the role of network topology on extremism propagation with the relative agreement opinion dynamics abstract in deffuant et al j artif soc soc simulation 5 2002 4 we proposed a simple model of opinion dynamics which we used to simulate the influence of extremists in a population simulations were run without any specific interaction structure and varying the simulation parameters we observed different attractors such as predominance of centrism or of extremism we even observed in certain conditions that the whole population drifts to one extreme of the opinion even if initially there are an equal number of extremists at each extreme of the opinion axis in the present paper we study the influence of the social networks on the presence of such a dynamical behavior in particular we use small world networks with variable connectivity and randomness of the connections we find that the drift to a single extreme appears only beyond a critical level of connectivity which decreases when the randomness increases

geometric formulation of mechanical systems with one sided constraints a geometric formulation of analytical mechanics of systems with holonomic and holonomic one sided constraints is proposed this is an extension of work by tulczyjew 1986 in which non holonomic constraints were also considered but not the one sided case the introduction of lagrange multipliers to give explicit equations of motion was not considered by tulczyjew either this allows the authors to study problems of motion of particles hitting moving walls they verify that the conditions of collision are satisfied

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

new reductions of the kadomtsev petviashvili and two dimensional toda lattice hierarchies via symmetry constraints new types of reductions of the kadomtsev petviashvili kp hierarchy and the two dimensional toda lattice 2dtl hierarchy are considered on the basis of sato s approach within this approach these hierarchies are represented by infinite sets of equations for potentials u1 u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp and the 2dtl hierarchies are studied under constraints of the following type n 1n nsn u1 u2 u3 x where sn are symmetries for these hierarchies n are arbitrary constants and is an arbitrary linear functional of the quantity it is shown that for the kp hierarchy these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 u3 un many known systems and several new ones are among them symmetry reductions for the 2dtl hierarchy give rise both to finite dimensional dynamical systems and 1 1 dimensional discrete systems some few results for the modified kp hierarc

spectral properties of the laplacian of multiplex networks omez et al phys rev lett 110 028701 2013 some of the authors proposed a framework for the study of diffusion processes in such networks here we extend the previous framework to deal with general configurations in several layers of networks and analyze the behavior of the spectrum of the laplacian of the full multiplex we derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues this spectrum is used to gain insight into physical phenomena on top of multiplex specifically diffusion processes and synchronizability

darboux transformations for the isospectral and nonisospectral mkp equation abstract darboux transformations for the isospectral and nonisospectral modified kadomtsev petviashvili mkp equations are investigated in the isospectral case it is an auto darboux transformation however in the nonisospectral case it is not auto darboux tranformation

vandermonde like determinants and n fold darboux backlund transformations we define vandermonde like determinants and analyze their structure the resulting scheme is well suited to achieve a remarkable compactness and transparency in n soliton formulas or more generally in formulas for n fold darboux or backlund transformations

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

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

structural measures for multiplex networks many real world complex systems consist of a set of elementary units connected by relationships of different kinds all such systems are better described in terms of multiplex networks where the links at each layer represent a different type of interaction between the same set of nodes rather than in terms of single layer networks in this paper we present a general framework to describe and study multiplex networks whose links are either unweighted or weighted in particular we propose a series of measures to characterize the multiplexicity of the systems in terms of i basic node and link properties such as the node degree and the edge overlap and reinforcement ii local properties such as the clustering coefficient and the transitivity and iii global properties related to the navigability of the multiplex across the different layers the measures we introduce are validated on a genuinely multiplex data set of indonesian terrorists where information among 78 individuals are recorded with respect to mutual trust common operations exchanged communications and business relationships

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

dynamics of bounded confidence opinion in heterogeneous social networks concord against partial antagonism abstract bounded confidence models of opinion dynamics in social networks have been actively studied in recent years in particular opinion formation and extremism propagation along with other aspects of social dynamics in this work after an analysis of limitations of the deffuant weisbuch dw bounded confidence relative agreement model we propose the mixed model that takes into account two psychological types of individuals concord agents c agents are friendly people they interact in a way that their opinions always get closer agents of the other psychological type show partial antagonism in their interaction pa agents opinion dynamics in heterogeneous social groups consisting of agents of the two types was studied on different social networks erdos renyi random graphs small world networks and complete graphs limit cases of the mixed model pure c and pa societies were also studied we found that group opinion formation is qualitatively almost independent of the topology of networks used in this work opinion fragmentation polarization and consensus are observed in the mixed model at different proportions of pa and c agents depending on the value of initial opinion tolerance of agents as for the opinion formation and arising of dissidents the opinion dynamics of the c agents society was found to be similar to that of the dw model except for the rate of opinion convergence nevertheless mixed societies showed dynamics and bifurcation patterns notably different to those of the dw model the influence of biased initial conditions over opinion formation in heterogeneous social groups was also studied versus the initial value of opinion uncertainty varying the proportion of the pa to c agents bifurcation diagrams showed an impressive evolution of collective opinion in particular radical changes of left to right consensus or vice versa at an opinion uncertainty value equal to 0 7 in the model with the pa c mixture of population near 50 50

