ieee transactions on signal processing, vol. 61, no. 22...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013 5675

Distributed Adaptive Networks: A GraphicalEvolutionary Game-Theoretic View

Chunxiao Jiang, Member, IEEE, Yan Chen, Member, IEEE, and K. J. Ray Liu, Fellow, IEEE

Abstract—Distributed adaptive filtering has been consideredas an effective approach for data processing and estimation overdistributed networks. Most existing distributed adaptive filteringalgorithms focus on designing different information diffusionrules, regardless of the nature evolutionary characteristic of adistributed network. In this paper, we study the adaptive networkfrom the game theoretic perspective and formulate the distributedadaptive filtering problem as a graphical evolutionary game. Withthe proposed formulation, the nodes in the network are regardedas players and the local combiner of estimation information fromdifferent neighbors is regarded as different strategies selection.We show that this graphical evolutionary game framework is verygeneral and can unify the existing adaptive network algorithms.Based on this framework, as examples, we further propose twoerror-aware adaptive filtering algorithms. Moreover, we usegraphical evolutionary game theory to analyze the informationdiffusion process over the adaptive networks and evolutionarilystable strategy of the system. Finally, simulation results are shownto verify the effectiveness of our analysis and proposed methods.

Index Terms—Adaptive filtering, graphical evolutionary game,distributed estimation, adaptive networks, data diffusion.

I. INTRODUCTION

R ECENTLY, the concept of adaptive filter networkderived from the traditional adaptive filtering was

emerging, where a group of nodes cooperatively estimatesome parameters of interest from noisy measurements [1].Such a distributed estimation architecture can be applied tomany scenarios, such as wireless sensor networks for envi-ronment monitoring, wireless Ad-hoc networks for militaryevent localization, distributed cooperative sensing in cognitiveradio networks and so on [2], [3]. Compared to the classicalcentralized architecture, the distributed one is not only morerobust when the center node may be dysfunctional, but alsomore flexible when the nodes are with mobility. Therefore,distributed adaptive filter network has been considered as

Manuscript received April 22, 2013; revised July 01, 2013 and August 23,2013; accepted August 24, 2013. Date of current version October 14, 2013. Theassociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Hing Cheung So. This work was partly funded by project61371079, 61271267, and 60932005 supported by NSFC China, PostdoctoralScience Foundation funded project.C. Jiang is with the Department of Electrical and Computer Engineering, Uni-

versity of Maryland, College Park, MD 20742 USA, and also with the Depart-ment of Electronic Engineering, Tsinghua University, Beijing 100084, China(e-mail: [email protected]).Y. Chen and K. J. Ray Liu are with the Department of Electrical and

Computer Engineering, University of Maryland, College Park, MD 20742USA (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2013.2280444

an effective approach for the implementation of data fusion,diffusion and processing over distributed networks [4].In a distributed adaptive filter network, at every time instant, node receives a set of data that satisfies a linearregression model as follow

(1)

where is a deterministic but unknown vector,is a scalar measurement of some random process is the

regression vector at time with zero mean and covariancematrix , and is the random noisesignal at time with zero mean and variance . Note that theregression data and measurement process are temporallywhite and spatially independent, respectively and mutually. Theobjective for each node is to use the data set toestimate parameter .In the literatures, many distributed adaptive filtering algo-

rithms have been proposed for the estimation of parameter .The incremental algorithms, in which node updates , i.e.,the estimation of , through combining the observed data setsof itself and node , were proposed, e.g., the incrementalLMS algorithm [5]. Unlike the incremental algorithms, the dif-fusion algorithms allow node to combine the data sets from allneighbors, e.g., diffusion LMS [6], [7] and diffusion RLS [8].Besides, the projection-based adaptive filtering algorithms weresummarized in [9], e.g., the projected subgradient algorithm[10] and the combine-project-adapt algorithm [11]. In [12], theauthors considered the node’s mobility and analyzed the mobileadaptive networks.While achieving promising performance, these traditional

distributed adaptive filtering algorithms mainly focused ondesigning different information combination rules or diffu-sion rules among the neighborhood by utilizing the networktopology information and/or nodes’ statistical information.For example, the relative degree rule considers the degreeinformation of each node [8], and the relative degree-variancerule further incorporates the variance information of each node[6]. However, most of the existing algorithms are somehowintuitively designed to achieve some specific objective, sort oflike bottom-up approaches to the distributed adaptive networks.There is no existing work that offers a design philosophy toexplain why combination and/or diffusion rules are developedand how they are related in a unified view. Is there a generalframework that can reveal the relationship among the existingrules and provide fundamental guidance for better design ofdistributed adaptive filtering algorithms? In our quest to answerthe question, we found that in essence the parameter updatingprocess in distributed adaptive networks follows similarly the

1053-587X © 2013 IEEE

5676 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 22, NOVEMBER 15, 2013

evolution process in natural ecological systems. Therefore,based on the graphical evolutionary game, in this paper, wepropose a general framework that can offer a unified view ofexisting distributed adaptive algorithms, and provide possibleclues for new future designs. Unlike the traditional bottom-upapproaches that focus on some specific rules, our frameworkprovide a top-down design philosophy to understand the fun-damental relationship of distributed adaptive networks.The main contributions of this paper are summarized as fol-

lows.1) We propose a graphical evolutionary game theoretic frame-work for the distributed adaptive networks, where nodes inthe network are regarded as players and the local combi-nation of estimation information from different neighborsis regarded as different strategies selection. We show thatthe proposed graphical evolutionary theoretic frameworkcan unify existing adaptive filtering algorithms as specialcases.

2) Based on the proposed framework, as examples, we fur-ther design two simple error-aware distributed adaptive fil-tering algorithms. When the noise variance is unknown,our proposed algorithm can achieve similar performancecompared with existing algorithms but with lower com-plexity, which immediately shows the advantage of theproposed general framework.

3) Using the graphical evolutionary game theory, we analyzethe information diffusion process over the adaptive net-work, and derive the diffusion probability of informationfrom good nodes.

4) We prove that the strategy of using information from goodnodes is evolutionarily stable strategy either in completegraphs or incomplete graphs.

The rest of this paper is organized as follows. We sum-marize the existing works in Section II. In Section III, wedescribe in details how to formulate the distributed adaptivefiltering problem as a graphical evolutionary game. We thendiscuss the information diffusion process over the adaptivenetwork in Section IV, and further analyze the evolutionarilystable strategy in Section V. Simulation results are shown inSection VI. Finally, we draw conclusions in Section VII.

