temperature quench of the ising model on two types of random networks in both networks which are embedded in a one dimensional space the first neighbor connections exist and the average degree is 4 per node in random model a the second neighbor connections are rewired with a probability p while in random model b additional connections between neighbors at a euclidean distance l l 1 are introduced with a probability p l proportionally l we find that for both models the dynamics leads to freezing such that the system gets locked in a disordered state the point at which the disorder of the nonequilibrium steady state is maximum is located the behavior of dynamical quantities such as residual energy order parameter and persistence are discussed and compared overall the behavior of physical quantities are similar although subtle differences are observed due to the difference in the nature of randomness

phase transition of a one dimensional ising model with distance dependent connections the critical behavior of the ising model on a one dimensional network which has long range connections at distances l 1 with the probability theta l approximately l m is studied by using monte carlo simulations through analyzing the ising model on networks with different m values this paper discusses the impact of the global correlation which decays with the increase of m on the phase transition of the ising model adding the analysis of the finite size scaling of the order parameter m it is observed that in the whole range of 0 m 2 a finite temperature transition exists and the critical exponents show consistence with mean field values which indicates a mean field nature of the phase transition

stochastic bifurcations in the nonlinear parallel ising model we investigate the phase transitions of a nonlinear parallel version of the ising model characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one this model arises in problems of opinion formation the mean field approximation shows chaotic oscillations by changing the couplings or the connectivity the spatial model shows bifurcations in the average magnetization similar to that seen in the mean field approximation induced by the change of the topology after rewiring short range to long range connection as predicted by the small world effect these coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins induced by long range couplings similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution degree of asynchronism of the updating we also examined the effects of inhomogeneity mixing ferromagnetic and antiferromagnetic coupling which induces an unexpected bifurcation diagram with a bubbling behavior as also happens for dilution

surface preserving robust watermarking of 3 d shapes this paper describes a new statistical approach for watermarking mesh representations of 3 d graphical objects a robust digital watermarking method has to mitigate among the requirements of watermark invisibility robustness embedding capacity and key security the proposed method employs a mesh propagation distance metric procedure called the fast marching method fmm which defines regions of equal geodesic distance width calculated with respect to a reference location on the mesh each of these regions is used for embedding a single bit the embedding is performed by changing the normalized distribution of local geodesic distances from within each region two different embedding methods are used by changing the mean or the variance of geodesic distance distributions geodesic distances are slightly modified statistically by displacing the vertices in their existing triangle planes the vertex displacements performed according to the fmm ensure a minimal surface distortion while embedding the watermark code robustness to a variety of attacks is shown according to experimental results

critical growth problems for polyharmonic operators we prove that critical growth problems for polyharmonic operators admit nontrivial solutions for a wide class of lower order perturbations of the critical term the results highlight the phenomenon of bifurcation of the critical dimensions discovered by pucci and serrin moreover we show that another bifurcation seems to appear for nonresonant dimensions

small world phenomena and the statistics of linear polymers a regular lattice in which the sites can have long range connections at a distance l with a probabilty p l l in addition to the short range nearest neighbour connections shows small world behaviour for 0 c in the most appropriate physical example of such a system namely the linear polymer network the exponent is related to the exponents of the corresponding n vector model in the n 0 limit and its value is less than c still the polymer networks do not show small world behaviour here we show that this is due to a small value constraint on the number q of long range connections per monomer in the network in the general q space we obtain a phase boundary separating regions with and without small world behaviour and show that the polymer network falls marginally in the regular lattice region

computation of volume and surface body moments abstract the paper presents explicit formulae for calculation of surface moments for polytopes in r n we describe also efficient algorithms for calculation of 3d body volume and surface moments the algorithms are based on explicit formulae for moment calculation and take advantages of shape polygonal representation they use only coordinates of the body vertices and faces orientation

ising model in small world networks the ising model in small world networks generated from two and three dimensional regular lattices has been studied monte carlo simulations were carried out to characterize the ferromagnetic transition appearing in these systems in the thermodynamic limit the phase transition has a mean field character for any finite value of the rewiring probability p which measures the disorder strength of a given network for small values of p both the transition temperature and critical energy change with p as a power law in the limit stackrel ensuremath rightarrow p 0 the heat capacity at the transition temperature diverges logarithmically in two dimensional 2d networks and as a power law in 3d

the moment calculation of polyhedra abstract this paper proposes a novel approach to calculate polyhedra moments at first the gaussian theorem is used to convert a volume integral into a surface one by which we decrease the moment computational complexity from o n 3 to o n 2 then an iterative method is proposed to calculate the surface integral finally some comparison shows that the computational complexity of the new method is independent of voxel numbers of polyhedra therefore the new moment computation is more efficient

