phase transitions in dynamical random graphs we study a large time limit of a markov process whose states are finite graphs the number of the vertices is described by a supercritical branching process and the dynamics of edges is determined by the rates of appending and deleting we find a phase transition in our model similar to the one in the random graph model g n p we derive a formula for the line of critical parameters which separates two different phases one is where the size of the largest component is proportional to the size of the entire graph and another one where the size of the largest component is at most logarithmic with respect to the size of the entire graph in the supercritical phase we find the asymptotics for the size of the largest component

maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

seed size strongly affects cascades on random networks the average avalanche size in the watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance the dependence of cascade size upon the mean degree z of the network is known to exhibit several transitions these are typically continuous at low z and discontinuous at high z here it is demonstrated that the low z transition may in fact be discontinuous in certain parameter regimes connections between these results and the zero temperature random field ising model on random graphs are discussed

clustering in complex networks i general formalism we develop a full theoretical approach to clustering in complex networks a key concept is introduced the edge multiplicity that measures the number of triangles passing through an edge this quantity extends the clustering coefficient in that it involves the properties of two and not just one vertices the formalism is completed with the definition of a three vertex correlation function which is the fundamental quantity describing the properties of clustered networks the formalism suggests different metrics that are able to thoroughly characterize transitive relations a rigorous analysis of several real networks which makes use of this formalism and the metrics is also provided it is also found that clustered networks can be classified into two main groups the weak and the strong transitivity classes in the first class edge multiplicity is small with triangles being disjoint in the second class edge multiplicity is high and so triangles share many edges as we shall see in the following paper the class a network belongs to has strong implications in its percolation properties

percolation critical exponents in scale free networks we study the behavior of scale free networks having connectivity distribution p k k 2l close to the percolation threshold we show that for networks with 3 l 4 known to undergo a transition at a finite threshold of dilution the critical exponents are different than the expected mean field values of regular percolation in infinite dimensions networks with 2 l 3 possess only a percolative phase nevertheless we show that in this case percolation critical exponents are well defined near the limit of extreme dilution where all sites are removed and that also then the exponents bear a strong l dependence the regular mean field values are recovered only for l 4

random graphs with hidden color we propose and investigate a unifying class of sparse random graph models based on a hidden coloring of edge vertex incidences extending an existing approach random graphs with a given degree distribution in a way that admits a nontrivial correlation structure in the resulting graphs the approach unifies a number of existing random graph ensembles within a common general formalism and allows for the analytic calculation of observable graph characteristics in particular generating function techniques are used to derive the size distribution of connected components clusters as well as the location of the percolation threshold where a giant component appears

percolation and epidemics in random clustered networks the social networks that infectious diseases spread along are typically clustered because of the close relation between percolation and epidemic spread the behavior of percolation in such networks gives insight into infectious disease dynamics a number of authors have studied percolation or epidemics in clustered networks but the networks often contain preferential contacts in high degree nodes we introduce a class of random clustered networks and a class of random unclustered networks with the same preferential mixing percolation in the clustered networks reduces the component sizes and increases the epidemic threshold compared to the unclustered networks

multiple percolation transitions in a configuration model of a network of networks recently much attention has been paid to the study of the robustness of interdependent and multiplex networks and in particular the networks of networks the robustness of interdependent networks can be evaluated by the size of a mutually connected component when a fraction of nodes have been removed from these networks here we characterize the emergence of the mutually connected component in a network of networks in which every node of a network layer is connected with q its randomly chosen replicas in some other networks and is interdependent of these nodes with probability r we find that when the superdegrees q of different layers in a network of networks are distributed heterogeneously multiple percolation phase transition can occur we show that depending on the value of r these transition are continuous or discontinuous

cascades on clique based graphs we present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of highly clustered random graphs introduced by gleeson j p gleeson phys rev e 80 036107 2009 a condition for the existence of global cascades is also derived applications of this approach include analyses of percolation and watts s model we show how our techniques can be used to study the effects of in group bias in cascades on social networks

