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

generalized voterlike model on activity driven networks with attractiveness we study the behavior of a generalized consensus dynamics on a temporal network of interactions the activity driven network with attractiveness in this temporal network model agents are endowed with an intrinsic activity a ruling the rate at which they generate connections and an intrinsic attractiveness b modulating the rate at which they receive connections the consensus dynamics considered is a mixed voter and moran dynamics each agent either in state 0 or 1 modifies his or her state when connecting with a peer thus an active agent copies his or her state from the peer with probability p or imposes his or her state to him or her with the complementary probability 1 ensuremath p applying a heterogeneous mean field approach we derive a differential equation for the average density of voters with activity a and attractiveness b in state 1 which we use to evaluate the average time to reach consensus and the exit probability defined as the probability that a single agent with activity a and attractiveness b eventually imposes his or her state to a pool of initially unanimous population in the opposite state we study a number of particular cases finding an excellent agreement with numerical simulations of the model interestingly we observe a symmetry between voter and moran dynamics in pure activity driven networks and their static integrated counterparts that exemplifies the strong differences that a time varying network can impose on dynamical processes

fast fixation with a generic network structure we investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics spatially structured wright fisher models ecology hubbell type models linguistics the utterance selection model and opinion dynamics the voter model as special cases these models all have absorbing states of fixation where all the nodes are in the same state earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the degree distribution of the network here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics while holding the degree distribution constant in particular we show that the mean time to fixation can be accelerated even on homogeneous networks when certain nodes are very much more likely to be copied from than copied to we further show that there is a complex interplay between degree distribution and asymmetry when they may covary and that the results are robust to correlations in the network or the initial condition

localization transition lifschitz tails and rare region effects in network models effects of heterogeneity in the suspected infected susceptible model on networks are investigated using quenched mean field theory the emergence of localization is described by the distributions of the inverse participation ratio and compared with the rare region effects appearing in simulations and in the lifschitz tails the latter in the linear approximation is related to the spectral density of the laplacian matrix and to the time dependent order parameter i show that these approximations indicate correctly griffiths phases both on regular one dimensional lattices and on small world networks exhibiting purely topological disorder i discuss the localization transition that occurs on scale free networks at 3 degree exponent

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

discrete opinion dynamics on networks based on social influence a model of opinion dynamics based on social influence on networks was studied the opinion of each agent can have integer values i 1 2 i and opinion exchanges are restricted to connected agents it was found that for any i 2 and self confidence parameter 0 u 1 when u is a degree independent constant the weighted proportion qi of the population that hold a given opinion i is a martingale and the fraction qi of opinion i will gradually converge to qi the tendency can slow down with the increase of degree assortativity of networks when u is degree dependent qi does not possess the martingale property however qi still converges to it in both cases for a finite network the states of all agents will finally reach consensus further if there exist stubborn persons in the population whose opinions do not change over time it was found that for degree independent constant u both qi and qi will converge to fixed proportions which only depend on the distribution of initial obstinate persons and naturally the final equilibrium state will be the coexistence of diverse opinions held by the stubborn people the analytical results were verified by numerical simulations on barabasi albert ba networks the model highlights the influence of high degree agents on the final consensus or coexistence state and captures some realistic features of the diffusion of opinions in social networks

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

separating temporal and topological effects in walk based network centrality the recently introduced concept of dynamic communicability is a valuable tool for ranking the importance of nodes in a temporal network two metrics broadcast score and receive score were introduced to measure the centrality of a node with respect to a model of contagion based on time respecting walks this article examines the temporal and structural factors influencing these metrics by considering a versatile stochastic temporal network model we analytically derive formulas to accurately predict the expectation of the broadcast and receive scores when one or more columns in a temporal edge list are shuffled these methods are then applied to two publicly available data sets and we quantify how much the centrality of each individual depends on structural or temporal influences from our analysis we highlight two practical contributions a way to control for temporal variation when computing dynamic communicability and the conclusion that the broadcast and receive scores can under a range of circumstances be replaced by the row and column sums of the matrix exponential of a weighted adjacency matrix given by the data

