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

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

nonequilibrium phase transition due to isolation of communities we introduce a simple model of a growing system with m competing communities the model corresponds to the phenomenon of defeats suffered by social groups living in isolation a nonequilibrium phase transition is observed when at critical time t c the first isolated cluster occurs in the one dimensional system the volume of the new phase i e the number of the isolated individuals increases with time as z ensuremath sim t 3 for a large number of possible communities the critical density of filled space is equal to ensuremath rho c m n 1 3 where n is the system size a similar transition is observed for erd ifmmode mbox h o else h o fi s r enyi random graphs and barab asi albert scale free networks analytical results are in agreement with numerical simulations

effects of attachment preferences on coevolution of opinions and networks abstract in the coevolution of network structures and opinion formation we investigate the effects of a mixed population with distinctive relinking preferences on both the convergence time and the network structures it has been found that a heterogeneous network structure is easier to be reached with more high degree preferential hdp nodes there exists high correlation between the convergence time and the network heterogeneity the heterogeneous degree distribution caused by preferential attachment accelerates the convergence to a consensus state and the shortened convergence time inhibits the occurrence of the following disquieting situation that occurs in a continuously evolving network with preferential attachment and long time evolvement most of the nodes would become separated and only a few leaders would have immediate neighbors analytical calculations based on mean field theory reveal that both the transition point p t r and the consensus time depend upon the standard deviation of the degree distribution d p t r increases while decreases with the rise of d functions of p t r k k 1 and a n k d 2 k are found theoretical analyses are in accordance with simulation data

clustering of tag induced subgraphs in complex networks abstract we study the behavior of the clustering coefficient in tagged networks the rich variety of tags associated with the nodes in the networks we considered provide additional relevant information about the entities represented by the nodes such specific features can be important for practical applications like searching in the networks here we examine how the clustering coefficient changes when narrowing the network to a subgraph associated with a given tag and how it correlates with various other properties of the subgraph a further question addressed in our paper is how the clustering coefficient of the individual nodes is affected by the tags on the node we argue that this kind of analysis can be useful when one is aimed at a detailed description of the structure of large complex systems

zero temperature glauber dynamics on small world networks the zero temperature glauber dynamics of the ferromagnetic ising model on small world networks rewired from a two dimensional square lattice has been studied by numerical simulations for increasing disorder in finite networks the nonequilibrium dynamics becomes faster so that the ground state is found more likely for any finite value of the rewiring probability p the likelihood of reaching the ground state goes to zero in the thermodynamic limit similarly to random networks the spin correlation r is found to decrease with distance as r exp r being a correlation length scaling with p as p 0 73 these results are compared with those obtained earlier for addition type small world networks

analytical calculation of fragmentation transitions in adaptive networks in adaptive networks fragmentation transitions have been observed in which the network breaks into disconnected components we present an analytical approach for calculating the transition point in general adaptive network models using the example of an adaptive voter model we demonstrate that the proposed approach yields good agreement with numerical results

solution of voter model dynamics on annealed small world networks an analytical study of the behavior of the voter model on the small world topology is performed in order to solve the equations for the dynamics we consider an annealed version of the watts strogatz ws network where long range connections are randomly chosen at each time step the resulting dynamics is as rich as on the original ws network a temporal scale tau separates a quasistationary disordered state with coexisting domains from a fully ordered frozen configuration tau is proportional to the number of nodes in the network so that the system remains asymptotically disordered in the thermodynamic limit

a best possible improvement of wald s equation that d 0 0 and d cx 2 x 0 put a emax1 n always exists and est l so that est o iff l 0 let s eiissni and s emaxl liskii as a consequence of these facts it follows that if xn are independent and have mean zero then eiistii 00 and est 0 whenever es oo in the mean zero case s 4sn and so in fact est oo implies est 0 this constitutes a best possible improvement of wald s equation