critical semilinear biharmonic equations in rn an existence theorem is obtained for a fourth order semilinear elliptic problem in r n involving the critical sobolev exponent n 4 n 4 n 4 a preliminary result is that the best constant in the sobolev embedding l 2n n 4 r n is attained by all translations and dilations of 1 x 2 4 n 2 the best constant is found to be

geography in a scale free network model we offer an example of a network model with a power law degree distribution p k approximately k alpha for nodes but which nevertheless has a well defined geography and a nonzero threshold percolation probability for alpha 2 the range of real world contact networks this is different from p c 0 for alpha 3 results for the original well mixed scale free networks in our lattice based scale free network individuals link to nearby neighbors on a lattice even considerable additional small world links do not change our conclusion of nonzero thresholds when applied to disease propagation these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering

polyharmonic kirchhoff problems involving exponential non linearity of choquard type with singular weights abstract in this work we study the higher order kirchhoff type choquard equation k c involving a critical exponential non linearity and singular weights we prove the existence of solution to k c using mountain pass lemma in light of moser trudinger and singular adams moser inequalities in the second part of the paper using the nehari manifold technique and minimization over its suitable subsets we prove the existence of at least two solutions to the kirchhoff type choquard equation p m involving convex concave type non linearity

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

disorder induced phase transition in kinetic models of opinion dynamics abstract we propose a model of continuous opinion dynamics where mutual interactions can be both positive and negative different types of distributions for the interactions all characterized by a single parameter p denoting the fraction of negative interactions are considered results from exact calculation of a discrete version and numerical simulations of the continuous version of the model indicate the existence of a universal continuous phase transition at p p c below which a consensus is reached although the order disorder transition is analogous to a ferromagnetic paramagnetic phase transition with comparable critical exponents the model is characterized by some distinctive features relevant to a social system

on the quasilinear elliptic problem with a critical hardy sobolev exponent and a hardy term abstract in the present paper a quasilinear elliptic problem with a critical sobolev exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions for the problem is obtained the result depends crucially on the parameters p t s and

epidemics disorder and percolation abstract spatial models for spread of an epidemic may be mapped onto bond percolation we point out that with disorder in the strength of contacts between individuals patchiness in the spread of the epidemic is very likely and the criterion for epidemic outbreak depends strongly on the disorder because the critical region of the corresponding percolation model is broadened in some networks the percolation threshold is zero if another kind of disorder is present namely divergent fluctuations in the number of contacts we give an example a network with a well defined geography where this is not necessarily so and discuss whether real infection networks are likely to have this property

infinitely many solutions for some nonlinear elliptic problems in symmetrical domains on considere le probleme aux valeurs limites u u u u 2 2 0 u h 0 1 ou r n est un domaine borne n 3 2 2n n 2 est l exposant critique pour le plongement de sobolev et est un parametre positif reel on demontre l existence d une infinite de solutions quand a de bonnes symetries

multiple solutions for an elliptic system on bounded and unbounded domains abstract in this paper we present some results of multiplicity for the following system u a x u b x v h u u v in v b x u c x v h v u v in u v d o 1 2 where may be either a bounded domain or the whole r n the nonlinearity h is a 2 homogeneous function and hu hv h

the susceptible infected recovered model on a euclidean network we consider a susceptible infected recovered sir epidemic model on a euclidean network in one dimension in which nodes at a distance l are connected with probability p l l in addition to nearest neighbors the topology of the network changes as is varied and its effect on the sir model is studied r t the recovered fraction of population up to time t and the total duration of the epidemic are calculated for different values of the infection probabilities q and a threshold behavior is observed for all up to 2 0 above the threshold value q qc the saturation value rsat attains a finite value both rsat and show scaling behavior in a finite system of size n and qc is constant for 0 qc the results are compared to static percolation phenomena and also to mean field results for finite systems discussions on the properties of the euclidean network are made in the light of the present results

solutions of elliptic equations with critical sobolev exponent in dimension three theorem 1 for every a 72 4 there exists a solution of i the previously known results concerning this problem are the following 1 if 1 5 0 i has no solution this follows from pohozaev s identity see lo and from the principle of unique continuation when a 0 we will suppose in the following that i 0 2 every positive solution of i is radial see gidas et al 9 3 let a be the first eigenvalue of a in b with dirichlet boundary condition and p1 the associated eigenfunction then for every a 2 i i has no positive solution this result follows from multiplying i by cp 4 for every a e n2 4 rr there exists a positive solution of i see brezis and nirenberg 5 5 for i 5 n2 4 i has no radial solution see 5 6 for every k e n if 1 e zk with zk k t n k2n2 i has a radial solution with k 1 zeros see atkinson and peletier 2 31 7 the analogue of theorem 1 in dimension n 2 4 holds for general domains see capozzi et al 6 and ambrosetti and struwe l our method is very different and we do not know whether theorem 1 is valid for general domains in ir3

