aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

lower and upper bounds for the allocation problem and other nonlinear optimization problems we demonstrate the impossibility of strongly polynomial algorithms for the allocation problem in the comparison model and in the algebraic tree computation model that follow from lower bound results consequently there are no strongly polynomial algorithms for nonlinear concave separable optimization over a totally unimodular constraint matrix this is in contrast to the case when the objective is linear we present scaling based algorithms that use a greedy algorithm as a subroutine the algorithms are polynomial for the allocation problem and its extensions and are also optimal for the sample allocation problem and the generalized upper bounds allocation problem in that the complexity meets the lower bound derived from the comparison model for other extensions of the allocation problem the scaling based algorithms presented here are the fastest known these algorithms are also polynomial time algorithms for solving with e accuracy the allocation problem and its extension in continuous variables

opinion dynamics of modified hegselmann krause model in a group based population with heterogeneous bounded confidence abstract continuous opinion dynamics in a group based population with heterogeneous bounded confidences is considered in this paper a slightly modified hegselmann krause model is proposed and agents are classified into three categories open minded moderate minded and closed minded agents while the whole population is divided into three subgroups accordingly we study how agents of each category and the population size can affect opinion dynamics it is observed that the number of final opinion clusters is dominated by the closed minded agents open minded agents cannot contribute to forming opinion consensus and the existence of open minded agents may diversify the final opinions instead for the fixed population size and proportion of closed minded agents the relative size of the largest final opinion cluster varies along concave parabola like curve as the proportion of open minded agents increases and there is a tipping point when the number of open minded agents is almost equal to that of moderate minded agents for the fixed proportion of the three categories in the population as the population size becomes larger the number of final opinion clusters will reach a plateau some of the results are different from the previous studies

incremental stochastic subgradient algorithms for convex optimization this paper studies the effect of stochastic errors on two constrained incremental subgradient algorithms the incremental subgradient algorithms are viewed as decentralized network optimization algorithms as applied to minimize a sum of functions when each component function is known only to a particular agent of a distributed network first the standard cyclic incremental subgradient algorithm is studied in this the agents form a ring structure and pass the iterate in a cycle when there are stochastic errors in the subgradient evaluations sufficient conditions on the moments of the stochastic errors are obtained that guarantee almost sure convergence when a diminishing step size is used in addition almost sure bounds on the algorithm s performance with a constant step size are also obtained next the markov randomized incremental subgradient method is studied this is a noncyclic version of the incremental algorithm where the sequence of computing agents is modeled as a time nonhomogeneous markov chain such a model is appropriate for mobile networks as the network topology changes across time in these networks convergence results and error bounds for the markov randomized method in the presence of stochastic errors for diminishing and constant step sizes are obtained

on the gegenbauer transformation the gegenbauer transformation g k is defined for 1 2 k 0 1 2 by where if being the gegenbauer polynomial of index and degree k and l 0 k is the tchebichef polynomial of degree k the transformation is studied on the spaces l p denned by the norm and its boundedness and range on these spaces is determined and inversion formulae are found

an algorithmic approach to collective behavior the emergence of collective structure from the decentralized interaction of autonomous agents remains with notable exceptions a mystery while powerful tools from dynamics and statistical mechanics have been brought to bear sometimes with great success an algorithmic perspective has been lacking viewing collective behavior through the lens of natural algorithms offers potential benefits this article examines the merits and challenges of an algorithmic approach to the emergence of collective order

inertial hegselmann krause systems we derive an energy bound for inertial hegselmann krause hk systems which are a variant of the classic hk model that allows agents to change weights as they please at each step we use the bound to prove the convergence of hk systems with static agents which settles a widely believed conjecture this paper also introduces anchored hk systems and show their equivalence to the symmetric heterogeneous model

locally monotonic regression the concept of local monotonicity appears in the study of the set of root signals of the median filter and provides a measure of the smoothness of the signal the median filter is a suboptimal smoother under this measure of smoothness since a filter pass does necessarily yield a locally monotonic output even if a locally monotonic output does result there is no guarantee that it will possess other desirable properties such as optimal similarity to the original signal locally monotonic regression is a technique for the optimal smoothing of finite length discrete real signals under such a criterion a theoretical framework in which the existence of locally monotonic regression is proved and algorithms for their computation are given regression is considered as an approximation problem in r sub n the criterion of approximation is derived from a semimetric and the approximating set is the collection of signals sharing the property of being locally monotonic