global culture a noise induced transition in finite systems we analyze the effect of cultural drift modeled as noise in axelrod s model for the dissemination of culture the disordered multicultural frozen configurations are found not to be stable this general result is proven rigorously in d51 where the dynamics is described in terms of a lyapunov potential in d52 the dynamics is governed by the average relaxation time t of perturbations noise at a rate r t 21 induces monocultural configurations whereasr t 21 sustains disorder in the thermodynamic limit the relaxation time diverges and global polarization persists in spite of a dynamics of local convergence

topology induced coarsening in language games we investigate how very large populations are able to reach a global consensus out of local microscopic interaction rules in the framework of a recently introduced class of models of semiotic dynamics the so called naming game we compare in particular the convergence mechanism for interacting agents embedded in a low dimensional lattice with respect to the mean field case we highlight that in low dimensions consensus is reached through a coarsening process that requires less cognitive effort of the agents with respect to the mean field case but takes longer to complete in one dimension the dynamics of the boundaries is mapped onto a truncated markov process from which we analytically computed the diffusion coefficient more generally we show that the convergence process requires a memory per agent scaling as n and lasts a time n 1 2 d in dimension d 4 the upper critical dimension while in mean field both memory and time scale as n 3 2 for a population of n agents we present analytical and numerical evidence supporting this picture

coevolution of glauber like ising dynamics and topology we study the coevolution of a generalized glauber dynamics for ising spins with tunable threshold and of the graph topology where the dynamics takes place this simple coevolution dynamics generates a rich phase diagram in the space of the two parameters of the model the threshold and the rewiring probability the diagram displays phase transitions of different types spin ordering percolation and connectedness at variance with traditional coevolution models in which all spins of each connected component of the graph have equal value in the stationary state we find that for suitable choices of the parameters the system may converge to a state in which spins of opposite sign coexist in the same component organized in compact clusters of like signed spins mean field calculations enable one to estimate some features of the phase diagram

a method of approximating expectations of functions of sums of independent random variables let x 1 x 2 cdots be a sequence of independent random variables with s n sum n i 1 x i fix alpha 0 let phi cdot be a continuous strictly increasing function on lbrack 0 infty such that phi 0 0 and phi cx leq c alpha phi x for all x 0 and all c geq 2 suppose a is a real number and j is a finite nonempty subset of the positive integers in this paper we are interested in approximating e max j in j phi a s j we construct a number b j a from the one dimensional distributions of the x s such that the ratio e max j in j phi a s j phi b j a is bounded above and below by positive constants which depend only on alpha bounds for these constants are given

φ moment inequalities for independent and freely independent random variables abstract this paper is devoted to the study of moments of sums of independent freely independent random variables more precisely let f k k 1 n be a sequence of positive symmetrically distributed independent random variables and let be an orlicz function with 2 condition we provide an equivalent expression for the quantity e k 1 n f k in term of the sum of disjoint copies of the sequence f k k 1 n we also prove an analogous result in the setting of free probability furthermore we provide an equivalent characterization of sup 1 k n x k for positive freely independent random variables and also present some new results on free johnson schechtman inequalities in the quasi banach symmetric operator space

the effect of zealots on the rate of consensus achievement in complex networks abstract in this study we investigate the role of zealots on the result of voting process on both scale free and watts strogatz networks we observe that inflexible individuals are very effective in consensus achievement and also in the rate of ordering process in complex networks zealots make the magnetization of the system to vary exponentially with time we obtain that on sf networks increasing the zealots population z exponentially increases the rate of consensus achievement the time needed for the system to reach a desired magnetization shows a power law dependence on z as well we obtain that the decay time of the order parameter shows a power law dependence on z we also investigate the role of zealots degree on the rate of ordering process and finally we analyze the effect of network s randomness on the efficiency of zealots moving from a regular to a random network the re wiring probability p r w increases we show that with increasing p r w the efficiency of zealots for reducing the consensus achievement time increases the rate of consensus is compared with the rate of ordering for different re wiring probabilities of ws networks