evolutionary dynamics on degree heterogeneous graphs the evolution of two species with different fitness is investigated on degree heterogeneous graphs the population evolves either by one individual dying and being replaced by the offspring of a random neighbor voter model dynamics or by an individual giving birth to an offspring that takes over a random neighbor node invasion process dynamics the fixation probability for one species to take over a population of n individuals depends crucially on the dynamics and on the local environment starting with a single fitter mutant at a node of degree k the fixation probability is proportional to k for voter model dynamics and to 1 k for invasion process dynamics

detecting community structure using biased random merging abstract decomposing a network into small modules or communities is beneficial for understanding the structure and dynamics of the network one of the most prominent approaches is to repeatedly join communities together in pairs with the greatest increase in modularity so that a dendrogram that shows the order of joins is obtained then the community structure is acquired by cutting the dendrogram at the levels corresponding to the maximum modularity however there tends to be multiple pairs of communities that share the maximum modularity increment and the greedy agglomerative procedure may only merge one of them although the modularity function typically admits a lot of high scoring solutions the greedy strategy may fail to reach any of them in this paper we propose an enhanced data structure in order to enable diverse choices of merging operations in community finding procedure this data structure is actually a max heap equipped with an extra array that stores the maximum modularity increments and the corresponding community pairs is merged in the next move by randomly sampling elements in this head array additional diverse community structures can be efficiently extracted the head array is designed to host the community pairs corresponding to the most significant increments in modularity so that the modularity structures obtained out of the sampling exhibit high modularity scores that are on the average even greater than what the cnm algorithm produces our method is tested on both real world and computer generated networks

coalescing random walks and voter model consensus times on the torus in mathbb z d voter models on z d and on the torus of site n in z d are considered the second ones get trapped with probability 1 as t tends to infinity at all 0 s or all 1 s the asymptotic growth of these trapping times are examined as n tends to infinity this is done with limit thewems for coalescing random walk systems on the torus of side n in z d

infrequent social interaction can accelerate the spread of a persuasive idea we study the spread of a persuasive new idea through a population of continuous time random walkers in one dimension the idea spreads via social gatherings involving groups of nearby walkers who act according to a biased majority rule after each gathering the group takes on the new idea if more than a critical fraction 1 2 1 2 of them already hold it otherwise they all reject it the boundary of a domain where the new idea has taken hold expands as a traveling wave in the density of new idea holders our walkers move by levy motion and we compute the wave velocity analytically as a function of the frequency of social gatherings and the exponent of the jump distribution when this distribution is sufficiently heavy tailed then counter to intuition the idea can propagate faster if social gatherings are held less frequently when jumps are truncated a critical gathering frequency can emerge which maximizes propagation velocity we explore our model by simulation confirming our analytical results

propagation on networks an exact alternative perspective by generating the specifics of a network structure only when needed on the fly we derive a simple stochastic process that exactly models the time evolution of susceptible infectious dynamics on finite size networks the small number of dynamical variables of this birth death markov process greatly simplifies analytical calculations we show how a dual analytical description treating large scale epidemics with a gaussian approximation and small outbreaks with a branching process provides an accurate approximation of the distribution even for rather small networks the approach also offers important computational advantages and generalizes to a vast class of systems

sequential algorithm for fast clique percolation in complex network research clique percolation introduced by palla der enyi and vicsek nature london 435 814 2005 is a deterministic community detection method which allows for overlapping communities and is purely based on local topological properties of a network here we present a sequential clique percolation algorithm scp to do fast community detection in weighted and unweighted networks for cliques of a chosen size this method is based on sequentially inserting the constituent links to the network and simultaneously keeping track of the emerging community structure unlike existing algorithms the scp method allows for detecting k clique communities at multiple weight thresholds in a single run and can simultaneously produce a dendrogram representation of hierarchical community structure in sparse weighted networks the scp algorithm can also be used for implementing the weighted clique percolation method recently introduced by farkas et al new j phys 9 180 2007 the computational time of the scp algorithm scales linearly with the number of k cliques in the network as an example the method is applied to a product association network revealing its nested community structure

