Distributed Adaptive Networks: A Graphical Evolutionary Game-Theoretic View

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1 Distibuted Adaptive Netwoks: A Gaphical Evolutionay Gae-Theoetic View Chunxiao Jiang, Mebe, IEEE, Yan Chen, Mebe, IEEE, and K. J. Ray Liu, Fellow, IEEE axiv:.45v [cs.gt] Sep 3 Abstact Distibuted adaptive filteing has been consideed as an effective appoach fo data pocessing and estiation ove distibuted netwoks. Most existing distibuted adaptive filteing algoiths focus on designing diffeent infoation diffusion ules, egadless of the natue evolutionay chaacteistic of a distibuted netwok. In this pape, we study the adaptive netwok fo the gae theoetic pespective and foulate the distibuted adaptive filteing poble as a gaphical evolutionay gae. With the poposed foulation, the nodes in the netwok ae egaded as playes and the local cobine of estiation infoation fo diffeent neighbos is egaded as diffeent stategies selection. We show that this gaphical evolutionay gae faewok is vey geneal and can unify the existing adaptive netwok algoiths. Based on this faewok, as exaples, we futhe popose two eo-awae adaptive filteing algoiths. Moeove, we use gaphical evolutionay gae theoy to analyze the infoation diffusion pocess ove the adaptive netwoks and evolutionaily stable stategy of the syste. Finally, siulation esults ae shown to veify the effectiveness of ou analysis and poposed ethods. Index Tes Adaptive filteing, gaphical evolutionay gae, distibuted estiation, adaptive netwoks, data diffusion. I. INTRODUCTION Recently, the concept of adaptive filte netwok deived fo the taditional adaptive filteing was eeging, whee a goup of nodes coopeatively estiate soe paaetes of inteest fo noisy easueents []. Such a distibuted estiation achitectue can be applied to any scenaios, such as wieless senso netwoks fo envionent onitoing, wieless Ad-hoc netwoks fo ilitay event localization, distibuted coopeative sensing in cognitive adio netwoks and so on [], [3]. Copaed to the classical centalized achitectue, the distibuted one is not only oe obust when the cente node ay be dysfunctional, but also oe flexible when the nodes ae with obility. Theefoe, distibuted adaptive filte netwok has been consideed as an effective appoach fo the ipleentation of data fusion, diffusion and pocessing ove distibuted netwoks [4]. In a distibuted adaptive filte netwok, at evey tie instant t, node i eceives a set of data {d i t,u i,t } that satisfies a Copyight c 3 IEEE. Pesonal use of this ateial is peitted. Howeve, peission to use this ateial fo any othe puposes ust be obtained fo the IEEE by sending a equest to pubs-peissions@ieee.og. Chunxiao Jiang is with Depatent of Electical and Copute Engineeing, Univesity of Mayland, College Pak, MD 74, USA, and also with Depatent of Electonic Engineeing, Tsinghua Univesity, Beijing 84, P. R. China e-ail: chx.jiang@gail.co. Yan Chen, and K. J. Ray Liu ae with Depatent of Electical and Copute Engineeing, Univesity of Mayland, College Pak, MD 74, USA e-ail: yan@ud.edu, kjliu@ud.edu. linea egession odel as follow d i t = u i,t w +v i t, whee w is a deteinistic but unknown M vecto, d i t is a scala easueent of soe ando pocess d i, u i,t is the M egession vecto at tie t with zeo ean and covaiance atix R ui = E u i,t u i,t >, and vi t is the ando noise signal at tie t with zeo ean and vaiance σ i. Note that the egession data u i,t and easueent pocess d i ae tepoally white and spatially independent, espectively and utually. The objective fo each node is to use the data set {d i t,u i,t } to estiate paaete w. In the liteatues, any distibuted adaptive filteing algoiths have been poposed fo the estiation of paaete w. The inceental algoiths, in which node i updates w, i.e., the estiation of w, though cobining the obseved data sets of itself and node i, wee poposed, e.g., the inceental LMS algoith [5]. Unlike the inceental algoiths, the diffusion algoiths allow node i to cobine the data sets fo all neighbos, e.g., diffusion LMS [6], [7] and diffusion RLS [8]. Besides, the pojection-based adaptive filteing algoiths wee suaized in [9], e.g., the pojected subgadient algoith [] and the cobine-pojectadapt algoith []. In [], the authos consideed the node s obility and analyzed the obile adaptive netwoks. While achieving poising pefoance, these taditional distibuted adaptive filteing algoiths ainly focused on designing diffeent infoation cobination ules o diffusion ules aong the neighbohood by utilizing the netwok topology infoation and/o nodes statistical infoation. Fo exaple, the elative degee ule consides the degee infoation of each node [8], and the elative degee-vaiance ule futhe incopoates the vaiance infoation of each node [6]. Howeve, ost of the existing algoiths ae soehow intuitively designed to achieve soe specific objective, sot of like botto-up appoaches to the distibuted adaptive netwoks. Thee is no existing wok that offes a design philosophy to explain why cobination and/o diffusion ules ae developed and how they ae elated in a unified view. Is thee a geneal faewok that can eveal the elationship aong the existing ules and povide fundaental guidance fo bette design of distibuted adaptive filteing algoiths? In ou quest to answe the question, we found that in essence the paaete updating pocess in distibuted adaptive netwoks follows siilaly the evolution pocess in natual ecological systes. Theefoe, based on the gaphical evolutionay gae, in this pape, we popose a geneal faewok that can offe a unified view of existing distibuted adaptive algoiths, and

2 povide possible clues fo new futue designs. Unlike the taditional botto-up appoaches that focus on soe specific ules, ou faewok povide a top-down design philosophy to undestand the fundaental elationship of distibuted adaptive netwoks. The ain contibutions of this pape ae suaized as follows. We popose a gaphical evolutionay gae theoetic faewok fo the distibuted adaptive netwoks, whee nodes in the netwok ae egaded as playes and the local cobination of estiation infoation fo diffeent neighbos is egaded as diffeent stategies selection. We show that the poposed gaphical evolutionay theoetic faewok can unify existing adaptive filteing algoiths as special cases. Based on the poposed faewok, as exaples, we futhe design two siple eo-awae distibuted adaptive filteing algoiths. When the noise vaiance is unknown, ou poposed algoith can achieve siila pefoance copaed with existing algoiths but with lowe coplexity, which iediately shows the advantage of the poposed geneal faewok. 3 Using the gaphical evolutionay gae theoy, we analyze the infoation diffusion pocess ove the adaptive netwok, and deive the diffusion pobability of infoation fo good nodes. 