coherence thresholds in models of language change and evolution the effects of noise dynamics and network of interactions a simple model of language evolution proposed by komarova niyogi and nowak is characterized by a payoff in communicative function and by an error in learning that measure the accuracy in language acquisition the time scale for language change is generational and the model s equations in the mean field approximation are a particular case of the replicator mutator equations of evolutionary dynamics in well mixed populations this model exhibits a critical coherence threshold i e a minimal accuracy in the learning process is required to maintain linguistic coherence in this work we analyze in detail the effects of different fitness based dynamics driving linguistic coherence and of the network of interactions on the nature of the coherence threshold by performing numerical simulations and theoretical analyses of three different models of language change in finite populations with two types of structure fully connected networks and regular random graphs we find that although the threshold of the original replicator mutator evolutionary model is robust with respect to the structure of the network of contacts the coherence threshold of related fitness driven models may be strongly affected by this feature

naming games in two dimensional and small world connected random geometric networks we investigate a prototypical agent based model the naming game on two dimensional random geometric networks the naming game a baronchelli et al j stat mech theory exp 2006 p06014 is a minimal model employing local communications that captures the emergence of shared communication schemes languages in a population of autonomous semiotic agents implementing the naming games with local broadcasts on random geometric graphs serves as a model for agreement dynamics in large scale autonomously operating wireless sensor networks further it captures essential features of the scaling properties of the agreement process for spatiallyembedded autonomous agents among the relevant observables capturing the temporal properties of the agreement process we investigate the cluster size distribution and the distribution of the agreement times both exhibiting dynamic scaling we also present results for the case when a small density of long range communication links are added on top of the random geometric graph resulting in a small world like network and yielding a significantly reduced time to reach global agreement we construct a finite size scaling analysis for the agreement times in this case

scale free brain functional networks functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites analysis of the resulting networks in different tasks shows that a the distribution of functional connections and the probability of finding a link versus distance are both scale free b the characteristic path length is small and comparable with those of equivalent random networks and c the clustering coefficient is orders of magnitude larger than those of equivalent random networks all these properties typical of scale free small world networks reflect important functional information about brain states

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

solution of the voter model by spectral analysis an exact spectral analysis of the markov propagator for the voter model is presented for the complete graph and extended to the complete bipartite graph and uncorrelated random networks using a well defined martingale approximation in diffusion dominated regions of phase space which is almost everywhere for the voter model this method is applied to compute analytically several key quantities such as exact expressions for the m time step propagator of the voter model all moments of consensus times and the local times for each macrostate this spectral method is motivated by a related method for solving the ehrenfest urn problem and by formulating the voter model on the complete graph as an urn model comparisons of the analytical results from the spectral method and numerical results from monte carlo simulations are presented to validate the spectral method

breakthroughs in the theory of macdonald polynomials in 1998 i g macdonald 1 introduced a remarkable new basis for the space of symmetric functions the elements of this basis are denoted where is a partition and p q are two free parameters the s which are now called macdonald polynomials specialize to many of the well known bases for the symmetric functions by suitable choices of the parameters q and t in fact we can obtain in this manner the schur functions the hall littlewood symmetric functions the jack symmetric functions the zonal symmetric functions the zonal spherical functions and the elementary and monomial symmetric functions

consensus formation on coevolving networks groups formation and structure we study the effect of adaptivity on a social model of opinion dynamics and consensus formation we analyse how the adaptivity of the network of contacts between agents to the underlying social dynamics affects the size and topological properties of groups and the convergence time to the stable final state we find that while on static networks these properties are determined by percolation phenomena on adaptive networks the rewiring process leads to different behaviors adaptive rewiring fosters group formation by enhancing communication between agents of similar opinion though it also makes possible the division of clusters we show how the convergence time is determined by the characteristic time of link rearrangement we finally investigate how the adaptivity yields nontrivial correlations between the internal topology and the size of the groups of agreeing agents

combinatorial theory of macdonald polynomials i proof of haglund s formula abstract haglund recently proposed a combinatorial interpretation of the modified macdonald polynomials h we give a combinatorial proof of this conjecture which establishes the existence and integrality of h as corollaries we obtain the cocharge formula of lascoux and schutzenberger for hall littlewood polynomials a formula of sahi and knop for jack s symmetric functions a generalization of this result to the integral macdonald polynomials j a formula for h in terms of lascoux leclerc thibon polynomials and combinatorial expressions for the kostka macdonald coefficients k when is a two column shape