update rules and interevent time distributions slow ordering versus no ordering in the voter model we introduce a general methodology of update rules accounting for arbitrary interevent time iet distributions in simulations of interacting agents we consider in particular update rules that depend on the state of the agent so that the update becomes part of the dynamical model as an illustration we consider the voter model in fully connected random and scale free networks with an activation probability inversely proportional to the time since the last action where an action can be an update attempt an exogenous update or a change of state an endogenous update we find that in the thermodynamic limit at variance with standard updates and the exogenous update the system orders slowly for the endogenous update the approach to the absorbing state is characterized by a power law decay of the density of interfaces observing that the mean time to reach the absorbing state might be not well defined the iet distributions resulting from both update schemes show power law tails

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

small but slow world how network topology and burstiness slow down spreading while communication networks show the small world property of short paths the spreading dynamics in them turns out slow here the time evolution of information propagation is followed through communication networks by using empirical data on contact sequences and the susceptible infected model introducing null models where event sequences are appropriately shuffled we are able to distinguish between the contributions of different impeding effects the slowing down of spreading is found to be caused mainly by weight topology correlations and the bursty activity patterns of individuals

on the p calculus and linear logic abstract we detail abramsky s proofs as processes paradigm for interpreting classical linear logic cll girard 1987 into a synchronous version of the calculus recently proposed by milner 1992 1993 the translation is given at the abstract level of proof structures we give a detailed treatment of information flow in proof nets and show how to mirror various evaluation strategies for proof normalization we also give soundness and completeness results for the process calculus translations of various fragments of cll the paper also gives a self contained introduction to some of the deeper proof theory of cll and its process interpretation

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

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

susceptibility of the ising model on the scale free network with a cayley tree like structure abstract we derive the exact expression for the zero field susceptibility of each spin of the ising model on the scale free sf network having the degree distribution p k k with the cayley tree like structure the system shows that i the zero field susceptibility of a spin in the interior part diverges below the transition temperature of the sf network with the bethe lattice like structure t c for 3 while it diverges at any finite temperature for 3 and ii the surface part diverges below the divergence temperature of the sf network with the cayley tree like structure t s for 3 while it diverges at any finite temperature for 3

spreading dynamics following bursty human activity patterns we study the susceptible infected model with power law waiting time distributions p ensuremath tau ensuremath tau ensuremath ensuremath alpha as a model of spreading dynamics under heterogeneous human activity patterns we found that the average number of new infections n t at time t decays as a power law in the long time limit n t t ensuremath ensuremath beta leading to extremely slow prevalence decay we also found that the exponent in the spreading dynamics ensuremath beta is related to that in the waiting time distribution ensuremath alpha in a way depending on the interactions between agents but insensitive to the network topology these observations are well supported by both the theoretical predictions and the long prevalence decay time in real social spreading phenomena our results unify individual activity patterns with macroscopic collective dynamics at the network level

spectral analysis and slow spreading dynamics on complex networks the susceptible infected susceptible sis model is one of the simplest memoryless systems for describing information or epidemic spreading phenomena with competing creation and spontaneous annihilation reactions the effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean field qmf approximations in scale free networks qmf can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix therefore it can provide predictions on possible rare region effects thus on the occurrence of slow dynamics i compare qmf results of sis with simulations on various large dimensional graphs in particular i show that for erd s renyi graphs this method predicts correctly the occurrence of rare region effects it also provides a good estimate for the epidemic threshold in case of percolating graphs griffiths phases emerge if the graph is fragmented or if we apply a strong exponentially suppressing weighting scheme on the edges the latter model describes the connection time distributions in the face to face experiments in case of a generalized barabasi albert type of network with aging connections strong rare region effects and numerical evidence for griffiths phase dynamics are shown the dynamical simulation results agree well with the predictions of the spectral analysis applied for the weighted adjacency matrices