4 We pove that the stategy of using infoation fo good nodes is evolutionaily stable stategy eithe in coplete gaphs o incoplete gaphs. The est of this pape is oganized as follows. We suaize the existing woks in Section II. In Section III, we descibe in details how to foulate the distibuted adaptive filteing poble as a gaphical evolutionay gae. We then discuss the infoation diffusion pocess ove the adaptive netwok in Section IV, and futhe analyze the evolutionaily stable stategy in Section V. Siulation esults ae shown in Section VI. Finally, we daw conclusions in Section VII. II. RELATED WORKS Let us conside an adaptive filte netwok with N nodes. If thee is a fusion cente that can collect infoation fo all nodes, then global centalized optiization ethods can be used to deive the optial updating ule fo the paaete w, whee w is a deteinistic but unknown M vecto fo estiation, as shown in the left pat of Fig.. Fo exaple, in the global LMS algoith, the paaete updating ule can be witten as [6] N w t+ = w t +µ d i t u i,t w t, i= u i,t whee µ is the step size and { } denotes coplex conjugation opeation. With, we can see that the centalized LMS algoith equies the infoation of {d i t,u i,t } acoss the whole netwok, which is geneally ipactical. Moeove, such a centalized achitectue highly elies on the fusion cente and will collapse when the fusion cente is dysfunctional o soe data links ae disconnected. d i t,ut i node i Fig.. Fusion Cente w d i t,ut i node i w Left: centalized odel. Right: distibuted odel. If thee is no fusion cente in the netwok, then each node needs to exchange infoation with the neighbos to update the paaete as shown in the ight pat of Fig.. In the liteatue, seveal distibuted adaptive filteing algoiths have been intoduced, such as distibuted inceental algoiths [5], distibuted LMS [6], [7], and pojection-based algoiths [], []. These distibuted algoiths ae based on the classical adaptive filteing algoiths, whee the diffeence is that nodes can use infoation fo neighbos to estiate the paaete w. Taking one of the distibuted LMS algoiths, Adaptthen-Cobine Diffusion LMS ATC [6], as an exaple, the paaete updating ule fo node i is χ i,t+ = w i,t +µ i o ij u j,t d j t u j,t w j,t, j N i w i,t+ = 3 a ij χ j,t+, j N i whee N i denotes the neighboing nodes set of node i including node i itself, o ij and a ij ae linea weights satisfying the following conditions o ij = a ij =, if j / N i, N N 4 o ij =, a ij =. j= j= In a pactical scenaio, since the exchange of full aw data {d i t,u i,t } aong neighbos is costly, the weight o ij is usually set as o ij =, if j i, as in [6]. In such a case, fo node i with degee n i including node i itself, i.e., the cadinality of set N i and neighbou set {i,i,...,i ni }, we can wite the geneal paaete updating ule as w i,t+ = A i,t+ Fw i,t,fw i,t,...,fw ini,t, = A i,t+ jfw j,t, 5 j N i whee F can be any adaptive filteing algoith, e.g. Fw i,t = w i,t + µu i,t d it u i,t w i,t fo the LMS algoith,a i,t+ epesents soe specific linea cobination ule. The 5 gives a geneal fo of existing distibuted adaptive filteing algoiths, whee the cobination ule A i,t+ ainly deteines the pefoance. Table I suaizes the existing cobination ules, whee fo all ules A i,t+ j =, if j / N i. Fo Table I, we can see that the weights of the fist fou cobination ules ae puely based on the netwok topology. The disadvantage of such topology-based ules is that, they ae

3 3 TABLE I DIFFERENT COMBINATION RULES. Nae Rule: A i j = Unifo [][3] Maxiu degee [8][4] Laplacian [5][6] Relative degee [8] Relative degee-vaiance [6] Metopolis [6][7] Hastings [7], fo all j N n i i, fo j i, N n i, fo j = i. N fo j i n ax, n i, fo j = i. n ax n j, fo all j N k N n i i k n j σ j k N n i k σ, fo all j N i k, fo j i, ax{ N i, N j } k i A ik, fo j = i. σj ax{ N i σ i, N j σ j }, fo j i, k i A ik, fo j = i. sensitive to the spatial vaiation of signal and noise statistics acoss the netwok. The elative degee-vaiance ule shows bette ean-squae pefoance than othes, which, howeve, equies the knowledge of all neighbos noise vaiances. As discussed in Section I, all these distibuted algoiths ae only focusing on designing the cobination ules. Nevetheless, a distibuted netwok is just like a natual ecological syste and the nodes ae just like individuals in the syste, which ay spontaneously follow soe natue evolutionay ules, instead of soe specific atificially pedefined ules. Besides, although vaious kinds of cobination ules have been developed, thee is no geneal faewok which can eveal the unifying fundaentals of distibuted adaptive filteing pobles. In the sequel, we will use gaphical evolutionay gae theoy to establish a geneal faewok to unify existing algoiths and give insights of the distibuted adaptive filteing poble. III. GRAPHICAL EVOLUTIONARY GAME FORMULATION A. Intoduction of Gaphical Evolutionay Gae Evolutionay gae theoy EGT is oiginated fo the study of ecological biology [8], which diffes fo the classical gae theoy by ephasizing oe on the dynaics and stability of the whole population s stategies [9], instead of only the popety of the equilibiu. EGT has been widely used to odel uses behavios in iage pocessing [], as well as counication and netwoking aea [][], such as congestion contol [3], coopeative sensing [4], coopeative pee-to-pee PP steaing [5] and dynaic spectu access [6]. In these liteatues, evolutionay gae has been shown to be an effective appoach to odel the dynaic social inteactions aong uses in a netwok. EGT is an effective appoach to study how a goup of playes conveges to a stable equilibiu afte a peiod of Fig = Gaphical evolutionay gae odel stategic inteactions. Such an equilibiu stategy is defined as the Evolutionaily Stable Stategy ESS. Fo an evolutionay gae with N playes, a stategy pofile a = a,...,a N, whee a i X and X is the action space, is an ESS if and only if, a a, a satisfies following [9]: U i a i, a i U ia i, a i, 6 if U i a i, a i = U i a i, a i, U i a i, a i < U i a i, a i, 7 whee U i stands fo the utility of playe i and a i denotes the stategies of all playes othe than playe i. We can see that the fist condition is the Nash equilibiu NE condition, and the second condition guaantees the stability of the stategy. Moeove, we can also see that a stict NE is always an ESS. If all playes adopt the ESS, then no utant stategy could invade the population unde the influence of natual selection. Even if a sall pat of playes ay not be ational and take out-of-equilibiu stategies, ESS is still a locally stable state. Let us conside an evolutionay gae with stategies X = {,,...,}. The utility atix, U, is an atix, whose enties, u ij, denote the utility fo stategy i vesus stategy j. The population faction of stategy i is given by p i, whee by f i = i=p i =. The fitness of stategy i is given j= p ju ij. Fo the aveage fitness of the whole population, we haveφ = i= p if i. The Wight-Fishe odel has been widely adopted to let a goup of playes convege to the ESS [7], whee the stategy updating equation fo each playe can be witten as p i t+ = p itf i t. 8 φt Note that one assuption in the Wight-Fishe odel is that when the total population is sufficiently lage, the faction of playes using stategy i is equal to the pobability of one individual playe using stategy i. Fo 8, it can be seen that the stategy updating pocess in the evolutionay gae is siila to the paaete updating pocess in adaptive filte poble. It is intuitive that we can use evolutionay gae to foulate the distibuted adaptive filte poble. The classical evolutionay gae theoy consides a population of M individuals in a coplete gaph. Howeve, in any scenaios, playes spatial locations ay lead to an incoplete gaph stuctue. Gaphical evolutionay gae theoy is intoduced to study the stategies evolution in such a

