Fast Consensus Seeking on Networks with Antagonistic Interactions

It is well known that all agents in amultiagent system can asymptotically converge to a common value based on consensus protocols. Besides, the associated convergence rate depends on the magnitude of the smallest nonzero eigenvalue of Laplacian matrix L. In this paper, we introduce a superposition system to superpose to the original system and study how to change the convergence rate without destroying the connectivity of undirected communication graphs. Andwe find the result if the eigenvectorx of eigenvalueλ has two identical entriesxi = xj, then theweight and existence of the edge eij donot affect themagnitude ofλ, which is the argument of this paper. By taking advantage of the inequality of eigenvalues, conditions are derived to achieve the largest convergence rate with the largest delaymargin, and, at the same time, the corresponding topology structure is characterized in detail. In addition, amethod of constructing invalid algebraic connectivity weights is proposed to keep the convergence rate unchanged. Finally, simulations are given to demonstrate the effectiveness of the results.


Introduction
In recent years, many issues related to consensus have been studied [1][2][3][4][5][6][7][8][9][10].This is due to the large number of potential applications of consensus, ranging from engineering and computer science to biology, ecology, and social science.In [1], Jadbabaie proposed a nearest neighbor law to coordinate the control of the whole system, so that agents can reach a common final value.Olfati-Saber and Murray proposed a simple consensus protocol to achieve average consensus for undirected graphs and balanced digraphs in [2].Based on quasi-consensus, Cai proposed an approach for clustering in [3].And Liu proposed several necessary and sufficient conditions for consensus of second-order multiagent systems under directed topologies [4].For high-order linear timeinvariant singular multiagent systems with constant time delays, admissible output consensus design problem was investigated in [5].For hybrid multiagent systems, necessary and sufficient conditions were also developed for the consensus [6,7].
The smallest nonzero eigenvalue  2 () of Laplacian  is the algebraic connectivity, which was first proposed by Fiedler [11].Olfati-Saber and Murray conducted a preliminary discussion on the convergence rate of consensus and proposed the concept of communication cost [2].The algebraic connectivity increases as the communication cost goes up.An approach of minimizing the guaranteed cost was given in [12].For a stable system, we always expect that the system has a larger convergence rate.Based on the consensus protocol, the convergence rate becomes larger as the algebraic connectivity of the system increases.
The convergence rate problem of consensus had been studied in [13][14][15][16][17][18][19][20].The fast consensus and the optimization problem of convergence rate have been studied extensively [13].The convergence rate of consensus was studied under the condition of weighted average in [14].Jin proposed a multihop relay protocol to make a larger algebraic connectivity [15].In [16], Olfati-Saber studied the fast consensus for smallworld networks through randomly rewiring edges.In [18], Yang studied the optimization problem of convergence rate for second-order systems with time delays by the frequencydomain method.Yang proposed a new variable not only to measure the situation of optimization, but also to cope with the tradeoff between  2 () and   ().Xiao developed the 2 Complexity results of delay margin by analyzing   () in [17].An optimal consensus protocol minimizing team cost function was proposed in [19], and an optimal synchronization protocol was designed for the largest convergence rate and minimal steady state error when the protocol is perturbed by an additive noise in [20].
To improve the convergence rate of consensus in multiagent systems, we need to change the algebraic connectivity  2 ().The variation of algebraic connectivity depends on the variation of weights and topologies of communication graphs.The algebraic connectivity can be improved by taking advantage of algorithms and graph theory, which was explored in [21][22][23][24].The convergence rate is susceptible to the perturbation, which will further affect the stability of the system [25][26][27][28].Based on former researchers' works [29][30][31][32][33][34][35][36][37], we propose a concept of invalid algebraic connectivity weights (IACW), which is shown not only to be resistant to the perturbation but also can avoid unnecessary waste of costs.There are three methods to get a larger convergence rate: (1) changing the protocol; (2) changing the weights of a system; (3) changing the topology of a system.In this paper, we take methods (2) and (3).In particular, we investigate how the convergence rate varies when a superposition system is joined to the original system and present a detailed characterization for the variation of convergence rate.The results are developed by analyzing the variation of eigenvalues and eigenvectors of Laplacian matrix.Under fixed costs, we give the most optimal case, which can make the convergence rate and delay margin the largest.For a complete graph, if the cost is fixed, then  2 () reaches maximum and   () reaches minimum.Thus, the convergence rate and delay margin achieve the biggest magnitude.Since the topology is unique, this achieves the optimization of convergence rate and delay margin.In addition, the method and conditions are put forward for the construction of invalid algebraic connectivity weights.
In a multiagent network system, not all agents are cooperative.There are some agents which are competing.The cooperative and competitive relationship is represented, respectively, by positive weights and negative weights.Positive and negative weights have opposite effects on system performance.In some circumstances, negative weights are also needed.For example, in a system with time delay, the negative weights can reduce the magnitude of   ( L), which results in a larger delay margin.Hence, the fast consensus with antagonistic interactions is considered in the paper.
The rest of the paper is organized as follows.In Section 2, basic definitions, properties, and system model are addressed.In Section 3, we study the variation of convergence rate when a superposition system is superposed on the original system, and characterize the most optimal case of convergence rate under fixed cost.In Section 4, results are derived for the identification of invalid algebraic connectivity weights and the associated construction of IACW.In Section 5, simulation results are presented to show the effectiveness of the approach.Finally, conclusions are given in Section 6.

