Spectral Properties of the Grounded Laplacian Matrix with Applications to Consensus in the Presence of Stubborn Agents

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1 Spectral Properte of the Grounded Laplacan Matrx wth Applcaton to Conenu n the Preence of Stubborn Agent Mohammad Pran and Shreya Sundaram Abtract We tudy lnear conenu and opnon dynamc n network that contan tubborn agent. Prevou work ha hown that the convergence rate of uch dynamc gven by the mallet egenvalue of the grounded Laplacan nduced by the tubborn agent. Buldng on th, we defne a noton of centralty for each node n the network baed upon the mallet egenvalue obtaned by removng that node from the network. We how that th centralty can devate from other well known centralte. We then characterze certan properte of the mallet egenvalue and correpondng egenvector of the grounded Laplacan n term of the graph tructure and the expected aborpton tme of a random walk on the graph. I. INTRODUCTION Collectve behavor n network of agent ha been tuded n a varety of communte ncludng ocology, phyc, bology, economc, computer cence and engneerng [], [2]. A topc that ha receved partcular nteret that of opnon dynamc and conenu n network, where the agent repeatedly update ther opnon or tate va nteracton wth ther neghbor [3] [5]. For certan clae of nteracton dynamc, varou condton have been provded on the network topology that guarantee convergence to a common tate [6] [9]. Ade from provdng condton under whch convergence occur, the queton of what value the agent converge to alo of mportance. In partcular, the ablty of certan ndvdual agent n the network to excevely nfluence the fnal value can be vewed a both a beneft and a drawback, dependng on whether thoe agent are vewed a leader or adverare. The effect of ndvdual agent ntal value on the fnal conenu value ha been tuded n [0] []. When a ubet of agent fully tubborn (.e., they refue to update ther value), t ha been hown that under a certan cla of lnear update rule, the value of all other agent aymptotcally converge to a convex combnaton of the tubborn agent value [2]. Gven the ablty of ndvdual to nfluence lnear opnon dynamc by keepng ther value contant, a natural metrc to conder the peed at whch the populaton converge to the fnal value for a gven et of tubborn agent or leader. The convergence rate dctated by pectral properte of certan matrce; n contnuou-tme dynamc, th the grounded Laplacan matrx [3]. There are varou recent work that nvetgate the leader electon problem, where the goal to elect a et of leader (or tubborn agent) to maxmze the Th materal baed upon work upported by the Natural Scence and Engneerng Reearch Councl of Canada. The author are wth the Department of Electrcal and Computer Engneerng at the Unverty of Waterloo. E-mal: {mpran, undara}@uwaterloo.ca convergence rate [2], [4] [6]. Smlarly, one can conder the problem of leader electon problem n network n the preence of noe [7], where the man goal to mnmze the teady tate error covarance of the follower [8]. In th paper, we contnue the nvetgaton of convergence rate n contnuou-tme lnear conenu dynamc wth tubborn agent. Specfcally, gven the key role of the grounded Laplacan n th ettng, we provde a characterzaton of certan pectral properte of th matrx, addng to the lterature on uch matrce [2] [4], [9]. We defne a natural centralty metrc, termed groundng centralty, baed on the mallet egenvalue of the grounded Laplacan nduced by each node. Th capture the mportance of the node n the network va the rate of convergence t would nduce n the conenu dynamc f choen a a tubborn agent. We how that th centralty metrc can devate from other common centralty metrc uch a cloene centralty, betweenne centralty and degree centralty. We then provde bound on the mallet egenvalue of the grounded Laplacan, along wth certan properte of the aocated egenvector. Fnally, we compare the mallet egenvalue to the maxmum expected aborpton tme for a random walk on the underlyng graph. We provde bound baed on the devaton n component of the egenvector for the mallet egenvalue. Our reult on the pectrum of the grounded Laplacan are of ndependent nteret, wth applcaton to varou ettng [3], [9]. The tructure of the paper a follow. In Secton III we ntroduce the mathematcal formulaton of the conenu problem n the preence of tubborn agent, and how the role of the grounded Laplacan matrx. We ntroduce the noton of groundng centralty n Secton IV. Secton V ntroduce ome pectral properte of the grounded Laplacan matrx for general graph a well a ome unque properte for pecal graph uch a tree. In Secton VI we make a connecton to aborbng random walk on Markov chan and provde bound on the mallet egenvalue of the grounded Laplacan n term of the pread of the component n the aocated egenvector. Secton VII conclude the paper. II. DEFINITIONS AND NOTATION We ue G(V, E) to denote a weghted and undrected graph where V the et of vertce (or node) and E V V the et of edge. The neghbor of vertex v V n graph G are gven by the et N = {v j V (v, v j ) E}. The weght aocated wth an edge connectng vertex v V to vertex v j V denoted by w j. We take the weght to be nonnegatve and ymmetrc (.e., w j = w j ).

2 We defne d = v j N w j ; n unweghted graph where each w j {0, }, d mply denote the degree of vertex v. We wll be conderng a nonempty ubet of vertce S V to be tubborn, and aume wthout lo of generalty that the tubborn agent are placed lat n an orderng of the agent. We ay a vertex v V \S an α-vertex f N S, and ay v a β-vertex otherwe. Vertce n V \ S wll alo be called follower. A. Conenu Model III. BASIC MODELS Conder a network of agent decrbed by the connected and undrected graph G(V, E), repreentng the tructure of the ytem, and a et of dfferental equaton decrbng the nteracton between each par. In the tudy of conenu and contnuou-tme opnon dynamc [7], each agent v V tart wth an ntal calar tate (or opnon) y (t), whch t contnuouly update a a functon of the tate of t neghbor. A commonly tuded veron of thee dynamc nvolve a lnear update rule of the form ẏ (t) = v j N w j (y j (t) y (t)), () where w j the weght agned by node v to the tate of t neghbor v j. Aggregatng the tate of all of the node nto the vector Y (t) = [ y (t) y 2 (t) y n (t) ] T, equaton () can be wrtten a Ẏ = LY, (2) where L the weghted graph Laplacan defned a L = D A wth degree matrx D = dag(d, d 2,..., d n ) and weghted adjacency matrx A contanng the weght w j n entry (, j). For an undrected graph G wth ymmetrc weght (.e., w j = w j for all j), L a ymmetrc matrx wth real egenvalue that can be ordered equentally a 0 = λ λ 2 λ n 2 where = max v j N w j. When the graph G connected, the econd mallet egenvalue λ 2 potve and all node aymptotcally reach conenu on the value y eq = ct Y (0) where c the left egenvector of L c T correpondng to λ [7]. B. Conenu In The Preence of Stubborn Agent Aume that there a ubet S V of agent whoe opnon are kept contant throughout tme,.e., v S, y R uch that y (t) = y t R 0. Such agent are known a tubborn agent or leader (dependng on the context) [2], [4]. In th cae the dynamc (2) can be wrtten n the matrx form ] [ ] [ ] [ẎF (t) L L = 2 YF (t), (3) Ẏ S (t) L 2 L 22 Y S (t) where Y F and Y S are the tate of the follower and tubborn agent, repectvely. Snce the tubborn agent keep ther value contant, the matrce L 2 and L 22 are zero. It can be hown that the tate of each follower aymptotcally converge to a convex combnaton of the value of the tubborn agent, and that the rate of convergence aymptotcally gven by the mallet egenvalue of L [2]. The matrx L called the grounded Laplacan formed by removng the row and column correpondng to the tubborn agent from L. In the ret of the paper, the grounded Laplacan formed by removng the row and column of L correpondng to the vertce n et S V wll be denoted by L gs, and L g f S = {v }. When the et S fxed and clear from the context, we wll mply ue the notaton L g to denote the grounded Laplacan. For any gven et S, we denote the mallet egenvalue of the grounded Laplacan matrx by λ(l gs ) or mply λ. Remark : When the graph G connected, the grounded Laplacan matrx a potve defnte matrx and t nvere a nonnegatve matrx (.e., a matrx whoe element are nonnegatve) [20]. From the Perron-Frobenu (P-F) theorem, the egenvector aocated wth the mallet egenvalue of the grounded Laplacan can be choen to be nonnegatve (elementwe). Furthermore, when the tubborn agent not a cut vertex, the egenvector aocated wth the mallet egenvalue can be choen to have all element potve. There have been varou recent nvetgaton of graph properte that mpact the convergence rate for a gven et of tubborn agent, leadng to the development of algorthm to fnd approxmately optmal et of tubborn agent to maxmze the convergence rate [2], [4], [6]. Motvated by the fact that the mallet egenvalue of the grounded Laplacan domnate the convergence rate, n the next ecton we defne a noton of groundng centralty for each node n the network, correpondng to how quckly that node would lead the ret of the network to teady tate f choen to be tubborn. We how va an example that groundng centralty can devate from other common centralty metrc n network, and then n ubequent ecton, we provde bound on the groundng centralty baed on graph theoretc properte. IV. GROUNDING CENTRALITY There are varou metrc for evaluatng the mportance of ndvdual node n a network. Common example nclude eccentrcty (the larget dtance from the gven node to any other node), cloene centralty (the um of the dtance from the gven node to all other node n the graph), degree centralty (the degree of the gven node) and betweenne centralty (the number of hortet path between all node that pa through the gven node) [2] [22]. In addton to the above centralty metrc (whch are purely baed on poton n the network), one can alo derve centralty metrc that pertan to certan clae of dynamc occurrng on the network. For example, [23] agned a centralty core to each node baed on t component n a left-egenvector of the ytem matrx. Smlarly, [24] tuded dcrete tme conenu dynamc and propoed centralty metrc to capture the nfluence of forceful agent. The dcuon on convergence rate nduced by each node n the lat ecton alo lend telf to a natural dynamcal centralty metrc, defned a follow.

3 Defnton : Conder a weghted graph G(V, E), wth weght w j for edge (, j) E. The groundng centralty of each vertex v V, denoted by I(), I() = λ(l g ). The et of groundng central vertce n the graph G gven by IC(G) = argmax v V λ(l g ). Accordng to the above defnton, a groundng central vertex v IC(G) a vertex that maxmze the aymptotc convergence rate f choen a a tubborn agent (or leader), over all poble choce of ngle tubborn agent. It wa hown n [2] and [6] that the convergence tme n a network n the preence of tubborn agent (or leader) upper bounded by an ncreang functon of the dtance from the tubborn agent to the ret of the network. In the cae of a ngle tubborn agent, the noton of dtance from that agent to the ret of the network mlar to that of cloene centralty and eccentrcty. Whle th a natural approxmaton for the groundng centralty (and ndeed play a role n the upper bound provded n thoe paper), there are graph where the groundng centralty can devate from other well known centralte, a hown below. Example : A broom tree, B n,, a tar S wth leaf vertce and a path of length n attached to the center of the tar, a llutrated n Fg. [25] Fg. : Broom tree wth = 4, n = 9. Defne the cloene central vertex a a vertex whoe ummaton of dtance to the ret of the vertce mnmum, the degree central vertex a a vertex wth maxmum degree n the graph and the center of the graph a a vertex wth mallet eccentrcty [22]. Conder the broom tree B 2 +,. By numberng the vertce a hown n Fg., for = 500, we fnd (numercally) that the groundng central vertex vertex 64, and the center of the graph 750. The cloene and degree and betweenne central vertce are located at the mddle of the tar (vertex 50). The devaton of the groundng central vertex from the other centralte and the center of th graph ncreae a ncreae. A dcued n the prevou ecton, the problem of characterzng the groundng centralty of vertce ung graphtheoretc properte an ongong area of reearch [2] [4], [6]. In the next ecton, we develop ome bound for λ(l g ) by tudyng certan pectral properte of the grounded Laplacan. V. SPECTRAL PROPERTIES OF THE GROUNDED LAPLACIAN MATRIX L g Theorem : Conder a graph G(V, E) wth a ngle tubborn agent v V. Defne w = max v N w to be the larget edge weght between any α-vertex and the tubborn agent. Then the mallet egenvalue of the weghted grounded Laplacan matrx L g atfe 0 λ w. Proof: The lower bound mply obtaned from the Cauchy nterlacng theorem, whch tate that 0 = λ (L) λ(l g ) λ 2 (L), where λ 2 (L) the econd mallet egenvalue of the graph Laplacan matrx and known a the algebrac connectvty of graph G [26]. [ ] For the upper bound, let x = T x x 2 x n denote the egenvector aocated wth the mallet egenvalue. The egenvector equaton for the -th vertex gven by d x w j x j = λx. (4) v j N \{v } Addng all of thee egenvector equaton (or equvalently, multplyng both de of the egenvector equaton correpondng to λ(l g ) by a vector contng of all ), we have T L g x = T λx. (5) The left hand de of (5) equal v j N w j x j and the rght hand de equal λ v x j V\{v } j. Thu (5) gve w j x j = λ x j. (6) v j N v j V\{v } Ung the fact that w = max vj N w j, N V \ {v } and all egenvector component are nonnegatve, equaton (6) yeld λ w v j N xj v j V\{v} xj, from whch the reult follow. The upper bound gven above for λ(l g ) tghter than the upper bound obtaned from the Cauchy nterlacng theorem. Th dfference become more apparent when we conder the fact that n certan graph (e.g., unweghted Erdo- Reny random graph), the algebrac connectvty grow unboundedly [27] whle λ(l g ) reman bounded. The above theorem alo provde the followng characterzaton of the mallet egenvalue of the grounded Laplacan n the cae of multple tubborn agent. Corollary : Gven a graph G(V, E) wth a et S V of tubborn agent, for each α-vertex v j, defne w Sj = w j. v N j S The mallet egenvalue of the grounded Laplacan L gs atfe 0 λ w where w = max vj V\S w Sj. Proof: When S =, the reult follow from Theorem. If S >, we contruct a new graph Ḡ wth vertex et (V \ S) {v }, wth a ngle tubborn agent v connected to all α-vertce n V \ S. For each α-vertex v j, defne the weght on the edge (v, v j ) a w j = w Sj defned n the corollary. Snce the row and column correpondng to the tubborn agent are removed to obtan the grounded Laplacan matrx, the grounded Laplacan for graph G the ame a the grounded Laplacan for graph Ḡ. Th convert the network wth multple tubborn agent nto a network wth a ngle tubborn agent and the reult follow from Theorem.

4 For the cae of graph wth homogeneou weght w, Theorem how that 0 λ w, and n partcular, λ for unweghted graph. The followng reult how that the upper bound reached f and only f the leader agent are drectly connected to every other agent. Propoton : Gven a connected graph G(V, E) wth homogeneou potve weght w and a tubborn agent v V, we have λ(l g ) = w f and only f N = V \ {v }. Proof: We tart by rearrangng (6) to obtan λ x j = (w λ) x j. (7) v j N v j V\{N {v }} The f part trvally true accordng to (7). In order to how the only-f part, we prove by contradcton. Suppoe λ = w and there ext a β-vertex n G (.e., a vertex that not a neghbor of v ). Accordng to (7) t egenvector component hould be zero and accordng to (4) the egenvector component of t neghbor are zero. If L g rreducble,.e., the tubborn agent v not a cut vertex, nce G connected, the egenvector wll be x = 0. If L g reducble,.e., the tubborn agent v a cut vertex, by removng v the graph G parttoned nto eparate component. If there ext a β-vertex n the component of G whoe egenvector element are nonzero, nce th component connected, by the ame argument all of the egenvector element n th component wll be zero. A the egenvector element of other component are zero we have x = 0. Throughout the ret of the paper we focu on the cae of a ngle tubborn agent v V and homogenou weght w =, and denote the reultng grounded Laplacan by L g (or L g ). Note that changng the weght w only cale the degree of the vertce and the egenvalue of L g, but doe not affect the egenvector correpondng to λ(l g ). We alo aume that the tubborn agent doe not connect to all of the other agent, a th cae trvally covered by the above reult. The followng reult wll be helpful for provdng a tghter bound on the mallet egenvalue of the grounded Laplacan for certan choce of tubborn agent. Propoton 2: Let x be the egenvector correpondng to the mallet egenvalue of L g. For each vertex v V \{v }, v j N \{v} xj defne a v = N \{v } f N \ {v } > 0, and a v = 0 otherwe. Then for each β-vertex v we have x > a v and for each α-vertex v we have x a v. Proof: Rearrangng the egenvector equaton (4), we v j N \{v} xj have x = d λ. From Theorem we know that 0 < λ. Thu we have v x j N \{v } j v < x = x j N \{v } j d d λ v x (8) j N \{v } j. d Snce we have a v = v j N \{v} xj v j N \{v} xj d, v V \ {N {v }} and a v = d, v N, accordng to (8) we have x > a v, v V \ {N {v }} and x a v, v N. In the pecal cae of N = {v }, v become an olated vertex after removng v. The block correpondng to vertex v n L g ha egenvalue, and thu not the block of L g contanng the mallet egenvalue (by Propoton and the aumpton that v doe not connect to all node). In th cae a v = 0 whch gve x = a v. Accordng to the above propoton, the egenvector element of an α-vertex le than the average value of t neghbor egenvector entre. Thu t doe not have the maxmum egenvector component among t neghbor. Smlarly the egenvector component of a β-vertex greater than the average value of t neghbor component and t doe not have the mnmum egenvector component among t neghbor. Corollary 2: For any β-vertex v, there a decreang equence of egenvector component tartng from v that end at an α-vertex. Proof: Snce each β-vertex ha a neghbor wth maller egenvector component, tartng from any β-vertex there a path contng of vertce that have decreang egenvector component. If th equence doe not fnh at an α-vertex t fnhe at a β-vertex. There ext another vertex n the neghborhood of that vertex wth maller egenvector component. Thu the decreang equence mut fnh at one of the α-vertce. Th lead to the followng corollary; a vertex ad to be n the -th layer f t hortet path to the tubborn agent ha length. Corollary 3: The mnmum egenvector component n layer and j, where > j, occur n layer j. Proof: Let v and v be the vertce wth mnmum egenvector component among the vertce n layer and j repectvely. Accordng to Corollary 2 there a path tartng from v and endng at an α-vertex, makng a decreang equence of egenvector component. Snce > j th path contan a vertex v n layer j. Thu accordng to Corollary 2 we have x v x v x v whch prove the clam. A a reult the global mnmum egenvector component belong to one of the α-vertce. The above reult lead to the followng tghter upper bound for λ(l g ), a compared to Theorem, for a pecal cla of graph whch contan tree. Propoton 3: If G a connected graph and the tubborn agent v ha d =, then λ V\{v. } Proof: Let k = V \ {v }, and let v k be the ngle α-vertex n G. Accordng to Corollary 3, t egenvector component maller than the egenvector component of all of the other k vertce n V \ {v } and accordng to (7) we have x k = λ λ (x λ(k ) + x x k ) λ x k, whch gve λ k. If v a cut vertex n G(V, E), removng t caue the graph G to be parttoned nto multple component. The grounded Laplacan L g block dagonal, where each block the grounded Laplacan matrx for one of the component.

5 In th cae the egenvalue of L g are the unon of the egenvalue of each block, and the mallet egenvalue of each block can be bounded a above, leadng to the followng corollary. Corollary 4: If the tubborn agent v a cut vertex and each component formed by removng v ha a ngle α- vertex, then λ k where k the ze of the larget component n the graph nduced by V \ {v }. A mentoned above, the cla of graph condered n Corollary 4 contan tree, becaue each vertex n a tree a cut vertex and all of t ncdent edge are cut edge. VI. BOUNDS ON GROUNDING CENTRALITY VIA ABSORPTION TIME The convergence properte of lnear conenu dynamc wth tubborn agent are cloely related to certan properte of random walk on graph, ncludng mxng tme, commute tme, and aborpton probablte [], [2], [4], [5]. In th ecton we dcu the relatonhp between groundng centralty and the expected aborpton tme of an aborbng random walk on the underlyng graph. To th end, we frt revew ome properte of the nvere of the grounded Laplacan matrx. A. Properte of the Invere of the Grounded Laplacan Matrx A dcued n Remark, when the graph G connected, for any v V, the nvere of the grounded Laplacan matrx L g (G) ext and a nonnegatve matrx. In th cae, an alternatve defnton for the groundng centralty of v from the one n Defnton that t the maxmum egenvalue of L g, wth IC(G) = argmn v V λ max (L g ). Snce the egenvector correpondng to the larget egenvalue of L g (G) the ame a the egenvector for the mallet egenvalue of L g (G), the properte dcued n Remark contnue to hold (.e., th egenvector can be choen to be nonnegatve, and trctly potve f v not a cut vertex). One of the conequence of the P-F theorem appled to that the larget egenvalue atfe L g λ max (L g ) max{[l g ] }, (9) where [L g ] the -th row of L g. Let x = [ x x 2 x n ] T be the egenvector correpondng to λ max (L g ). The element x n th egenvector aocated wth the -th vertex. A th egenvector ha nonnegatve element, we can normalze t uch that max x =. Let = x max x mn = x mn 0 be the dfference between the maxmum and the mnmum entre of egenvector x. Snce all of the element of L g and x are nonnegatve, and we have x ( ) = x mn elementwe, we have [L g ] [( )] [L g ] x = λ max (L g )x. Snce 0 x we have [L g ] [( )] λ max (L g ). Combned wth (9), th gve max{[l g ] [( )]} λ max (L g ) max {[L g ] }. (0) By mnmzng over all choce of tubborn agent v V from (0) we have mn max{[l g ] ( )} mn mn λ max (L g ) max{[l g ] }. () A 0 the upper bound and the lower bound of () approach mn λ max (L g ). Equaton (0) and () provde bound on the groundng centralty of each vertex n the graph and the groundng centralty of vertce n IC(G), repectvely. We now relate the bound n () to an aborbng random walk on the underlyng graph. B. Relatonhp of Groundng Centralty to an Aborbng Random Walk on Graph We tart wth the followng prelmnary defnton about aborbng Markov chan. Defnton 2: A Markov chan a equence of random varable X, X 2, X 3,... wth the property that gven the preent tate, the future and pat tate are ndependent. Mathematcally P r(x n+ = x X = x, X 2 = x 2,..., X n = x n ) = P r(x n+ = x X n = x n ). A tate x of a Markov chan called aborbng f t mpoble to leave t,.e., P r(x n+ = x X n = x ) =. A Markov chan aborbng f t ha at leat one aborbng tate and f from every tate t poble to go to an aborbng tate. A tate that not aborbng called a tranent tate [9]. If there are r aborbng tate and t tranent tate, the tranton matrx wll have the canoncal form P = [ Q R 0 I ], P n = [ Q n R 0 I ], (2) where Q t t, R t r and R t r are ome nonzero matrce, 0 r t a zero matrx and I r r an dentty matrx. The frt t tate are tranent and the lat r tate are aborbng. The probablty of gong to tate x j from tate x gven by entry p j of matrx P. Furthermore entry (, j) of the matrx P n the probablty of beng n tate x j after n tep when the chan tarted n tate x. The fundamental matrx for P gven by [9] N = Q j = (I Q). (3) j=0 The entry n j of N gve the expected number of tme tep that the proce n the tranent tate x j when t tart from the tranent tate x. Furthermore the -th entry of N the expected number of tep before the chan aborbed,

6 gven that the chan tart n the tate x. In the context of a random walk on a gven graph G(V, E) contanng one aborbng vertex v, the probablty of gong from tranent vertex v to the tranent vertex v j P j = A j /d where A the adjacency matrx and d the degree of v. Thu the matrx Q n (2) become Q = Dg A g where A g and D g are the grounded degree and grounded adjacency matrx, repectvely (obtaned by removng the row and column correpondng to the aborbng tate v from thoe matrce). To relate the aborbng walk to the grounded Laplacan, note that L g Comparng to (3), we have = (D g A g ) = (I Dg A g ) Dg. N = L g D g. (4) where the ndex denote that vertex v an aborbng tate. Th lead to the followng reult. Propoton 4: Gven graph G(V, E) and a tubborn agent v V, let d max and d mn denote the maxmum and mnmum degree of vertce n V \ {v }, repectvely. Then max{[n ] [( )]} λ max (L g ) d max max{[n ] }, (5) d mn where [N ] the expected aborpton tme of a random walk tartng at v V \ {v } wth aborbng vertex v. Proof: Subttutng (4) nto (0) gve max{[n ] [( )]} max{[n Dg ] [( )]} d max λ max (L g ) max{[n Dg ] } max{[n ] }, d mn whch prove the clam. Remark 2: Takng the mnmum over all poble choce of aborbng vertex n (5) gve mn max d max {[N ] [( )]} mn λ max (L g ) mn max{[n ] }. d mn The upper and lower bound approach each other n graph where d max and d mn are equal and 0 (.e., the mallet egenvector component goe to ). In th cae, the groundng central vertex become a vertex that f choen a the aborbng vertex, the maxmum expected aborpton tme n the random walk on G mnmzed. VII. CONCLUSION We analyzed pectral properte of the grounded Laplacan matrx n the context of lnear conenu dynamc wth tubborn agent. 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