4 4 TABLE II CORRESPONDENCE BETWEEN GRAPHICAL EGT AND DISTRIBUTED ADAPTIVE NETWORK. Initial population Selection fo bith Selection fo death Replace Gaphical EGT N Playes Pue stategy of playe i with n i neighbos {i,i,...,i ni } Distibuted adaptive netwok N Nodes in the netwok Node i cobines infoation fo one of its neighbos {i,i,...,i ni } a BD update ule. Initial population Selection fo death Selection fo bith Replace Mixed stategy of playe i with n i neighbos {p,p,...,p ni } Node i s cobine Weight {A i,a i,...,a i n i } b DB update ule. Mixed stategy update of playe i Cobine update of node i Equilibiu Convegence netwok state Initial population Selection fo update Selection fo iitation Iitation c IM update ule. finite stuctued population [8], whee each vetex epesents a playe and each edge epesents the epoductive elationship between valid neighbos, i.e., θ ij denotes the pobability that the stategy of node i will eplace that of node j, as shown in Fig.. Gaphical EGT focuses on analyzing the ability of a utant gene to ovetake a goup of finite stuctued esidents. One of the ost ipotant eseach issues in gaphical EGT is how to copute the fixation pobability, i.e., the pobability that the utant will eventually ovetake the whole stuctued population [9]. In the following, we will use gaphical EGT to foulate the dynaic paaete updating pocess in a distibuted adaptive filte netwok. B. Gaphical Evolutionay Gae Foulation In gaphical EGT, each playe updates stategy accoding to his/he fitness afte inteacting with neighbos in each ound. Siilaly, in distibuted adaptive filteing, each node updates its paaete w though incopoating the neighbos infoation. In such a case, we can teat the nodes in a distibuted filte netwok as playes in a gaphical evolutionay gae. Fo node i with n i neighbos, it has n i pue stategies {i,i,...,i ni }, whee stategy j eans updating w i,t+ using the updated infoation fo its neighboj,a i,t+ j. We can see that 5 epesents the adoption of ixed stategy. In such a case, the paaete updating in distibuted adaptive filte netwok can be egaded as the stategy updating in gaphical EGT. Table II suaizes the coespondence between the teinologies in gaphical EGT and those in distibuted adaptive netwok. We fist discuss how playes stategies ae updated in gaphical EGT, which is then applied to the paaete updating in distibuted adaptive filteing. In gaphical EGT, the fitness of a playe is locally deteined fo inteactions with all adjacent playes, which is defined as [3] f = α B +α U, 9 whee B is the baseline fitness, which epesents the playe s inheent popety. Fo exaple, in a distibuted adaptive netwok, a node s baseline fitness can be intepeted as the Fig. 3. Thee diffeent update ules, whee death selections ae shown in dak blue and bith selections ae shown in ed. quality of its noise vaiance. U is the playe s utility which is deteined by the pedefined utility atix. The paaete α epesents the selection intensity, i.e., the elative contibution of the gae to fitness. The case α epesents the liit of weak selection [3], while α = denotes stong selection, whee fitness equals utility. Thee ae thee diffeent stategy updating ules fo the evolution dynaics, called as bith-death BD, death-bith DB and iitation IM [3]. BD update ule: a playe is chosen fo epoduction with the pobability being popotional to fitness Bith pocess. Then, the chosen playe s stategy eplaces one neighbo s stategy unifoly Death pocess, as shown in Fig. 3-a. DB update ule: a ando playe is chosen to abandon his/he cuent stategy Death pocess. Then, the chosen playe adopts one of his/he neighbos stategies with the pobability being popotional to thei fitness Bith pocess, as shown in Fig. 3-b. IM update ule: each playe eithe adopts the stategy of one neighbo o eains with his/he cuent stategy, with the pobability being popotional to fitness, as shown in Fig. 3-c. These thee kinds of stategy updating ules can be atched to thee diffeent kinds of paaete updating algoiths in distibuted adaptive filteing. Suppose that thee ae N nodes in a stuctued netwok, whee the degee of node i is n i. We use N to denote the set of all nodes and N i to denote the neighbohood set of node i, including node i itself. Fo the BD update ule, the pobability that node i adopts stategy j, i.e., using updated infoation fo its neighbo node j, is P j = f j k N f k f j k N f k n j, whee the fist te is the pobability that the neighboing node j is chosen to epoduction, which is popotional

5 5 to its fitness f j, and the second te n j is the pobability that node i is chosen fo adopting stategy j. Note that the netwok topology infoation n j is equied to calculate. In such a case, the equivalent paaete updating ule fo node i can be witten by w i,t+ = j N i\{i} j N i\{i} f j k N f k n j f j k N f k Fw j,t + Fw i,t. Siilaly, fo the DB updating ule, we can obtain the coesponding paaete updating ule fo node i as w i,t+ = f j Fw j,t + n i j N i\{i} k N i f k f j Fw i,t. n i k N i f k j N i\{i} Fo the IM updating ule, we have w i,t+ = f j j N i k N i f k n j Fw j,t. 3 Note that, and 3 ae expected outcoe of BD, DB and IM updated ules, which can be efeed in [35], [37]. The pefoance of adaptive filteing algoith is usually evaluated by two easues: ean-squae deviation MSD and excess-ean-squae eo EMSE, which ae defined as MSD = E w t w, 4 EMSE = E ut w t w. 5 Using, and 3, we can calculate the netwok MSD and EMSE of these thee update ules accoding to [6]. C. Relationship to Existing Distibuted Adaptive Filteing Algoiths In Section II, we have suaized the existing distibuted adaptive filteing algoiths in 5 and Table I. In this subsection, we will show that all these algoiths ae the special cases of the IM update ule in ou poposed gaphical EGT faewok. Copae 5 and 3, we can see that diffeent fitness definitions ae coesponding to diffeent distibuted adaptive filteing algoiths in Table I. Fo the unifo ule, the fitness can be unifoly defined as f i = and using the IM update ule, we have w i,t+ = Fw j,t, 6 n i j N i which is equivalent to the unifo ule in Table I. Hee, the definition of f i = eans the adoption of fixed fitness and weak selection α <<. Fo the Laplacian ule, when updating the paaete of node i, the fitness of nodes in N i can be defined as {, fo j i, f j = 7 n ax n i +, fo j = i. TABLE III DIFFERENT FITNESS DEFINITIONS. Nae Fitness: f j = Unifo [][3] Maxiu degee [8][4] Laplacian [5][6] Relative degee [8] Relative degee-vaiance [6] Metopolis [6][7] Hastings [7], fo all j N i, fo j i, N n i +, fo j = i., fo j i n ax ni +, fo j = i. n j, fo all j N i n j σ j, fo all j N i k i A ik ax{ N i, N j } k i A ik σ j,i ax{ N i, N j } k j A jk σ i,j ax{ N i σ i, N j σ j } ax{ N i σ i, N j σ j } k j A jk Fo 7, we can see that each node gives oe weight to the infoation fo itself though enhancing its own fitness. Siilaly, fo the Relative-degee-vaiance ule, the fitness can be defined as f j = n j σ j, fo all j N i. 8 Fo the etopolis ule and Hastings ule, the coesponding fitness definitions ae based on stong selection odel α, whee utility plays a doinant ole in 9. Fo the etopolis ule, the utility atix of nodes can be defined as Node i Node i Node j i k i A ik ax{ N i, N j } Node j i ax{ N i, N j } k j A jk Fo the Hastings ule, the utility atix can be defined as Node i Node j i Node i k i A ik σ j,i ax{ N i σ i, Nj σ j } Node j i σ i,j ax{ N i σi, Nj σ j } k j A jk 9 Table III suaizes diffeent fitness definitions coesponding to diffeent cobination ules in Table I. Theefoe, we can see that the existing algoiths can be suaized into ou poposed gaphical EGT faewok with coesponding fitness definitions. D. Eo-awae Distibuted Adaptive Filteing Algoith To illustate ou gaphical EGT faewok, as exaples, we futhe design two distibuted adaptive algoiths by choosing diffeent fitness functions. As discussed in Section II, the existing distibuted adaptive filteing algoiths eithe ely on the pio knowledge of netwok topology o the equieent of additional netwok statistics. All of the ae not obust to a dynaic netwok, whee a node location ay change

6 6 and the noise vaiance of each node ay also vay with tie. Consideing these pobles, we popose eo-awae algoiths based on the intuition that neighbos with low ean-squae-eo MSE should be given oe weight while neighbos with high MSE should be given less weight. The instantaneous eo of node i, denoted by i, can be calculated by i,t = d i t u i,t w i,t, n n n n whee only local data{d i t,u i,t } ae used. The appoxiated MSE of node i, denoted by β i, can be estiated by following update ule in each tie slot, Fig. 4. Gaphical evolutionay gae odel. β i,t = ν i,t β i,t +ν i,t i,t, whee ν i,t is a positive paaete. We assue that nodes can exchange thei instantaneous MSE infoation with neighbos. Based on the estiated MSE, we design two kinds of fitness: exponential fo and powe fo as follows: Powe: f i = βi λ, 3 Exponential: f i = e λβi, 4 whee λ is a positive coefficient. Note that the fitness defined in 3 and 4 ae just two exaples of ou poposed faewok, while any othe fos of fitness can be consideed, e.g., f i = logλβ i. Using the IM update ule, we have w i,t+ = j N i w i,t+ = j N i λ βj,t k N i β λ k,t Fw j,t, 5 e λβj,t k N i e λβ k,t Fw j,t. 6 Fo 5 and 6, we can see that the poposed algoiths do not diectly depend on any netwok topology infoation. Moeove, they can also adapt to a dynaic envionent when the noise vaiance of nodes ae unknown o suddenly change, since the weights can be iediately adjusted accodingly. In [33], a siila algoith was also poposed based on the instantaneous MSE infoation, which is a special case of ou eo-awae algoith with powe fo of λ =. Note that the deteinistic coefficients ae adopted when ipleenting 5 and 6, instead of using ando cobining efficient with soe pobability. Howeve, the algoith can also be ipleented using a ando selection with pobabilities. Thee will be no pefoance loss since the expected outcoe is the sae, but the efficiency convegence speed will be lowe. In Section V, we will veify the pefoance of the poposed algoith though siulation. IV. DIFFUSION ANALYSIS In a distibuted adaptive filte netwok, thee ae nodes with good signals, i.e., lowe noise vaiance, as well as nodes with poo signals. The pincipal objective of distibuted adaptive filteing algoiths is to stiulate the diffusion of good signals to the whole netwok to enhance the netwok pefoances. In this section, we will use the EGT to analyze such a dynaic diffusion pocess and deive the close-fo expession fo the diffusion pobability. In the following diffusion analysis, we assue that all nodes have the sae egesso statistics R u, but diffeent noise statistics. In a gaphical evolutionay gae, the stuctued population ae eithe esidents o utants. An ipotant concept is the fixation pobability, which epesents the pobability that the utant will eventually ovetake the whole population [34]. Let us conside a local adaptive filte netwok as shown in Fig. 4, whee the hollow points denote coon nodes, i.e., nodes with coon noise vaiance σ ; and the solid points denote good nodes, i.e., nodes with a lowe noise vaianceσ.σ and σ satisfy that σ >> σ. Hee, we adopt the binay signal odel to bette eveal the diffusion pocess of good signals. If we egad the coon nodes as esidents and the good nodes as utants, the concept of fixation pobability in EGT can be applied to analyze the diffusion of good signals in the netwok. Accoding to the definition of fixation pobability, we define the diffusion pobability in a distibuted filte netwok as the pobability that a good signal can be adopted by all nodes to update paaetes in the netwok. A. Stategies and Utility Matix As shown in Fig. 4, fo the node at the cente, its neighbos include both coon nodes and good nodes. When the cente node updates its paaete w i, it has the following two possible stategies: { S, using infoation fo coon nodes, 7 S, using infoation fo good nodes. In such a case, we can define the utility atix as follow: S S S π σ,σ π σ,σ S π σ,σ π σ,σ u u =, 8 u 3 u 4 whee πx,y epesents the steady EMSE of node with noise vaiance x using infoation fo node with noise vaiance y. Fo exaple, πσ,σ is the steady EMSE of node with noise vaiance σ adopting stategy S, i.e., updating its w using infoation fo node with noise vaiance σ which in tun adopts stategy S. In ou diffusion analysis, we assue that only two playes ae inteacting with each othe at one tie instant, i.e., thee ae two nodes exchanging and cobining infoation with each othe at one tie instant. In such a case, the payoff atix is two-use case. Note that a node chooses one specific neighbo with soe pobability,

7 7 which is equivalent to the weight that the node gives to that neighbo. Since the steady EMSE πx, y in the utility atix is deteined by the infoation cobining ule, thee is no geneal expessions fo πx, y. Nevetheless, by intuition, we know that the steady EMSE of node with vaiance σ should be lage than that of node with vaiance σ since σ >> σ, and adopting stategy S should be oe beneficial than adopting stategy S since the node can obtain bette infoation fo othes, i.e., πσ,σ > πσ,σ > πσ,σ > πσ,σ. Theefoe, we assue that the utility atix defined in 8 has the quality as follow u < u 3 < u < u 4. 9 Hee, we use an exaple in [7] to have a close-fo expession fo πx, y to illustate and veify this intuition. Accoding to [7], with sufficiently sall step size µ, the optial πx,y can be calculated by πx,y = c σ +c x 4 σ, 3 c = µtru 4, c = µ ζ, 3 σ = x y x +y, σ = x y, whee ζ = col{ζ,...,ζ N } consists of the eigenvalues of R u ecall that R u is the covaiance atix of the obseved egession data u t. Accoding to 3 and 3, we have πσ,σ = c σ +c, 3 πσ,σ = c σ σ σ +σ σ +c σ, 33 σ πσ,σ = c σ σ σ +σ +c σ, 34 πσ,σ = c σ +c. 35 Supposeσ = τσ, though copaing 3-35, we can deive the condition fo πσ,σ > πσ,σ > πσ,σ > πσ,σ as follows µ < τtr uσ 4+τ ζ. 36 Accoding to [7], the deivation of optial πx,y in 3 and 3 is based on the assuption thatµis sufficiently sall. Theefoe, the condition of µ in 36 holds. In such a case, we can conclude that πσ,σ > πσ,σ > πσ,σ > πσ,σ, which iplies that u < u 3 < u < u 4. In the following, we will analyze the diffusion pocess of stategy S, i.e., the ability of good signals diffusing ove the whole netwok. We conside an adaptive filte netwok based on a hoogenous gaph with geneal degee n and adopt the IM update ule fo the paaete update [35]. Let p and p denote the pecentages of nodes using stategies S and S in the population, espectively. Let p, p, p and p denote the pecentages of edge, whee p eans the pecentage of edge on which both nodes use stategy S and S. Let q denote the conditional pobability of a node using stategy S given that the adjacent node is using stategy S, siila we have q, q and q. In such a case, we have p +p =, q X +q X =, 37 p XY = p Y q X Y, p = p, 38 whee X and Y ae eithe o. The equations iply that the state of the whole netwok can be descibed by only two vaiables, p and q. In the following, we will calculate the dynaics of p and q unde the IM update ule. B. Dynaics of p and q In ode to deive the diffusion pobability, we fist need to analyze the diffusion pocess of the syste. As discussed in the pevious subsection, the syste dynaics unde IM update ule can be epesented by paaetes p and q. Thus, in this subsection, we will fist analyze the dynaics of p and q to undestand the dynaic diffusion pocess of the adaptive netwok. Accoding to the IM update ule, a node using stategy S is selected fo iitation with pobability p. As shown in the left pat of Fig. 4, aong its n neighbos not including itself, thee ae n nodes using stategy S and n nodes using stategy S, espectively, whee n + n = n. The pecentage of such a configuation is n n q n q n. In such a case, the fitness of this node is f = α+αn u +n u, 39 whee the baseline fitness is noalized as. We can see that 39 includes the noalized baseline fitness and also the fitness fo utility, which is the standad definition of fitness used in the EGT filed, as shown in 9. Aong those n neighbos, the fitness of node using stategy S is [n q f = α+α + ] u 3 +n q u 4, 4 and the fitness of node using stategy S is [n q f = α+α + ] u +n q u. 4 In such a case, the pobability that the node using stategy S is eplaced by S is P = n f n f +n f +f. 4 Theefoe, the pecentage of nodes using stategy S, p, inceases by /N with pobability Pob p = n = p q n N qn n +n =n n n f. 43 n f +n f +f Meanwhile, the edges that both nodes use stategys incease by n, thus, we have Pob p = n n = p q n nn n qn n f. 44 n f +n f +f

8 8 Siila analysis can be applied to the node using stategy S. Accoding to the IM update ule, a node using stategy S is selected fo iitation with pobability p. As shown in the ight pat of Fig. 4, we also assue that thee ae n nodes using stategy S and n nodes using stategy S aong its n neighbos. The pecentage of such a phenoenon is n n q n q n. Thus, the fitness of this node is g = α+αn u +n u Aong those n neighbos, the fitness of node using stategy S is g = α+α n q u 3 + [ n q + ] u 4, 46 and the fitness of node using stategy S is g = α+α n q u + [ n q + ] u. 47 In such a case, the pobability that the node using stategy S is eplaced by S is n g P =. 48 n g +n g +g Theefoe, the pecentage of nodes using stategy S, p, deceases by /N with pobability Pob p = n = p q n N qn n +n =n n n g. 49 n g +n g +g Meanwhile, the edges that both nodes use stategy S decease by n, thus, we have Pob p = n n = p q n nn n qn n g. 5 n g +n g +g Cobining 43 and 49, we have the dynaic of p as ṗ = N Pob p = N N Pob p = N = αnn p Nn+ γ u +γ u +γ 3 u 3 +γ 4 u 4 +Oα,5 whee the second equality is accoding to Taylo s Theoe and weak selection assuption with α goes to zeo [36], and the paaetes γ, γ, γ 3 and γ 4 ae given as follows: γ = q [n q +q +3], 5 γ = q q [n q +q +] n,53 γ 3 = q +q [n q +q +]+ n, 54 γ 4 = q [n q +q +3]. 55 In 5, the dot notation ṗ epesents the dynaic of p, i.e., the vaiation of p within a tiny peiod of tie. In such a case, the utility obtained fo the inteactions is consideed as liited contibution to the oveall fitness of each playe. On one hand, the esults deived fo weak selection often eain as valid appoxiations fo lage selection stength [3]. On the othe hand, the weak selection liit has a long tadition in theoetical biology [37]. Moeove, the weak selection assuption can help achieve a close-fo analysis of diffusion pocess and bette eveal how the stategy diffuses ove the netwok. Siilaly, by cobining 44 and 5, we have the dynaics of p as n n ṗ = nn Pob p = n nn n = n n nn Pob p = n nn n = = p n+n +n q q +Oα.56 Besides, we can also have the dynaics of q as q = d p dt p p = +n q q +Oα. 57 n+n p C. Diffusion Pobability Analysis The dynaic equation ofp in 5 eflects the the dynaic of nodes updating w using infoation fo good nodes, i.e., the diffusion status of good signals in the netwok. A positive ṗ eans that good signals ae diffusing ove the netwok, while a negativeṗ eans that good signals have not been well adopted. The diffusion pobability of good signals is closely elated to the noise vaiance of good nodes σ. Intuitively, the lowe σ, the highe pobability that good signals can spead the whole netwok. In this subsection, we will analyze the close-fo expession fo the diffusion pobability. As discussed at the beginning of Section IV, the state of whole netwok can be descibed by only p and q. In such a case, 5 and 57 can be e-witten as functions of p and q ṗ = α Gp,q +Oα, 58 q = G p,q +Oα. 59 Fo 58 and 59, we can see that q conveges to equilibiu in a uch faste ate thanp unde the assuption of weak selection. At the steady state of q, i.e., q =, we have q q = n. 6 In such a case, the dynaic netwok will apidly convege onto the slow anifold, defined by G p,q =. Theefoe, we can assue that 6 holds in the whole convegence pocess of p. Accoding to and 6, we have q = p + n p, 6 q = n n p, 6 q = n n p, 63 q = n n p. 64