Consensus Protocol and Consensus State
Let G = {V, E, } be a weighted undirected graph, where V = {V 1 , . . ., V  } denotes a set of  nodes, E ⊆ V × V denotes the set of edges, and  ∈ R × is the adjacency matrix of undirected graph G.In this paper, if V  has a communication link with V  , then there is an edge   ∈ E between nodes V  and V  , with   =   ̸ = 0, ,  ∈ {1, . . ., }; and   is the weight between nodes V  and V  .Here, the self-loop of V, under which   = 0, ∀ = 1, . . ., , is not considered.
The first-order multiagent system is given by where   denotes the state of agent , which is the th component of ; N() denotes the set of adjacent agents of agent .The Laplacian matrix of G is represented by  = −, where  ∈ R × is the diagonal connectivity degree matrix of .So (1) can be written as The entries of  can be written as can be written as Assume that the communication graph consists of  nodes, and the spectrum of eigenvalues is  1 () ≤  2 () ⋅ ⋅ ⋅ ≤   ().The spectrum means that the eigenvalues are arranged according to a certain order.Let  = span{ 1 , . . .,   } denote a vector space, in which any vector can be represented as a linear combination of  1 , . . .,   ; i.e.,  = { |  =  1  1 + ⋅ ⋅ ⋅ +     }.Below ∪ Ã means that the entries of matrices  and Ã plus together.For Â = ∪ Ã, the entries of Â are â =   + ã , where   , ã are the entries of , Ã, respectively.
Definition 1.The system associated with graph G = {V, Ẽ, Ã}, which has the same node set V with the original graph G, is called a superposition system.Let G be superposed to G. We get a connected graph Ĝ = {V, E ∪ Ẽ, ∪ Ã}.
Then the system associated with Ĝ is called a superposed system.
We give the definition of superposition system to represent the situation of the adding edges such as rank( L), numbers, and weights of edges, so that we study the variation of superposed systems under different situations of adding edges.
In this paper, graph Ĝ is connected and the spectrums of eigenvalues associated with both L and L are identical with Complexity 3 , where L denotes the Laplacian matrix of G, 1 ≤ rank( L) ≤  − 1 and L denote the Laplacian matrix of Ĝ.In what follows,  − L ≥ 0 means that  − L is positive semidefinite.Although there are negative weights, L can still be positive semidefinite or negative semidefinite.