a semantic measure of the execution time in linear logic we give a semantic account of the execution time i e the number of cut elimination steps leading to the normal form of an untyped mell net we first prove that 1 a net is head normalizable i e normalizable at depth 0 if and only if its interpretation in the multiset based relational semantics is not empty and 2 a net is normalizable if and only if its exhaustive interpretation a suitable restriction of its interpretation is not empty we then give a semantic measure of execution time we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut link from the interpretations of the two cut free nets these results are inspired by similar ones obtained by the first author for the untyped lambda calculus

rare regions of the susceptible infected susceptible model on barabasi albert networks i extend a previous work to susceptible infected susceptible sis models on weighted barab asi albert scale free networks numerical evidence is provided that phases with slow power law dynamics emerge as the consequence of quenched disorder and tree topologies studied previously with the contact process i compare simulation results with spectral analysis of the networks and show that the quenched mean field qmf approximation provides a reliable relatively fast method to explore activity clustering this suggests that qmf can be used for describing rare region effects due to network inhomogeneities finite size study of the qmf shows the expected disappearance of the epidemic threshold ensuremath lambda c in the thermodynamic limit and an inverse participation ratio ensuremath sim 0 25 meaning localization in case of disassortative weight scheme contrarily for the multiplicative weights and the unweighted trees this value vanishes in the thermodynamic limit suggesting only weak rare region effects in agreement with the dynamical simulations strong corrections to the mean field behavior in case of disassortative weights explains the concave shape of the order parameter ensuremath rho ensuremath lambda at the transition point application of this method to other models may reveal interesting rare region effects griffiths phases as the consequence of quenched topological heterogeneities

aging and percolation dynamics in a non poissonian temporal network model we present an exhaustive mathematical analysis of the recently proposed non poissonian activity driven nopad model moinet et al phys rev lett 114 108701 2015 a temporal network model incorporating the empirically observed bursty nature of social interactions we focus on the aging effects emerging from the non poissonian dynamics of link activation and on their effects on the topological properties of time integrated networks such as the degree distribution analytic expressions for the degree distribution of integrated networks as a function of time are derived exploring both limits of vanishing and strong aging we also address the percolation process occurring on these temporal networks by computing the threshold for the emergence of a giant connected component highlighting the aging dependence our analytic predictions are checked by means of extensive numerical simulations of the nopad model

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

microscopic abrams strogatz model of language competition abstract the differential equation of abrams and strogatz for the competition between two languages is compared with agent based monte carlo simulations for fully connected networks as well as for lattices in one two and three dimensions with up to 10 9 agents in the case of socially equivalent languages agent based models and a mean field approximation give grossly different results

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

path lengths correlations and centrality in temporal networks in temporal networks where nodes interact via sequences of temporary events information or resources can only flow through paths that follow the time ordering of events such temporal paths play a crucial role in dynamic processes however since networks have so far been usually considered static or quasistatic the properties of temporal paths are not yet well understood building on a definition and algorithmic implementation of the average temporal distance between nodes we study temporal paths in empirical networks of human communication and air transport although temporal distances correlate with static graph distances there is a large spread and nodes that appear close from the static network view may be connected via slow paths or not at all differences between static and temporal properties are further highlighted in studies of the temporal closeness centrality in addition correlations and heterogeneities in the underlying event sequences affect temporal path lengths increasing temporal distances in communication networks and decreasing them in the air transport network

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

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

voter model on networks and the multivariate beta distribution in elections the vote shares or turnout rates show a strong spatial correlation the logarithmic decay with distance suggests that a two dimensional 2d noisy diffusive equation describes the system based on the study of u s presidential elections data it was determined that the fluctuations of vote shares also exhibit a strong and long range spatial correlation previously it was considered difficult to induce strong and long range spatial correlation of the vote shares without breaking the empirically observed narrow distribution we demonstrate that a voter model on networks shows such a behavior in the model there are many voters in a node who are affected by the agents in the node and by the agents in the linked nodes a multivariate wright fisher diffusion equation for the joint probability density of the vote shares is derived the stationary distribution is a multivariate generalization of the beta distribution in addition we also estimate the equilibrium values and the covariance matrix of the vote shares and obtain a correspondence with a multivariate normal distribution this approach largely simplifies the calibration of the parameters in the modeling of elections