opinion evolution analysis for short range and long range deffuant weisbuch models abstract in this paper we propose and then analyze two generalized deffuant weisbuch dw models the generalized models extend the conventional dw model by taking multiple choices in two different ways first we demonstrate the almost sure convergence of the agent opinions for the short range multi choice dw dynamics when only the opinions within confidence regions may be count in then we analyze dynamical behavior about the long range multi choice dw model when some opinions out of the confidence ranges are considered with a weighted combination moreover both theoretical and simulation results show that the dynamical behaviors of the two models are totally different

a fast scaling algorithm for minimizing separable convex functions subject to chain constraints we consider the problem of minimizing j nc j x j subject to the following chain constraints x 1 x 2 x 3 x n where c j x j is a convex function of x j for each j n 1 2 n this problem is a generalization of theisotonic regression problems with complete order an important class of problems in regression analysis that has been examined extensively in the literature we refer to this problem as thegeneralized isotonic regression problem in this paper we focus on developing a fast scaling algorithm to obtain an integer solution of the generalized isotonic regression problem letu denote the difference between an upper bound on an optimal value of x n and a lower bound on an optimal value of x 1 under the assumption that the evaluation of any function c j x j takeso 1 time we show that the generalized isotonic regression problem can be solved ino n logu time this improves by a factor of n the previous best running time ofo n 2 logu to solve the same problem in addition when our algorithm is specialized to theisotonic median regression problem where c j x j c j x j a ji for specified values ofc js anda js the algorithm obtains a real valued optimal solution ino n logn time this time bound matches the best available time bound to solve the isotonic median regression problem but our algorithm uses simpler data structures and may be easier to implement

nearly isotonic regression we consider the problem of approximating a sequence of data points with a nearly isotonic or nearly monotone function this is formulated as a convex optimization problem that yields a family of solutions with one extreme member being the standard isotonic regression fit we devise a simple algorithm to solve for the path of solutions which can be viewed as a modified version of the well known pool adjacent violators algorithm and computes the entire path in o n log n operations n being the number of data points in practice the intermediate fits can be used to examine the assumption of monotonicity nearly isotonic regression admits a nice property in terms of its degrees of freedom at any point along the path the number of joined pieces in the solution is an unbiased estimate of its degrees of freedom we also extend the ideas to provide nearly convex approximations

pairwise stochastic bounded confidence opinion dynamics heavy tails and stability unlike traditional graph based linear dynamics where agents exchange opinions with their neighbors in a static social graph regardless of their differences in opinions the bounded confidence opinion dynamics models exchange between agents with similar opinions we generalize the bounded confidence opinion dynamics model by incorporating pairwise stochastic interactions probabilistic influencing based on opinion differences and the self or endogenous evolutions of the agent opinions which are represented by random processes the opinion exchanges resulting from influencing have pairwise contraction effects whereas endogenous motions have an expansive effect for instance of a diffusive nature we analytically characterize the conditions under which this stochastic dynamics is stable in an appropriate sense in the diffusive case the presence of heavy tailed influence functions with a pareto exponent of 2 is critical for stability for a pair of agents an influence function maps opinion differences to probabilities of influence moreover this model sheds light on dynamics that combine aspects of graph based and bounded confidence dynamics

a stabilization theorem for dynamics of continuous opinions abstract a stabilization theorem for processes of opinion dynamics is presented the theorem is applicable to a wide class of models of continuous opinion dynamics based on averaging like the models of hegselmann krause and weisbuch deffuant the analysis detects self confidence as a driving force of stabilization

randomized optimal consensus of multi agent systems in this paper we formulate and solve a randomized optimal consensus problem for multi agent systems with stochastically time varying interconnection topology the considered multi agent system with a simple randomized iterating rule achieves an almost sure consensus meanwhile solving the optimization problem min z r email protected i 1 nf i z in which the optimal solution set of objective function f i can only be observed by agent i itself at each time step simply determined by a bernoulli trial each agent independently and randomly chooses either taking an average among its neighbor set or projecting onto the optimal solution set of its own optimization component both directed and bidirectional communication graphs are studied connectivity conditions are proposed to guarantee an optimal consensus almost surely with proper convexity and intersection assumptions the convergence analysis is carried out using convex analysis we compare the randomized algorithm with the deterministic one via a numerical example the results illustrate that a group of autonomous agents can reach an optimal opinion by each node simply making a randomized trade off between following its neighbors or sticking to its own opinion at each time step