Variation of Convergence Rate
The changing of algebraic connectivity relies on the variation of eigenvalues of Laplacian .So the following lemma is introduced.
By Hermam Weyl theorem, some significant inequalities can be derived.This is essential to the development of results.In what follows, let  = ,  = L,  +  = L.
The superposition system has the same node set with the original system, and its edges can be constructed arbitrarily.We aim to analyze the variations of convergence rate after joining different superposition systems.For a system with undirected graphs, the following theorem can be achieved after joining a superposition system.Theorem 3.For an undirected connected graph G, if L is a positive semidefinite matrix, then If L is negative semidefinite, then Only for  2 () with multiplicity 1, the necessary and sufficient condition for equalities in (7) and ( 8) to be true is that there is a vector  such that  =  2 (), L = 0, L =  2 ( L). Proof which guarantees that there is a unit vector  ∈  1 ∩  2 ∩  3 .So we obtain the following inequality: For  = 2,  = 1, let L be positive semidefinite; it follows that Denote  1 = span{ 1 , . . .,  + },  2 = span{ 1 , . . .,  − },  3 = span{  , . . .,   },  ∈ {1, . . ., },  ∈ {0, 1, . . .,  − }.Then which guarantees that there is a unit vector  ∈  1 ∩  2 ∩  3 .Thus, For  = 2,  = 0, let L be negative semidefinite; one has If  2 () is not a repeated eigenvalue, then  2 () =  2 ( L) only if there is an eigenvector  associated with  2 () satisfying L = 0.In case that there is no eigenvector  satisfying L = 0, then  2 ( L) is larger than  2 ().So there are no vectors , ,  with == such that    +   L =  2 ()+0 =  2 ( L).Thus the necessary and sufficient condition for equalities in (7) and ( 8) to be true is that there is a vector  satisfying  =   (), L = 0, and L =  2 ( L).
Assume that  2 () is a repeated eigenvalue with multiplicity .If  2 () or  1+ () changes and the other eigenvalues remain unchanged, L and  share  − 1 eigenvalues, while Complexity the corresponding eigenvectors of L are different from those of .Since  has  − 1 identical eigenvalues with L, there is an eigenvector  which has components satisfying   =   , such that L = 0,  ̸ = , ,  ∈ {1, . . ., }.In this case, however, the components of the corresponding eigenvector of  are different from each other.That is, there are different vectors , ,  such that    +   L =  2 () +  2 ( L) =  2 ( L) =   L.
Theorem 3 shows that the convergence rate of a system can be increased or decreased by adding a superposition system.This is always true for the positive semidefinite matrix L or negative semidefinite matrix L, no matter what is the sign of the weights of L. It is based on the analysis whether the equalities in ( 7) and ( 8) hold, we can figure out the variations of the convergence rate.In what follows, more specific instructions will be given on how to affect the convergence rate with respect to the problems and situations encountered during the process.
Then, based on ( 15), if L is negative semidefinite, we have Proof.Let L be a positive semidefinite matrix.For  = 1 in (13), and  −1 ( L) = 0, we have By ( 15) and ( 17), Assume that there are repeated eigenvalues of Laplacian  associated with graph G, which are  +1 () =  +2 () = ⋅ ⋅ ⋅ =  + (),  = 1, . . .,  − , and there exists one vector   ∈ X satisfying L  ̸ = 0. Note that not for every vector  ∈ X,    +   L =    holds.So there is one of repeated eigenvalues, the value of which is improved; i.e., In case L is negative semidefinite, the same line of arguments yields that Proposition 4 shows if the eigenvector  of one eigenvalue  has two identical entries   =   , adding an edge   does not affect the eigenvalue , which means the weight, and even the existence of   does not affect .For an eigenvalue  with multiplicity , the corresponding eigenvector has  − 1 eigenvectors with two identical entries at any positions, so that if rank() = 1, no matter the position of the adding edge, there is only one  change; that is, the multiplicity of  of superposed system is  − 1.It is easy to know that if the multiplicity of  is , there is at least one eigenvector of  that has  − 1 pairs of identical entries; that is, if rank( L) < , the superposed system has the eigenvalue .

Changes of Convergence Rate
Definition 6.For an entry   of the adjacency matrix , ,  ∈ {1, . . ., }, if the increase of its value does not affect the algebraic connectivity of the communication graph, we call these weighted entries invalid algebraic connectivity weights (IACW).
For an invalid algebraic connectivity weight   , if there is a perturbation on one of associated nodes V  and V  so that the value of   decreases, the magnitude of each eigenvalue except 0 and  2 () decreases firstly, while the delay margin of the system increases.Thus, in this case, the stability and the convergence of the system can be protected to a certain extent.
If there is an invalid algebraic connectivity weight in the system, then L = 0, where  is an eigenvector of  2 () satisfying    +   L =  2 ( L) =   L.All the eigenvector    corresponding to the smallest nonzero eigenvalue must contain two components   and   with   =   ,  ̸ = , ,  ∈ {1, . . ., }.Below is a further explanation.
Due to the existence of repeated eigenvalue, the convergence rate does not necessarily change when a superposition system is superposed to the original system.