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

fragmentation transitions in multistate voter models adaptive models of opinion formation among humans can display a fragmentation transition where a social network breaks into disconnected components here we investigate this transition in a class of models with arbitrary number of opinions in contrast to previous work we do not assume that opinions are equidistant or arranged on a one dimensional conceptual axis our investigation reveals detailed analytical results on fragmentations in a three opinion model which are confirmed by agent based simulations furthermore we show that in certain models the number of opinions can be reduced without affecting the fragmentation points

a combinatorial formula for nonsymmetric macdonald polynomials we give a combinatorial formula for the nonsymmetric macdonald polynomials e x q t the formula generalizes our previous combinatorial interpretation of the integral form symmetric macdonald polynomials j x q t we prove the new formula by verifying that it satisfies a recurrence due to knop and sahi that characterizes the nonsymmetric macdonald polynomials

hierarchy and polysynchrony in an adaptive network we describe a simple adaptive network of coupled chaotic maps the network reaches a stationary state frozen topology for all values of the coupling parameter although the dynamics of the maps at the nodes of the network can be nontrivial the structure of the network shows interesting hierarchical properties and in certain parameter regions the dynamics is polysynchronous nodes can be divided in differently synchronized classes but contrary to cluster synchronization nodes in the same class need not be connected to each other these complicated synchrony patterns have been conjectured to play roles in systems biology and circuits the adaptive system we study describes ways whereby this behavior can evolve from undifferentiated nodes

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

a probabilistic interpretation of the macdonald polynomials the two parameter macdonald polynomials are a central object of algebraic combinatorics and representation theory we give a markov chain on partitions of k with eigenfunctions the coefficients of the macdonald polynomials when expanded in the power sum polynomials the markov chain has stationary distribution a new two parameter family of measures on partitions the inverse of the macdonald weight rescaled the uniform distribution on cycles of permutations and the ewens sampling formula are special cases the markov chain is a version of the auxiliary variables algorithm of statistical physics properties of the macdonald polynomials allow a sharp analysis of the running time in natural cases a bounded number of steps suffice for arbitrarily large k

voter model with non poissonian interevent intervals recent analysis of social communications among humans has revealed that the interval between interactions for a pair of individuals and for an individual often follows a long tail distribution we investigate the effect of such a non poissonian nature of human behavior on dynamics of opinion formation we use a variant of the voter model and numerically compare the time to consensus of all the voters with different distributions of interevent intervals and different networks compared with the exponential distribution of interevent intervals i e the standard voter model the power law distribution of interevent intervals slows down consensus on the ring this is because of the memory effect in the power law case the expected time until the next update event on a link is large if the link has not had an update event for a long time on the complete graph the consensus time in the power law case is close to that in the exponential case regular graphs bridge these two results such that the slowing down of the consensus in the power law case as compared to the exponential case is less pronounced as the degree increases

voter model dynamics in complex networks role of dimensionality disorder and degree distribution we analyze the ordering dynamics of the voter model in different classes of complex networks we observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks we also find that when there is no ordering in the system the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity the existence of hubs i e highly connected nodes in the network modifies the linear system size scaling law of the survival time the size of an ordered domain is sensitive to the network disorder and the average degree decreasing with both however it seems not to depend on network size and on the heterogeneity of the degree distribution

ribbon tableaux hall littlewood functions quantum affine algebras and unipotent varieties we introduce a new family of symmetric functions which are q analogs of products of schur functions defined in terms of ribbon tableaux these functions can be interpreted in terms of the fock space representation fq of uq sln and are related to hall littlewood functions via the geometry of flag varieties we present a series of conjectures and prove them in special cases the essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of fq using ribbon tableaux