II. RELATED WORKS

Let us consider an adaptive filter network with nodes. Ifthere is a fusion center that can collect information from allnodes, then global (centralized) optimization methods can beused to derive the optimal updating rule for the parameter ,where is a deterministic but unknown vector for es-timation, as shown in the left part of Fig. 1. For example, inthe global LMS algorithm, the parameter updating rule can bewritten as [6]

(2)

where is the step size and denotes complex conjugationoperation. With (2), we can see that the centralized LMS algo-rithm requires the information of across the wholenetwork, which is generally impractical. Moreover, such a cen-

Fig. 1. Left: centralized model. Right: distributed model.

tralized architecture highly relies on the fusion center and willcollapse when the fusion center is dysfunctional or some datalinks are disconnected.If there is no fusion center in the network, then each node

needs to exchange information with the neighbors to update theparameter as shown in the right part of Fig. 1. In the literature,several distributed adaptive filtering algorithms have been in-troduced, such as distributed incremental algorithms [5], dis-tributed LMS [6], [7], and projection-based algorithms [10],[11]. These distributed algorithms are based on the classicaladaptive filtering algorithms, where the difference is that nodescan use information from neighbors to estimate the parameter. Taking one of the distributed LMS algorithms, Adapt-then-

Combine Diffusion LMS (ATC) [6], as an example, the param-eter updating rule for node is

(3)

where denotes the neighboring nodes set of node (includingnode itself), and are linear weights satisfying the fol-lowing conditions

(4)

In a practical scenario, since the exchange of full raw dataamong neighbors is costly, the weight is usually

set as , if , as in [6]. In such a case, for nodewith degree (including node itself, i.e., the cardinality of set) and neighbor set , we can write the general

parameter updating rule as

(5)

where can be any adaptive filtering algorithm, e.g.,for the LMS algo-

rithm, represents some specific linear combinationrule. The (5) gives a general form of existing distributed adap-tive filtering algorithms, where the combination rulemainly determines the performance. Table I summarizes theexisting combination rules, where for all rules ,if .

JIANG et al.: DISTRIBUTED ADAPTIVE NETWORKS 5677

TABLE IDIFFERENT COMBINATION RULES

From Table I, we can see that the weights of the first fourcombination rules are purely based on the network topology.The disadvantage of such topology-based rules is that, they aresensitive to the spatial variation of signal and noise statisticsacross the network. The relative degree-variance rule showsbetter mean-square performance than others, which, however,requires the knowledge of all neighbors’ noise variances. As dis-cussed in Section I, all these distributed algorithms are only fo-cusing on designing the combination rules. Nevertheless, a dis-tributed network is just like a natural ecological system and thenodes are just like individuals in the system, which may sponta-neously follow some nature evolutionary rules, instead of somespecific artificially predefined rules. Besides, although variouskinds of combination rules have been developed, there is no gen-eral framework which can reveal the unifying fundamentals ofdistributed adaptive filtering problems. In the sequel, we willuse graphical evolutionary game theory to establish a generalframework to unify existing algorithms and give insights of thedistributed adaptive filtering problem.

III. GRAPHICAL EVOLUTIONARY GAME FORMULATION

A. Introduction of Graphical Evolutionary Game

Evolutionary game theory (EGT) is originated from thestudy of ecological biology [18], which differs from the clas-sical game theory by emphasizing more on the dynamics andstability of the whole population’s strategies [19], instead ofonly the property of the equilibrium. EGT has been widelyused to model users’ behaviors in image processing [20], aswell as communication and networking area [21], [22], suchas congestion control [23], cooperative sensing [24], coopera-tive peer-to-peer (P2P) streaming [25] and dynamic spectrumaccess [26]. In these literatures, evolutionary game has beenshown to be an effective approach to model the dynamic socialinteractions among users in a network.

EGT is an effective approach to study how a group of playersconverges to a stable equilibrium after a period of strategic in-teractions. Such an equilibrium strategy is defined as the Evolu-tionarily Stable Strategy (ESS). For an evolutionary game withplayers, a strategy profile , where

and is the action space, is an ESS if and only if,satisfies following [19]:

(6)

(7)

where stands for the utility of player and denotes thestrategies of all players other than player . We can see that thefirst condition is the Nash equilibrium (NE) condition, and thesecond condition guarantees the stability of the strategy. More-over, we can also see that a strict NE is always an ESS. If allplayers adopt the ESS, then no mutant strategy could invade thepopulation under the influence of natural selection. Even if asmall part of players may not be rational and take out-of-equi-librium strategies, ESS is still a locally stable state.Let us consider an evolutionary game with strategies

. The utility matrix, , is an matrix,whose entries, , denote the utility for strategy versusstrategy . The population fraction of strategy is given by, where . The fitness of strategy is given

by . For the average fitness of the wholepopulation, we have . The Wright-Fisher modelhas been widely adopted to let a group of players converge tothe ESS [27], where the strategy updating equation for eachplayer can be written as

(8)

Note that one assumption in the Wright-Fisher model is thatwhen the total population is sufficiently large, the fraction ofplayers using strategy is equal to the probability of one indi-vidual player using strategy . From (8), it can be seen that thestrategy updating process in the evolutionary game is similar tothe parameter updating process in adaptive filter problem. It isintuitive that we can use evolutionary game to formulate the dis-tributed adaptive filter problem.The classical evolutionary game theory considers a popula-

tion of individuals in a complete graph. However, in manyscenarios, players’ spatial locations may lead to an incompletegraph structure. Graphical evolutionary game theory is intro-duced to study the strategies evolution in such a finite struc-tured population [28], where each vertex represents a playerand each edge represents the reproductive relationship betweenvalid neighbors, i.e., denotes the probability that the strategyof node will replace that of node , as shown in Fig. 2. Graph-ical EGT focuses on analyzing the ability of a mutant gene toovertake a group of finite structured residents. One of the mostimportant research issues in graphical EGT is how to computethe fixation probability, i.e., the probability that the mutant willeventually overtake the whole structured population [29]. In thefollowing, we will use graphical EGT to formulate the dynamic


Fig. 2. Graphical evolutionary game model.