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

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

conditions for the emergence of scaling in the inter event time of uncorrelated and seasonal systems abstract inter event times have been studied across various disciplines in search for correlations in this paper we show analytical and numerical evidence that at the population level a power law can be obtained by assuming poissonian agents with different characteristic times and at the individual level by assuming poissonian agents that change the rates at which they perform an event in a random or deterministic fashion the range in which we expect to see this behavior and the possible deviations from it are studied by considering the shape of the rate distribution

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

measuring burstiness for finite event sequences characterizing inhomogeneous temporal patterns in natural and social phenomena is important to understand underlying mechanisms behind such complex systems and hence even to predict and control them temporal inhomogeneities in event sequences have been described in terms of bursts that are rapidly occurring events in short time periods alternating with long inactive periods the bursts can be quantified by a simple measure called the burstiness parameter which was introduced by goh and barabasi europhys lett 81 48002 2008 euleej0295 507510 1209 0295 5075 81 48002 the burstiness parameter has been widely used due to its simplicity which however turns out to be strongly affected by the finite number of events in the time series as the finite size effects on burstiness parameter have been largely ignored we analytically investigate the finite size effects of the burstiness parameter then we suggest an alternative definition of burstiness that is free from finite size effects and yet simple using our alternative burstiness measure one can distinguish the finite size effects from the intrinsic bursty properties in the time series we also demonstrate the advantages of our burstiness measure by analyzing empirical data sets

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

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

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

non validity of index law in fractional calculus a fractional differential operator with markovian and non markovian properties abstract we presented an analysis of evolutions equations generated by three fractional derivatives namely the riemann liouville caputo fabrizio and the atangana baleanu fractional derivatives for each evolution equation we presented the exact solution for time variable and studied the semigroup principle the riemann liouville fractional operator verifies the semigroup principle but the associate evolution equation does not the caputo fabrizio fractional derivative does not satisfy the semigroup principle but surprisingly the exact solution satisfies very well all the principle of semigroup however the atangana baleanu for small time is the stretched exponential derivative which does not satisfy the semigroup as operators for a large time the atangana baleanu derivative is the same with riemann liouville fractional derivative thus satisfies semigroup principle as an operator the solution of the associated evolution equation does not satisfy the semigroup principle as riemann liouville with the connection between semigroup theory and the markovian processes we found out that the atangana baleanu fractional derivative has at the same time markovian and non markovian processes we concluded that the fractional differential operator does not need to satisfy the semigroup properties as they portray the memory effects which are not always markovian we presented the exact solutions of some evolutions equation using the laplace transform in addition to this we presented the numerical solution of a nonlinear equation and show that the model with the atangana baleanu fractional derivative has random walk for small time we also observed that the mittag leffler function is a good filter than the exponential and power law functions which makes the atangana baleanu fractional derivatives powerful mathematical tools to model complex real world problems

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

mathematical analysis of dengue fever outbreak by novel fractional operators with field data abstract an epidemiological model is proposed for three different types of novel fractional order derivative operators known as the caputo the caputo fabrizio and the atangana baleanu caputo operators the classical model is fractionalized while taking care of the dimensional consistency for each ordinary differential equation of the model the true field data obtained through some authentic sources for the dengue fever outbreak in cape verde islands in 2009 is the major motivation behind analysis of the present research study the proposed model being nonlinear possesses the possibility of having no closed form solution it is therefore existence and uniqueness for the solutions of the models are investigated via fixed point theory the residuals computed via least squares approach for all types of cases under consideration reveal the better performance of the models under fractional order derivative operators proving that the dynamics of the disease dengue virus can be well understood if non local effects are taken into consideration within the model based upon the results obtained the efficiency rates of the fractional order operators under the caputo caputo fabrizio and the atangana baleanu caputo are higher than that of the existing classical model

generalized master equations for non poisson dynamics on networks the traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network this implicitly assumes that the edges are governed by poisson processes which is not typically the case in empirical temporal networks accordingly we examine the effects of non poisson inter event statistics on the dynamics of edges and we apply the concept of a generalized master equation to the study of continuous time random walks on networks we show that this equation reduces to the standard rate equations when the underlying process is poissonian and that its stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate we conduct numerical simulations and also derive analytical results for the stationary solution under the assumption that all edges have the same waiting time distribution we discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature

a poissonian explanation for heavy tails in e mail communication abstract patterns of deliberate human activity and behavior are of utmost importance in areas as diverse as disease spread resource allocation and emergency response because of its widespread availability and use e mail correspondence provides an attractive proxy for studying human activity recently it was reported that the probability density for the inter event time between consecutively sent e mails decays asymptotically as with 1 the slower than exponential decay of the inter event time distribution suggests that deliberate human activity is inherently non poissonian here we demonstrate that the approximate power law scaling of the inter event time distribution is a consequence of circadian and weekly cycles of human activity we propose a cascading nonhomogeneous poisson process that explicitly integrates these periodic patterns in activity with an individual s tendency to continue participating in an activity using standard statistical techniques we show that our model is consistent with the empirical data our findings may also provide insight into the origins of heavy tailed distributions in other complex systems complex systems human activity hypothesis testing point process

analytically solvable autocorrelation function for weakly correlated interevent times long term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions such temporal correlations can be understood not only by heterogeneous interevent times iets but also by correlations between iets in contrast to the role of heterogeneous iets on the autocorrelation function little is known about the effects due to the correlations between iets to rigorously study these effects we derive an analytical form of the autocorrelation function for the arbitrary iet distribution in the case with weakly correlated iets where the farlie gumbel morgenstern copula is adopted for modeling the joint probability distribution function of two consecutive iets our analytical results are confirmed by numerical simulations for exponential and power law iet distributions for the power law case we find a tendency of the steeper decay of the autocorrelation function for the stronger correlation between iets our analytical approach enables us to better understand long term temporal correlations induced by the correlations between iets

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

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

random walks on activity driven networks with attractiveness virtually all real world networks are dynamical entities in social networks the propensity of nodes to engage in social interactions activity and their chances to be selected by active nodes attractiveness are heterogeneously distributed here we present a time varying network model where each node and the dynamical formation of ties are characterized by these two features we study how these properties affect random walk processes unfolding on the network when the time scales describing the process and the network evolution are comparable we derive analytical solutions for the stationary state and the mean first passage time of the process and we study cases informed by empirical observations of social networks our work shows that previously disregarded properties of real social systems such as heterogeneous distributions of activity and attractiveness as well as the correlations between them substantially affect the dynamical process unfolding on the network

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

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

the role of power decay exponential decay and mittag leffler function s waiting time distribution application of cancer spread abstract the use of fractional differential operators riemann liouvillend caputo to model real world problems leads to a waiting time distribution governs by the power law this operators obeys the so called index law of the classical mechanic and also possess an artificial singularity around the initial or starting point of the process there are failures or limitations attached to the power law fractional differential operator its capacity of obeying the index law leads waiting time distribution with no crossover behavior which is a limitation for describing real world with statistical resetting and also spread or diffusion of some material in different scales or some kind of anomalous spread like that of cancer the second big problem is the singularity induced by the power law decay function this singularity leads bad memory at the zero starting point at this point the prediction suggested by the model are not reliable as they can sometime misleading due to influence of the artificial singularity the introduction of non singular kernel is welcome in the field of fractional calculus as they brought news weapons in this field to accurately model real world problems in particular diffusion or spread in different states we welcome them in fractional calculus for two reasons they do not lead to inclusion of artificial singularity into the mathematical models in this case the history of the dynamical process the second reason is their ability of describing the two different waiting times distribution which is an ideal waiting time distribution as such is observed in many biological phenomena such as the spread of cancer this crossover behavior of this new fractional differential operators is due to their capacity of not obeying the index law imposed in fractional calculus the numerical simulations presented in this paper are in perfect agreement with the conclusion of the study done in atangana 2018 where it was proved that the index law is irrelevant in fractional calculus as the most important fact is the effect of the fractional differential operator to the associate evolution equation we conclude in this paper that fractional differential operators not obeying index law are suitable mathematical tools to model real world problems

head linear reduction and pure proof net extraction abstract proof net calculus introduced in girard 1987 has been extended in the paradigm of scott s domains equation d d d which generates a logical point of view for pure calculus in donos 1989 methodologically speaking in this paper the proof theoretic counterpart of bohm s theorem given in bohm 1968 for pure calculus is proposed as extension of the curry howard paradigm technically speaking as the extraction of a subterm using the reduction is possible also subnet extraction can be internalized by cut elimination using proof nets realizes managing and better understanding the procedure of extraction