Theorem 7. If a superposition system related undirected graph
G is superposed to an undirected connected graph G, a composite connected graph Ĝ is generated for the corresponding superposed system.Then the convergence rate of the superposed system changes as follows.
(i) If there are invalid algebraic connectivity weights in G, the adjacency matrix Ã of the superposition system only consists of invalid algebraic connectivity weights, and L is positive semidefinite, then the superposed system converges at a rate equal to the original system; i.e.,  2 ( L) =  2 ().

Proof.
(i) If a system associated with the original graph G has an invalid algebraic connectivity weight, then all the eigenvectors corresponding to the smallest nonzero eigenvalue of  contain two components   ,   with   =   ,  ̸ = , ,  ∈ {1, . . ., }.Ã only consists of invalid algebraic connectivity weights if the superposition system only establishes a connection between V  and V  , such that there is an eigenvector  corresponding to  2 () with L = 0,    +   L =  2 ()+0 =  2 ( L) =   L.Therefore, the convergence rate of the superposed system remains unchanged.
Example 8.The graph associated with the original system is shown as Figure 2(e).The superposition system G1 only establishes connections among nodes 3, 4, and 5. Thus, rank( L1 ) = 3.The superposed system Ĝ1 is shown as Figure 2(b).Superpose another superposition system G2 to Ĝ1 , where G2 is shown as Figure 2(c) and only establishes the connection between nodes 1 and 2, which means rank( L2 ) = 1.The superposed system Ĝ2 is shown as Figure 2(d)., L1 , and L2 are given as follows.
Because the same elements may exist in the eigenvector  of  2 (), superposition systems do not always enhance the convergence rate.In case  2 > 1, the convergence rate changes only if the entries of  associated with the edges of supposition system are not all equal.For example, if only the entries  3 ,  4 ,  5 of  are equal and the others are different from each other, then the superposition system with the connections among nodes 2, 3, 4, and 5 can enhance the convergence rate, while the superposition system with only the connections among nodes 3, 4, and 5 will not have such effect.
Algorithm 1 gives a method of finding a superposition system to enhance the convergence rate.In Algorithm,  =  + 1 changes the label of node , which makes different nodes connect to node  or nodes ,  + 1, . that are not all equal; otherwise,  is a zero vector.So there is a superposition system that can increase the rate of convergence.Proposition 11.For an undirected connected graph G, if the smallest nonzero eigenvalue of  is repeated, the invalid algebraic connectivity weights can be constructed in any two nodes.
Proof.The invalid algebraic connectivity weights exist if and only if, for an eigenvector  corresponding to the smallest nonzero eigenvalue of , there are two components   ,   of  satisfying   =   ,  ̸ = , ,  ∈ {1, . . ., }.
For the positive semidefinite matrix L+ or negative semidefinite matrix L− , let rank( L+ ) = rank( L− ) = 1.If  2 () is a repeated eigenvalue, only one of repeated eigenvalues of L will change.Then for the superposition system, the nonzero entries of Ã+ or Ã− are invalid algebraic connectivity weights.All the eigenvectors corresponding to  2 () of L contain components   ,   with   =   , where (, ) represents the position of each nonzero weights associated with Ã+ and Ã− .