TABLE IICORRESPONDENCE BETWEEN GRAPHICAL EGT AND DISTRIBUTED ADAPTIVE

NETWORK

parameter updating process in a distributed adaptive filter net-work.

B. Graphical Evolutionary Game Formulation

In graphical EGT, each player updates strategy according tohis/her fitness after interacting with neighbors in each round.Similarly, in distributed adaptive filtering, each node updatesits parameter through incorporating the neighbors’ informa-tion. In such a case, we can treat the nodes in a distributed filternetwork as players in a graphical evolutionary game. For nodewith neighbors, it has pure strategies ,where strategy means updating using the updated in-formation from its neighbor . We can see that (5)represents the adoption of mixed strategy. In such a case, theparameter updating in distributed adaptive filter network can beregarded as the strategy updating in graphical EGT. Table IIsummarizes the correspondence between the terminologies ingraphical EGT and those in distributed adaptive network.We first discuss how players’ strategies are updated in graph-

ical EGT, which is then applied to the parameter updating indistributed adaptive filtering. In graphical EGT, the fitness of aplayer is locally determined from interactions with all adjacentplayers, which is defined as [30]

(9)

where is the baseline fitness, which represents the player’sinherent property. For example, in a distributed adaptive net-work, a node’s baseline fitness can be interpreted as the quality

Fig. 3. Three different update rules, where death selections are shown in darkblue and birth selections are shown in red. (a) BD update rule. (b) DB updaterule. (c) IM update rule.

of its noise variance. is the player’s utility which is deter-mined by the predefined utility matrix. The parameter repre-sents the selection intensity, i.e., the relative contribution of thegame to fitness. The case represents the limit of weakselection [31], while denotes strong selection, where fit-ness equals utility. There are three different strategy updatingrules for the evolution dynamics, called as birth-death (BD),death-birth (DB) and imitation (IM) [32].• BD update rule: a player is chosen for reproductionwith the probability being proportional to fitness (Birthprocess). Then, the chosen player’s strategy replaces oneneighbor’s strategy uniformly (Death process), as shownin Fig. 3(a).

• DB update rule: a random player is chosen to abandonhis/her current strategy (Death process). Then, the chosenplayer adopts one of his/her neighbors’ strategies withthe probability being proportional to their fitness (Birthprocess), as shown in Fig. 3(b).

• IM update rule: each player either adopts the strategy ofone neighbor or remains with his/her current strategy, withthe probability being proportional to fitness, as shown inFig. 3(c).

These three kinds of strategy updating rules can be matchedto three different kinds of parameter updating algorithms in dis-tributed adaptive filtering. Suppose that there are nodes in astructured network, where the degree of node is . We useto denote the set of all nodes and to denote the neighborhoodset of node , including node itself.For the BD update rule, the probability that node adopts

strategy , i.e., using updated information from its neighbornode , is

(10)


where the first term is the probability that the neigh-

boring node is chosen to reproduction, which is proportionalto its fitness , and the second term is the probability thatnode is chosen for adopting strategy . Note that the networktopology information is required to calculate (10). In sucha case, the equivalent parameter updating rule for node can bewritten by

(11)

Similarly, for the DB updating rule, we can obtain the corre-sponding parameter updating rule for node as

(12)

For the IM updating rule, we have

(13)

Note that (11), (12) and (13) are expected outcome of BD, DBand IM updated rules, which can be referred in [35], [37].The performance of adaptive filtering algorithm is usually

evaluated by two measures: mean-square deviation (MSD) andexcess-mean-square error (EMSE), which are defined as

(14)

(15)

Using (11), (12) and (13), we can calculate the network MSDand EMSE of these three update rules according to [6].

C. Relationship to Existing Distributed Adaptive FilteringAlgorithms

In Section II, we have summarized the existing distributedadaptive filtering algorithms in (5) and Table I. In this subsec-tion, we will show that all these algorithms are the special casesof the IM update rule in our proposed graphical EGT frame-work. Compare (5) and (13), we can see that different fitnessdefinitions are corresponding to different distributed adaptivefiltering algorithms in Table I. For the uniform rule, the fitnesscan be uniformly defined as and using the IM updaterule, we have

(16)

which is equivalent to the uniform rule in Table I. Here, thedefinition of means the adoption of fixed fitness and

TABLE IIIDIFFERENT FITNESS DEFINITIONS

weak selection . For the Laplacian rule, when updatingthe parameter of node , the fitness of nodes in can be definedas

(17)

From (17), we can see that each node gives more weight to theinformation from itself through enhancing its own fitness. Sim-ilarly, for the Relative-degree-variance rule, the fitness can bedefined as

(18)

For the metropolis rule and Hastings rule, the correspondingfitness definitions are based on strong selection model ,where utility plays a dominant role in (9). For the metropolisrule, the utility matrix of nodes can be defined as

(19)

For the Hastings rule, the utility matrix can be defined as

(20)

Table III summarizes different fitness definitions correspondingto different combination rules in Table I. Therefore, we cansee that the existing algorithms can be summarized into ourproposed graphical EGT framework with corresponding fitnessdefinitions.


D. Error-Aware Distributed Adaptive Filtering Algorithm

To illustrate our graphical EGT framework, as examples, wefurther design two distributed adaptive algorithms by choosingdifferent fitness functions. As discussed in Section II, the ex-isting distributed adaptive filtering algorithms either rely on theprior knowledge of network topology or the requirement of ad-ditional network statistics. All of them are not robust to a dy-namic network, where a node location may change and the noisevariance of each node may also vary with time. Consideringthese problems, we propose error-aware algorithms based onthe intuition that neighbors with low mean-square-error (MSE)should be given more weight while neighbors with high MSEshould be given less weight. The instantaneous error of node ,denoted by , can be calculated by

(21)

where only local data are used. The approximatedMSE of node , denoted by , can be estimated by followingupdate rule in each time slot,

(22)

where is a positive parameter. We assume that nodes canexchange their instantaneous MSE information with neighbors.Based on the estimated MSE, we design two kinds of fitness:exponential form and power form as follows:

(23)

(24)

where is a positive coefficient. Note that the fitness defined in(23) and (24) are just two examples of our proposed framework,while many other forms of fitness can be considered, e.g.,

. Using the IM update rule, we have

(25)

(26)

From (25) and (26), we can see that the proposed algorithmsdo not directly depend on any network topology information.Moreover, they can also adapt to a dynamic environment whenthe noise variance of nodes are unknown or suddenly change,since the weights can be immediately adjusted accordingly.In [33], a similar algorithm was also proposed based on theinstantaneous MSE information, which is a special case of ourerror-aware algorithm with power form of . Note thatthe deterministic coefficients are adopted when implementing(25) and (26), instead of using random combining efficientwith some probability. However, the algorithm can also beimplemented using a random selection with probabilities. Therewill be no performance loss since the expected outcome is thesame, but the efficiency (convergence speed) will be lower.In Section V, we will verify the performance of the proposedalgorithm through simulation.