minimizers of cost functions involving nonsmooth data fidelity terms application to the processing of outliers we present a theoretical study of the recovery of an unknown vector x in mathbb r p such as a signal or an image from noisy data y in mathbb r q by minimizing with respect to x a regularized cost function cal f x y psi x y alpha phi x where psi is a data fidelity term phi is a smooth regularization term and alpha 0 is a parameter typically psi x y ax y 2 where a is a linear operator the data fidelity terms psi involved in regularized cost functions are generally smooth functions only a few papers make an exception to this and they consider restricted situations nonsmooth data fidelity terms are avoided in image processing in spite of this we consider both smooth and nonsmooth data fidelity terms our goal is to capture essential features exhibited by the local minimizers of regularized cost functions in relation to the smoothness of the data fidelity term in order to fix the context of our study we consider psi x y sum i psi a i tx y i where a i t are the rows of a and psi is cal c m on mathbb r setminus 0 we show that if psi 1 0 psi 1 0 then typical data y give rise to local minimizers hat x of cal f y which fit exactly a certain number of the data entries there is a possibly large set hat h of indexes such that a i t hat x y i for every i in hat h in contrast if psi is smooth on mathbb r for almost every y the local minimizers of cal f y do not fit any entry of y thus the possibility that a local minimizer fits some data entries is due to the nonsmoothness of the data fidelity term this is a strong mathematical property which is useful in practice by way of application we construct a cost function allowing aberrant data outliers to be detected and to be selectively smoothed our numerical experiments advocate the use of nonsmooth data fidelity terms in regularized cost functions for special purposes in image and signal processing

on approximations and ergodicity classes in random chains we study the limiting behavior of a random dynamic system driven by a stochastic chain our interest is in the chains that are not necessarily ergodic but are decomposable into ergodic classes to investigate the conditions under which the ergodic classes of a model can be identified we introduce and study an l1 approximation and infinite flow graph of the model we show that the l1 approximations of random chains preserve certain limiting behavior using the l1 approximations we show how the connectivity of the infinite flow graph is related to the structure of the ergodic groups of the model our main result of this paper provides conditions under which the ergodicity groups of the model can be identified by considering the connected components in the infinite flow graph we provide two applications of our main result to random networks namely broadcast over time varying networks and networks with random link failure

adaptive decision dynamics bifurcations multistability and chaos abstract in this paper we propose a model describing the dynamical process of decision and opinion formation of two economic homogeneous interacting and boundedly rational agents the decisional process represented in our model is given by an adaptive adjustment mechanism in which two agents take into account the difference between their own opinion and the opinion of the other agent the smaller that difference the larger the weight given to the comparison of the opinions by means of an auxiliary variable describing the distance between the opinions we obtain a one dimensional dynamical system for which we investigate via analytical and numerical tools the stability of the unique steady state its bifurcations as well as the existence of a globally absorbing interval and of chaotic dynamics we also investigate multistability phenomena i e the presence of coexisting attractors finally we relax the assumption of homogeneity between agents and we show that there is a strong correspondence between the dynamic behaviors in the scenarios with and without homogeneity

opinion dynamics with decaying confidence application to community detection in graphs we study a class of discrete time multi agent systems modeling opinion dynamics with decaying confidence we consider a network of agents where each agent has an opinion at each time step the agents exchange their opinion with their neighbors and update it by taking into account only the opinions that differ from their own less than some confidence bound this confidence bound is decaying an agent gives repetitively confidence only to its neighbors that approach sufficiently fast its opinion essentially the agents try to reach an agreement with the constraint that it has to be approached no slower than a prescribed convergence rate under that constraint global consensus may not be achieved and only local agreements may be reached the agents reaching a local agreement form communities inside the network in this paper we analyze this opinion dynamics model we show that communities correspond to asymptotically connected components of the network and give an algebraic characterization of communities in terms of eigenvalues of the matrix defining the collective dynamics finally we apply our opinion dynamics model to address the problem of community detection in graphs we propose a new formulation of the community detection problem based on eigenvalues of normalized laplacian matrix of graphs and show that this problem can be solved using our opinion dynamics model we consider three examples of networks and compare the communities we detect with those obtained by existing algorithms based on modularity optimization we show that our opinion dynamics model not only provides an appealing approach to community detection but that it is also effective

the convergence of bird flocking we bound the time it takes for a group of birds to stabilize in a standard flocking model each bird averages its velocity with its neighbors lying within a fixed radius we resolve the worst case complexity of this natural algorithm by providing asymptotically tight bounds on the time to equilibrium we reduce the problem to two distinct questions in computational geometry and circuit complexity