Optimization of Convergence
Rate.In a stable system, it is always expected that there is a larger convergence rate to take less time to get the stable state.In case all the weights of a communication graph are increased, the values of nonzero eigenvalues will be improved as a whole, and accordingly the stability can be achieved with a larger rate.From a practical point of view, however, this will increase the cost of realization.Even so, we still hope to achieve the rapidity of convergence under fixed cost.Below, let () denote the cost for achieving consensus.With () being fixed, we hope to find the most optimal topology to achieve the largest convergence rate.
For the communication delay   between nodes V  and V  , we consider the case   = .Then system (2) along with protocol (1) can be written as Lemma 13 (see [2]).Consider a network of integrator agents with identical communication time delay  in all links.Assume that the network topology is fixed, undirected, and connected.Then, protocol (22) with   =  globally asymptotically solves the average consensus problem if and only if either of the following equivalent conditions is satisfied.
By Theorem 12,  2 () is determined as long as the system achieves the largest convergence rate under fixed cost.Since the value of  2 () is determined, it is expected to construct topologies to achieve the largest convergence rate, Complexity i.e., seeking   , where   is the Laplacian having an eigenvalue max   2 ().If the graph of a multiagent system is a complete graph and all of its weights are identical, then the system achieves the largest convergence rate under fixed cost ∑  =1   =  > 0,  2 () = ∑  =1   /( − 1).In this case,  2 () achieves the biggest magnitude and   () achieves the smallest magnitude.By Lemma 13, the delay margin  * is decided only by   ().If there is a communication delay in the system and the   () achieves the smallest magnitude, then the delay margin achieves the biggest magnitude.
Remark 15.If a node V  is connected with all the other nodes, and the weights between V  and the others are identical, then  =   −   is an eigenvalue of .
For a system with time delays , the convergence rate is also affected by time delays .
Corollary 16.For a fixed topology, if the time delay  increases and  <  * , the convergence rate of the system reduces.
Example 17.The original system is shown as Figure 3(a), and the system with the largest convergence rate under the same cost is shown as Figure 3(b).The corresponding Laplacian matrices are For the eigenvalues of original system,  2 () = 1.7273,  5 () = 9.438, and the delay margin is  * = 0.1661.
All the nonzero eigenvalues of the system with the largest convergence rate are 5, and the delay margin of this system is  * = 0.3141.Proof.Since there is a nontrivial cell with  nodes and V  ∈ V is fixed, we construct the superposition system by selecting any node V  in the nontrivial cell to establish one communication link with the given V  .The dynamic equations of the nodes in the nontrivial cell are