9 9 Theefoe, the diffusion pocess can be chaacteized by only p. Thus, we can focus on the dynaics of p to deive the diffusion pobability, which is given by following Theoe. Theoe : In a distibuted adaptive filte netwok which can be chaacteized by a N-node egula gaph with degee n, suppose thee ae coon nodes with noise vaiance σ and good nodes with noise vaiance σ, whee each coon node has connection edge with only one good node. If each node updates its paaete w using the IM update ule, the diffusion pobability of the good signal can be appoxiated by P diff = n+ + αnn 6n+ 3ξ u +ξ u +ξ 3 u 3 +ξ 4 u 4, 65 whee the paaetes ξ, ξ, ξ 3 and ξ 4 ae as follows: ξ = n 5n+3, ξ = n n 3, 66 ξ 3 = n +n 3, ξ 4 = n +4n Poof: See Appendix. Using Theoe, we can calculate the diffusion pobability of the good signals ove the netwok, which can be used to evaluate the pefoance of an adaptive filte netwok. Siilaly, the diffusion dynaics and pobabilities unde BD and DB update ules can also be deived using the sae analysis. The following theoe shows an inteesting esult, which is based on an ipotant theoe in [9], stating that evolutionay dynaics unde BD, DB, and IM ae equivalent fo undiected egula gaphs. Theoe : In a distibuted adaptive filte netwok which can be chaacteized by a N-node egula gaph with degee n, suppose thee ae coon nodes with noise vaiance σ and good nodes with noise vaianceσ, whee each coon node has connection edge with only one good node. If each node updates its paaete w using the IM update ule, the diffusion pobabilities of good signals unde BD and DB update ules ae sae with that unde the IM update ule. V. EVOLUTIONARILY STABLE STRATEGY In the last section, we have analyzed the infoation diffusion pocess in an adaptive netwok unde the IM update ule, and deived the diffusion pobability of stategy S that using infoation fo good nodes. On the othe hand, consideing that if the whole netwok has aleady chosen to adopt this favoable stategy S, is the cuent state a stable netwok state, even though a sall faction of nodes adopt the othe stategy S? In the following, we will answe these questions using the concept of evolutionaily stable stategy ESS in evolutionay gae theoy. As discussed in Section III-A, the ESS ensues that one stategy is esistant against invasion of anothe stategy [38]. In ou syste odel, it is obvious that S, i.e., using infoation fo good nodes, is the favoable stategy and a desied ESS in the netwok. In this section, we will check whethe stategy S is evolutionaily stable. A. ESS in Coplete Gaphs We fist discuss whethe stategy S is an ESS in coplete gaphs, which is shown by the following theoe. Theoe 3: In a distibuted adaptive filte netwok that can be chaacteized by coplete gaphs, stategy S is always an ESS stategy. Poof: In a coplete gaph, each node eets evey othe node equally likely. In such a case, accoding to the utility atix in 8, the aveage utilities of using stategies S and S ae given by U = p u +p u, 68 U = p u 3 +p u 4, 69 whee p and p ae the pecentages of population using stategies S and S, espectively. Conside the scenaio that the ajoity of the population adopt stategys, while a sall faction of the population adopt S which is consideed as invasion, p = ǫ. In such a case, accoding to the definition of ESS in 7, stategy S is evolutionay stable if U > U fo p,p = ǫ, ǫ, i.e., ǫu 3 u + ǫu 4 u >. 7 Fo ǫ, the left hand side of 7 is positive if and only if u 4 > u o u 4 = u and u 3 > u. 7 The 7 gives the sufficient evolutionay stable condition of stategy S. In ou syste, we have u 4 > u > u 3 > u, which eans that 7 always holds. Theefoe, stategy S is always an ESS if the adaptive filte netwok is a coplete gaph. B. ESS in Incoplete Gaphs Let us conside an adaptive filte netwok which can be chaacteized by an incoplete egula gaph with degee n. The following theoe shows that stategy S is always an ESS in such an incoplete gaph. Theoe 4: In a distibuted adaptive filte netwok which can be chaacteized by a egula gaph with degeen, stategy S is always an ESS stategy. Poof: Using the pai appoxiation ethod [3], the eplicato dynaics of stategiess ands on a egula gaph of degee n can be appoxiated siply by ṗ = p p u +p u φ, 7 ṗ = p p u 3 +p u 4 φ, 73 whee φ = p p u +p p u +u 3 +p p u 4 is the aveage utility, and u, u, u 3 and u 4 ae given as follows: u = u, u = u +u, u 3 = u 74 3 u, u 4 = u 4. The paaete u depends on the thee update ules IM, BD and DB, which is given by [3] IM: u = n+3u +u u 3 n+3u 4, 75 n+3n BD: u = n+u +u u 3 n+u 4, 76 n+n DB: u = u +u u 3 u n

10 TR i u Node i Node i Fig. 5. Netwok infoation fo siulation, including netwok topology fo nodes left, tace of egesso covaiance TR u ight top and noise vaiance σ i ight botto. In such a case, the equivalent utility atix is S S S u u +u S u 3 u. 78 u 4 Accoding to 7, the evolutionay stable condition fo stategy S is u 4 > u +u. 79 Since u < u 3 < u < u 4, we have u < fo all thee update ules. In such a case, 79 always holds, which eans that stategy S is always an ESS stategy. This copletes the poof of the theoe. Tansient netwok EMSE db Relative degee algoith [8] Hastings algoith [7] Adaptive cobine algoith [7] Relative degee-vaiance algoith [6] Poposed eo-awae algoith with powe fo Poposed eo-awae algoith with exponential fo Tie Index a Netwok EMSE. VI. SIMULATION RESULTS In this section, we develop siulations to copae the pefoances of diffeent adaptive filteing algoiths, as well as to veify the deivation of infoation diffusion pobability and the analysis of ESS. A. Mean-squae Pefoances The netwok topology used fo siulation is shown in the left pat of Fig. 5, whee andoly nodes ae andoly located. The signal and noise powe infoation of each node ae also shown in the ight pat of Fig. 5, espectively. In the siulation, we assue that the egessos with size M = 5, ae zeo-ean Gaussian and independent in tie and space. The unknown vecto is set to be w = 5 / and the step size of the LMS algoith at each node i is set as µ i =.. All the siulation esults ae aveaged ove 5 independent unnings. All the pefoance copaisons ae conducted aong six diffeent kinds of distibuted adaptive filteing algoiths as follows: Relative degee algoith [8]; Hastings algoith [7]; Adaptive cobine algoith [7]; Relative degee-vaiance algoith [6]; Poposed eo-awae algoith with powe fo; Poposed eo-awae algoith with exponential fo. Aong these algoiths, the adaptive cobine algoith [7] and ou poposed eo-awae algoith ae based on dynaic cobines weights, which ae updated in each tie slot. The Tansient netwok MSD db Fig Relative degee algoith [8] Hastings algoith [7] Adaptive cobine algoith [7] Relative degee-vaiance algoith [6] Poposed eo-awae algoith with powe fo Poposed eo-awae algoith with exponential fo Tie Index b Netwok MSD. Tansient pefoances copaison with known noise vaiances. diffeence is the updating ule, whee the adaptive cobine algoith in [7] uses optiization and pojection ethod, and ou poposed algoiths use the appoxiated EMSE infoation. In the fist copaison, we assue that the noise vaiance of each node is known by the Hastings and elative degee-vaiance algoiths. Fig. 6 shows the tansient netwok-pefoance copaison esults aong six kinds of algoiths in tes of EMSE and MSD. Unde the siila convegence ate, we can see that the elative degee-vaiance