Fig. 4. Graphical evolutionary game model.

IV. DIFFUSION ANALYSIS

In a distributed adaptive filter network, there are nodes withgood signals, i.e., lower noise variance, as well as nodes withpoor signals. The principal objective of distributed adaptive fil-tering algorithms is to stimulate the diffusion of good signalsto the whole network to enhance the network performances. Inthis section, we will use the EGT to analyze such a dynamicdiffusion process and derive the close-form expression for thediffusion probability. In the following diffusion analysis, we as-sume that all nodes have the same regressor statistics , butdifferent noise statistics.In a graphical evolutionary game, the structured population

are either residents or mutants. An important concept is the fixa-tion probability, which represents the probability that the mutantwill eventually overtake the whole population [34]. Let us con-sider a local adaptive filter network as shown in Fig. 4, where thehollow points denote common nodes, i.e., nodes with commonnoise variance ; and the solid points denote good nodes, i.e.,nodes with a lower noise variance . and satisfy that

. Here, we adopt the binary signal model to betterreveal the diffusion process of good signals. If we regard thecommon nodes as residents and the good nodes as mutants, theconcept of fixation probability in EGT can be applied to analyzethe diffusion of good signals in the network. According to thedefinition of fixation probability, we define the diffusion proba-bility in a distributed filter network as the probability that a goodsignal can be adopted by all nodes to update parameters in thenetwork.

A. Strategies and Utility Matrix

As shown in Fig. 4, for the node at the center, its neighborsinclude both common nodes and good nodes. When the centernode updates its parameter , it has the following two possiblestrategies:

(27)

In such a case, we can define the utility matrix as follow:

(28)

where represents the steady EMSE of node with noisevariance using information from node with noise variance. For example, is the steady EMSE of node with


noise variance adopting strategy , i.e., updating itsusing information from node with noise variance which inturn adopts strategy . In our diffusion analysis, we assumethat only two players are interacting with each other at one timeinstant, i.e., there are two nodes exchanging and combining in-formation with each other at one time instant. In such a case,the payoff matrix is two-user case. Note that a node chooses onespecific neighbor with some probability, which is equivalent tothe weight that the node gives to that neighbor.Since the steady EMSE in the utility matrix is deter-

mined by the information combining rule, there is no general ex-pressions for . Nevertheless, by intuition, we know thatthe steady EMSE of node with variance should be larger thanthat of node with variance since , and adoptingstrategy should be more beneficial than adopting strategysince the node can obtain better information from others, i.e.,

. There-fore, we assume that the utility matrix defined in (28) has thequality as follow

(29)

Here, we use an example in [17] to have a close-form expressionfor to illustrate and verify this intuition. According to[17], with sufficiently small step size , the optimal canbe calculated by

(30)

(31)

where consists of the eigenvalues of(recall that is the covariance matrix of the observed regres-sion data ). According to (30) and (31), we have

(32)

(33)

(34)

(35)

Suppose , through comparing (32)–(35), we can de-rive the condition for

as follows

(36)

According to [17], the derivation of optimal in (30) and(31) is based on the assumption that is sufficiently small.Therefore, the condition of in (36) holds. In such a case, wecan conclude that

, which implies that .In the following, we will analyze the diffusion process of

strategy , i.e., the ability of good signals diffusing over thewhole network. We consider an adaptive filter network basedon a homogenous graph with general degree and adopt theIM update rule for the parameter update [35]. Let and

denote the percentages of nodes using strategies and inthe population, respectively. Let and de-note the percentages of edge, where means the percentageof edge on which both nodes use strategy and . Letdenote the conditional probability of a node using strategygiven that the adjacent node is using strategy , similar wehave and . In such a case, we have

(37)

(38)

where and are either or . The equations (37)–(38) implythat the state of the whole network can be described by only twovariables, and . In the following, we will calculate thedynamics of and under the IM update rule.

B. Dynamics of and

In order to derive the diffusion probability, we first need toanalyze the diffusion process of the system. As discussed in theprevious subsection, the system dynamics under IM update rulecan be represented by parameters and . Thus, in thissubsection, we will first analyze the dynamics of andto understand the dynamic diffusion process of the adaptive net-work. According to the IM update rule, a node using strategyis selected for imitation with probability . As shown in the leftpart of Fig. 4, among its neighbors (not including itself), thereare nodes using strategy and nodes using strategy ,respectively, where . The percentage of such a con-

figuration is . In such a case, the fitness of this

node is

(39)

where the baseline fitness is normalized as 1. We can see that(39) includes the normalized baseline fitness and also the fitnessfrom utility, which is the standard definition of fitness used inthe EGT filed, as shown in (9). Among those neighbors, thefitness of node using strategy is

(40)

and the fitness of node using strategy is

(41)

In such a case, the probability that the node using strategy isreplaced by is

(42)

Therefore, the percentage of nodes using strategy , in-creases by with probability

(43)


Meanwhile, the edges that both nodes use strategy increaseby , thus, we have

(44)

Similar analysis can be applied to the node using strategy. According to the IM update rule, a node using strategyis selected for imitation with probability . As shown

in the right part of Fig. 4, we also assume that there arenodes using strategy and nodes using strategyamong its neighbors. The percentage of such a phenomenon

is . Thus, the fitness of this node is

(45)

Among those neighbors, the fitness of node using strategyis

(46)

and the fitness of node using strategy is

(47)

In such a case, the probability that the node using strategyis replaced by is

(48)

Therefore, the percentage of nodes using strategy , de-creases by with probability

(49)

Meanwhile, the edges that both nodes use strategy decreaseby , thus, we have

(50)

Combining (43) and (49), we have the dynamic of as

(51)

where the second equality is according to Taylor’s Theorem andweak selection assumption with goes to zero [36], and theparameters and are given as follows:

(52)

(53)

(54)

(55)