generalized canonical transformations for time dependent systems we introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space we prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable the theory of generating functions as well as that of hamilton jacobi is developed some further applications are developed

lagrangian submanifolds and higher order mechanical systems higher order lagrangian and hamiltonian systems time dependent or independent are interpreted in terms of lagrangian submanifolds of sympletic higher order tangent bundles the relation between both formalisms is given

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

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

comparison of voter and glauber ordering dynamics on networks we study numerically the ordering process of two very simple dynamical models for a two state variable on several topologies with increasing levels of heterogeneity in the degree distribution we find that the zero temperature glauber dynamics for the ising model may get trapped in sets of partially ordered metastable states even for finite system size and this becomes more probable as the size increases voter dynamics instead always converges to full order on finite networks even if this does not occur via coherent growth of domains the time needed for order to be reached diverges with the system size in both cases the ordering process is rather insensitive to the variation of the degreee distribution from sharply peaked to scale free

constrained time dependent lagrangian systems and lagrangian submanifolds in this paper a description a la tulczyjew of constrained time dependent dynamical systems in terms of special symplectic manifolds is first developed next this description is combined with the constraint algorithm for time dependent constrained systems

integrable hamiltonian systems related to the polynomial eigenvalue problem the independent integrals of motion in involution for the hamiltonian system related to the second order polynomial eigenvalue problem are constructed by using relevant recursion formula the hierarchy of hamiltonian systems obtained from the above problem and the time part of the lax pair are shown to be completely integrable and they are shown to commute with each other furthermore their solution solves the evolution equation associated with the lax pair

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

vandermonde like determinants representations of darboux transformations and explicit solutions for the modified kadomtsev petviashvili equation abstract recently the 2 1 dimensional modified kadomtsev petviashvili mkp equation was decomposed into two known 1 1 dimensional soliton equations by dai and geng h h dai x g geng j math phys 41 2000 7501 in the present paper a systematic and simple method is proposed for constructing three kinds of explicit n fold darboux transformations and their vandermonde like determinants representations of the two known 1 1 dimensional soliton equations based on their lax pairs as an application of the darboux transformations three explicit multi soliton solutions of the two 1 1 dimensional soliton equations are obtained in particular six new explicit soliton solutions of the 2 1 dimensional mkp equation are presented by using the decomposition the explicit formulas of all the soliton solutions are also expressed by vandermonde like determinants which are remarkably compact and transparent

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

lagrangian submanifolds and dynamics on lie algebroids in some previous papers a geometric description of lagrangian mechanics on lie algebroids has been developed in this topical review we give a hamiltonian description of mechanics on lie algebroids in addition we introduce the notion of a lagrangian submanifold of a symplectic lie algebroid and we prove that the lagrangian hamiltonian dynamics on lie algebroids may be described in terms of lagrangian submanifolds of symplectic lie algebroids the lagrangian hamiltonian formalism on lie algebroids permits us to deal with lagrangian hamiltonian functions not defined necessarily on tangent cotangent bundles thus we may apply our results to the projection of lagrangian hamiltonian functions which are invariant under the action of a symmetry lie group as a consequence we obtain that lagrange poincare hamilton poincare equations are the euler lagrange hamilton equations associated with the corresponding atiyah algebroid moreover we prove that lagrange poincare hamilton poincare equations are the local equations defining certain lagrangian submanifolds of symplectic atiyah algebroids

multi solitonic solutions for the variable coefficient variant boussinesq model of the nonlinear water waves abstract for the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth of the water we consider a variable coefficient variant boussinesq vcvb model with symbolic computation we construct the connection between the vcvb model and a variable coefficient ablowitz kaup newell segur vcakns system under certain constraints using the n fold darboux transformation of the vcakns system we present two sets of multi solitonic solutions for the vcvb model which are expressed in terms of the vandermonde like and double wronskian determinants respectively dynamics of those solutions are analyzed and graphically discussed such as the parallel solitonic waves shape changing collision head on collision fusion fission behavior and elastic fusion coupled interaction

regularity properties of fiber derivatives associated with higher order mechanical systems the aim of this work is to study fiber derivatives associated to lagrangian and hamiltonian functions describing the dynamics of a higher order autonomous dynamical system more precisely given a function in t t k 1 q we find necessary and sufficient conditions for such a function to describe the dynamics of a kth order autonomous dynamical system thus being a kth order hamiltonian function then we give a suitable definition of hyper regularity for these higher order hamiltonian functions in terms of their fiber derivative in addition we also study an alternative characterization of the dynamics in lagrangian submanifolds in terms of the solutions of the higher order euler lagrange equations

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

effects of heterogeneous convergence rate on consensus in opinion dynamics abstract the deffuant model has attracted much attention in the study of opinion dynamics here we propose a modified version by introducing into the model a heterogeneous convergence rate which is dependent on the opinion difference between interacting agents and a tunable parameter we study the effects of heterogeneous convergence rate on consensus by investigating the probability of complete consensus the size of the largest opinion cluster the number of opinion clusters and the relaxation time we find that the decrease of the convergence rate is favorable to decreasing the confidence threshold for the population to always reach complete consensus and there exists optimal resulting in the minimal bounded confidence threshold moreover we find that there exists a window before the threshold of confidence in which complete consensus may be reached with a nonzero probability when is not too large we also find that within a certain confidence range decreasing the convergence rate will reduce the relaxation time which is somewhat counterintuitive

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

consensus formation on adaptive networks the structure of a network can significantly influence the properties of the dynamical processes that take place on them while many studies have been paid to this influence much less attention has been devoted to the interplay and feedback mechanisms between dynamical processes and network topology on adaptive networks adaptive rewiring of links can happen in real life systems such as acquaintance networks where people are more likely to maintain a social connection if their views and values are similar in our study we consider different variants of a model for consensus formation our investigations reveal that the adaptation of the network topology fosters cluster formation by enhancing communication between agents of similar opinion although it also promotes the division of these clusters the temporal behavior is also strongly affected by adaptivity while on static networks it is influenced by percolation properties on adaptive networks both the early and late time evolutions of the system are determined by the rewiring process the investigation of a variant of the model reveals that the scenarios of transitions between consensus and polarized states are more robust on adaptive networks

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

a generalized norton hoff model and the prandtl reuss law of plasticity the aim of this article is to study the quasistatic evolution of a three dimensional elastic perfectly plastic solid which satisfies the prandtl reuss law the evolution of the field of stresses which solves a time dependent variational inequality and that of the field of displacements u have been described in previous works 15 26 35 36 37 but it was not shown there that and u satisfy indeed the prandtl reuss constitutive law in this article we find and u in a class of functions which are sufficiently regular for the prandtl reuss law to make sense and we prove that and u satisfy the constitutive law this result is attained by considering the elastic perfectly plastic model as the limit of a family of elastic visco plastic models like those of norton and hoff the norton hoff type models which we introduce depend on a viscosity parameter 0 we study the perturbed models i e 0 fixed and then we pass to the limit 0

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

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

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

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

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

higher order lagrangian systems geometric structures dynamics and constraints in order to study the connections between lagrangian and hamiltonian formalisms constructed from a perhaps singular higher order lagrangian some geometric structures are constructed intermediate spaces between those of lagrangian and hamiltonian formalisms partial ostrogradski s transformations and unambiguous evolution operators connecting these spaces are intrinsically defined and some of their properties studied equations of motion constraints and arbitrary functions of lagrangian and hamiltonian formalisms are thoroughly studied in particular all the lagrangian constraints are obtained from the hamiltonian ones once the gauge transformations are taken into account the true number of degrees of freedom is obtained both in the lagrangian and hamiltonian formalisms and also in all the intermediate formalisms herein defined

1 soliton solution of the generalized kp equation with generalized evolution this paper obtains the 1 soliton solution of three variants of the generalized kp equation with generalized evolution the solitary wave ansatz is used to carry out the integration of such equations the parameter domain is also identified in the process the numerical simulations are also obtained in this paper

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

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

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

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures

opinion clusters in a modified hegselmann krause model with heterogeneous bounded confidences and stubbornness abstract in opinion dynamics with continuous opinion bounded confidence is a critical parameter agents can interact with each other only when the opinion difference between them is less than the bounded confidence larger bounded confidence always leads to fewer opinion clusters stubbornness characterizing the insistence of an agent on her own opinion is thought to only affect the transition time in this work a modified hegselmann krause model with heterogeneous population is investigated where agents in different same subpopulation have different same bounded confidence and stubbornness we find that due to the interaction among subpopulations increasing the stubbornness of agents in the subpopulation with the largest bounded confidence favors fewer opinion clusters and the expansion of the largest cluster we also find that increasing the bounded confidence of a subpopulation leads to fewer clusters and a larger largest cluster provided that all the others have large bounded confidence while one subpopulation is with a small bounded confidence there exist an optimal bounded confidence of another subpopulation for the smallest number of opinion clusters and that for the largest size of the largest cluster