min max time consensus tracking with communication guarantee in this paper a collection of double integrator agents with bounded inputs is considered communication is possible between any two agents only if the inter agent distance is less than a fixed threshold a special node referred to as the leader generates an unknown reference trajectory to which all the other agents are required to converge in the shortest possible time assuming the initial communication graph to be connected a directed spanning tree rooted at the leader is identified using a local algorithm the dynamics of any two agents connected by an edge in the selected tree are modeled as a time optimal pursuit evasion game while maintaining the inter agent communication link using the corresponding feedback saddle point strategies local min max time control laws for each pair of agents are formulated for the selected tree the proposed collection of local min max time control strategies is shown to be the communication preserving global min max time strategy

reaching consensus on rumors abstract an important contribution in sociophysics is the galam s model of rumors spreading this model provides an explanation of rumors spreading in a population and explains some interesting social phenomena such as the diffusion of hoaxes in this paper the model has been reformulated as a markov process highlighting the stochastic nature of the phenomena this formalization allows us to derive conditions for consensus to be reached and for the existence of some interesting phenomena such as the emergence of impasses the proposed formulation allows a deeper and more comprehensive analysis of the diffusion of rumors

convergence properties of the heterogeneous deffuant weisbuch model the deffuant weisbuch dw model is a bounded confidence opinion dynamics model that has attracted much recent interest despite its simplicity and appeal the dw model has proved technically hard to analyze and its most basic convergence properties easy to observe numerically are only conjectures this paper solves the convergence problem for the heterogeneous dw model with the weighting factor not less than 1 2 we establish that for any positive confidence bounds and initial values the opinion of each agent will converge to a limit value almost surely and the convergence rate is exponential in mean square moreover we show that the limiting opinions of any two agents either are the same or have a distance larger than the confidence bounds of the two agents finally we provide some sufficient conditions for the heterogeneous dw model to reach consensus

isotonic median regression a linear programming approach the isotonic median regression problem arises in statistics it is known that the isotonic median regression problem with respect to a complete order may be solved by a pool adjacent violators algorithm in this paper we show that this algorithm is a dual method for solving a linear programming formulation of the problem the linear programming approach provides additional insight into the algorithm as well as a simple proof of its validity we also analyze the computational complexity of the algorithm and discuss its significance from the standpoint of linear programming theory

the total s energy of a multiagent system we introduce the total s energy of a multiagent system with time dependent links this provides a new analytical perspective on bidirectional agreement dynamics which we use to bound the convergence rates of dynamical systems for synchronization flocking opinion dynamics and social epistemology

opinion fluctuations and disagreement in social networks we study a tractable opinion dynamics model that generates long run disagreements and persistent opinion fluctuations our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents 1 regular agents who update their beliefs according to information that they receive from their social neighbors and 2 stubborn agents who never update their opinions and might represent leaders political parties or media sources attempting to influence the beliefs in the rest of the society when the society contains stubborn agents with different opinions the belief dynamics never lead to a consensus among the regular agents instead beliefs in the society fail to converge almost surely the belief profile keeps on fluctuating in an ergodic fashion and it converges in law to a nondegenerate random vector the structure of the graph describing the social network and the location of the stubborn agents within it shape the opinion dynamics the expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the kolmogorov backward equation of a continuous time markov chain on the graph with absorbing states corresponding to the stubborn agents and hence to converge to a harmonic vector with every regular agent s value being the weighted average of its neighbors values and boundary conditions corresponding to the stubborn agents beliefs expected cross products of the agents beliefs allow for a similar characterization in terms of coupled markov chains on the graph describing the social network we prove that in large scale societies which are highly fluid meaning that the product of the mixing time of the markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large a condition of homogeneous influence emerges whereby the stationary beliefs marginal distributions of most of the regular agents have approximately equal first and second moments

on the range of the hankel and extended hankel transforms abstract the range of the hankel and extended hankel transforms on some spaces of functions is described the paley wiener theorem for the hankel transform is also obtained