small world networks links with long tailed distributions small world networks swn obtained by randomly adding to a regular structure additional links al are of current interest in this paper we explore based on physical models a new variant of swn in which the probability of realizing an al depends on the chemical distance between the connected sites we assume a power law probability distribution and study random walkers on the network focusing especially on their probability of being at the origin we connect the results to l evy flights which follow from a mean field variant of our model

phase transitions in a network with a range dependent connection probability we consider a one dimensional network in which the nodes at euclidean distance l can have long range connections with a probability p l approximately l delta in addition to nearest neighbor connections this system has been shown to exhibit small world behavior for delta 2 above which its behavior is like a regular lattice from the study of the clustering coefficients we show that there is a transition to a random network at delta 1 the finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicates 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 behavior of a single quantity the average bond length the phase transitions in all dimensions are nontrivial in nature

the critical neumann problem of kirchhoff type in this paper we study the critical neumann problem of kirchhoff type by using the variational method and the concentration compactness argument the existence and multiplicity of nontrivial solutions are obtained

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

efficient 3d geometric and zernike moments computation from unstructured surface meshes this paper introduces and evaluates a fast exact algorithm and a series of faster approximate algorithms for the computation of 3d geometric moments from an unstructured surface mesh of triangles being based on the object surface reduces the computational complexity of these algorithms with respect to volumetric grid based algorithms in contrast it can only be applied for the computation of geometric moments of homogeneous objects this advantage and restriction is shared with other proposed algorithms based on the object boundary the proposed exact algorithm reduces the computational complexity for computing geometric moments up to order n with respect to previously proposed exact algorithms from n9 to n6 the approximate series algorithm appears as a power series on the rate between triangle size and object size which can be truncated at any desired degree the higher the number and quality of the triangles the better the approximation this approximate algorithm reduces the computational complexity to n3 in addition the paper introduces a fast algorithm for the computation of 3d zernike moments from the computed geometric moments with a computational complexity n4 while the previously proposed algorithm is of order n6 the error introduced by the proposed approximate algorithms is evaluated in different shapes and the cost benefit ratio in terms of error and computational time is analyzed for different moment orders

a critical elliptic problem for polyharmonic operators abstract in this paper we study the existence of solutions for a critical elliptic problem for polyharmonic operators we prove the existence result in some general domain by minimizing on some infinite dimensional finsler manifold for some suitable perturbation of the critical nonlinearity when the dimension of domain is larger than critical one for the critical dimensions we prove also the existence of solutions in domains perforated with the small holes some unstable solutions are obtained at higher level sets by coron s topological method provided that the minimizing solution does not exist

fast and exact computation of cartesian geometric moments using discrete green s theorem abstract green s theorem evaluates a double integral over a planar object by a simple integration along the boundary of the object it can be used in fast computation of cartesian geometric moments of two dimensional binary objects since the object shape is totally determined by the boundary li and shen pattern recognition 24 807 813 1991 used an approximation of green s theorem to deal with discrete objects this method is fast but not accurate in this paper we present a discrete version of green s theorem which evaluates a double sum over a two dimensional discrete object by a simple summation along the discrete boundary of the object by using the discrete green s theorem we propose a new algorithm for fast computation of the geometric moments the new algorithm is faster than the previous methods and gives exactly the same results as if a double sum were used the importance of the exact computation is discussed by examining the accuracy of hu s moment invariants and some other moment based shape features which are frequently used in various shape description and pattern recognition tasks a fast method for computing moments of regions in gray level images using the discrete green s theorem is also presented we show that the moments of this type are useful in structural texture analysis

size of outbreaks near the epidemic threshold the spread of infectious diseases near the epidemic threshold is investigated scaling laws for the size and the duration of outbreaks originating from a single infected individual in a large susceptible population are obtained the maximal size of an outbreak n ensuremath ast scales as n 2 3 with n the population size this scaling law implies that the average outbreak size n scales as n 1 3 moreover the maximal and the average duration of an outbreak grow as t ensuremath ast ensuremath sim n 1 3 and t ensuremath sim text ln phantom rule 0 2em 0ex n respectively