parabolic kazhdan lusztig polynomials for hermitian symmetric pairs we study the parabolic kazhdan lusztig polynomials for hermitian symmetric pairs in particular we show that these polynomials are always either zero or a monic power of q and that they are combinatorial invariants

transmission of information and herd behavior an application to financial markets we propose a model for stochastic formation of opinion clusters modeled by an evolving network and herd behavior to account for the observed fat tail distribution in returns of financial price data the only parameter of the model is h the rate of information dispersion per trade which is a measure of herding behavior for h below a critical h the system displays a power law distribution of the returns with exponential cutoff however for hg h an increase in the probability of large returns is found and may be associated with the occurrence of large crashes

nonequilibrium transitions in complex networks a model of social interaction we analyze the nonequilibrium order disorder transition of axelrod s model of social interaction in several complex networks in a small world network we find a transition between an ordered homogeneous state and a disordered state the transition point is shifted by the degree of spatial disorder of the underlying network the network disorder favoring ordered configurations in random scale free networks the transition is only observed for finite size systems showing system size scaling while in the thermodynamic limit only ordered configurations are always obtained thus in the thermodynamic limit the transition disappears however in structured scale free networks the phase transition between an ordered and a disordered phase is restored

ising model on networks with an arbitrary distribution of connections we find the exact critical temperature t c of the nearest neighbor ferromagnetic ising model on an equilibrium random graph with an arbitrary degree distribution p k we observe an anomalous behavior of the magnetization magnetic susceptibility and specific heat when p k is fat tailed or loosely speaking when the fourth moment of the distribution diverges in infinite networks when the second moment becomes divergent t c approaches infinity the phase transition is of infinite order and size effect is anomalously strong

analytic treatment of consensus achievement in the single type zealotry voter model abstract we introduce zealots of one opinion in the voter model on a complete graph and examine how they affect consensus achievement using first step analysis for markov chains to obtain an exact solution we find that the mean consensus time scales with the population size n increasing the number of zealots z will decrease the consensus time in a power law fashion for large z the mean magnetization was also analytically obtained and was found to contain an exponential dependence on z the dynamics for the complete graph are qualitatively similar to those obtained in another study for the barabasi albert network in general the existence of zealots serves to hasten consensus except in the case where only a few zealots oppose the vast majority

the coevolution of individual economic characteristics and socioeconomic networks abstract the dynamics of individual characteristics of economic agents is modeled with the link structure influenced by this dynamics links between agents with similar characteristics are more stable than those between agents with vastly different characteristics a simple scaling law describes the number of distinct surviving characteristic realizations as a function of the number of agents and the number of possible distinct characteristics realizations with the chosen specification the investigated properties do not essentially differ from those found for analogous sociophysics models with a fixed network structure

self organization of complex networks as a dynamical system to understand the dynamics of real world networks we investigate a mathematical model of the interplay between the dynamics of random walkers on a weighted network and the link weights driven by a resource carried by the walkers our numerical studies reveal that under suitable conditions the co evolving dynamics lead to the emergence of stationary power law distributions of the resource and link weights while the resource quantity at each node ceaselessly changes with time we analyze the network organization as a deterministic dynamical system and find that the system exhibits multistability with numerous fixed points limit cycles and chaotic states the chaotic behavior of the system leads to the continual changes in the microscopic network dynamics in the absence of any external random noises we conclude that the intrinsic interplay between the states of the nodes and network reformation constitutes a major factor in the vicissitudes of real world networks

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

order of magnitude bounds for expectations involving quadratic forms let x 1 x 2 ldots x n be independent mean zero random variables and let a ij 1 leq i j leq n be an array of constants with a ii equiv 0 we present a method of obtaining the order of magnitude of e phi sum 1 leq i j leq n a ij x ix j for any such x i and a ij and any nonnegative symmetric convex function phi with phi 0 0 such that for some integer k geq 0 phi x 2 k is convex and simultaneously phi x 2 k 1 is concave on lbrack 0 infty the approximation is based on decoupling inequalities valid for all such mean zero x i and reals a ij and a certain further independentization procedure

slow dynamics and rare region effects in the contact process on weighted tree networks we show that generic slow dynamics can occur in the contact process on complex networks with a tree like structure and a superimposed weight pattern in the absence of additional nontopological sources of quenched disorder the slow dynamics is induced by rare region effects occurring on correlated subspaces of vertices connected by large weight edges and manifests in the form of a smeared phase transition we conjecture that more sophisticated network motifs could be able to induce griffiths phases as a consequence of purely topological disorder

ordering dynamics in the voter model with aging abstract we study theoretically and numerically the voter model with memory dependent dynamics at the mean field level the internal age related to the time an individual spends holding the same state is added to the set of binary states of the population such that the activation probability p i for attempting a change of state has an explicit dependence on the internal age i we derive a closed set of integro differential equations describing the time evolution of the fraction of individuals with a given state and internal age and from it we obtain analytical results characterizing the behavior of the system close to the absorbing states in general different internal age dependent activation probabilities p i have different effects on the dynamics in the case of aging i e p i being a decreasing function of its argument i either the system reaches consensus or it gets trapped in a frozen state depending on whether the limiting value p is zero or positive and on the rate at which p i approaches p moreover when the system reaches consensus the fraction x t of nodes not holding the consensus state may decay exponentially with time or as a power law depending again on the specific details of the functional form of p i for the case of anti aging when the activation probability p i is an increasing function of i the system always reaches a steady state with coexistence of opinions exact conditions for having one or another behavior together with the equations and explicit expressions for the power law exponents are provided

a model of the effects of authority on consensus formation in adaptive networks impact on network topology and robustness abstract opinions of individuals in real social networks are arguably strongly influenced by external determinants such as the opinions of those perceived to have the highest levels of authority in order to model this we have extended an existing model of consensus formation in an adaptive network by the introduction of a parameter representing each agent s level of authority based on their opinion relative to the overall opinion distribution we found that introducing this model along with a randomly varying opinion convergence factor significantly impacts the final state of converged opinions and the number of interactions required to reach that state we also determined the relationship between initial and final network topologies for this model and whether the final topology is robust to node removals our results indicate firstly that the process of consensus formation with a model of authority consistently transforms the network from an arbitrary initial topology to one with distinct measurements in mean shortest path clustering coefficient and degree distribution secondly we found that subsequent to the consensus formation process the mean shortest path and clustering coefficient are less affected by both random and targeted node disconnection speculation on the relevance of these results to real world applications is provided

dynamics of majority rule in two state interacting spin systems we introduce a two state opinion dynamics model where agents evolve by majority rule in each update a group of agents is specified whose members then all adopt the local majority state in the mean field limit where a group consists of randomly selected agents consensus is reached in a time that scales ln n where n is the number of agents on finite dimensional lattices where a group is a contiguous cluster the consensus time fluctuates strongly between realizations and grows as a dimension dependent power of n the upper critical dimension appears to be larger than 4 the final opinion always equals that of the initial majority except in one dimension

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

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

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 recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

combinatorics of the two species asep and koornwinder moments abstract in previous work 12 13 14 the first and third authors introduced staircase tableaux which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process asep and for the moments of the askey wilson weight function the fact that the asep and askey wilson moments are related at all is unexpected and is due to 45 the asep is a model of particles hopping on a one dimensional lattice of n sites with open boundaries particles can enter and exit at both left and right borders it was introduced around 1970 34 43 and is cited as a model for both traffic flow and translation in protein synthesis meanwhile the askey wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials so from this previous work we have the relationship asep staircase tableaux askey wilson moments the askey wilson polynomials can be viewed as the one variable case of the multivariate koornwinder polynomials which are also known as the macdonald polynomials attached to the non reduced affine root system c n c n it is natural then to ask whether one can generalize the relationships among the asep askey wilson moments and staircase tableaux in such a way that koornwinder moments replace askey wilson moments in 15 we made a precise link between koornwinder moments and the two species asep a generalization of the asep which has two species of particles with different weights in this article we introduce rhombic staircase tableaux and show that we have the relationship 2 species asep rhombic staircase tableaux koornwinder moments in particular we give formulas for the stationary distribution of the two species asep and for koornwinder moments in terms of rhombic staircase tableaux download download high res image 75kb download download full size image

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

early fragmentation in the adaptive voter model on directed networks we consider voter dynamics on a directed adaptive network with fixed out degree distribution a transition between an active phase and a fragmented phase is observed this transition is similar to the undirected case if the networks are sufficiently dense and have a narrow out degree distribution however if a significant number of nodes with low out degree is present then fragmentation can occur even far below the estimated critical point due to the formation of self stabilizing structures that nucleate fragmentation this process may be relevant for fragmentation in current political opinion formation processes

graph fission in an evolving voter model we consider a simplified model of a social network in which individuals have one of two opinions called 0 and 1 and their opinions and the network connections coevolve edges are picked at random if the two connected individuals hold different opinions then with probability 1 one imitates the opinion of the other otherwise i e with probability the link between them is broken and one of them makes a new connection to an individual chosen at random i from those with the same opinion or ii from the network as a whole the evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions letting be the fraction of voters holding the minority opinion after the evolution stops we are interested in how depends on and the initial fraction u of voters with opinion 1 in case i there is a critical value c which does not depend on u with u for c and 0 for c u u but for c u we have u 1 2 using simulations and approximate calculations we explain why these two nearly identical models have such dramatically different phase transitions

a symmetrization desymmetrization procedure for uniformly good approximation of expectations involving arbitrary sums of generalized u statistics let be a symmetric function nondecreasing on 0 and satisfying a 2 growth condition x 1 y 1 x 2 y 2 x n y n be independent random vectors such that for each 1 i n either y i x i or y i is independent of all the other variates and the marginal distributions of x i and y j are otherwise arbitrary let f ij x y 1 i j n be any array of real valued measurable functions we present a method of obtaining the order of magnitude of e f ij x i y j 1 i j n the proof employs a double symmetrization introducing independent copies x i y j of x i y j and moving from summands of the form f ij x i yj to what we call f ij s x i yj x i y j substitution of fixed constants x i and y j for x i and y j results in f ij s x i y j x i y j which equals f ij x i y j adjusted by a sum of quantities of first order separately in x i and y j introducing further explicit first order adjustments call them g 1ij x i x y and g 2ij y j x y it is proved that e 1 i j n f ij s x i y j x i y j g 1ij x i x y g 2ij y j x y e 1 i j n f ij s x i y j x i y j 2 f s x y x y where the latter is an explicitly computable quantity for any x 0 and y which come within a factor of two of minimizing f s x y x y it is shown that e 1 i j n f ij x i y j max f s x y x 0 y 0 e 1 i j n f ij x i y j 0 f ij x i 0 y j f ij x i 0 y j 0 g 1ij x i x i 0 y j 0 g 2ij y j x i 0 y j 0 wich is computable approximable in terms of the underlying random variables these results extend to the expectation of of a sum of functions of k components

exact results for kinetics of catalytic reactions the kinetics of an irreversible catalytic reaction on substrate of arbitrary dimension is examined in the limit of infinitesimal reaction rate reaction controlled limit we solve the dimer dimer surface reaction model or voter model exactly in arbitrary dimension d the density of reactive interfaces is found to exhibit a power law decay for d 2 and a slow logarithmic decay in two dimensions we discuss the relevance of these results for the monomer monomer surface reaction model

multiopinion coevolving voter model with infinitely many phase transitions we consider an idealized model in which individuals changing opinions and their social network coevolve with disagreements between neighbors in the network resolved either through one imitating the opinion of the other or by reassignment of the discordant edge specifically an interaction between x and one of its neighbors y leads to x imitating y with probability 1 and otherwise i e with probability x cutting its tie to y in order to instead connect to a randomly chosen individual building on previous work about the two opinion case we study the multiple opinion situation finding that the model has infinitely many phase transitions in the large graph limit with infinitely many initial opinions moreover the formulas describing the end states of these processes are remarkably simple when expressed as a function of 1