strategic influence in social networks we consider a model of influence with a set of non strategic agents and two strategic agents the non strategic agents have initial opinions and are linked through a simply connected network they update their opinions as in the degroot model the two strategic agents have fixed opinions 1 and 0 respectively and are characterized by the magnitude of the impact they can exert on non strategic agents each strategic agent forms a link with one non strategic agent in order to alter the average opinion that eventually emerges in the network this procedure defines a zero sum game whose players are the two strategic agents and whose strategy set is the set of non strategic agents we focus on the existence and the characterization of equilibria in pure strategy in this setting simple examples show that the existence of a pure strategy equilibrium does depend on the structure of the network the characterization of equilibrium we obtain emphasizes on the one hand the influenceability of target agents and on the other hand their centrality whose natural measure in our context defines a new concept related to betweenness centrality that we call intermediacy we also show that in the case where the two strategic agents have the same impact symmetric equilibria emerge as natural solutions whereas in the case where the impacts are uneven the strategic players generally have differentiated equilibrium targets the high impacts agent focusing on centrality and the low impact agent on influenceability

product of random stochastic matrices the paper deals with the convergence properties of the products of random row stochastic matrices the limiting behavior of such products is studied from a dynamical system point of view in particular by appropriately defining a dynamic associated with a given sequence of random row stochastic matrices we prove that the dynamics admits a class of time varying lyapunov functions including a quadratic one then we discuss a special class of stochastic matrices a class p which plays a central role in this work we then study cut balanced chains and using some geometric properties of these chains we characterize the stability of a subclass of cut balanced chains as a special consequence of this stability result we obtain an extension of a central result in the non negative matrix theory stating that for any aperiodic and irreducible row stochastic matrix a the limit limk ak exists and it is a rank one stochastic matrix we show that a generalization of this result holds not only for sequences of stochastic matrices but also for independent random sequences of such matrices

establishing consistent and realistic reorder intervals in production distribution systems the objective of this paper is to present a model and algorithm that can be used to find consistent and realistic reorder intervals for each item in large scale production distribution systems we assume such systems can be represented by directed acyclic graphs demand for each end item is assumed to occur at a constant and continuous rate production is instantaneous and no backorders are allowed both fixed setup costs and echelon holding costs are charged at each stage we limit our attention to nested and stationary policies furthermore we restrict the reorder interval for each stage to be a power of 2 times a base planning period the model that results from these assumptions is an integer nonlinear programming problem the optimal solution can be found using the proposed algorithm which is a polynomial time algorithm a real world example is given to illustrate the procedure

reaching a consensus in a dynamically changing environment a graphical approach this paper presents new graph theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments the concepts of rooted strongly rooted and neighbor shared are defined and conditions are derived for compositions of sequences of directed graphs to be of these types the graph of a stochastic matrix is defined and it is shown that under certain conditions the graph of a sarymsakov matrix and a rooted graph are one and the same as an illustration of the use of the concepts developed in this paper graph theoretic conditions are obtained which address the convergence question for the leaderless version of the widely studied vicsek consensus problem

diffusive influence systems influence systems seek to model how influence broadly defined spreads across a dynamic network we build a general analytical framework which we then use to prove that while turing complete influence dynamics of the diffusive type is almost surely asymptotically periodic in addition to resolving the dynamics of a widely used family of multiagent systems we introduce a general renormalization method for the bifurcation analysis of multiagent systems

on integral transformations with g function kernels integral transformations t of the formal form where g is a meijer g function are studied on the spaces r of functions f such that results are obtained on the boundedness representation and range of the transformations and on when they have an inverse of the same formal form

well posedness of the limiting equation of a noisy consensus model in opinion dynamics abstract this paper establishes the global well posedness of the nonlinear fokker planck equation for a noisy version of the hegselmann krause model the equation captures the mean field behavior of a classic multiagent system for opinion dynamics we prove the global existence uniqueness nonnegativity and regularity of the weak solution we also exhibit a global stability condition which delineates a forbidden region for consensus formation this is the first nonlinear stability result derived for the hegselmann krause model

a faster algorithm solving a generalization of isotonic median regression and a class of fused lasso problems many applications in the areas of production signal processing economics bioinformatics and statistical learning involve a given partial order on certain parameters and a set of noisy observations of the parameters the goal is to derive estimated values of the parameters that satisfy the partial order that minimize the loss of deviations from the given observed values a prominent application of this setup is the well known isotonic regression problem where the partial order is a total order a commonly studied isotonic regression problem is the isotonic median regression imr where the loss function is a sum of absolute value functions on the deviations of the estimated values from the observation values we present here the most efficient algorithm to date for a generalization of imr with any convex piecewise linear loss function as well as for variant models that include ell 1 norm penalty separation regularization functions in the objective on the violation of the total order constraints suc