connectivity degrees in the threshold preferential attachment model abstract in this paper we present a study of the connectivity degrees of the threshold preferential attachment model a generalization of the barabasi albert model to heterogeneous complex networks the threshold model incorporates the states of the nodes in its preferential linking rule and assumes that the affinity between network nodes follows an inverse relationship with the distance between their states we numerically analyze the connectivity degrees of the model studying the influence of the main parameters on the distribution of connectivity degrees and its statistics the average degree and highest degree of the network we show that such statistics exhibit markedly different behaviors in the dependence on the model parameters particularly as regards the interaction threshold nevertheless we show that the two statistics converge in the limit of null threshold and often exhibit scaling that can be described by power laws of the model parameters

on the behavior over long time intervals of finite difference and finite element approximations to first order hyperbolic systems abstract standard error estimates for approximations to hyperbolic equations are only valid over a finite time interval this paper considers the behavior of the error in the two most common discretization schemes over the entire interval 0 t finite difference approximations are analyzed for approximating the cauchy problem for hyperbolic systems it is recalled that if the system is dissipative error estimates that are global in t can be obtained for strictly conservative systems the role of local energy decay is pointed out and used to obtain global in t local in x error estimates finite element methods are considered for nonlinear initial boundary value problems for hyperbolic systems it is proven that if the nonlinear term is locally lipschitz and the equation is dissipative when linearized about the true solution then global in t convergence follows with the expected rates

general formalism for inhomogeneous random graphs we present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models where vertices come in different types and the probability of realizing an edge depends on the types of its terminal vertices this approach provides a general framework for the analysis of a large class of models the generic phase structure is derived using generating function techniques and relations to other classes of models are pointed out

the maximum likelihood prior consider an estimate theta of a parameter theta based on repeated observations from a family of densities f theta evaluated by the kullback leibler loss function k theta theta int log f theta f theta f theta the maximum likelihood prior density if it exists is the density for which the corresponding bayes estimate is asymptotically negligibly different from the maximum likelihood estimate the bayes estimate corresponding to the maximum likelihood prior is identical to maximum likelihood for exponential families of densities in predicting the next observation the maximum likelihood prior produces a predictive distribution that is asymptotically at least as close in expected truncated kullback leibler distance to the true density as the density indexed by the maximum likelihood estimate it frequently happens in more than one dimension that maximum likelihood corresponds to no prior density and in that case the maximum likelihood estimate is asymptotically inadmissible and may be improved upon by using the estimate corresponding to a least favorable prior as in brown the asymptotic risk for an arbitrary estimate near maximum likelihood is given by an expression involving derivatives of the estimator and of the information matrix admissibility questions for these near ml estimates are determined by the existence of solutions to certain differential equations

a stable finite element method for initial boundary value problems for first order hyperbolic systems a nonstandard finite element method for initial boundary value problems for first order hyperbolic systems in one space dimension with general boundary conditions is analyzed the method can be considered as a generalization of the box scheme we first establish a stability result for the method and then derive several error estimates u x t a x ux x t b x u x t f x t 1 1 o 0 t g kxu l t u x 0 u0 x for x t e i x j 0 1 x 0 where t 0 here aix and 5 x are n x n matrices we assume that aix has n real eigenvalues xx x a2 x a x and that there exist a constant t0 0 and an integer m 0 t0 for all xg furthermore we assume that for each x e there is a nonsingular matrix qix such that

interdependent networks with identical degrees of mutually dependent nodes we study a problem of failure of two interdependent networks in the case of identical degrees of mutually dependent nodes we assume that both networks a and b have the same number of nodes n connected by the bidirectional dependency links establishing a one to one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links i e their degrees coincide this implies that both networks have the same degree distribution p k we call such networks correspondently coupled networks ccns we assume that the nodes in each network are randomly connected we define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes that belong to the mutual giant component remain functional we assume that initially a 1 p fraction of nodes are randomly removed because of an attack or failure and find analytically for an arbitrary p k the fraction of nodes p that belong to the mutual giant component we find that the system undergoes a percolation transition at a certain fraction p p c which is always smaller than p c for randomly coupled networks with the same p k we also find that the system undergoes a first order transition at p c 0 if p k has a finite second moment for the case of scale free networks with 2 0 finally we find that the robustness of ccn increases with the broadness of their degree distribution

cascades on a class of clustered random networks we present an analytical approach to determining the expected cascade size in a broad range of dynamical models on the class of random networks with arbitrary degree distribution and nonzero clustering introduced previously in m e j newman phys rev lett 103 058701 2009 a condition for the existence of global cascades is derived as well as a general criterion that determines whether increasing the level of clustering will increase or decrease the expected cascade size applications examples of which are provided include site percolation bond percolation and watts threshold model in all cases analytical results give excellent agreement with numerical simulations