Construction of Invalid Algebraic Connectivity Weights
where the entries of the columns in  1 ∈ R × and  2 ∈ R ×(−−) are all equal.In the superposition system, an arbitrary node V  ∈   is selected to establish one communication link with the fixed V  .Then the dynamic equations remain unchanged when the positions of  +1 , . . .,  + are exchanged, and the positions of ẋ +1 , . . ., ẋ + remain unchanged.In this situation, the topology of the system remains unchanged.The discussion above means that the variations of eigenvalues associated with  are equal since there is only one link between V  and any node of   in the superposition system.Note that the variation of algebraic connectivity is also identical.Then, if V  ∈   , the variation of the algebraic connectivity is unrelated to the choosing of V  .Thus, if V  ∈   is selected to construct a superposition system, its convergence rate is identical with L , and accordingly  2 ( L ) =  2 ( L ), where  ̸ = ,  ̸ = ,  ̸ = .
Example 20.A graph with two nontrivial cells is shown as Figure 4.The corresponding Laplacian  is The expression of Laplacian  implies that there is a partition  1 = {1},  2 = {2, 3},  3 = {4, 5}.Consider the case that rank( L) = 1, which means that there is an edge with weight 1 in the graph associated with the superposition system.Let us establish the connection between any two nodes, and let L denote the corresponding matrix which has only one link between nodes V  and V  .L denotes the Laplacian matrix of the superposed system corresponding to The variations of the other eigenvalues of L are the same.
Let   denote an eigenvalue of a submatrix which is generated by selecting the rows and columns corresponding to the nodes of the nontrivial cell   .  =   −   =   −   , with ,  indicating that   only relies on   , ,  ∈   .The  V denotes the vector which only contains two nonzero entries, and the two entries are opposite from each other.
Corollary 21.For an equitable weights partition  with a nontrivial cell   , the multiplicity of Proof.Since Laplacian  is symmetric, the geometrical and algebraic multiplicity of each eigenvalue of  is equal to each other.Hence, the number of linearly independent eigenvectors  of   is the multiplicity of   .Note that  − rank(   − ) is equal to the number of linearly independent solutions of (  − ) = 0.It follows that  − rank(  − ) is equal to the number of linearly independent vectors  which satisfies  =   , (  ) =  − rank(   − ).The structure Complexity form of   − implies that the row entries of  corresponding to the nodes in   are equal.Therefore,  − 1 ≤ (  ).
Remark 22.The number of eigenvalues   is no less than the number of the nontrivial cells   .
It is worth mentioning that, for an equitable weights partition, the submatrix corresponding to a nontrivial cell   consisting of  nodes only has the eigenvalue   with its multiplicity satisfying (  ) =  − 1. There, however, will be such a situation (  ) >  − 1, which is the consequence of the nontrivial cell   working with the other cells.
Proposition 23.For an equitable weights partition, if there is a nontrivial cell with  nodes, the number of linearly independent a V in the eigenvectors corresponding to   is  − 1.
Example 24.A graph with two nontrivial cells is shown in Figure 5.The corresponding Laplacian  is = diag {0, 9, 10, 13, 13, 15} .The expression of  implies that there is a partition  with  1 = {1},  2 = {2, 3},  3 = {4, 5, 6}.The entries of diagonal matrix   are the eigenvalues of , and the columns of  are the eigenvectors corresponding to   .Note that V 4 and V 5 are  V.It can be seen that 9 is an eigenvalue of the submatrix obtained by selecting the rows and columns corresponding to the nodes in  2 , and its corresponding eigenvector is a Faria vector.The eigenvalue 10 is not associated with any submatrix corresponding to any cell.The second and third entry of the eigenvector corresponding to eigenvalue 10 are equal, and the same thing happens to the fourth, fifth, and sixth entry.It is the same situation to eigenvalue 15 with the second and third entry of its eigenvector being equal as well as the fourth, fifth, and sixth entry.The two eigenvectors of eigenvalue 13  (iii) If () ̸ =   −   , let us consider the entries of 's of (), where these entries correspond to the nodes in   .Assume that these entries are not equal.Then vectors   are still eigenvectors associated with () after the positions of these unequal entries are exchanged.Let   denote the linear combination of  and   .The entries of   corresponding to the nodes in the nontrivial cell   are equal, and the others corresponding to the trivial cell  ¬ are 0, which contradicts with the assumption.Therefore, the entries in the eigenvectors corresponding to () ̸ =   −   are equal.
The equitable weights partition leads to some special perspectives on the eigenvalues and eigenvectors of Laplacian matrix, which allows us to propose an explicit method for constructing IACW.
Theorem 26.For an equitable weights partition of graph G, suppose that there is a nontrivial cell   with  nodes.Then the following statements hold for IACW.

Convergence Rate under Almost Equitable
Weights Partition.Different from the equitable weights partition, the weights in the nontrivial cell   are not equal if the almost equitable weights partition is taken into account.So the latter makes a part of properties lost compared to the equitable weights partition.Thus, this case is more complex.For the almost equitable weights partition, since the weights between the nodes in nontrivial cell   are different,   =   −   is no longer true.As a consequence,   cannot be characterized exactly.
Lemma 27.For an undirected graph G, there is an eigenvector  of a nonzero eigenvalue of  so that   1  = 0, where 1  ∈ R  denotes the column vector with all entries taking 1.
In case there is a partition with a nontrivial cell   ,  can be decomposed in accordance with the nontrivial cell, where   corresponds to the nodes in the nontrivial cell, which indicates the weights between nodes in   and its degree.=   yield that the weights between the nodes in the nontrivial cell   are IACW.
When the nontrivial cells existed, there are IACWs in a system.Thus, we can construct the nontrivial cell to get the IACWs.And it is easy to know that the IACWs are changed, when the case  2 () =   transforms into  2 () ̸ =   .The cases of the IACWs are identical for equitable weights partition and almost equitable weights partition.