In (51), the dot notation represents the dynamic of , i.e.,the variation of within a tiny period of time. In such a case,the utility obtained from the interactions is considered as limitedcontribution to the overall fitness of each player. On one hand,the results derived from weak selection often remain as validapproximations for larger selection strength [31]. On the otherhand, the weak selection limit has a long tradition in theoreticalbiology [37]. Moreover, the weak selection assumption can helpachieve a close-form analysis of diffusion process and betterreveal how the strategy diffuses over the network. Similarly, bycombining (44) and (50), we have the dynamics of as

(56)

Besides, we can also have the dynamics of as

(57)

C. Diffusion Probability Analysis

The dynamic equation of in (51) reflects the dynamicof nodes updating using information from good nodes, i.e.,the diffusion status of good signals in the network. A posi-tive means that good signals are diffusing over the net-work, while a negative means that good signals have notbeen well adopted. The diffusion probability of good signals isclosely related to the noise variance of good nodes . Intu-itively, the lower , the higher probability that good signalscan spread the whole network. In this subsection, we will ana-lyze the close-form expression for the diffusion probability.As discussed at the beginning of Section IV, the state of whole

network can be described by only and . In such a case,(51) and (57) can be re-written as functions of and

(58)

(59)

From (58) and (59), we can see that converges to equi-librium in a much faster rate than under the assumption ofweak selection. At the steady state of , i.e., ,we have

(60)


In such a case, the dynamic network will rapidly converge ontothe slow manifold, defined by . Therefore,we can assume that (60) holds in the whole convergence processof . According to (37)–(38) and (60), we have

(61)

(62)

(63)

(64)

Therefore, the diffusion process can be characterized by only. Thus, we can focus on the dynamics of to derive the

diffusion probability, which is given by following Theorem 1.Theorem 1: In a distributed adaptive filter network which can

be characterized by a -node regular graph with degree , sup-pose there are common nodes with noise variance and goodnodes with noise variance , where each common node hasconnection edge with only one good node. If each node updatesits parameter using the IM update rule, the diffusion proba-bility of the good signal can be approximated by

(65)

where the parameters and are as follows:

(66)

(67)

Proof: See Appendix.Using Theorem 1, we can calculate the diffusion probability

of the good signals over the network, which can be used to eval-uate the performance of an adaptive filter network. Similarly, thediffusion dynamics and probabilities under BD and DB updaterules can also be derived using the same analysis. The followingtheorem shows an interesting result, which is based on an impor-tant theorem in [29], stating that evolutionary dynamics underBD, DB, and IM are equivalent for undirected regular graphs.Theorem 2: In a distributed adaptive filter network which can

be characterized by a -node regular graph with degree , sup-pose there are common nodes with noise variance and goodnodes with noise variance , where each common node hasconnection edge with only one good node. If each node updatesits parameter using the IM update rule, the diffusion proba-bilities of good signals under BD and DB update rules are samewith that under the IM update rule.

V. EVOLUTIONARILY STABLE STRATEGY

In the last section, we have analyzed the information diffu-sion process in an adaptive network under the IM update rule,and derived the diffusion probability of strategy that usinginformation from good nodes. On the other hand, consideringthat if the whole network has already chosen to adopt this fa-vorable strategy , is the current state a stable network state,even though a small fraction of nodes adopt the other strategy? In the following, we will answer these questions using the

concept of evolutionarily stable strategy (ESS) in evolutionary

game theory. As discussed in Section III-A, the ESS ensuresthat one strategy is resistant against invasion of another strategy[38]. In our system model, it is obvious that , i.e., using in-formation from good nodes, is the favorable strategy and a de-sired ESS in the network. In this section, we will check whetherstrategy is evolutionarily stable.

A. ESS in Complete Graphs

We first discuss whether strategy is an ESS in completegraphs, which is shown by the following theorem.Theorem 3: In a distributed adaptive filter network that can

be characterized by complete graphs, strategy is always anESS strategy.

Proof: In a complete graph, each node meets every othernode equally likely. In such a case, according to the utility ma-trix in (28), the average utilities of using strategies andare given by

(68)

(69)

where and are the percentages of population using strate-gies and , respectively. Consider the scenario that themajority of the population adopt strategy , while a smallfraction of the population adopt which is considered as in-vasion, . In such a case, according to the definition ofESS in (7), strategy is evolutionary stable if for

, i.e.,

(70)

For , the left hand side of (70) is positive if and only if

`` '' `` '' (71)

The (71) gives the sufficient evolutionary stable condition ofstrategy . In our system, we have ,which means that (71) always holds. Therefore, strategy isalways an ESS if the adaptive filter network is a complete graph.

B. ESS in Incomplete Graphs

Let us consider an adaptive filter network which can be char-acterized by an incomplete regular graph with degree . Thefollowing theorem shows that strategy is always an ESS insuch an incomplete graph.Theorem 4: In a distributed adaptive filter network which can

be characterized by a regular graph with degree , strategyis always an ESS strategy.

Proof: Using the pair approximation method [32], thereplicator dynamics of strategies and on a regular graphof degree can be approximated simply by

(72)

(73)

where is the averageutility, and and are given as follows:

(74)


The parameter depends on the three update rules (IM, BD andDB), which is given by [32]

(75)

(76)

(77)

In such a case, the equivalent utility matrix is

(78)

According to (71), the evolutionary stable condition forstrategy is

(79)

Since , we have for all threeupdate rules. In such a case, (79) always holds, which meansthat strategy is always an ESS strategy. This completes theproof of the theorem.

VI. SIMULATION RESULTS

In this section, we develop simulations to compare the per-formances of different adaptive filtering algorithms, as well asto verify the derivation of information diffusion probability andthe analysis of ESS.

A. Mean-Square Performances

The network topology used for simulation is shown in theleft part of Fig. 5, where 20 randomly nodes are randomly lo-cated. The signal and noise power information of each node arealso shown in the right part of Fig. 5, respectively. In the sim-ulation, we assume that the regressors with size , arezero-mean Gaussian and independent in time and space. Theunknown vector is set to be and the step size ofthe LMS algorithm at each node is set as . All thesimulation results are averaged over 500 independent runnings.All the performance comparisons are conducted among six dif-ferent kinds of distributed adaptive filtering algorithms as fol-lows:• Relative degree algorithm [8];• Hastings algorithm [17];• Adaptive combiner algorithm [7];• Relative degree-variance algorithm [6];• Proposed error-aware algorithm with power form;• Proposed error-aware algorithm with exponential form.