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

dynamical random graphs with memory we study the large time dynamics of a markov process whose states are finite but unbounded graphs the number of vertices is described by a supercritical branching process and the edges follow a certain mean field dynamics determined by the rates of appending and deleting the older an edge is the lesser is the probability that it is still in the graph the lifetime of any edge is distributed exponentially we call its mean value common for all edges a parameter of memory since it shows for how long the system keeps a particular connection between the vertices in the graph we show that our model provides a bridge between two well known models when the parameter of memory goes to infinity this is a generalized model of random growth and when this parameter is zero i e no memory our model behaves as a random graph thus by introducing a general class of dynamical graphs we have a unified overview on rather different models and the relations between them we find all the critical values of the parameters at which our model exhibits phase transitions and describe the properties of the phase diagram finally we compare and discuss the efficiency of the corresponding networks

random graphs containing arbitrary distributions of subgraphs traditional random graph models of networks generate networks that are locally treelike meaning that all local neighborhoods take the form of trees in this respect such models are highly unrealistic most real networks having strongly nontreelike neighborhoods that contain short loops cliques or other biconnected subgraphs in this paper we propose and analyze a class of random graph models that incorporates general subgraphs allowing for nontreelike neighborhoods while still remaining solvable for many fundamental network properties among other things we give solutions for the size of the giant component the position of the phase transition at which the giant component appears and percolation properties for both site and bond percolation on networks generated by the model

analysis of a threshold model of social contagion on degree correlated networks we analytically determine when a range of abstract social contagion models permit global spreading from a single seed on degree correlated undirected random networks we deduce the expected size of the largest vulnerable component a network s tinderboxlike critical mass as well as the probability that infecting a randomly chosen individual seed will trigger global spreading in the appropriate limits our results naturally reduce to standard ones for models of disease spreading and to the condition for the existence of a giant component recent advances in the distributed infinite seed case allow us to further determine the final size of global spreading events when they occur to provide support for our results we derive exact expressions for key spreading quantities for a simple yet rich family of random networks with bimodal degree distributions

edge union of networks on the same vertex set random network generators such as erd s r nyi watts strogatz and barab si albert models are used as models to study real world networks let g1 v e1 and g2 v e2 be two such networks on the same vertex set v this paper studies the degree distribution and clustering coefficient of the resultant networks g v e1 e2

long paths and cycles in dynamical graphs we study the large time dynamics of a markov process whose states are finite directed graphs the number of the vertices is described by a supercritical branching process and the edges follow a certain mean field dynamics determined by the rates of appending and deleting we find sufficient conditions under which asymptotically a s the order of the largest component is proportional to the order of the graph a lower bound for the length of the longest directed path in the graph is provided as well we derive an explicit formula for the limit as time goes to infinity of the expected number of cycles of a given finite length finally we study the phase diagram

how clustering affects the bond percolation threshold in complex networks the question of how clustering nonzero density of triangles in networks affects their bond percolation threshold has important applications in a variety of disciplines recent advances in modeling highly clustered networks are employed here to analytically study the bond percolation threshold in comparison to the threshold in an unclustered network with the same degree distribution and correlation structure the presence of triangles in these model networks is shown to lead to a larger bond percolation threshold i e clustering increases the epidemic threshold or decreases resilience of the network to random edge deletion

stable and unstable numerical boundary conditions for galerkin approximations to hyperbolic systems abstract the galerkin method for first order hyperbolic systems is considered this method is seen to be unstable in an example because of improper treatment of the boundary conditions in the discrete problem the galerkin method is modified to be stable for any well posed initial boundary value problem in one dimension it is shown that the previous estimates of the error in this modified method can be improved when the system admits an energy conserving norm a system in two space dimensions in a region with a corner is considered it is shown how the construction of stable methods for such problems can be reduced to a very simple linear programming problem