Simulation Results
When a superposition system is superposed to an original system, some weights of the original graph are changed  accordingly.In order to verify the variation of the convergence rate on topologies and weights, we do simulations for Example 8.
Figures 9, 10, and 11 are the simulations of Example 17, where the initial states are the same as Example 8.For  = 0.19, the state response is shown in Figure 9.It can be seen that the system is unstable since the time delay  is larger than the delay margin  * = 0.1661.For graph G  , however, the system can achieve consensus with  = 0.19 as shown in Figure 10, which means that the system with graph G  has a larger delay margin  * than the system with graph G in Example 17.The corresponding topological structures in  Precisely because the topology corresponding to Figure 9 is different from that corresponding to Figure 10, the system associated with Figure 9 is unstable at the delay  while Figure 10 is stable at the same delay.When  = 0.14, the system associated with graph G  achieves a faster consensus than the system at  = 0.19, which is shown in Figure 11.

Conclusions and Future Work
In this paper, we proposed a superposition system which was superposed to the original system to explore the variation of convergence rate.By analyzing the eigenvector of  2 (), results were derived on checking whether the convergence rate can be changed.When the Laplacian L of the superposition system only consists of invalid algebraic connectivity weights, it was proven that the convergence rate remains unchanged.Otherwise, the convergence rate changes.We gave the most optimal case of the convergence rate under fixed cost, which makes the convergence rate the largest and the system more stable.Finally, we proposed a method of constructing invalid algebraic connectivity weights to make systems resistant in a certain extent to the perturbation.In addition, based on the equitable weights partition and almost equitable weights partition, we analyzed the changes of eigenvalues and eigenvectors to discover the variation of convergence rate.In future work, the optimization of convergence rate and convergence rate on directed graphs will be studied.Although, explicit results have been derived for the convergence rate by taking advantage of the proposed concept of superposition systems, the optimization of convergence rate still needs further study.In the future work, convergence rate on directed graphs will also be studied.

Corollary 9 .
For an undirected connected graph G with the multiplicity of  2 () being  2 , let r2 represent the multiplicity of  2 () of L, where the Laplacian L of the superposition system G is positive semidefinite and rank( L) = ,  ∈ {1, . . .,  − 1}.

Figure 3 :
Figure 3: (a) Original system G; (b) the largest convergence rate system G  .

Figure 4 :
Figure 4: A graph with nontrivial cells.

Figure 5 :
Figure 5: A graph with nontrivial cells.
(i) If  2 () ̸ =   , the weights between the nodes in   are IACW.(ii) If  2 () =   , and  − rank(   − ) =  − 1, the weights except those between the nodes in the nontrivial cell   are IACW.Proof.By Lemma 25, if  2 () ̸ =   , the entries in the eigenvector  of  2 () corresponding to the nodes in   are equal, and accordingly the weights between the nodes in   are IACW.If  2 () =   , and  − rank(   − ) =  − 1, then Lemma 25 means that the th and the th entry in the eigenvectors of  2 () are opposite, and the others are 0. Thus the weights except those in the nontrivial cell   are IACW.

