how community structure influences epidemic spread in social networks abstract two key features of social networks are the community structure and the high clustering coefficient for understanding their influences on dynamical processes we present a model with both an adjustable clustering coefficient and an adjustable degree of community this model has an invariant degree distribution when its clustering coefficient is being adjusted we find that the efficiency of epidemic spreading in this model depends mainly on the degree of community and decreases with increase of the degree of community for a fixed degree of community the efficiency will decrease with increase of the clustering coefficient numerical simulations have confirmed the theoretic analysis

a limiting result for the structure of collisions between many independent diffusions we study a system of a large number n of independent diffusions undergoing collisions in a roughly constant mean free path regime we derive a propagation of chaos result when n goes to infinity concerning the limit behavior of the trajectory of a particle its first time of collision with all the other particles and the trajectory of the colliding particle

growing scale free networks with tunable clustering we extend the standard scale free network model to include a triad formation step we analyze the geometric properties of networks generated by this algorithm both analytically and by numerical c

simulation of stopped diffusions in this work we study standard euler updates for simulating stopped diffusions as an immediate application we discuss the computation of first exit times of diffusions from a domain we focus on one dimensional situations and show how the ideas for the simulation of killed diffusions can be adapted to this problem in particular we give a fully implementable algorithm to compute the first exit time from an interval numerically the brownian motion case is treated in detail and extensions to general diffusions are given special emphasis is given to numerical experiments for every ansatz we include numerical experiments confirming the conjectured accuracy of our methods our algorithm is of order one in a weak sense comparisons with other algorithms are shown results that are superior to those obtained with other methods are presented when approximating a first hitting time distribution the results obtained with our algorithm are much better than those achieved with other methods

scale free networks from self organization we show how scale free degree distributions can emerge naturally from growing networks by using random walks for selecting vertices for attachment this result holds for several variants of the walk algorithm and for a wide range of parameters the growth mechanism is based on using local graph information only so this is a process of self organisation the standard mean field equations are an excellent approximation for network growth using these rules we discuss the effects of finite size on the degree distribution and compare analytical results to simulated networks finally we generalise the random walk algorithm to produce weighted networks with power law distributions of both weight and degree

finiteness and fluctuations in growing networks we study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks we first determine the exact degree distribution of finite networks by generating function approaches the resulting distributions exhibit an unusual finite size scaling behaviour and they are also sensitive to the initial conditions we argue that fluctuations in the number of nodes of degree k become gaussian for fixed degree as the size of the network diverges we also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution

strong matching preclusion number of graphs abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings the strong matching preclusion number or simply smp number s m p g of a graph g is the minimum number of vertices and or edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings this is an extension of the matching preclusion problem and has been introduced by park and ihm in this paper we first study the smp number of some special graph classes and give some sharp upper and lower bounds of smp number next graphs with large and small smp number are characterized respectively in the end we investigate the nordhaus gaddum type relations on smp number

dynamics of directed graphs the world wide web we introduce and simulate a growth model of the world wide web based on the dynamics of outgoing links that is motivated by the conduct of the agents in the real web to update outgoing links re directing them towards constantly changing selected nodes emergent statistical correlation between the distributions of outgoing and incoming links is a key feature of the dynamics of the web the growth phase is characterized by temporal fractal structures which are manifested in the hierarchical organization of links we obtain quantitative agreement with the recent empirical data in the real web for the distributions of in and out links and for the size of the connected component in a fully grown network of n nodes we study the structure of connected clusters of nodes that are accessible along outgoing links from a randomly selected node the distributions of size and depth of the connected clusters with a giant component exhibit supercritical behavior by decreasing the control parameter average fraction of updated and added links per time step towards c n 10 the web can resume a critical structure with no giant component in it we find a different universality class when the updates of links are not allowed i e for 0 corresponding to the network of science citations

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

scale free networks generated by random walkers we present a simple mechanism for generating undirected scale free networks using random walkers where the network growth is determined by choosing parent vertices by sequential random walks we show that this mechanism produces scale free networks with degree exponent 3 and clustering coefficients depending on random walk length the mechanism can be interpreted in terms of preferential attachment without explicit knowledge of node degrees

complex scale free networks with tunable power law exponent and clustering abstract we introduce a network evolution process motivated by the network of citations in the scientific literature in each iteration of the process a node is born and directed links are created from the new node to a set of target nodes already in the network this set includes m ambassador nodes and l of each ambassador s descendants where m and l are random variables selected from any choice of distributions p l and q m the process mimics the tendency of authors to cite varying numbers of papers included in the bibliographies of the other papers they cite we show that the degree distributions of the networks generated after a large number of iterations are scale free and derive an expression for the power law exponent in a particular case of the model where the number of ambassadors is always the constant m and the number of selected descendants from each ambassador is the constant l the power law exponent is 2 l 1 l for this example we derive expressions for the degree distribution and clustering coefficient in terms of l and m we conclude that the proposed model can be tuned to have the same power law exponent and clustering coefficient of a broad range of the scale free distributions that have been studied empirically

probabilistically induced domain decomposition methods for elliptic boundary value problems monte carlo as well as quasi monte carlo methods are used to generate only few interfacial values in two dimensional domains where boundary value elliptic problems are formulated this allows for a domain decomposition of the domain a continuous approximation of the solution is obtained interpolating on such interfaces and then used as boundary data to split the original problem into fully decoupled subproblems the numerical treatment can then be continued implementing any deterministic algorithm on each subdomain both monte carlo or quasi monte carlo simulations and the domain decomposition strategy allow for exploiting parallel architectures scalability and natural fault tolerance are peculiarities of the present algorithm examples concern helmholtz and poisson equations whose probabilistic treatment presents additional complications with respect to the case of homogeneous elliptic problems without any potential term and source