exact solutions for social and biological contagion models on mixed directed and undirected degree correlated random networks we derive analytic expressions for the possibility probability and expected size of global spreading events starting from a single infected seed for a broad collection of contagion processes acting on random networks with both directed and undirected edges and arbitrary degree degree correlations our work extends previous theoretical developments for the undirected case and we provide numerical support for our findings by investigating an example class of networks for which we are able to obtain closed form expressions

dynamic random networks in dynamic populations we consider a random network evolving in continuous time in which new nodes are born and old may die and where undirected edges between nodes are created randomly and may also disappear the node population is markovian and so is the creation and deletion of edges given the node population each node is equipped with a random social index and the intensity at which a node creates new edges is proportional to the social index and the neighbour is either chosen uniformly or proportional to its social index in a modification of the model we derive properties of the network as time and the node population tends to infinity in particular the degree distribution is shown to be a mixed poisson distribution which may exhibit a heavy tail e g power law if the social index distribution has a heavy tail the limiting results are verified by means of simulations and the model is fitted to a network of sexual contacts

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

cascade of failures in coupled network systems with multiple support dependence relations we study both analytically and numerically the cascade of failures in two coupled network systems a and b where multiple support dependence relations are randomly built between nodes of networks a and b in our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network we assume that networks a and b have i sizes n a and n b ii degree distributions of connectivity links p a k and p b k iii degree distributions of support links p a k and p b k and iv random attack removes 1 r a n a and 1 r b n b nodes form the networks a and b respectively we find the fractions of nodes a and b which remain functional giant component at the end of the cascade process in networks a and b in terms of the generating functions of the degree distributions of their connectivity and support links in a special case of erd s renyi networks with average degrees a and b in networks a and b respectively and poisson distributions of support links with average degrees a and b in networks a and b respectively a r a 1 exp a b 1 exp a a and b r b 1 exp b a 1 exp b b in the limit of a and b both networks become independent and our model becomes equivalent to a random attack on a single erd s renyi network we also test our theory on two coupled scale free networks and find good agreement with the simulations

giant strongly connected component of directed networks we describe how to calculate the sizes of all giant connected components of a directed graph including the strongly connected one in particular the world wide web is a directed network the results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in and out degree distribution p k i k o we show that if p k i k o does not factorize the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in and out components the calculations of the relative sizes of all the giant components are demonstrated using the simplest examples we explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one

social contagions on multiplex networks with heterogeneous population abstract in this paper we study the effects of heterogeneous population on the dynamics of social contagions on multiplex networks we assume a fraction of f nodes with a higher adoption threshold t 1 and the remaining fraction of 1 f nodes with adoption threshold 1 a social contagion model is proposed to describe the social contagions in which a susceptible node adopting the contagion only when its received accumulated information is larger than the adoption threshold in either subnetwork with an edge based compartmental approach and extensive numerical simulations we find that the system exhibits a continuous phase transition for small values of f while shows a hybrid phase transition for relatively large values of f and t for homogeneous multiplex networks the hybrid phase transition occurs while there is only a continuous phase transition for heterogeneous multiplex networks our theoretical predictions agree well with numerical simulations

tuning clustering in random networks with arbitrary degree distributions we present a generator of random networks where both the degree dependent clustering coefficient and the degree distribution are tunable following the same philosophy as in the configuration model the degree distribution and the clustering coefficient for each class of nodes of degree k are fixed ad hoc and a priori the algorithm generates corresponding topologies by applying first a closure of triangles and second the classical closure of remaining free stubs the procedure unveils an universal relation among clustering and degree degree correlations for all networks where the level of assortativity establishes an upper limit to the level of clustering maximum assortativity ensures no restriction on the decay of the clustering coefficient whereas disassortativity sets a stronger constraint on its behavior correlation measures in real networks are seen to observe this structural bound

exact minimax estimation of the predictive density in sparse gaussian models we consider estimating the predictive density under kullback leibler loss in an l0 sparse gaussian sequence model explicit expressions of the first order minimax risk along with its exact constant asymptotically least favorable priors and optimal predictive density estimates are derived compared to the sparse recovery results involving point estimation of the normal mean new decision theoretic phenomena are seen suboptimal performance of the class of plug in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk we find that minimax optimal strategies lie outside the gaussian family but can be constructed with threshold predictive density estimates novel minimax techniques involving simultaneous calibration of the sparsity adjustment and the risk diversification mechanisms are used to design optimal predictive density estimates