Among these algorithms, the adaptive combiner algorithm [7]and our proposed error-aware algorithm are based on dynamiccombiners (weights), which are updated in each time slot. Thedifference is the updating rule, where the adaptive combiner al-gorithm in [7] uses optimization and projection method, and ourproposed algorithms use the approximated EMSE information.In the first comparison, we assume that the noise variance

of each node is known by the Hastings and relative degree-variance algorithms. Fig. 6 shows the transient network-per-formance comparison results among six kinds of algorithms in

Fig. 5. Network information for simulation, including network topology for 20nodes (left), trace of regressor covariance (right top) and noise variance(right bottom).

Fig. 6. Transient performances comparison with known noise variances. (a)Network EMSE. (b) Network MSD.

terms of EMSE and MSD. Under the similar convergence rate,we can see that the relative degree-variance algorithm performsthe best. The proposed algorithm with exponential form per-forms better than the relative degree algorithm. With the powerform fitness, the proposed algorithm can achieve similar perfor-mance, if not better than, compared with adaptive combiner al-gorithm, and both algorithms performs better than all other algo-rithms except the relative degree-variance algorithm. However,as discussed in Section 2, the relative degree-variance algorithm


Fig. 7. Steady performances comparison with known noise variances. (a)Node’s EMSE. (b) Node’s MSD.

requires noise variance information of each node, while ourproposed algorithm does not. Fig. 7 shows the correspondingsteady-state performances of each node for six kinds of dis-tributed adaptive filtering algorithms in terms of EMSE andMSD. Since the steady-state result is for each node, besides av-eraging over 500 independent runnings, we average at each nodeover 100 time slots after the convergence. We can see that thecomparison results of steady-state performances are similar tothose of the transient performances.In the second comparison, we assume that the noise vari-

ance of each node is unknown, but can be estimated by themethod proposed in [17]. Figs. 8 and 9 show the transient andsteady-state performances for six kinds of algorithms in termsof EMSE and MSD under similar convergence rate. Since thenoise variance estimation requires additional complexity, wealso simulate the Hastings and relative degree-variance algo-rithms without variance estimation for fair comparison, wherethe noise variance is set as the network average variance, whichis assumed to be prior information. Comparing with Fig. 7, wecan see that when the noise variance information is not avail-able, the performance degradation of relative degree-variancealgorithm is significant, about 0.5 dB (12% more error) evenwith noise variance estimation, while the performance of Hast-ings algorithm degrades only a little since it relies less on thenoise variance information. From Fig. 8(b), we can clearly seethat when the variance estimation method is not adopted, ourproposed algorithm with power form achieves the best perfor-mance. When the variance estimation method is adopted, theperformances of our proposed algorithm with power form, therelative degree-variance and the adaptive combiner algorithm

Fig. 8. Transient performances comparison with unknown noise variances. (a)Network EMSE. (b) Network MSD.

are similar, all of which perform better than other algorithms.Nevertheless, the complexity of both relative degree-variancealgorithm with variance estimation and the adaptive combineralgorithm are higher than that of our proposed algorithm withpower form. Such results immediately show the advantage ofthe proposed general framework.We should notice that more al-gorithms with better performances under certain criteria can bedesigned based on the proposed framework by choosing moreproper fitness functions.

B. Diffusion Probability

In this subsection, we develop simulation to verify the dif-fusion probability analysis in Section IV. For the simulationsetup, three types of regular graphs are generated with degree

, 4 and 6, respectively, as shown in Fig. 10(a). All thesethree types of graphs are with nodes, where eachnode’s trace of regressor covariance is set to be, the common nodes’s noise variance is set as

and the good node’s noise variance is set as .In the simulation, the network is initialized with the state thatall common nodes choosing strategy . Then, at each time


Fig. 9. Steady performances comparison with unknown noise variances. (a)Node’s EMSE. (b) Node’s MSD.

Fig. 10. Diffusion probabilities under three types of regular graphs. (a) Regulargraph structures with degree , 4 and 6. (b) Diffusion probability.

Fig. 11. Strategy updating process in a 10 10 grid network with degreeand number of nodes .

step, a randomly chosen node’s strategy is updated accordingto the IM rules under weak selection , as illus-trated in Section III-B. The update steps are repeated until ei-ther strategy has reached fixation or the number of stepshas reach the limit. The diffusion probability is calculated by thefraction of runs where strategy reached fixation out ofruns. Fig. 10(b) shows the simulation results, from which wecan see that all the simulated results are basically accord withthe corresponding theoretical results and the gaps are due to theapproximation during the derivations.Moreover, we can see thatthe diffusion probability of good signal decreases along with theincrease of its noise variance, i.e., better signal has better diffu-sion capability.

C. Evolutionarily Stable Strategy

To verify that strategy is an ESS in the adaptive network,we further simulate the IM update rule on a 10 10 grid net-work with degree and number of nodes , asshown in Fig. 11 where the hollow points represent commonnodes and the solid nodes represent good nodes. In the simu-lation, all the settings are same with those in the simulation ofdiffusion probability in Section VI.B, except the initial networksetting. The initial network state is set that the majority of nodesadopt strategy denoted with black color (including bothhollow and solid nodes) in Fig. 11, and only a very small per-centage of nodes use strategy denoted with red color. Fromthe strategy updating process of the whole network illustrated inFig. 11, we can see that the network finally abandons the unfa-vorable strategy , which verifies the stability of strategy .

VII. CONCLUSION

In this paper, we proposed an evolutionary game theoreticframework to offer a very general view of the distributed adap-tive filtering problems and unify existing algorithms. Based onthis framework, as examples, we further designed two error-aware adaptive filtering algorithms. Using the graphical evo-lutionary game theory, we analyzed the information diffusionprocess in the network under the IM update rule, and proved thatthe strategy of using information from nodes with good signalis always an ESS. We would like to emphasize that, unlike thetraditional bottom-up approaches, the proposed graphical evo-lutionary game theoretic framework provides a top-down design


philosophy to understand the fundamentals of distributed adap-tive algorithms. Such a top-down design philosophy is very im-portant to the field of distributed adaptive signal process, sinceit offers a unified view of the formulation and can inspire morenew distributed adaptive algorithms to be designed in the future.

APPENDIXPROOF OF THEOREM 1

Proof: First, let us define as the mean of the incre-ment of per unit time given as follows

(80)

where the second step is derived by substituting (60)–(64) into(51) and the parameters and are given as follows:

(81)

(82)

We then define as the variance of the increment ofper unit time, which can be calculated by

(83)

where can be computed by

(84)

In such a case, can be approximated by

(85)

Suppose the initial percentage of good nodes in the network is. Let us define as the probability that these good

signals can finally be adopted by the whole network, i.e., allnodes can update their own using information from goodnodes. According to the backward Kolmogorov equation [39],

satisfies following differential equation

(86)

With the weak selection assumption, we can have the approxi-mate solution of as

(87)

Let us consider the worst initial system state that eachcommon node has connection with only one good node, i.e.,

, we have

(88)

By substituting (81) and (82) into (88), we can have the close-form expression for the diffusion probability in (65). This com-pletes the proof of the theorem.Remark: From (87), we can see that there are two terms con-

stituting the expression of diffusion probability: the initial per-centage of strategy (the initial system state) and thesecond term representing the changes of system state after be-ginning, in which determines whether is increasing ordecreasing along with the system updating. If , i.e.,the diffusion probability is even lower than the initial percentageof strategy , the information from good nodes are shrinkingover the network, instead of spreading. Therefore,is more favorable for the improvement of the adaptive networkperformance.

REFERENCES

[1] V. D. Blondel, J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis,“Distributed processing over adaptive networks,” in Proc. Adapt. Sens.Array Process. Workshop, Lexington, MA, USA, Jun. 2006, pp. 1–3.

[2] D. Li, K. D. Wong, Y. H. Hu, and A. M. Sayed, “Detection, classifica-tion, and tracking of targets,” IEEE Signal Process. Mag., vol. 19, no.2, pp. 17–29, 2002.

[3] F. C. R. Jr, M. L. R. de Campos, and S. Werner, “Distributed co-operative spectrum sensing with selective updating,” in Proc. Eur.Signal Process. Conf. (EUSIPCO), Bucharest, Romania, Aug. 2012,pp. 474–478.

[4] S. Haykin and K. J. R. Liu, Handbook on Array Processing and SensorNetworks. New York, NY, USA: IEEE-Wiley, 2009.

[5] C. G. Lopes and A. H. Sayed, “Incremental adaptive strategies overdistributed networks,” IEEE Trans. Signal Process., vol. 55, no. 8, pp.4064–4077, 2007.

[6] F. S. Cattivelli and A. H. Sayed, “Diffusion LMS strategies for dis-tributed estimation,” IEEE Trans. Signal Process., vol. 58, no. 3, pp.1035–1048, 2010.

[7] N. Takahashi, I. Yamada, and A. H. Sayed, “Diffusion least-meansquares with adaptive combiners: Formulation and performanceanalysis,” IEEE Trans. Signal Process., vol. 58, no. 9, pp. 4795–4810,2010.

[8] F. S. Cattivelli, C. G. Lopes, and A. H. Sayed, “Diffusion recursiveleast-squares for distributed estimation over adaptive networks,” IEEETrans. Signal Process., vol. 56, no. 5, pp. 1865–1877, 2008.

[9] S. Theodoridis, K. Slavakis, and I. Yamada, “Adaptive learning in aworld of projections,” IEEE Signal Process. Mag., vol. 28, no. 1, pp.97–123, 2011.

[10] R. L. G. Cavalcante, I. Yamada, and B. Mulgrew, “An adaptive pro-jected subgradient approach to learning in diffusion networks,” IEEETrans. Signal Process., vol. 57, no. 7, pp. 2762–2774, 2009.

[11] S. Chouvardas, K. Slavakis, and S. Theodoridis, “Adaptive robust dis-tributed learning in diffusion sensor networks,” IEEE Trans. SignalProcess., vol. 59, no. 10, pp. 4692–4707, 2011.

[12] S.-Y. Tu and A. H. Sayed, “Mobile adaptive networks,” IEEE J. Sel.Topics Signal Process., vol. 5, no. 4, pp. 649–664, 2011.

[13] V. D. Blondel, J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis,“Convergence in multiagent coordination, consensus, and flocking,” inProc. Joint 44th IEEE Conf. Decision Contr. Eur. Contr. Conf. (CDC-ECC), Seville, Spain, Dec. 2005, pp. 2996–3000.

[14] L. Xiao, S. Boyd, and S. Lall, “A scheme for robust distributed sensorfusion based on average consensus,” in Proc. Inf. Process. Sens. Netw.(IPSN), Los Angeles, CA, USA, Apr. 2005, pp. 63–70.

[15] D. S. Scherber and H. C. Papadopoulos, “Locally constructed algo-rithms for distributed computations in Ad Hoc networks,” in Proc.Inf. Process. Sens. Netw. (IPSN), Berkeley, CA, USA, Apr. 2004, pp.11–19.

[16] L. Xiao and S. Boyd, “Fast linear iterations for distributed averaging,”Syst. Contr. Lett., vol. 53, no. 1, pp. 65–78, 2004.


[17] X. Zhao and A. H. Sayed, “Performance limits for distributed estima-tion over LMS adaptive networks,” IEEE Trans. Signal Process., vol.60, no. 10, pp. 5107–5124, 2012.

[18] J. M. Smith, Evolution and the Theory of Games. Cambridge, U.K.:Cambridge Univ. Press, 1982.

[19] R. Cressman, Evolutionary Dynamics and Extensive Form Games.Cambridge, MA, USA: MIT Press, 2003.

[20] Y. Chen, Y. Gao, and K. J. R. Liu, “An evolutionary game-theoreticapproach for image interpolation,” in Proc. IEEE ICASSP, 2011, pp.989–992.

[21] K. J. R. Liu and B. Wang, Cognitive Radio Networking and Security:A Game Theoretical View. Cambridge, U.K.: Cambridge Univ. Press,2010.

[22] B.Wang, Y.Wu, andK. J. R. Liu, “Game theory for cognitive radio net-works: An overview,” Comput. Netw., vol. 54, no. 14, pp. 2537–2561,2010.

[23] E. H. Watanabe, D. Menasché, E. Silva, and R. M. Leão, “Modelingresource sharing dynamics of VoIP users over a WLAN using a game-theoretic approach,” in Proc. IEEE INFOCOM, 2008, pp. 915–923.

[24] B. Wang, K. J. R. Liu, and T. C. Clancy, “Evolutionary cooperativespectrum sensing game: How to collaborate?,” IEEE Trans. Commun.,vol. 58, no. 3, pp. 890–900, 2010.

[25] Y. Chen, B. Wang, W. S. Lin, Y. Wu, and K. J. R. Liu, “Coop-erative peer-to-peer streaming: An evolutionary game-theoreticapproach,” IEEE Trans. Circuit Syst. Video Technol., vol. 20, no. 10,pp. 1346–1357, 2010.

[26] C. Jiang, Y. Chen, Y. Gao, andK. J. R. Liu, “Joint spectrum sensing andaccess evolutionary game in cognitive radio networks,” IEEE Trans.Wireless Commun., vol. 12, no. 5, pp. 2470–2483, 2013.

[27] R. Fisher, The Genetical Theory of Natural Selection. NewYork, NY,USA: Clarendon, 1930.

[28] E. Lieberman, C. Hauert, and M. A. Nowak, “Evolutionary dynamicson graphs,” Nature, vol. 433, pp. 312–316, 2005.

[29] P. Shakarian, P. Roos, andA. Johnson, “A review of evolutionary graphtheory with applications to game theory,” Biosyst., vol. 107, no. 2, pp.66–80, 2012.

[30] M. A. Nowak and K. Sigmund, “Evolutionary dynamics of biologicalgames,” Science, vol. 303, pp. 793–799, 2004.

[31] H. Ohtsuki, M. A. Nowak, and J. M. Pacheco, “Breaking the sym-metry between interaction and replacement in evolutionary dynamicson graphs,” Phys. Rev. Lett., vol. 98, no. 10, p. 108106, 2007.

[32] H. Ohtsukia and M. A. Nowak, “The replicator equation on graphs,” J.Theor. Biol., vol. 243, pp. 86–97, 2006.

[33] X. Zhao and A. H. Sayed, “Clustering via diffusion adaptation overnetworks,” in Proc. Int. Workshop Cognitive Inf. Process., Spain, May2012, pp. 1–6.

[34] M. Slatkin, “Fixation probabilities and fixation times in a subdividedpopulation,” J. Theor. Biol., vol. 35, no. 3, pp. 477–488, 1981.

[35] H. Ohtsuki, C. Hauert, E. Lieberman, and M. A. Nowak, “A simplerule for the evolution of cooperation on graphs and social networks,”Nature, vol. 441, pp. 502–505, 2006.

[36] F. Fu, L. Wang, M. A. Nowak, and C. Hauert, “Evolutionary dynamicson graphs: Efficient method for weak selection,” Phys. Rev., vol. 79,no. 4, p. 046707, 2009.

[37] G. Wild and A. Traulsen, “The different limits of weak selection andthe evolutionary dynamics of finite populations,” J. Theor. Biol., vol.247, no. 2, pp. 382–390, 2007.

[38] H. Ohtsukia and M. A. Nowak, “Evolutionary stability on graphs,” J.Theor. Biol., vol. 251, no. 4, pp. 698–707, 2008.

[39] W. J. Ewense, Mathematical Population Genetics: Theoretical Intro-duction. New York, NY, USA: Springer, 2004.

Chunxiao Jiang (S’09–M’13) received the B.S.degree in information engineering from BeijingUniversity of Aeronautics and Astronautics (Bei-hang University) in 2008 and the Ph.D. degree fromTsinghua University (THU), Beijing, in 2013, bothwith the highest honors. During 2011–2012, he vis-ited the Signals and Information Group (SIG) at theDepartment of Electrical and Computer Engineering(ECE), University of Maryland (UMD), supportedby China Scholarship Council (CSC) for one year.He is currently a Research Associate with the ECE

Department, UMD, with Prof. K. J. Ray Liu, and also a postdoctoral researcher

with the EE Department, THU. His research interests include the applicationsof game theory and queuing theory in wireless communication and networkingand social networks.Dr. Jiang received the Beijing Distinguished Graduated Student Award, Chi-

nese National Fellowship and Tsinghua Outstanding Distinguished DoctoralDissertation in 2013.

Yan Chen (S’06–M’11) received the Bachelor’s de-gree from University of Science and Technology ofChina in 2004, the M.Phil. degree from Hong KongUniversity of Science and Technology (HKUST) in2007, and the Ph.D. degree from University of Mary-land, College Park, in 2011. From 2011 to 2013, hehas been a Postdoctoral research associate with theDepartment of Electrical and Computer Engineering,University of Maryland, College Park.Currently, he is a Principal Technologist at Origin

Wireless Communications. He is also affiliated withSignal and Information Group of the University of Maryland, College Park. Hiscurrent research interests are in social learning and networking, behavior anal-ysis and mechanism design for network systems, multimedia signal processing,and communication.Dr. Chen received the University of Maryland Future Faculty Fellowship

in 2010, Chinese Government Award for outstanding students abroad in 2011,University of Maryland ECE Distinguished Dissertation Fellowship HonorableMention in 2011, and was the Finalist of A. James Clark School of EngineeringDeans Doctoral Research Award in 2011.

K. J. Ray Liu (F’03) was named a DistinguishedScholar-Teacher of University of Maryland, Col-lege Park, in 2007, where he is Christine KimEminent Professor of Information Technology. Heleads the Maryland Signals and Information Groupconducting research encompassing broad areas ofsignal processing and communications with recentfocus on cooperative and cognitive communications,social learning and network science, informationforensics and security, and green information andcommunications technology.

Dr. Liu is the recipient of numerous honors and awards including the IEEESignal Processing Society Technical Achievement Award and DistinguishedLecturer. He also received various teaching and research recognitions fromthe University of Maryland including the university-level Invention of theYear Award; and Poole and Kent Senior Faculty Teaching Award, OutstandingFaculty Research Award, and Outstanding Faculty Service Award, all from A.James Clark School of Engineering. An ISI Highly Cited Author, he is a Fellowof AAAS. He is President of IEEE Signal Processing Society where he hasserved as Vice President—publications and Board of Governors. He was theEditor-in-Chief of the IEEE SIGNAL PROCESSING MAGAZINE and the foundingEditor-in-Chief of EURASIP Journal on Advances in Signal Processing.

ieee transactions on signal processing, vol. 61, no. 22...

Documents