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

what big data tells sampling the social network by communication channels big data has become the primary source of understanding the structure and dynamics of the society at large scale the network of social interactions can be considered as a multiplex where each layer corresponds to one communication channel and the aggregate of all of them constitutes the entire social network however usually one has information only about one of the channels or even a part of it which should be considered as a subset or sample of the whole here we introduce a model based on a natural bilateral communication channel selection mechanism which for one channel leads to consistent changes in the network properties for example while it is expected that the degree distribution of the whole social network has a maximum at a value larger than one we get a monotonically decreasing distribution as observed in empirical studies of single channel data we also find that assortativity may occur or get strengthened due to the sampling method we analyze the far reaching consequences of our findings

mathematical modelling and analysis of love dynamics a fractional approach abstract a range of new fractional order love dynamics is modelled in this paper using the caputo power law caputo fabrizio exponential decay law and atangana baleanu mittag leffler law fractional operators we utilize the new fractional adams bashforth schemes for the approximation of these derivatives this method was developed with the standard differentiation technique by applying the fundamental theorem of calculus and taking the difference of two times levels at t t n t n 1 the stability analysis applies the theory of classical order differential equations and dynamical system the existence and uniqueness of solution of fractional dynamics is proved by adopting the fixed point theorem numerical results presented for various values justify the theoretical findings it was observed that modelling of interpersonal and romantic love affairs with fractional derivative could exhibit some strange emotional attractors

new approaches to the fractional dynamics of schistosomiasis disease model abstract in this paper schistosomiasis fractional order dynamic model is examined via exponential law kernel sense and mittag leffler kernel in liouville caputo sense some special solutions for two operators are obtained using the iterative scheme through laplace transform and sumudu picard integration technique the uniqueness and existence of solution for both operators are established the numerical solutions for both operators approve that the desirable results can be obtained when alpha value is less than 1

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

voter models on heterogeneous networks we study simple interacting particle systems on heterogeneous networks including the voter model and the invasion process these are both two state models in which in an update event an individual changes state to agree with a neighbor for the voter model an individual imports its state from a randomly chosen neighbor here the average time t n to reach consensus for a network of n nodes with an uncorrelated degree distribution scales as n ensuremath mu 1 2 ensuremath mu 2 where ensuremath mu k is the k text th moment of the degree distribution quick consensus thus arises on networks with broad degree distributions we also identify the conservation law that characterizes the route by which consensus is reached parallel results are derived for the invasion process in which the state of an agent is exported to a random neighbor we further generalize to biased dynamics in which one state is favored the probability for a single fitter mutant located at a node of degree k to overspread the population char22 the fixation probability char22 is proportional to k for the voter model and to 1 k for the invasion process

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

an algebraic view of the bohm out technique using an algebraic representation of closed normal forms in calculus the bohm s theorem is rephrased as an equality predicate between elements of a term algebra the presented algebraic interpretation gives new insight into the bohm out technique and allows for original applications of the method

a new approach for solving integro differential equations of variable order abstract in this work we consider a class of nonlinear integro differential equations of variable order existence uniqueness and stability results are discussed for solving the considered equations operational matrices based on the shifted legendre polynomials are used first we approximate the unknown function and its derivatives in terms of the shifted legendre polynomials then by substituting these approximations into the equation and using the properties of the shifted legendre polynomials together with the collocation points the main problem is reduced to a system of nonlinear algebraic equations an error bound is proved for the approximate solution obtained by the proposed method finally some illustrative examples are included to show the efficiency and accuracy of the proposed scheme