11 Steady EMSE db Steady MSD db Fig Relative degee algoith [8] Hastings algoith [7] Adaptive cobine algoith [7] Relative degee-vaiance algoith [6] Poposed eo-awae algoith with powe fo Poposed eo-awae algoith with exponential fo Node Index a Node s EMSE Node Index b Node s MSD. Steady pefoances copaison with known noise vaiances. algoith pefos the best. The poposed algoith with exponential fo pefos bette than the elative degee algoith. With the powe fo fitness, the poposed algoith can achieve siila pefoance, if not bette than, copaed with adaptive cobine algoith, and both algoiths pefos bette than all othe algoiths except the elative degee-vaiance algoith. Howeve, as discussed in Section, the elative degee-vaiance algoith equies noise vaiance infoation of each node, while ou poposed algoith does not. Fig. 7 shows the coesponding steady-state pefoances of each node fo six kinds of distibuted adaptive filteing algoiths in tes of EMSE and MSD. Since the steady-state esult is fo each node, besides aveaging ove 5 independent unnings, we aveage at each node ove tie slots afte the convegence. We can see that the copaison esults of steady-state pefoances ae siila to those of the tansient pefoances. In the second copaison, we assue that the noise vaiance of each node is unknown, but can be estiated by the ethod poposed in [7]. Fig. 8 and Fig. 9 show the tansient and steady-state pefoances fo six kinds of algoiths in tes of EMSE and MSD unde siila convegence ate. Since the noise vaiance estiation equies additional coplexity, we also siulate the Hastings and elative degee-vaiance algoiths without vaiance estiation fo fai copaison, whee the noise vaiance is set as the netwok aveage vaiance, which is assued to be pio infoation. Copaing with Fig. 7, we can see that when the noise vaiance infoation is not available, the pefoance degadation of elative degee-vaiance algoith is significant, about.5db % oe eo even with noise vaiance estiation, while the pefoance of Hastings algoith degades only a little since it elies less on the noise vaiance infoation. Fo Fig. 8-b, Tansient netwok EMSE db Tansient netwok MSD db Fig Relative degee algoith [8] Hasting algoith [7] with noise estiation Hasting algoith [7] without noise estiation Adaptive cobine algoith [7] Relative degee-vaiance algoith [6] with noise estiation Relative degee-vaiance algoith [6] withiout noise estiation Poposed eo-awae algoith with powe fo Poposed eo-awae algoith with exponential fo Tie Index a Netwok EMSE. Relative degee algoith [8] Hasting algoith [7] with noise estiation Hasting algoith [7] without noise estiation Adaptive cobine algoith [7] Relative degee-vaiance algoith [6] with noise estiation Relative degee-vaiance algoith [6] withiout noise estiation Poposed eo-awae algoith with powe fo Poposed eo-awae algoith with exponential fo Tie Index b Netwok MSD. Tansient pefoances copaison with unknown noise vaiances. we can clealy see that when the vaiance estiation ethod is not adopted, ou poposed algoith with powe fo achieves the best pefoance. When the vaiance estiation ethod is adopted, the pefoances of ou poposed algoith with powe fo, the elative degee-vaiance and the adaptive cobine algoith ae siila, all of which pefo bette than othe algoiths. Nevetheless, the coplexity of both elative degee-vaiance algoith with vaiance estiation and the adaptive cobine algoith ae highe than that of ou poposed algoith with powe fo. Such esults iediately show the advantage of the poposed geneal faewok. We should notice that oe algoiths with bette pefoances unde cetain citeia can be designed based on the poposed faewok by choosing oe pope fitness functions. B. Diffusion Pobability In this subsection, we develop siulation to veify the diffusion pobability analysis in Section IV. Fo the siulation setup, thee types of egula gaphs ae geneated with degee n = 3, 4 and 6, espectively, as shown in Fig. -a. All these thee types of gaphs ae with N = nodes, whee each node s tace of egesso covaiance is set to be TR u =, the coon nodes s noise vaiance is set as σ =.5 and

12 -3.5 Relative degee algoith [8] Hastings algoith [7] with noise estiation Hastings algoith [7] without noise estiation Adaptive cobine algoith [7] Relative degee-vaiance algoith [6] with noise estiation Relative degee-vaiance algoith [6] withiout noise estiation Poposed eo-awae algoith with powe fo Poposed eo-awae algoith with exponential fo a Regula gaph stuctues with degee n = 3, 4 and 6. Steady EMSE db Steady EMSE db Node Index a Node s EMSE Node Index b Node s MSD. Fig.. Diffusion Pobability Noise Vaiance of Good Nodes b Diffusion pobability. n=3 Theoetical Results n=3 Siulated Results n=4 Theoetical Results n=4 Siulated Results n=6 Theoetical Results n=6 Siulated Results Diffusion pobabilities unde thee types of egula gaphs. Fig. 9. Steady pefoances copaison with unknown noise vaiances. the good node s noise vaiance is set as σ [.,.8]. In the siulation, the netwok is initialized with the state that all coon nodes choosing stategy S. Then, at each tie step, a andoly chosen node s stategy is updated accoding to the IM ules unde weak selection w =., as illustated in Section III-B. The update steps ae epeated until eithe stategy S has eached fixation o the nube of steps has each the liit. The diffusion pobability is calculated by the faction of uns whee stategy S eached fixation out of 6 uns. Fig. -b shows the siulation esults, fo which we can see that all the siulated esults ae basically accod with the coesponding theoetical esults and the gaps ae due to the appoxiation duing the deivations. Moeove, we can see that the diffusion pobability of good signal deceases along with the incease of its noise vaiance, i.e., bette signal has bette diffusion capability. C. Evolutionaily Stable Stategy To veify that stategy S is an ESS in the adaptive netwok, we futhe siulate the IM update ule on a gid netwok with degee n = 4 and nube of nodes N =, as shown in Fig. whee the hollow points epesent coon nodes and the solid nodes epesent good nodes. In the siulation, all the settings ae sae with those in the siulation of diffusion pobability in Section VI-B, except the initial netwok setting. The initial netwok state is set that the ajoity of nodes adopt stategy S denoted with black colo including both hollow and solid nodes in Fig., and only a vey sall pecentage of nodes use stategy S denoted with ed colo. Fo the stategy updating pocess of the whole netwok illustated in Fig., we can see that the netwok finally abandons the unfavoable stategy S, which veifies the stability of stategy S. VII. CONCLUSION In this pape, we poposed an evolutionay gae theoetic faewok to offe a vey geneal view of the distibuted adaptive filteing pobles and unify existing algoiths. Based on this faewok, as exaples, we futhe designed two eo-awae adaptive filteing algoiths. Using the gaphical evolutionay gae theoy, we analyzed the infoation diffusion pocess in the netwok unde the IM update ule, and poved that the stategy of using infoation fo nodes with good signal is always an ESS. We would like to ephasize that, unlike the taditional botto-up appoaches, the poposed gaphical evolutionay gae theoetic faewok povides a top-down design philosophy to undestand the fundaentals of distibuted adaptive algoiths. Such a top-down design philosophy is vey ipotant to the field of distibuted adaptive signal pocess, since it offes a unified view of the foulation and can inspie oe new distibuted adaptive algoiths to be designed in the futue. APPENDIX PROOF OF THEOREM Poof: Fist, let us define p as the ean of the inceent of p pe unit tie given as follows p = ṗ /N αnn n n+ p p ap +b. 8

13 3 Fig.. Stategy updating pocess in a gid netwok with degee n = 4 and nube of nodes N =. whee the second step is deived by substituting 6-64 into 5 and the paaetes a and b ae given as follows: a= n n+3u u u 3 +u 4, 8 b= n n+3u 3u +n +n 3u 3 +n+3u 4.8 We then define vp as the vaiance of the inceent of p pe unit tie, which can be calculated by whee p p = N p vp = ṗ, 83 /N can be coputed by Pob p = N + Pob p = N = nn N n n+ p p +Oα. 84 In such a case, vp can be appoxiated by vp N nn n n+ p p. 85 Suppose the initial pecentage of good nodes in the netwok is p. Let us define Hp as the pobability that these good signals can finally be adopted by the whole netwok, i.e., all nodes can update thei own w using infoation fo good nodes. Accoding to the backwad Kologoov equation [39], Hp satisfies following diffeential equation = p dhp + vp dp d Hp dp. 86 With the weak selection assuption, we can have the appoxiate solution of Hp as Hp = p + αn 6n+ p p a+3b+ap. 87 Let us conside the wost initial syste state that each coon node has connection with only one good node, i,e., p =, we have n+ H n+ n+ + αnn 6n+ 3a+3b. 88 By substituting 8 and 8 into 88, we can have the closefo expession fo the diffusion pobability in 65. This copletes the poof of the theoe. Reak: Fo 87, we can see that thee ae two tes constituting the expession of diffusion pobability: the initial pecentage of stategy S, p the initial syste state and the second te epesenting the changes of syste state afte beginning, in which a + 3b deteines whethe p is inceasing o deceasing along with the syste updating. If a+3b <, i.e., the diffusion pobability is even lowe than the initial pecentage of stategy S, the infoation fo good nodes ae shinking ove the netwok, instead of speading. Theefoe, a+3b > is oe favoable fo the ipoveent of the adaptive netwok pefoance. REFERENCES [] V. D. Blondel, J. M. Hendickx, A. Olshevsky, and J. N. Tsitsiklis, Distibuted pocessing ove adaptive netwoks, in Poc. Adaptive Senso Aay Pocessing Wokshop, Lexington, MA, Jun. 6, pp. 3. [] D. Li, K. D. Wong, Y. H. Hu, and A. M. Sayed, Detection, classification, and tacking of tagets, IEEE Signal Pocess. Mag., vol. 9, no., pp. 7 9,. [3] F. C. R. J, M. L. R. de Capos, and S. Wene, Distibuted coopeative spectu sensing with selective updating, in Poc. Euopean Signal Pocessing Confeence EUSIPCO, Buchaest, Roania, Aug., pp [4] S. Haykin and K. J. R. Liu, Handbook on Aay Pocessing and Senso Netwoks. New Yok: IEEE-Wiley, 9. [5] C. G. Lopes and A. H. Sayed, Inceental adaptive stategies ove distibuted netwoks, IEEE Tans. Signal Pocess., vol. 55, no. 8, pp , 7. [6] F. S. Cattivelli and A. H. Sayed, Diffusion LMS stategies fo distibuted estiation, IEEE Tans. Signal Pocess., vol. 58, no. 3, pp ,. [7] N. Takahashi, I. Yaada, and A. H. Sayed, Diffusion least-ean squaes with adaptive cobines: Foulation and pefoance analysis, IEEE Tans. Signal Pocess., vol. 58, no. 9, pp ,. [8] F. S. Cattivelli, C. G. Lopes, and A. H. Sayed, Diffusion ecusive leastsquaes fo distibuted etiation ove adaptive netwoks, IEEE Tans. Signal Pocess., vol. 56, no. 5, pp , 8. [9] S. Theodoidis, K. Slavakis, and I. Yaada, Adaptive leaning in a wold of pojections, IEEE Signal Pocess. Mag., vol. 8, no., pp. 97 3,. [] R. L. G. Cavalcante, I. Yaada, and B. Mulgew, An adaptive pojected subgadient appoach to leaning in diffusion netwoks, IEEE Tans. Signal Pocess., vol. 57, no. 7, pp , 9. [] S. Chouvadas, K. Slavakis, and S. Theodoidis, Adaptive obust distibuted leaning in diffusion senso netwoks, IEEE Tans. Signal Pocess., vol. 59, no., pp ,.

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Chunxiao Jiang S 9-M 3 eceived his B.S. degee in infoation engineeing fo Beijing Univesity of Aeonautics and Astonautics Beihang Univesity in 8 and the Ph.D. degee fo Tsinghua Univesity THU, Beijing in 3, both with the highest honos. Duing -, he visited the Signals and Infoation Goup SIG at Depatent of Electical & Copute Engineeing ECE of Univesity of Mayland UMD, suppoted by China Scholaship Council CSC fo one yea. D. Jiang is cuently a eseach associate in ECE depatent of UMD with Pof. K. J. Ray Liu, and also a post-docto in EE depatent of THU. His eseach inteests include the applications of gae theoy and queuing theoy in wieless counication and netwoking and social netwoks. D. Jiang eceived the Beijing Distinguished Gaduated Student Awad, Chinese National Fellowship and Tsinghua Outstanding Distinguished Doctoal Dissetation in 3. Yan Chen S 6-M eceived the Bachelo s degee fo Univesity of Science and Technology of China in 4, the M. Phil degee fo Hong Kong Univesity of Science and Technology HKUST in 7, and the Ph.D. degee fo Univesity of Mayland College Pak in. Fo to 3, he is a Postdoctoal eseach associate in the Depatent of Electical and Copute Engineeing at Univesity of Mayland College Pak. Cuently, he is a Pincipal Technologist at Oigin Wieless Counications. He is also affiliated with Signal and Infoation Goup of Univesity of Mayland College Pak. His cuent eseach inteests ae in social leaning and netwoking, behavio analysis and echanis design fo netwok systes, ultiedia signal pocessing and counication. D. Chen eceived the Univesity of Mayland Futue Faculty Fellowship in, Chinese Govenent Awad fo outstanding students aboad in, Univesity of Mayland ECE Distinguished Dissetation Fellowship Honoable Mention in, and was the Finalist of A. Jaes Clak School of Engineeing Deans Doctoal Reseach Awad in. K. J. Ray Liu F 3 was naed a Distinguished Schola-Teache of Univesity of Mayland, College Pak, in 7, whee he is Chistine Ki Einent Pofesso of Infoation Technology. He leads the Mayland Signals and Infoation Goup conducting eseach encopassing boad aeas of signal pocessing and counications with ecent focus on coopeative and cognitive counications, social leaning and netwok science, infoation foensics and secuity, and geen infoation and counications technology. D. Liu is the ecipient of nueous honos and awads including IEEE Signal Pocessing Society Technical Achieveent Awad and Distinguished Lectue. He also eceived vaious teaching and eseach ecognitions fo Univesity of Mayland including univesity-level Invention of the Yea Awad; and Poole and Kent Senio Faculty Teaching Awad, Outstanding Faculty Reseach Awad, and Outstanding Faculty Sevice Awad, all fo A. Jaes Clak School of Engineeing. An ISI Highly Cited Autho, D. Liu is a Fellow of IEEE and AAAS. D. Liu is Pesident of IEEE Signal Pocessing Society whee he has seved as Vice Pesident Publications and Boad of Goveno. He was the Edito-in- Chief of IEEE Signal Pocessing Magazine and the founding Edito-in-Chief of EURASIP Jounal on Advances in Signal Pocessing.