Figure 6 :
Figure 6: States of graph G.
Definition 18.  = { 1 , ...,   } is called an equitable weights partition if each node in cell   has the same weight with the nodes in   , ∀,  ∈ {1, ..., }.If each node in   has the same weight with the nodes in   the partition  is called an almost equitable weights partition, where ∀ ̸ = , ,  ∈ {1, ..., }.We denote the cardinality of cell   with |  |.A cell   is nontrivial if it contains more than one node; otherwise it is trivial.Equitable weights partition can be used in what follows to analyze the convergence rate of consensus, and we can construct the nontrivial cells to get the IACWs.4.1.Convergence Rate under Equitable Weights Partition.For a nontrivial cell   in an equitable weights partition, suppose that the number of nodes in   is |  | = .We have the following lemma.If L , L are superposition systems only constructed, respectively, by V  and V  and V  and V  ; then  2 ( L ) =  2 ( L ), where V  , V  ∈   , and V  ∈ V is a node given in 1 ,  2 , . . .,   2  }, where  i denotes a  V,  ∈ {1, 2, . . .,  2  }, and the nonzero entries in   is corresponding to the nodes in   .Since   2  consists of  2   V, then rank(  2 are both  V.For a connected graph G, if there exists a nontrivial cell   with  nodes, then the following claims hold under equitable weights partition.(i)If− rank(   − ) =  −1, the eigenvectors of   are all  V.(ii) If  − rank(   − ) >  − 1, the entries in the eigenvector  of   corresponding to any other nontrivial cell  −1 are equal to each other as long as the eigenvector is not a  V.If there is a nontrivial cell   containing nodes V  and V  , let us set   =   −   .Then rank(   − ) < , that is, |  −| = 0, where   is an eigenvalue of Laplacian .In case  = {0, . . ., 0,  ℎ , 0 . . ., 0, − ℎ , 0, . . ., 0}  ,  =   , where  ℎ =  ℎ ,  ℎ and  ℎ are nonzero constants, which denote the ℎ and ℎ entries of vector , respectively.If  − rank(   − ) =  − 1, it follows from Lemma 19 that the position exchanging of nodes in the nontrivial cell does not affect the structure of .As a consequence, the eigenvectors of   are also unaffected.Thus, when  ℎ exchanges its position with − ℎ , the original eigenvector  is replaced by −, and the entries except the ℎ and ℎ entries in eigenvectors  are 0.That is, there are  − 1 linearly independent eigenvectors of  V corresponding to the  − 1 eigenvalues   .(ii) If −rank(  −) ≥ , and the entries in eigenvector  corresponding to  −1 are equal, then, there are more than  − 1 V.Thus the entries of 's of   are equal.
Each column of   is equal to the other columns, which indicates the weights between nodes in   and nodes in other cells.¬correspondsto the trivial cells.Remark 28.For an equitable weights partition,   has an eigenvalue   =   −   with multiplicity  − 1 if there is a nontrivial cell.For an almost equitable weights partition,   only has an eigenvalue with multiplicity 1 which can be characterized.=   1 1    +   , where  1 denotes the first column of   , and 1  is the vector with proper dimension and all entries taking 1.   is the Laplacian matrix of a system constructed by the nodes in the nontrivial cell   .For almost equitable weights partitions,   is the general Laplacian matrix.So the nonzero eigenvalues cannot be characterized.Because of the existence of   1 1    , there is an eigenvalue   1 1  associated with   , and the entries in the eigenvector of   1 1  are all equal.For a matrix   ∈ R × , suppose that there is only one nontrivial cell associated with   .Then the Laplacian matrices  and   share  − 1 common eigenvalues   .Moreover, for matrix , its 's associated with   take   as a subvector, where   is the eigenvector of   associated with   ; that is,   = {   , 0, . .., 0}.Proof.Assume that the nontrivial cell   contains  nodes, and   =   1 1    +   .Then     =   1 1    +     =     , where     =     ,   =   1 1  +   .Let   = {   , 0, . . ., 0}, with the positions of entries 0 corresponding to the nodes in  ¬ .Since   1  = 0, it follows that [  ,   ] =     , [  ,  ¬ ] = 0 ¬ ,  =   .If   =   1 1  , we see that   is a vector with all of its entries taking 1.Thus    1  ̸ = 0; that is,      ̸ = 0, [  ,  ¬ ] ̸ = 0 ¬ .Therefore,   does not share the eigenvalue   1 1  with .So  and   share  − 1 common eigenvalues, and   = { , 0, . . ., 0}.
Lemma 29 explains the affections caused by nontrivial cells under equitable weights and almost equitable weights partition.The existence of nontrivial cells in   makes that part of eigenvalues and eigenvectors of  rely on   .For an almost equitable weights partition of graph G, suppose that there is a nontrivial cell   .If  2 () =   , the weights except those in   are all IACW.If  2 () ̸ =   , the weights in   are IACW.Proof.For an almost equitable weights partition with a nontrivial cell   ,   is an eigenvalue of submatrix obtained by selecting the rows and columns corresponding to the nodes in   .By Lemma 29, if  2 () =   , the entries of vector  except those corresponding to the nodes in   are 0, where  is the eigenvector corresponding to   .Therefore, the weights of  ¬ are IACW.Let   = [  ,  ¬ ]  , where   is the eigenvector of   .The entries of every column in   are equal to the others.Then    ¬ = 1  , where  is a constant.If  2 () ̸ =   , and the entries of   are different, then [  ,   ] =     + 1  =   ,     =   − c1  .If the entries of   are different, the entries of   − 1  are also different.Thus  =   and  2 () ̸