growing network with local rules preferential attachment clustering hierarchy and degree correlations the linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power law degree distributions it is then quite important to determine if this mechanism is the consequence of a general principle based on local rules in this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules it is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks these conclusions are based on both analytical and numerical results for different local rules including some models already proposed in the literature

spatially self organized resilient networks by a distributed cooperative mechanism abstract the robustness of connectivity and the efficiency of paths are incompatible in many real networks we propose a self organization mechanism for incrementally generating onion like networks with positive degree degree correlations whose robustness is nearly optimal as a spatial extension of the generation model based on cooperative copying and adding shortcut we show that the growing networks become more robust and efficient through enhancing the onion like topological structure on a space the reasonable constraint for locating nodes on the perimeter in typical surface growth as a self propagation does not affect these properties of the tolerance and the path length moreover the robustness can be recovered in the random growth damaged by insistent sequential attacks even without any remedial measures

efficient numerical solution of stochastic differential equations using exponential timestepping we present an exact timestepping method for brownian motion that does not require gaussian random variables to be generated time is incremented in steps that are exponentially distributed random variables boundaries can be explicitly accounted for at each timestep the method is illustrated by numerical solution of a system of diffusing particles

distribution characteristics of weighted bipartite evolving networks abstract motivated by an evolving model of online bipartite networks we introduce a model of weighted bipartite evolving networks in this model there are two disjoint sets of nodes called user node set and object node set edges only exist between two disjoint sets edge weights represent the usage amount between a couple of user node and object node this model not only clinches the bipartite networks internal mechanism of network growth but also takes into account the object strength deterioration over time step user strength and object strength follow power law distributions respectively the weighted bipartite evolving networks have scare free property in certain situations numerical simulations results agree with the theoretical analyses

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

multidimensional exponential timestepping with boundary test exponential timestepping algorithms are efficient for exit time problems because a boundary test can be performed at the end of each timestep giving high order convergence in numerical evaluation of mean exit times successive time increments are independent random variables with an exponential distribution we show how to perform exact timestepping for brownian motion in more than one dimension and consider hitting times of curved surfaces taking as examples the exit times from a circle from a ball from an ellipse and from the region bounded by the two lines of a hyperbola we report the results of numerical experiments performed with the algorithms developed in this work

exponential timestepping with boundary test for stochastic differential equations we present new numerical methods for scalar stochastic differential equations successive time increments are independent random variables with an exponential distribution we perform numerical experiments using a double well potential exponential timestepping algorithms are efficient for escape time problems because a simple boundary test can be performed at the end of each step

conditional matching preclusion for hypercube like interconnection networks the conditional matching preclusion number of a graph with n vertices is the minimum number of edges whose deletion results in a graph without an isolated vertex that does not have a perfect matching if n is even or an almost perfect matching if n is odd we develop some general properties on conditional matching preclusion and then analyze the conditional matching preclusion numbers for some hl graphs hypercube like interconnection networks

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

measuring the preferential attachment mechanism in citation networks abstract in this paper we investigated the preferential attachment mechanism pam by considering the dynamic property in papers in degree k for three citation networks we found that the past citations obtained in different years will have different influences on papers attachment rate k t we proposed two methods to consider these different influences one is the gradually vanishing memory preferential attachment mechanism gmpam based on weighted past citations the other is the short term memory preferential attachment mechanism smpam based on citations obtained in the recent one year period experiments showed that smpam is simpler and more universal in practice we can just calculate the citations to papers in the recent one year period to study the papers attachment property

emergence of the small world architecture in neural networks by activity dependent growth abstract in this paper we propose a model describing the growth and development of neural networks based on the latest achievements of experimental neuroscience the model is based on two evolutionary equations the first equation is for the evolution of the neurons state and the second is for the growth of axon tips by using the model we demonstrated the neuronal growth process from disconnected neurons to fully connected three dimensional networks for the analysis of the network s connections structure we used the random graphs theory methods it is shown that the growth in neural networks results in the formation of a well known small world network model the analysis of the connectivity distribution shows the presence of a strictly non gaussian but no scale free degree distribution for the in degree node distribution in terms of the graphs theory this study developed a new model of dynamic graph

simulating diffusion processes in discontinuous media in this article we propose new monte carlo techniques for moving a diffusive particle in a discontinuous media in this framework we characterize the stochastic process that governs the positions of the particle the key tool is the reduction of the process to a skew brownian motion sbm in a zone where the coefficients are locally constant on each side of the discontinuity the new position of the particle after a constant time step is sampled from the exact distribution of the sbm process at the considered time to do so we propose two different but equivalent algorithms a two steps simulation with a stop at the discontinuity and a one step direct simulation of the sbm dynamic some benchmark tests illustrate their effectiveness

exact asymptotics for the probability of exit from a domain and applications to simulation we study the asymptotics of the exit probability p x s e t where is the exit time from an open set and p x s e is the law of a diffusion process with a small parameter e multiplying the diffusion coefficient we consider the case of the brownian bridge in many dimensions this choice being motivated by applications to numerical simulation the method uses recent results reducing the problem to the solution of a system of linear first order pde s

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge