Decentralised minimaltime consensus


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1 Decentralised inialtie consensus Y Yuan, GB Stan, M Barahona, L Shi and J Gonçalves Abstract This study considers the discretetie dynaics of a networ of agents that exchange inforation according to the nearestneighbour protocol under which all agents are guaranteed to reach consensus asyptotically We present a fully decentralised algorith that allows any agent to copute the consensus value of the whole networ in finite tie using only the inial nuber of successive values of its own history We show that this inial nuber of steps is related to a Jordan bloc decoposition of the networ dynaics and present an algorith to obtain the inial nuber of steps in question by checing a ran condition on a Hanel atrix of the local observations Furtherore, we prove that the inial nuber of steps is related to other algebraic and graph theoretical notions that can be directly coputed fro the Laplacian atrix of the graph and fro the underlying graph topology I INTRODUCTION Fuelled by applications in a variety of fields, there has been a recent surge of interest in consensus dynaics In its ost basic forulation, the consensus proble studies the linear discretetie dynaics of a networ of agents that exchange inforation according to the nearestneighbour averaging rule The consensus proble has broad iplications beyond the analysis and design of collective behaviour in ultiagent systes Various applications can be cast in this fraewor, including swaring and flocing [1], [2], distributed coputing [3], agreeent in social networs [4], [5] or synchronisation of coupled oscillators [6], [7], [8] The design of efficient distributed consensus algoriths is a current focus of active research in the Control literature Under broad assuptions, wellnown results [9], [1], [11] give conditions to ensure that the state of each agent reaches the consensus value asyptotically Fro a practical point of view, however, requiring an infinite or arbitrarily long tie to obtain the final consensus value of the syste is unsatisfactory A finitetie protocol is designed in [12] The principles for the coputation of the asyptotic final value of the networ in finite tie were introduced in [13] In [14], we extended those results and studied the inial nuber of discretetie steps required by an arbitrarily chosen agent to copute the asyptotic final value of the networ without any prior nowledge of the syste dynaics Iportantly, the inforation used for that purpose was solely based on the accuulation of the successive state values of the agent under consideration, and, consequently, the corresponding algorith was truly decentralised Corresponding author: Ye Yuan and Jorge Gonçalves are with the Control Group, Departent of Engineering, University of Cabridge GuyBart Stan is with the Departent of Bioengineering, Iperial College London GuyBart Stan is also affiliated with the Centre for Synthetic Biology and Innovation, Iperial College London Mauricio Barahona is with Departent of Matheatics, Iperial College London Ling Shi is with the Departent of Electronic and Coputer Engineering, the Hong Kong University of Science and Technology This paper presents a characterisation of our results for decentralised inialtie consensus Firstly, we introduce an algorith that allows any agent in a consensusguaranteed networ to copute the consensus value using one less step than in [14] This algorith relies on the analysis of the ran of a Hanel atrix constructed fro local observations at any chosen node Furtherore, we show that the inial nuber of steps is lined to a global property of the networ: the degree of a specific atrix polynoial This provides us with an algebraic characterisation of the local convergence to consensus in ters of properties of the Laplacian atrix of the graph Finally, we show that the inial nuber of steps required to copute the consensus value fro local observations of any chosen node can also be characterised in ters of a cobinatorial graph theoretical property: the inial external equitable partition of the graph with respect to that node Throughout the paper we illustrate our results with relevant exaples to highlight how our fraewor can establish a lin between the spectral and graph theoretical properties of a networ of interacting agents and the inialtie solution of distributed decisionaing probles Notation: The notation in this paper is standard For a atrix A R M N, A[i, j] R denotes the eleent in the i th row and j th colun, A[i, :] R 1 N denotes its i th row, A[:,j] R M 1 denotes its j th colun, and A[i 1 : i 2,j 1 : j 2 ] denotes the subatrix of A defined by the rows i 1 to i 2 and the coluns j 1 to j 2 For a colun vector α R N 1, α[i] denotes its i th eleent We denote by e T r = [,,, 1 r th,,,] R 1 N Furtherore, I N denotes the identity atrix of diension N II CONSENSUS DYNAMICS: FORMULATION AND PREVIOUS RESULTS A Forulation of the proble Consider a directed unweighted graph denoted by G = (V, E,W), where V = {ν 1,,ν n } is the set of nodes, E V V is the set of edges, and W = {W [i, j]} i,j=1,,n is the corresponding n by n adjacency atrix, with W [i, j] =1 when there is a lin fro j to i, and W [i, j] =when there is no lin fro j to i Let x[i] R denote the state of node i, which ight represent a physical quantity such as altitude, position, teperature, voltage, etc The classical consensus proble on a networ of continuoustie integrator agents is defined by the following dynaics [1]: ẋ(t) = Lx(t), where L R n n is the Laplacian atrix induced by the topology G L is defined as L[i, i] = n l=i W [i, l], i = 1,,n and L[i, j] = W [i, j], i = j
2 Here, we consider the associated discretetie consensus dynaics on a networ: x +1 =(I n L) x Ax y = e T r x = x [r], (1) where x R n and is the sapling tie Without loss of generality, we concentrate on the case where the easurable output y R corresponds to the local state of an arbitrarily chosen agent labelled r B Global asyptotic convergence to distributed consensus (see [9], [1]): Let d ax = ax i L[i, i] denote the axial node outdegree of the graph G If the networ has a rooted directed spanning tree (or is connected in the case of an undirected graph) over tie, and the sapling tie is such that < <1/d ax, then the discretetie version of the classical consensus protocol given in (1) ensures global asyptotic convergence to consensus in the sense that li x = c T x 1n 1 where 1 n 1 is a colun vector with all coponents equal to 1, and c T is a constant row vector In other words, the values of all nodes converge asyptotically to the sae linear cobination of the initial node values x Algebraic characterisation of distributed asyptotic consensus [15]:: When c T 1 = 1, the iteration given by (1) achieves distributed consensus if and only if: A1 A has a siple eigenvalue at 1, and all other eigenvalues have a agnitude strictly less than 1 A2 The left and right eigenvectors of A corresponding to the eigenvalue 1 are c T and 1, respectively C Finitetie coputation of the final consensus value [13] Recent wor by Sundara and Hadjicostis [13] showed that it is possible to obtain the final value of the consensus dynaics in a finite nuber of steps Their result hinges on the use of the inial polynoial associated with the consensus dynaics (1) in conjunction with the final value theore Definition 1 (Minial polynoial of a atrix): The inial polynoial of atrix A R n n is the unique onic polynoial q(t) t D+1 + D i= α it i with inial degree D +1 that satisfies q(a) = Given the explicit solution of the linear syste in (1) with initial state x, it follows fro the definition of the inial polynoial that the dynaics in (1) satisfies the linear regression equation: x +D+1 + α D x +D + + α 1 x +1 + α x =, N (2) Siilarly, the regression equation for y = x [r], the easurable output at node r, is deterined by the inial polynoial of the corresponding atrix observability pair [A, e T r ] Definition 2 (Minial polynoial of a atrix pair): The inial polynoial associated with the atrix pair [A, e T r ] denoted by q r (t) t Dr+1 + D r i= α(r) i t i is the unique onic polynoial of inial degree D r +1 D +1 that satisfies e T r q r (A) = Again, it is straightforward to show that: y +Dr+1+α (r) D r y +Dr ++α (r) 1 y +1+α (r) y =, N (3) Therefore each node r will be associated with a particular length of the regression (D r + 1) which is upper bounded by the degree of the inial polynoial of the dynaical atrix A Consider now the Ztransfor of y 1 : Y (z) = Dr+1 i=1 α (r) i i 1 = y z i q r (z) H(z) q r (z) (4) Under the assuptions specified in Section IIB, the inial polynoial q r (t) does not possess any unstable root except for one single root located at 1 We can then define the following polynoial: p r (z) q Dr r(z) z 1 β i z i (5) i= The application of the final value theore [16] then gives the consensus value φ =li(z 1)Y (z) = H(1) z 1 p r (1) = yt D r β 1 T β where y T D r = y y 1 y Dr and β(dr+1) 1 is the vector of coefficients of the polynoial p r (z) defined in eq (5) Based on these results, an algorith to obtain the consensus value was proposed in [13] The proposed algorith was distributed but not entirely local, in the sense that a local calculation is repeated over n independent iterations (where n is the total nuber of nodes of the networ) and at each iteration, it requires each node to store its own values for n +1 steps Hence, a total of n(n + 1) successive values of x[r] are required for the calculation of φ D Minialtie, decentralised coputation of the final consensus value The ain purpose of this paper is to characterise the coputation in inial tie of the final consensus value φ using only the output observations y = x [r] of the node r alone We foralise and iprove here our previous results [14] and show that, for a general arbitrary initial condition, except for a set of initial conditions with Lebesgue easure zero [18], the consensus value can be obtained fro local observations in a inial nuber of steps that does not depend explicitly on the total size of the graph In our fraewor, the inial nuber of steps is coputed in a truly decentralised anner by checing a ran condition of a Hanel atrix constructed exclusively fro local output observations We also provide a graph theoretical characterisation of this local property in ters of the inial external equitable partition of the graph This characterisation provides insight into which properties 1 This follows fro the tieshift property of the Ztransfor: Z(x +n )=z n X(z) n 1 l= zn l x l where X(z) =Z(x ) (6)
3 of the graph contribute to the disparity in the ability of the different nodes to copute the global consensus value fro local inforation III MINIMAL TIME CONSENSUS AND THE JORDAN BLOCK DECOMPOSITION OF THE CONSENSUS DYNAMICS Given the linear syste in (1) and an initial state x, it follows fro above that there always exist scalars d d(r, x ) N and a,,a d R such that the following linear regression equation is satisfied N x +d+1 [r]+a d x +d [r]++a 1 x +1 [r]+a x [r] = (7) Fro the definitions above, it is clear that D r +1 is the inial length of recursion: D r +1=in d N ax {d(r, x )+1: eq (7) holds } x R n Rear 1: Aong the any recursions of length d that are not necessarily inial, (D r + 1) appears as a inax over the space of (d, x ) When d +1=D r +1, the coefficients a i in (7) correspond to α (r) i, the coefficients of the inial polynoial of the atrix pair [A, e T r ] in (3) In this section, we give an algebraic characterisation of the inial nuber of steps D r +1 based on the projection of the Jordan bloc decoposition of A on e T r Our ai is to obtain the coefficients α (r) i in (3) fro data, so that we can copute future outputs recursively Consider the standard Jordan decoposition: where A = SJS 1 where (8) S = s 1 s 2 s n (9) J = diag {J 1 (λ 1 ),J 2 (λ 2 ),,J l (λ l )} (1) λ i 1 λ i 1 J i (λ i )= λ i 1 λ i n i n i, (11) and s i, the coluns of the non singular atrix S, are the generalised eigenvectors of A [22] The atrix A has l (possibly degenerate) eigenvalues λ i, each of the associated with a Jordan bloc of size n i, such that l i=1 n i = n Without loss of generality, we assue that the blocs are ordered according to decreasing size: n 1 n 2 n l Using eq (8), the linear dynaics (1) can be rewritten as follows: x [r] =e T r A x = e T r S J S 1 x σ T J χ, (12) where the vectors σ T = σ1 T σ2 T σl T (13) 1 n χ T = χ T 1 χ T 2 χ T l (14) 1 n are partitioned according to the Jordan blocs in (8), eg, σ1 T = σ 11 σ 1ni and χ T 1 = χ 11 χ 1ni Here, J = diag J1 (λ 1 ),J2 (λ 2 ),,Jl (λ l ) has the well nown structure [17]: 1 Ji (λ i )= λ i Ji (), (15) i=1 = = where Ji () is the th power of a Jordan bloc, as defined in (11) The output dynaics (12) then becoes: l 1 x [r] = λ i σ T i Ji () χ i (16) Note that, because of its Jordan bloc structure, the atrix Ji () induces a strict shift on the vector χ i for n i Therefore, if ax i {n i }, we have: x [r] = l n i 1 i=1 = l n i 1 i=1 = λ i n i σ ij χ ij+ (17) j=1 λ i g i (18) However, soe of the g i ight be zero (we ight even have situations where all the coefficients associated with a particular eigenvalue are zero) so that the dynaics of node r can be written as: x [r] = l r n r i 1 i=1 = λ i g i (19) where n r i n i and l r l Here, {λ 1,,λ lr } is an ordered subset of distinct eigenvalues fro the original Jordan bloc decoposition As a consequence, the degree of the characteristic polynoial that underlies the length of the recursion for node r is: l r i=1 n r i = D r +1 Eq (19) can be rewritten as a dot product: g 1 g 2 x [r] =v r () T g r v1 T () v2 T () vl T r () where v T i () λ i 1 λ 1 i g T i g i g i(n r i 1) g lr n n λ r r i 1 i +1 i 1 n r i Based upon the decoposition of confluent Vanderonde atrices introduced in [19], it is easy to see that v T i () =e T i J i (λ i ) where J i (λ i ) is a Jordan bloc of size n r i as defined in (11) and e T i = 1 is the unit vector of the sae 1 n r i length The dynaics (12) can thus be rewritten in ters of a Jordan decoposition of reduced diensionality as follows: x [r] =E T r J r g r,, (2)
4 where E T r e T 1 e T l r 1 (D r+1) and J r diag {J 1 (λ 1 ), J 2 (λ 2 ),,J lr (λ lr )} (21) are partitioned according to the l r blocs Fro the analysis above, we have the following lea Lea 1: Consider the discretetie LTI syste (1) The inial polynoial associated with x[r], as given in Definition 2, is the characteristic polynoial of the atrix J r in eq (2) which has order D r +1= l r i=1 nr i The final consensus value φ can be coputed fro eq (6) based on the coefficients of the inial polynoial of the pair [A, e T r ] and the successive values of x[r] Proof: The Jordan atrix J r in eq (2) has the property that each of its Jordan bloc has distinct eigenvalues Hence, the inial polynoial of [A, e T r ] is the sae as the characteristic polynoial of [J r,e T r ] (see [17]): e T r q r (A) = e T r q r (J r ) Therefore, the inial polynoial possesses the following explicit for: det(j r ti) = l r i=1 (t λ i) nr i = t Dr+1 + α Dr t Dr + + α 1 t + α, and has degree D r +1 This latter relationship also shows that D r +1= l r i=1 nr i Rear 2: Lea 1 states that instead of an n diensional Jordan bloc for J of x [r], as in eq (12), the general expression of x [r] can be written in ters of a saller D r +1diensional Jordan atrix J r, as in eq (2) Rear 3: The inial integer value D r +1 necessary for the recursion (7) to hold for alost any initial condition x is given by the degree of the inial polynoial of the observability pair [A, e T r ] (see [14]) In other words, eq (7) holds for a randoly chosen initial state x, except for a set of initial conditions of Lebesgue easure zero [18] IV DECENTRALISED MINIMALTIME CONSENSUS COMPUTATION ALGORITHM In the decentralised proble, we assue that node r does not have access to any external inforation such as the total nuber of agents n in the networ, the local counication lins around node r or the state values or nuber of its neighbours In [14], we showed that for the general discretetie LTI syste (1), 2D r +3 successive discretetie steps are needed by agent r to copute the final value in a fully decentralised anner If the counication networ is welldesigned for consensus (ie, Assuptions A1 and A2 are satisfied and asyptotic convergence to consensus is guaranteed), we hereby propose an algorith that coputes the final value using 2D r +2 successive discretetie steps, ie, one fewer step than [14] Proble 1 (Decentralised proble): Consider the discretetie LTI dynaics in eq (1) where an arbitrarily chosen state x[r] is observed and assue that the conditions for consensus (Assuptions A1 and A2) are satisfied The decentralised proble is to copute the asyptotic value of this state φ =li x [r] using only its own previously observed values y = x [r] Consider the vector of successive discretetie values at node r, X,1,,2 [r] ={x [r], x 1 [r],,x 2 [r]}, and its associated Hanel atrix: x [r] x 1 [r] x [r] x 1 [r] x 2 [r] x +1 [r] Γ{X,1,,2 [r]} Z x [r] x +1 [r] x 2 [r] (22) We also define the vector of differences between successive values of x[r]: X,1,,2 [r] ={x 1 [r] x [r],,x 2+1 [r] x 2 [r]} The following algorith then allows us to copute the final consensus value in a inial nuber of steps Algorith 1 Decentralised inialtie consensus value coputation Data: Successive observations of x i [r], i =, 1, Result: Final consensus value: φ Step 1: Increase the diension of the square Hanel atrix Γ{ X,1,,2 [r]} until it loses ran and store the first defective Hanel atrix Step 2: The ernel β = β β Dr 1 1 T of the first defective Hanel atrix gives the coefficients of eq (6) Step 3: Copute the final consensus value φ using eq (6) To understand Algorith 1, consider a Vanderonde factorisation [19] of the Hanel atrix (22): Γ{X,1,,2 [r]} = V (,)T r V T (,), (23) in which we have defined the confluent Vanderonde atrix Er T J r V (,) (+1) (Dr+1) =, (24) Er T Jr in ters of the eleents defined in eq (21) As shown in [19], the (D r + 1) (D r + 1) bloc diagonal atrix E T r T r = diag{t r,1,,t r,lr }, T r,i R nr i nr i, has the following syetric upper antidiagonal for: t i t i T r,i = t i, t i where t i and are deterined fro the values of y Without loss of generality, consider λ 1 =1so that T r,1 R We then have Γ{ X,1,,2 [r]} =Γ{X 1,2,,2+1 [r]} Γ{X,1,,2 [r]} = VT r diag{λ 1,,λ lr }V T VT r V T = VT r diag{,λ 2 1,,λ lr 1}V T = V diag{, (λ 2 1)T r,2,,(λ lr 1)T r,lr }V T = V diag{(λ 2 1)T r,2,,(λ lr 1)T r,lr }V T,
5 Fig 1 Underlying topology for Exaple 1 with sapling tie =1/6 where V = V [2 : +1, 2:D r + 1] Fro the last equation, it is easy to see that for Γ{ X,1,,2 [r]} to be defective, one ust have D r +1 Theore 1: Consider the syste in (1) and assue that the conditions for consensus (Assuptions A1 and A2) are satisfied Then the inial nuber of successive discretetie values, starting fro step i, for the arbitrarily chosen state x[r], is 2(D r + 1) δ r in{i, δ r }, where δ r is the nuber of zero roots in q r (t) = Proof: Cobining the above derivations and perforing a proof siilar to the one presented in [Corollary 1, [14]] (by taing z x +1 [r] x [r] as y in that Corollary) yields the result More elaborate versions of the results presented here can be obtained by odifying the odel in eq (1) so as to encopass ore coplex situations, eg, tiedelays in the odel, noise in the observations or pacet drops in the observations Due to space constraints, we will not address the here and present the in ore detail in a future paper In the present paper, we only focus on the ideal odel in eq (1) For siplicity of exposition, we further ae the following assuption in the rest of this paper: 2 A3 The atrix A in eq (1) does not possess any eigenvalue at Under Assuption A3, Theore 1 establishes that the inial nuber of steps for node r to copute the final consensus value is 2D r +2 Exaple 1: Consider the networ topology in Fig 1 under dynaics (1) with A I n L and a sapling tie = 1/6 The topology is undirected and connected and A satisfies assuptions A1, A2, and A3 Therefore the final value of each node is the average of the initial state values For the randoly chosen initial state x = T, the final consensus value is thus We now apply Algorith 1 to node r =1 Step 1: We increase the diension of the square Hanel atrix Γ{ X,1,,2 [1]} until it loses ran This happens for 2 When A has soe eigenvalues at, the expression of the inial nuber of steps for node r to copute the final consensus value taes a ore coplicated for, see [14] =4 We then store the first defective Hanel atrix: Γ{ X,1,,8 [1]} = Step 2: The noralised ernel of the first defective Hanel atrix is β = T This gives the coefficients of eq (6) Step 3: We copute the final consensus value φ =26828 using eq (6) As shown here for node r =1, the value of φ obtained in a decentralised anner is equal to the average of the initial states Repeating this procedure for each of the six nodes gives the sae value φ However, the nuber of steps required by each node to copute the final consensus value φ differs This is suarised in Table I Ref [13] Our result Node 1 6 7=42 2 4=8 Node 2 6 7=42 2 4=8 Node 3 6 7=42 2 4=8 Node 4 6 7=42 2 5=1 Node 5 6 7=42 2 6=12 Node 6 6 7=42 2 6=12 TABLE I COMPARISON OF THE MINIMAL NUMBER OF SUCCESSIVE VALUES NEEDED BY EACH NODE TO COMPUTE THE FINAL CONSENSUS VALUE OF THE NETWORK IN FIG 1WITH n =6NODES While the ethod proposed in [13] requires a total of n(n + 1) successive values of x[r], our algorith shows that the inial nuber of successive values of x[r] is just 2(D r + 1) for alost all initial conditions Furtherore, our algorith is copletely decentralised, ie, our result does not require that the arbitrarily chosen state x[r] has any nowledge of the total nuber of nodes in the networ, n, or any other ind of global (centralised) inforation about the networ (contrary to what is assued in [13, Section V]) As can be noticed in Table I, soe nodes need fewer successive observations of their own state to copute the final consensus value of the whole networ In what follows, we call such nodes doinant nodes An iportant question arises at this point: given a consensusguaranteed networ, can we identify the doinant nodes? Below, we answer this question based on an algebraic characterisation of the inial nuber of steps which we then lin to a specific graph partition of the consensus networ around the chosen node V CHARACTERISATION ON THE MINIMAL NUMBER OF STEPS We now provide an answer to the question raised at the end of the last section fro two perspectives First, in Section V A, we provide an algebraic characterisation of the inial recursion length D r +1 for node r by perforing an analysis of the Laplacian of the graph Second, in Section VB, we
6 relate D r +1 to the nuber of cells in a special partition of the graph called the inial external equitable partition with respect to node r For siplicity of exposition, we only consider undirected graphs in the following sections, ie, we assue: A4 The atrices W, L, A in eq (1) are syetric A Algebraic characterisation An algebraic characterisation of the degree of the inial polynoial of [A, e T r ] can be obtained based on the Jordan bloc decoposition described in Section III The syetry of the Laplacian atrix in undirected graphs siplifies the analysis since the Jordan atrix in Eq (12) becoes diagonal The following Corollary provides a relationship between the inial nuber of successive values required by a node to copute the final consensus value of the networ and algebraic properties of the underlying graph Before presenting the ain result, we introduce the following notation, which will be used extensively in the reainder of the paper Definition 3 (Dcardinality of a set): Let Λ be a finite set, potentially containing repeated eleents, with cardinality card{λ} The dcardinality of the set, denoted dcard{λ}, is defined as the nuber of distinct eleents in the set Exaple 2: Let Λ={1, 2, 3, 1, 3, 5} Then card{λ} =6 and dcard{λ} =4 Our first algebraic characterisation of the inial recursion length at node r relates D r +1 to the nuber of distinct eigenvalues of the Laplacian atrix whose eigenvectors have nonzero coponents for node r, as given by the following Corollary Corollary 1: Consider the dynaics (1) where A is associated with an unweighted and undirected graph Denote the eigenvalues of the syetric atrix A by λ i and their corresponding right eigenvector by u i Let Λ={λ i (A) i = 1,,n} and Ψ r = {λ i (A) u i [r] =} Then D r +1=dcard{Λ/Ψ r }, where Λ/Ψ r is the relative copleent of Ψ r in Λ Proof: Since A is syetric, all the eigenvalues of A are real The proof then follows fro Lea 1 and the PBHtest [22] Consider now the following wellnown lea: Lea 2: [23, Theore 951] Let A be a syetric atrix in R n n and let R R n be such that R T R = I Define Θ = R T AR and let {v 1, v 2,, v } be an orthogonal set of eigenvectors for Θ such that Θv i = λ i (Θ)v i, where λ i (Θ) is the i th eigenvalue of Θ associated with the eigenvector v i Then, we have the following result: if λ i (Θ) = λ i (A) for i =1,,lthen, Rv i is an eigenvector of A with associated eigenvalue λ i (Θ) for i =1,,l Our second algebraic characterisation relates D r +1 with the nuber of eigenvalues shared by the Laplacian atrix and the rgrounded Laplacian atrix Theore 2: [21] Consider the syste in Eq (1) satisfying Assuptions A1 A4 The ran of the observability atrix for the pair [A, e T r ] is equal to n µ r, ie, D r +1=n µ r, where µ r is the nuber of eigenvalues shared between A and A r, where A r is the rgrounded Laplacian atrix, ie, the subatrix of A obtained by deleting the r th row and the r th colun Proof: Due to the page liitation, we refer the reader to the proof in [21] B Graphtheoretical characterisation In this section, we consider the following question: given an undirected networ, can we directly identify the doinant node(s) fro the graph without any algebraic coputation? We adopt definitions and notations fro [24] A partition of a graph G =(V, E) is defined as a apping fro vertices to subsets of vertices called cells: π : V {C 1,,C K } where C i V, i Let I(π) denote the iage of π, ie, I(π) ={C 1,,C K } and deg π (i, C j ) denote the nodetocell degree deg π (i, C j ) characterises the nuber of nodes in cell C j that share an edge with node v i under partition π: deg π (i, C j )=card { V π() =C j and (i, ) E} We define π 1 (C i )={j V π(j) =C i }, ie, the set of nodes that are apped to cell C i 3 In what follows, we use the concept of external equitable partition (EEP) [24] As we will show below, EEPs partition the graph into cells while neglecting the internal interconnection structure inside a cell We will show that the EEP with respect to a node is directly related to the inial nuber of steps necessary for this node to calculate the final consensus value Definition 4 (External equitable partition (EEP) [24]): A partition π of the set of nodes V consisting of s>1 cells {C 1,,C s } is external equitable if the nuber of neighbours C j of a vertex v C i depends only on the choice of C i and C j (i = j), ie, deg π (l, C j )=deg π (, C j ),, l π 1 (C i ) Definition 5 (Minial EEP with respect to a node): A partition π r of V consisting of cells {C 1,,C s } is external equitable with respect to node r if the partition is external equitable and the node r is in a cell alone, ie, π(v r )=v r The inial EEP of a graph with respect to node r, πr, is such that card{i(πr))} is inial Theore 3: Consider the syste in (1) Solely based on observations of node r, the inial length of recursion necessary to obtain the final consensus value is equal to the nuber of cells s r in πr, the inial external equitable partition with respect to node r, ie D r +1=card {I (πr)} s r (25) Proof: Without loss of generality, let r =1 We use a BreadthFirstSearch (BFS) algorith to label the cells, as follows We start fro node 1 (ie, cell 1) and explore all the neighbouring cells For each of those nearest cells, we consider their own neighbouring cells and so on, until we have labelled all the cells in the inial EEP with respect to cell 1 [21] 3 Note that π is not a onetoone apping but a onetoany apping However, we can still define a new function to ap bac fro C j to V We adopt this notation fro [24]
7 Consider now the bloc atrix obtained by peruting and partitioning A according to π1, the inial EEP with respect to node 1: A 11 A 12 A 1s1 A 21 A 22 A 2s1 A π 1 = A s11 A s12 A s1s 1 Here, A ii R li li contains the interconnections between any two nodes in cell Ci and l i denotes the nuber of nodes in cell Ci Hence, l 1 = 1 and s i=1 l i = n The offdiagonal subatrices A ij R li lj contain the interconnections between nodes in Ci and C j In particular, we will consider the following subatrices: A 1 A π 1 [2 : n, 2:n] f T 1 A π 1 [1, 1:n] = A 12 A 1j Note that there are only j neighbouring cells to cell 1, ie, A 1(j+1),,A 1s1 =for soe j>1 The observability atrix associated with the pair [A π 1,e T 1 ] is: 1 A 11 A 12 A 1s1 Ω=, (26) where is a placeholder representing a real value Let Ξ be the observability atrix associated with the pair [A 1, f1 T ] According to [2], [24], the ran of the observability atrix is equal to the diension of the following span ran(ξ) = dispan 1 r2 with r i = card {Ci } Hence, 1 r3,,, ran(ξ) = s 1 1, 1 rs1 fro whence it follows that 1 D 1 +1=ran(Ω) = ran Ξ = ran(ξ) + 1 = card {I (π 1)}, (27) Rear 4: Definition 5 iplies that that the nuber of cells in π r, s r, is greater or equal than the longest distance fro node r to all other nodes in the graph G, d(g,r) Therefore, D r +1 d(g,r) Rear 5: Theore 3 provides a lin between local observations, ie, the inial nuber of successive values that a node r needs to accuulate to copute the final consensus value of the networ) and a global property, ie, the underlying inial EEP of the networ with respect to node r Based on this theore, one can directly identify the doinant nodes in the networ without resorting to algebraic nuerical anipulations Exaple 3: As shown nuerically in Exaple 1, nodes 1, 2 and 3 are the doinant nodes since they only require 8 steps, ie, D r +1 = 4 for r = 1, 2, 3 It is easy to chec in Fig 2(a) that the inial external equitable partition with respect to these nodes has 4 cells Siilarly, Figs 2(b) and 2(c) show the inial EEPs for node 4 and for nodes 5 and 6, respectively The nuber of cells in the corresponding inial EEPs is consistent with the nuerical results in Exaple 1 which indicate that these nodes require respectively 1 and 12 successive values of their own state to copute the final consensus value of the networ according to Algorith 1 (a) 4cell based inial external equitable partition with respect to nodes 1, 2, 3 As illustrated in Exaple 1, nodes 1, 2, 3 require 2 4 = 8 steps to copute the final consensus value (b) 5cell based inial external equitable partition with respect to node 4 As illustrated in Exaple 1, node 4 requires 2 5 = 1 steps to copute the final consensus value (c) 6cell based inial external equitable partition with respect to nodes 5, 6 As shown in Exaple 1, nodes 5, 6 require 2 6 = 12 steps to copute the final consensus value Fig 2 Minial EEP with respect to the different nodes in Exaple 1 Different colours correspond to different cells (colour online)
8 VI CONCLUSION This paper forulates and analyses the decentralised inial tie consensus proble In contrast to other tools in the literature, our algorith coputes consensus fro the history of any node in a copletely decentralised, local anner The necessary inforation for any node is its own history and is therefore exclusively local The algorith does not require global nowledge, such as the total nuber of nodes in the syste, inforation about the neighbourhood of the node, or specific edge weights After characterising the inial nuber of steps required for any given node to copute the final consensus value, we provided algebraic, graphtheoretical and local inforative interpretations of the inial nuber of steps There are a nuber of interesting directions for future research in ters of networ design For instance, we are currently woring on the proble of coputing a inial EEP with respect to a node in polynoial tie Also it is iportant to ention that the EEPbased results provided here for undirected graphs can be extended to directed graphs at the price of a ore elaborate exposition Design of networ topologies that iniise algebraic connectivity was presented in [15], [25], [29] Instead of iniising the second sallest eigenvalue of a networ, we ai here at iniising the dcardinality of the Laplacian spectru An interesting question in this context is: given a constraint on the nuber of edges in the networ, what are the networ structures that iniise the dcardinality of the Laplacian spectru? Constructing Laplacian atrices with sall spectra has been intensively studied in the graph theoretic counity [26], [27] In the Appendix of [28], the author coputed all the Laplacian spectra for trees up to n = 1 vertices and connected graphs up to n =6vertices Interestingly, in a recent paper [25], the authors iniised the second sallest eigenvalue of a weighted Laplacian given a constraint on the nuber of edges in the graph It turned out in both exaples that the obtained optial Laplacian atrix had only 2 (resp 4) distinct eigenvalues for 5 (resp 1) node networs Future wor lies in forulating the optial inialtie consensus networ proble as a standard optiisation proble On the analysis part, future wor will consist in extending the odel in eq (1) so as to encopass ore coplex situations, eg, tiedelay in the odel, noise/quantisation error in the counication lins, pacet drop in the observations Yet another extension lies in the reconstruction of agentnetwor fro inial aount of observed data as illustrated in [3], [31] VII ACKNOWLEDGEMENT The authors than the anonyous reviewers for their constructive coents and insights Ye Yuan acnowledges the support fro Microsoft Research Cabridge through the PhD scholarship progra and the fruitful discussions fro Prof Richard Murray (Caltech) and Dr Svetlana Puzynina (Sobolev Institute of Matheatics) Jorge Gonçalves was supported in part by EPSRC grant nubers EP/G66477/1 and EP/I29753/1 REFERENCES [1] H Tanner, A Jadbabaie, and G J Pappas, Stable flocing of obile agents, part I: Fixed topology, in Proceedings of IEEE Conference of Decision and Control, 23 [2] F Cucer and S Sale, Eergent behavior in flocs, IEEE Transactions on Autoatic Control, 27 [3] D P Bertseas and J N Tsitsilis, Parallel and Distributed Coputation: Nuerical Methods, Englewood Cliffs, NJ: PrenticeHall, 1989 [4] R OlfatiSaber Ultrafast consensus in sallworld networs, in Proceedings of Aerican Control Conference, 25 [5] D Watts and S Strogatz, Collective dynaics of 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