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

impact of the heterogeneity of adoption thresholds on behavior spreading in complex networks abstract individuals always exhibit distinct personal profiles ranging from education social experiences to risk tolerance which induces a vastly different response in the behavior spreading dynamics to describe this phenomenon a widely used method is to assign a distinct adoption threshold for each individual which so far has eluded theoretical analysis in this paper we propose a behavior spreading model to describe the heterogeneity of individuals in adopting the behavior in which the adoption threshold follows the gaussian distribution based on the edge based compartmental theory the theoretical solutions of the behavior spreading size and the threshold are given for discontinuous and continuous phase transitions through theoretical analysis and extensive numerical simulations the impact of the heterogeneity of adoption thresholds on behavior spreading is demonstrated and we find that the phase transition depends on the mean and standard deviation of the adoption thresholds for small values of the mean adoption threshold the system exhibits a continuous phase transition for large values of the mean adoption threshold the phase transition is discontinuous continuous for the small large standard deviation of the adoption thresholds there is a remarkable agreement between theoretical predictions and numerical simulations

correlated random networks we develop a statistical theory of networks a network is a set of vertices and links given by its adjacency matrix c and the relevant statistical ensembles are defined in terms of a partition function z summation operator exp betah c the simplest cases are uncorrelated random networks such as the well known erdos renyi graphs here we study more general interactions h c which lead to correlations for example between the connectivities of adjacent vertices in particular such correlations occur in optimized networks described by partition functions in the limit beta infinity they are argued to be a crucial signature of evolutionary design in biological networks

properties of random graphs with hidden color we investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub coloring with or without the restriction to nondegenerate graphs the calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated cluster size statistics are derived with generating function techniques yielding a well defined percolation threshold explicit rules are derived for the enumeration of small subgraphs duality and redundancy is discussed and subclasses corresponding to commonly studied models are identified

deffuant model of opinion formation in one dimensional multiplex networks complex systems in the real world often operate through multiple kinds of links connecting their constituents in this paper we propose an opinion formation model under bounded confidence over multiplex networks consisting of edges at different topological and temporal scales we determine rigorously the critical confidence threshold by exploiting probability theory and network science when the nodes are arranged on the integers evolving in continuous time it is found that the existence of multiplexity impedes the convergence and that working with the aggregated or summarized simplex network is inaccurate since it misses vital information analytical calculations are confirmed by extensive numerical simulations

a shrinkage predictive distribution for multivariate normal observables summary we investigate shrinkage methods for constructing predictive distributions we consider the multivariate normal model with a known covariance matrix and show that there exists a shrinkage predictive distribution dominating the bayesian predictive distribution based on the vague prior when the dimension is not less than three kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function somiie key w ords invariance james stein estimator kullback leibler divergence stein s prior vague prior

stable galerkin methods for hyperbolic systems this paper considers the stability of the galerkin method for first order hyperbolic systems just as in finite difference theory many galerkin methods stable and convergent for the cauchy problem are unstable for hyperbolic boundary value problems because of improper treatment of the boundary conditions the usual galerkin method is seen to be unstable for a linear constant coefficient well posed hyperbolic system on 0 1 a method of treating boundary conditions is then proposed which will yield a stable and convergent method for any well posed linear hyperbolic system in one dimension this idea generalizes in a natural way to problems in more than one space dimension examples in one and two dimensions are considered this paper extends the work of gunzburger math comp 31 1977 pp 671 675 to the general hyperbolic system

clustering determines the dynamics of complex contagions in multiplex networks we present the mathematical analysis of generalized complex contagions in a class of clustered multiplex networks the model is intended to understand spread of influence or any other spreading process implying a threshold dynamics in setups of interconnected networks with significant clustering the contagion is assumed to be general enough to account for a content dependent linear threshold model where each link type has a different weight for spreading influence that may depend on the content e g product rumor political view that is being spread using the generating functions formalism we determine the conditions probability and expected size of the emergent global cascades this analysis provides a generalization of previous approaches and is especially useful in problems related to spreading and percolation the results present nontrivial dependencies between the clustering coefficient of the networks and its average degree in particular several phase transitions are shown to occur depending on these descriptors generally speaking our findings reveal that increasing clustering decreases the probability of having global cascades and their size however this tendency changes with the average degree there exists a certain average degree from which on clustering favors the probability and size of the contagion by comparing the dynamics of complex contagions over multiplex networks and their monoplex projections we demonstrate that ignoring link types and aggregating network layers may lead to inaccurate conclusions about contagion dynamics particularly when the correlation of degrees between layers is high

bond percolation on a class of clustered random networks analytical results are derived for the bond percolation threshold and the size of the giant connected component in a class of random networks with nonzero clustering the network s degree distribution and clustering spectrum may be prescribed and theoretical results match well with numerical simulations on both synthetic and real world networks

k core percolation on multiplex networks we generalize the theory of k text ensuremath mathrm core percolation on complex networks to mathbf k text ensuremath mathrm core percolation on multiplex networks where mathbf k ensuremath equiv k 1 k 2 k m multiplex networks can be defined as networks with vertices of one kind but m different types of edges representing different types of interactions for such networks the mathbf k text ensuremath mathrm core is defined as the largest subgraph in which each vertex has at least k i edges of each type i 1 2 m we derive self consistency equations to obtain the birth points of the mathbf k text ensuremath mathrm cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution to clarify our general results we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of erd ifmmode mbox h o else h o fi s r enyi and scale free multiplex networks we find hybrid phase transitions at the emergence points of mathbf k text ensuremath mathrm cores except the 1 1 core for which the transition is continuous we apply the mathbf k text ensuremath mathrm core decomposition algorithm to air transportation multiplex networks composed of two layers and obtain the size of k 1 k 2 cores

analysis of complex contagions in random multiplex networks we study the diffusion of influence in random multiplex networks where links can be of r different types and for a given content e g rumor product or political view each link type is associated with a content dependent parameter ci in 0 that measures the relative bias type i links have in spreading this content in this setting we propose a linear threshold model of contagion where nodes switch state if their perceived proportion of active neighbors exceeds a threshold namely a node connected to mi active neighbors and ki mi inactive neighbors via type i links will turn active if cimi ciki exceeds its threshold under this model we obtain the condition probability and expected size of global spreading events our results extend the existing work on complex contagions in several directions by i providing solutions for coupled random networks whose vertices are neither identical nor disjoint ii highlighting the effect of content on the dynamics of complex contagions and iii showing that content dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature

weak percolation on multiplex networks bootstrap percolation is a simple but nontrivial model it has applications in many areas of science and has been explored on random networks for several decades in single layer simplex networks it has been recently observed that bootstrap percolation which is defined as an incremental process can be seen as the opposite of pruning percolation where nodes are removed according to a connectivity rule here we propose models of both bootstrap and pruning percolation for multiplex networks we collectively refer to these two models with the concept of weak percolation to distinguish them from the somewhat classical concept of ordinary strong percolation while the two models coincide in simplex networks we show that they decouple when considering multiplexes giving rise to a wealth of critical phenomena our bootstrap model constitutes the simplest example of a contagion process on a multiplex network and has potential applications in critical infrastructure recovery and information security moreover we show that our pruning percolation model may provide a way to diagnose missing layers in a multiplex network finally our analytical approach allows us to calculate critical behavior and characterize critical clusters

generalized percolation in random directed networks we develop a general theory for percolation in directed random networks with arbitrary two point correlations and bidirectional edges that is edges pointing in both directions simultaneously these two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena equations for the percolation threshold and the sizes of the giant components are derived in the most general case we also present simulation results for a particular example of uncorrelated network with bidirectional edges confirming the theoretical predictions

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved