Advancing Convergence Speed of Distributed Consensus Time Synchronization Algorithms in Unmanned Aerial Vehicle Ad Hoc Networks

Abstract

Time synchronization is a critical prerequisite for unmanned aerial vehicle ad hoc networks (UANETs) to facilitate navigation and positioning, formation control, and data fusion. However, given the dynamic changes in UANETs, improving the convergence speeds of distributed consensus time synchronization algorithms with only local information poses a major challenge. To address this challenge, this study first establishes a convex model on the basis of graph theory and relevant theories of random matrices to approximate the original problem. Subsequently, three acceleration schemes for consensus algorithms are derived by minimizing the Frobenius norm of the iteration matrix. Additionally, this study provides a new upper bound for constant communication weights and discusses the limitations of existing metrics used to measure the convergence speeds of consensus algorithms. Finally, the proposed schemes are compared with existing ones through simulation. Our results indicate that the three proposed schemes can achieve faster convergence while maintaining high-precision synchronization in scenarios with static or known topological structures of networks. In scenarios where the topological structure of a UANET is time-varying and unknown, the scheme proposed in this paper achieves the fastest convergence speed.

1. Introduction

The widespread adoption of unmanned aerial vehicles (UAVs) across the military and civilian sectors has highlighted the inherent limitations faced by individual UAVs, including constraints related to energy, operational capabilities, and payload capacities, which impede their ability to execute complex tasks [1,2,3]. Consequently, the UAV ad hoc networks (UANETs), characterized by decentralized and distributed organizational paradigms, have become a focal point of contemporary research endeavors [4,5]. Many scholars have comprehensively investigated collaborative navigation, positioning, situational awareness, formation control, and data fusion within the domain of UANETs [6,7,8,9], with time synchronization among UAVs being a crucial prerequisite for such research endeavors [10,11,12].

However, the information and resources within UANETs inherently exhibit localization and dispersion. They are constrained by individual communication capabilities, information processing capacities, and operational contexts [8]. Consequently, achieving rapid and high-precision time synchronization among UAVs is a challenge. Typically, scholars employ consensus-based methodologies to realize time synchronization within UANETs [13]. In this context, consensus denotes the process whereby individuals within a UANET share information and interact according to predefined rules, ultimately converging to a unified clock reading over time [14,15].

1.1. Related Work

From the perspective of graph theory, the investigation into time synchronization within UANETs inherently revolves around the consensus problem [16]. Furthermore, insights from research conducted in domains such as sensor network time synchronization, the collaborative control of multi-agent systems, and UAV swarm flight can provide valuable guidance and references for addressing the consensus problem [17,18,19,20,21,22].

Initial research efforts centered on designing centralized tree-structured schemes to optimize consensus control within networks [23,24,25,26]. However, centralized approaches have become inadequate for meeting present application demands because of the increasingly complex dynamics of contemporary dynamic network topologies, alongside the considerations of algorithm robustness and scalability. Consequently, distributed consensus solutions have been developed [27,28]. Distributed consensus algorithms greatly reduce the demands on individual perception capabilities, the communication bandwidth, and computational resources, while offering benefits such as cost-effectiveness, robustness, adaptability, scalability, and stealthiness [22,29,30].

Apart from synchronization accuracy, another key performance metric of distributed consensus algorithms is convergence speed. It measures how quickly individual states in UANETs converge to the same value from different initial values. Numerous scholars have aimed to achieve rapid convergence in distributed consensus algorithms, starting with the communication topology network structure [31,32,33]. In their study of distributed formation shape control, Li et al., indicated that the convergence rate of the formation is only determined by the follower angles obtained by the target formation beforehand [34]. In these investigations, scholars often used graph theory to abstract the communication topology network structure. The eigenvalues of the Laplacian matrix contain crucial topological and dynamic information, greatly influencing the convergence speeds of distributed consensus algorithms [35,36]. To facilitate specific research on the convergence speeds of distributed consensus algorithms, scholars have typically utilized the second smallest eigenvalue of the Laplacian matrix corresponding to the communication topology network as a measure, also known as the algebraic connectivity of the graph [37,38]. However, the algebraic connectivity of the graph represents a lower bound rather than an exact evaluation metric for the convergence speed [39,40].

In existing research, efforts to enhance the convergence speeds of distributed consensus algorithms from a topological perspective can be categorized into two types: (1) reconstructing the communication topology and (2) optimizing the weights of communication topology links.

In the first approach, scholars primarily targeted improving the algebraic connectivity of graphs by modifying the actual communication topology of the network, despite the fact that maximizing algebraic connectivity has been proven to be an NP-hard problem [31]. Olfati-Saber et al., utilized rewiring operations based on small-world theory to enhance the algebraic connectivity of the network by over 1000 times while maintaining the same number of edges and nodes [32]. Amani et al., used local multiplicity to assess the impact of each node on various eigenvalues of the Laplacian matrix, ranking network nodes accordingly. They found that removing nodes based on this ranking can increase the algebraic connectivity of the graph [41]. Chen et al., demonstrated that the algebraic connectivity of the graph monotonically increases with an increase in edges with a constant number of nodes, but the eigenratio may vary [42]. They also noted that adding edges to cycles with more than four nodes decreases the eigenratio but maintains the algebraic connectivity [43]. In their subsequent studies, Chen et al., further revealed that fully homogeneous networks represent the optimal solution for synchronous performance when the numbers of nodes and edges remain fixed [44]. Despite the numerous studies on enhancing the algebraic connectivity of graphs, it is not the primary focus of this paper and, thus, will not be elaborated on further [45,46,47,48].

In the second approach, scholars optimized the convergence speeds of consensus algorithms by refining communication link weights when altering the communication topology structure was not feasible. These optimization schemes can be further categorized into two types on the basis of the information needed to determine communication link weights: global and local information acceleration schemes (see

Table 1. Global information acceleration schemes.

Schemes	$\mathbf{Construction}\mathbf{Method}\mathbf{of}\mathbf{the}\mathbf{Iteration}\mathbf{Matrix}\mathit{P}$
Best constant edge weight [31]	$P=I-ϵL,ϵ=\frac{2}{{\lambda }_2+{\lambda }_N}$
Optimal symmetric edge weights [31]	$\underset{s\in R}{min}s$ $s.t-sI\preccurlyeq P-\frac{{11}^{\mathit{T}}}{N}\preccurlyeq sI, P1=1,P=P^T$
Fastest mixing chain [49]	$\underset{s\in R}{min}s$ $s.t-sI\preccurlyeq P-\frac{{11}^{\mathit{T}}}{N}\preccurlyeq sI,P\ge 0,P1=1,P=P^T$
Least-mean-square weights [50]	$\underset{P}{min}{\displaystyle {\displaystyle \sum }_{i=2}^N}\frac{1}{1-{\lambda }_i{\left(P\right)}^2}$ $s.t \parallel P-\frac{{11}^{\mathit{T}}}{N}\parallel <1,P1=1,P=P^T$
Effective resistance minimization weights [51]	$\underset{P}{min} N{\displaystyle {\displaystyle \sum }_{i=2}^N}\frac{1}{1-{\lambda }_i\left(P\right)}$ $s.t P1=1,P\ge 0$

and

Table 2. Local information acceleration schemes.

Schemes	$\mathbf{Construction}\mathbf{Method}\mathbf{of}\mathbf{the}\mathbf{Iteration}\mathbf{Matrix}\mathit{P}$
Maximum-degree weight	$P=I-ϵL,ϵ=\frac{1}{d_{max}+1}$
Metropolis–Hastings weights [49]	$P_{ij}=\left\{\begin{array}{c}min\left(\frac{1}{d_i+1},\frac{1}{d_j+1}\right); iff \left(i,j\right)\in E and i\ne j\\ 1-{\displaystyle {\displaystyle \sum }_{j\in N_i}}{P}_{ij}; iff i=j\\ 0; otherwise\end{array}\right.$
Lazy Metropolis weights [52]	$P_{ij}=\left\{\begin{array}{c}\frac{1}{2}min\left(\frac{1}{d_i+1},\frac{1}{d_j+1}\right); iff \left(i,j\right)\in E and i\ne j\\ 1-\frac{1}{2}{\displaystyle {\displaystyle \sum }_{j\in N_i}}{P}_{ij}; iff i=j\\ 0; otherwise\end{array}\right.$

, respectively).

In Table 1 and Table 2, $P-\frac{{11}^{\mathit{T}}}{N}\preccurlyeq sI$ denotes that $sI-P+\frac{{11}^{\mathit{T}}}{N}$ is a positive semidefinite matrix, $P\ge 0$ denotes that all elements in $P$ are non-negative, and $\parallel P-\frac{{11}^{\mathit{T}}}{N}\parallel$ denotes the spectral norm of matrix $P-\frac{{11}^{\mathit{T}}}{N}$. Other symbols are defined in detail in Section 2.

In the construction of the iteration matrices P in the aforementioned acceleration schemes, local information acceleration schemes do not require each node to know the entire network topology, making them suitable for real-time dynamic scenarios [53].

Some scholars attempted to improve the convergence speed by leveraging stored information from nodes [54,55,56,57,58,59,60]. When dealing with synchronous state quantities represented by higher-order dynamic models, a promising research direction is enhancing convergence speed by incorporating information from past neighbor node steps.

1.2. Motivations

In practical working scenarios, UAVs often face constraints such as limited communication range, diverse working environments, and varied tasks. Consequently, constructing communication topologies to enhance convergence speed may not always be viable. Thus, investigating how to optimize network communication link weights under these constraints has significant practical value. This optimization aims to enable consensus algorithms to achieve rapid convergence while maintaining high-precision time synchronization.

However, existing research highlights several areas requiring further improvement. First, current methods for enhancing consensus algorithms’ convergence speeds by optimizing communication topology weights typically impose the constraint of a symmetric iteration matrix. This constraint limits the search space for optimal solutions. Second, the optimization objectives of acceleration schemes often focus on the spectral radii or norms of matrices. However, the convergence speeds of distributed consensus algorithms involve all eigenvalues of the iteration matrix, indicating potential alternative solutions to further enhance the convergence speed. Last, most current schemes for improving the convergence speeds of consensus algorithms through communication weight values rely on global information about the network topology. This reliance renders such algorithms impractical for scenarios with dynamically changing communication topologies.

Therefore, how individual nodes in UANETs can optimize communication weights in real time solely on the basis of local information remains to be thoroughly explored. Addressing this challenge would allow consensus algorithms to simultaneously improve convergence speed while preserving final convergence accuracy.

1.3. Statement of Contributions

This study addresses scenarios where improving convergence speed through the construction of the communication topology structures of UANETs is not feasible. This study establishes a convex model to approximate the original problem by minimizing the Frobenius norm of the iteration matrix. Three communication weight selection schemes are proposed to accelerate the convergence speed. This study includes numerical examples and simulations to validate the proposed schemes and compare them with existing algorithms in static and dynamic scenarios. The main contributions of this paper can be summarized as follows:

• A new upper bound for the constant weight $ϵ$ in the iteration matrix $P=I-ϵL$ of undirected graph consensus algorithms is proposed.
• The lack of a comprehensive and accurate theoretical index to describe the convergence speeds of consensus algorithms under different acceleration schemes and scenarios is highlighted.
• By establishing convex models similar to the original problem, three acceleration schemes for consensus algorithms are obtained and rigorously derived and validated through simulations. The three proposed schemes achieve faster convergence while maintaining synchronization accuracy compared with existing schemes for enhancing convergence speed in scenarios with static or known communication topologies.
• A scheme is put forward to update communication weights in real time for UANETs. This scheme minimizes the Frobenius norm of the iteration matrix. This scheme achieves the fastest convergence speed while maintaining high-precision time synchronization compared with existing schemes suitable for dynamic scenarios.

This paper is structured as follows. Section 2 covers graph theory, discrete consensus protocols, and relevant matrix theory. Section 3 introduces a new upper bound for the constant weight ϵ and elaborates the three proposed acceleration schemes and their theoretical derivations. Section 4 compares the proposed schemes with existing schemes through simulation and discusses the limitations of existing metrics for evaluating convergence speed. Section 5 concludes this study, and Section 6 outlines future research directions.

2. Preliminaries and Problem Formulation

2.1. Preliminaries

This section introduces the fundamental mathematical theories and models pertinent in the subsequent research [61,62].

2.1.1. Network Model

To study distributed consensus algorithms in UANETs, the communication topology of the entire system is abstracted as an undirected or balanced directed graph, denoted as $G=\left(V,E\right)$. Here, $V=\left\{1,2,\dots ,N\right\}$ represents the set of $N$ UAVs, each referred to as a node in the network, and $E$ represents the edge set. The edge $\left(i,j\right)\in E$ if and only if UAVs $i$ and $j$ can communicate with each other. The set of neighbors of node $i$ is denoted as $N_i$, while $d_i$ represents the degree of node $i$, which signifies the count of neighboring nodes. $d_{max}$ is defined as $d_{max}=\underset{i\in V}{max} d_i$, which represents the maximum degree in the network. If there exist $\left(i,j\right)\in E$ and $\left(j,k\right)\in E$, then nodes $i$ and $k$ are connected by a path. Moreover, if a path exists between any two nodes $i$ and $j$, the graph $G$ is classified as a connected graph.

The Laplacian matrix of graph $G$, denoted as $L$, is defined as $L=D-A$, where the adjacency matrix $A\in R^{N\times N}$ as follows:

$$A_{i,j}=\left\{\begin{array}{c}1; iff \left(i,j\right)\in E\\ 0; otherwise\end{array}\right.$$

(1)

Here, the degree matrix of $G$ is a diagonal matrix $D=\mathrm{diag}\left(\left[d_1,d_2,\dots ,d_N\right]\right)\in R^{N\times N}$. The Laplacian matrix of the graph can be defined alternatively using the incidence matrix $B$ as $L=I-B{B}^T$, where $B\in R^{N\times M}$ as follows:

$$B_{il}=\left\{\begin{array}{c}1; if edge l starts from node i\\ -1; if edge l ends at node i\\ 0; otherwise\end{array}\right.$$

(2)

The direction of each edge in the undirected graph can be chosen arbitrarily.

2.1.2. Consensus Protocol and Matrix Theory

In discrete consensus algorithms, under the assumption that communication is limited to first-order neighbors, the iterative form can typically be expressed as follows:

$$x_i\left(k+1\right)=x_i\left(k\right)+ϵ{\displaystyle \sum }_{j\in N_i}\left(x_j\left(k\right)-x_i\left(k\right)\right)$$

(3)

where $i\in V$, $k$ is the iteration count, $x_i\left(k\right)$ is the state value of node $i$ in the $k$th iteration, and $ϵ\in \left(0,1/d_{max}\right)$. Expressing Equation (3) in matrix form generates

$$X\left(k+1\right)=\left(I-ϵL\right)X\left(k\right)=PX\left(k\right)=P^{k+1}X\left(0\right)$$

(4)

where $X\left(k\right)={(x_1\left(k\right),x_2\left(k\right),\dots ,x_N\left(k\right))}^T$ is the state values of each node in the $k$th iteration, $X\left(0\right)$ is the initial states of the nodes, $I$ is the $N$-dimensional identity matrix, and $P\in R^{N\times N}$ is the doubly stochastic matrix, also known as the Perron matrix of the graph $G$ with parameter $ϵ$. The diagonal elements of $P$ correspond to the self-loops of the communication links of the nodes.

In UANETs, the communication topology is dynamic and time-varying because of the dynamic characteristics of the UAVs. To enhance the convergence speeds of consensus algorithms, different weights are assigned to each communication link, potentially disrupting the symmetry and non-negativity of the Perron matrix $P$. Therefore, Equation (4) is rewritten as follows:

$$X\left(k+1\right)=\left(I-H\left(k+1\right)\odot L\left(k+1\right)\right)X\left(k\right)=P\left(k+1\right)X\left(k\right)$$

(5)

where $\odot$ denotes the Hadamard product of matrices, and $H\left(k\right)={(h_{ij})}_{N\times N}\in R^{N\times N}$ is the communication weight matrix. $ϵL$ and $H\left(k+1\right)\odot L\left(k+1\right)$ are the weighted Laplacian matrix $\tilde{L}$.

Lemma 1.

The Laplacian matrix $L$ for an undirected graph $G$ is a real symmetric matrix exhibiting zero sums for rows and columns.

Lemma 2.

The Laplacian matrix L corresponding to an undirected connected graph G is a positive semidefinite matrix. The eigenvalues of L are arranged in increasing order of magnitude as follows:

$$0={\lambda }_1<{\lambda }_2\cdots \le {\lambda }_N$$

(6)

The second smallest nonzero eigenvalue ${\lambda }_2$ is referred to as the algebraic connectivity or spectral gap of the graph, and $rank\left(L\right)=N-1$.

Lemma 3.

The largest eigenvalue ${\lambda }_N$ of the Laplacian matrix of graph $G$ satisfies the following:

$${\lambda }_N\le \underset{i,j}{max}\left\{d_i+d_j-\left|N_i\cap N_j\right||\left(i,j\right)\in E\right\}\le N$$

(7)

where $\left|N_i\cap N_j\right|$ represents the number of common neighbors of nodes $i$ and $j$ [63].

Lemma 4.

(Schur’s inequality) Let $M={(m_{rs})}_{N\times N}\in R^{N\times N}$ have eigenvalues  ${\lambda }_{M,1},{\lambda }_{M,2},\cdots {\lambda }_{M,N}$. Then,

$${\displaystyle \sum }_{r=1}^N{\lambda }_{M,r}^2\le {\displaystyle \sum }_{r,s=1}^N{m}_{rs}^2=tr\left(M^{T}M\right)={\parallel M\parallel }_F^2$$

(8)

Here, $tr\left(·\right)$ is the trace of a matrix, and  ${\parallel ·\parallel }_F$ is the Frobenius norm of a matrix. The equality holds if and only if $M$ is a normal matrix, i.e.,  $M^{T}M=M{M}^T$, and real symmetric matrices are obviously normal matrices [64].

Lemma 5.

If a matrix is row diagonally dominant, its determinant is nonzero.

Lemma 6.

A Markov chain corresponding to an undirected graph composed of finite nodes with self-loops is necessarily an ergodic chain. Such an ergodic chain always possesses a unique stationary distribution, often referred to as the limit distribution [47].

2.2. Problem Formulation

As previously mentioned, distributed consensus algorithms iterate in the form of Equation (5), ultimately driving the state values of each node toward consensus. The consensus is expressed as follows:

$$\underset{k\to \infty }{lim}\left|x_i\left(k\right)-x_j\left(k\right)\right|=0, \forall i,j\in V$$

(9)

Here, the state values $x_i\left(k\right)$ of nodes hold different physical meanings depending on the specific application scenario. Examples are the velocities of nodes in unmanned aerial vehicle formation control, the clock information of nodes in ad hoc networks, the voltages of nodes in electrical grids, or the phases and frequencies of coupled oscillators.

This study uses the standard deviation $\zeta \left(k\right)$ of $X\left(k\right)$ as an evaluation metric, termed as synchronization precision to quantify the precision of achieving consensus among nodes in a UANET through a consensus algorithm. The theoretical values for the convergence of the algorithm can be derived for consensus algorithms in the form of $X\left(k+1\right)=\left(I-H\odot L\right)X\left(k\right)$. The concern of this study lies in the differences between the state values $x_i\left(k\right)$ after each convergence rather than deviations from theoretical values. Therefore, standard deviation is used instead of root mean square error.

To measure the convergence speed of the consensus algorithm, the convergence time $T_{P\left(k\right)}\left(\zeta \left(k\right),\sigma \right)$ is defined as the time when $\zeta \left(k\right)$ first becomes less than the permissible error $\sigma$. That is, $k=T_{P\left(k\right)}\left(\zeta \left(k\right),\sigma \right)$ is the smallest iteration number $k$ such that $\zeta \left(k\right)\le \sigma$.

To clarify, the assumptions made in this study are as follows:

Assumption 1.

The communication radius of each UAV in the UANET is equal. UAVs within the communication range communicate bidirectionally in the same period, so the communication topology network at any given time can be represented as an undirected or balanced directed graph $G$, with the corresponding Laplacian matrix $L$ being a real symmetric matrix.

Assumption 2.

At any given time, the network has no isolated nodes, i.e., the undirected graph $G$ is always connected.

3. Consensus Algorithm Accelerated Convergence Scheme

3.1. Constant Communication Weight

In this subsection, the most special and simplest form of consensus algorithm iteration is examined first, where all elements in $H\left(k+1\right)$ in Equation (5) are equal, resulting in Equation (4). In existing research, to constrain the iteration $P$ to be a non-negative matrix, the communication weight $ϵ$ is typically required to be within $\left(0,1/d_{max}\right)$. However, this constraint is not commonly employed in practical convergence analysis.

This section introduces a new upper bound for $ϵ$ and then presents a scheme for selecting $ϵ$ to enhance the convergence speed of the consensus algorithm.

Theorem 1.

In the iteration of $X\left(k+1\right)=\left(I-ϵL\right)X\left(k\right)$, a consensus algorithm in the form of $ϵ$ is within $\left(0,max\left(\frac{1}{d_{max}},\frac{2}{N}\right)\right)$.

Proof. 

When no nodes in the network have a degree greater than $N/2$, i.e., $\frac{1}{d_{max}}\ge 2/N$, can be directly obtained from the theory of Markov matrices.

When nodes in the network have a degree greater than $N/2$, i.e., $\frac{1}{d_{max}}<2/N$, according to Lemma 1 and Lemma 2, the matrix $P=\left(I-ϵL\right)$ can be diagonally similar to $P=Q\mathsf{\Sigma }{Q}^T$, where $\Sigma =\mathrm{diag}\left(\left[1-ϵ{\lambda }_1,1-ϵ{\lambda }_2,\dots ,1-ϵ{\lambda }_N\right]\right)$ is a diagonal matrix, $Q$ is an orthogonal matrix, and its column vectors $q_1,q_2,\cdots ,q_N$ are the eigenvectors corresponding to the eigenvalues, where $q_1={(\frac{1}{\sqrt{N}},\frac{1}{\sqrt{N}},\cdots ,\frac{1}{\sqrt{N}})}^T$. Substituting into Equation (4), the entire iteration process is as follows:

$$\begin{gathered} X\left(k\right)=\left(I-ϵL\right)X\left(k-1\right)=P^{k}X\left(0\right)=Q{\mathsf{\Sigma }}^k{Q}^{T}X\left(0\right) \\ =\left[q_1,q_2,\cdots ,q_N\right]\left[\begin{array}{c}{q_1}^{T}X\left(0\right)\\ {(1-ϵ{\lambda }_2)}^k{q_2}^{T}X\left(0\right)\\ ⋮\\ {(1-ϵ{\lambda }_N)}^k{q_N}^{T}X\left(0\right)\end{array}\right] \end{gathered}$$

(10)

To realize convergence, $\underset{i=2,\cdots N}{max}\left|1-ϵ{\lambda }_i\right|<1$ in Equation (10), i.e., $0<ϵ{\lambda }_i<2$, must be ensured.

Substituting into Lemma 3, $0<ϵ{\lambda }_i<ϵN<2$. When $0<ϵ<2/N$, Equation (10) converges, and Theorem 1 is proven. □

Scheme 1.

The Minimization of the Frobenius Norm Constant Weight: For the constant communication weight consensus algorithm in the form of $X\left(k+1\right)=\left(I-ϵL\right)X\left(k\right)$, the optimal solution for $ϵ$ can be obtained as  ${ϵ}_c^{*}=\frac{tr\left(D\right)}{tr\left(D^2\right)+tr\left(D\right)}$.

Proof. 

From Equation (10), the square of the two-norm of $X\left(k\right)$ follows that

$$\begin{array}{l}{\parallel X\left(k\right)\parallel }_2^2=X{\left(0\right)}^{T}Q{\mathsf{\Sigma }}^k{Q}^{T}Q{\mathsf{\Sigma }}^k{Q}^{T}X\left(0\right)\\ =\left[X{\left(0\right)}^T{q}_1,X{\left(0\right)}^T{q}_2,\cdots ,X{\left(0\right)}^T{q}_N\right]\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\begin{array}{c}0\\ {(1-ϵ{\lambda }_2)}^{2k}\\ 0\\ \cdots \end{array}\begin{array}{c}\cdots \\ \cdots \\ \ddots \\ 0\end{array}\begin{array}{c}0\\ 0⋮\\ 0\\ {(1-ϵ{\lambda }_N)}^{2k}\end{array}\right]\left[\begin{array}{c}{q_1}^{T}X\left(0\right)\\ {q_2}^{T}X\left(0\right)\\ ⋮\\ {q_N}^{T}X\left(0\right)\end{array}\right]\end{array}$$

(11)

As $k$ approaches infinity, the limit of ${\parallel X\left(k\right)\parallel }_2^2$ equals $\frac{1}{N}{({\displaystyle \sum }_{i=1}^N{x}_i\left(0\right))}^2$. This implies that the consensus algorithm has achieved convergence.

From the perspective of the squared modulus of the state vector $X\left(k\right)$, as the iteration count $k$ increases, the main factor affecting the convergence speed is $1+{(1-ϵ{\lambda }_2)}^{2k}+\cdots +{(1-ϵ{\lambda }_N)}^{2k}$, i.e., the sum of squares of the eigenvalues of matrix $P$. According to Lemma 4, the following optimization objective is established:

$$\begin{gathered} \underset{ϵ}{\min }{\parallel P\parallel }_F^2 \\ s.t P1=1,P^T=P \end{gathered}$$

(12)

Simplifying Equation (12) yields

$$\underset{ϵ}{min}{\parallel P\parallel }_F^2=\underset{ϵ}{min}{ϵ}^{2}tr\left(L^2\right)-2ϵtr\left(L\right)+N$$

(13)

Equation (13) is convex. Taking the derivative with respect to $ϵ$ and setting it as equal to zero yields

$${ϵ}_c^{*}=\frac{tr\left(L\right)}{tr\left(L^2\right)}=\frac{tr\left(D\right)}{tr\left(D^2-DA-AD+A^2\right)}=\frac{tr\left(D\right)}{tr\left(D^2\right)+tr\left(D\right)}$$

(14)

Thus, Scheme 1 is proven. □

Equation (14) indicates that ${ϵ}_c^{*}$ is easily computed and only requires knowledge of the degree matrix. When applied to scenarios where the topology structure is unknown, nodes must transmit their own degree information during communication, resulting in an additional communication burden close to the maximum-degree weight ${ϵ}_c^{max}=\frac{1}{d_{max}+1}$. In contrast, solving for the best constant edge weight ${ϵ}_c^{\lambda }=\frac{2}{{\lambda }_2+{\lambda }_N}$ requires knowledge of the entire Laplacian matrix. ${ϵ}_c^{*}$ and ${ϵ}_c^{\lambda }$ may lead to negative weights for self-feedback links, and the iteration matrix $P$ is no longer a doubly stochastic matrix.

However, in constant weight consensus algorithm acceleration schemes, apart from the self-feedback links of each node, the weights of the communication links throughout the network are all assigned the same value. This scenario limits further improvement in the convergence speed of the consensus algorithm.

3.2. Matrix Communication Weights

Previous research demonstrates that schemes using constant communication weights can only enhance the convergence speeds of consensus algorithms by adjusting the scalar $ϵ$. In this section, the scalar communication weight $ϵ$ is extended to the form of a communication weight matrix $H$. Formulations such as Equation (5) are used, where the iteration matrix is represented as $P=I-H\odot L$. The classification and study of $H$ are conducted on the basis of Lemma 2.4.

Scheme 2.

The Minimization of the Frobenius Norm Symmetric Weights: In the consensus algorithm with matrix communication weights in the form $X\left(k+1\right)=\left(I-H\odot L\right)X\left(k\right)$, subject to the constraint $H=H^T$, the optimal solution for  $H$ to minimize the Frobenius norm of matrix $P$ is denoted as $H_{sym}$.

$$H_{sym}=\left\{\begin{array}{c}\frac{1}{2}\left(D+A+2I\right)D{H}_{diag}=D_{diag}; \forall i=j\\ h_{ij}=1-\frac{h_{ii}{d}_i}{2}-\frac{h_{jj}{d}_j}{2}; \forall i\ne j\end{array}\right.$$

(15)

where $H_{diag}\in R^{N\times 1}$ is the column vector comprising the main diagonal elements of the $H_{sym}$ matrix, and $D_{diag}\in R^{N\times 1}$ is the column vector comprising the main diagonal elements of the $D$ matrix.

Proof. 

Following the same approach as used in Scheme 1, the Frobenius norm of matrix $P$ is employed as the optimization objective to establish the following optimization model:

$$\begin{gathered} \underset{H}{min}{\parallel I-H\odot L\parallel }_F^2 \\ s.t P1=1;P^{\mathit{T}}=P \end{gathered}$$

(16)

Simplifying the objective function yields

$${\parallel I-H\odot L\parallel }_F^2=tr\left({\left(I-H\odot L\right)}^T\left(I-H\odot L\right)\right)=N-2{\displaystyle \sum }_{i=1}^N{h}_{ii}{d}_i+{\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j\in N_i}{h}_{ij}^2+{\displaystyle \sum }_{i=1}^N{h}_{ii}^2{d}_i^2$$

(17)

From the constraint $P1=1$, it follows that

$$h_{ii}{d}_i={\displaystyle \sum }_{j\in N_i}{h}_{ij}\to h_{ii}=\frac{{{\displaystyle \sum }}_{j\in N_i}{h}_{ij}}{d_i}$$

(18)

Substituting Equation (18) into Equation (17) and taking the derivative with respect to $h_{ij}$ yield

$$\frac{\partial {\parallel I-H\odot L\parallel }_F^2}{\partial h_{ij}}=-4+4h_{ij}+2h_{ii}{d}_i+h_{jj}{d}_j$$

(19)

The objective function is convex. With Equation (19) equal to zero,

$$h_{ij}=1-\frac{h_{ii}{d}_i}{2}-\frac{h_{jj}{d}_j}{2}$$

(20)

Summing both sides over $j\in N_i$ yields

$$h_{ii}\left(\frac{1}{2}{d}_i^2+d_i\right)+\frac{1}{2}{\displaystyle \sum }_{j\in N_i}{h}_{jj}{d}_j=d_i$$

(21)

Consequently, establishing $N$ equations similar to Equation (21) for each node and arranging them in matrix form yield

$$\frac{1}{2}\left(D+A+2I\right)D{H}_{diag}=D_{diag}$$

(22)

This formulation simplifies the solution for $H_{diag}$ to solve a system of linear equations.

As the matrix $D+A+2I$ strictly dominates the row diagonal, according to Lemma 5, $det\left(D+A+2I\right)\ne 0$,

$$rank\left(B\right)=rank\left(B,D_{diag}\right)=N$$

(23)

where $B=D+A+2I$, $rank\left(·\right)$ is the rank of a matrix, and $rank\left(B,D_{diag}\right)$ is the rank of the augmented matrix for Equation (22).

By solving Equation (22), the unique solution for the main diagonal elements $h_{ii},i\in V$ of $H$ is obtained, whereas the remaining elements $h_{ij}$ can be derived through Equation (20). The communication weight matrix $H$ is uniquely determined and denoted as $H_{sym}$. Thus, Scheme 2 is established. Owing to the identical weights of symmetric links in this scheme, it is referred to as the Minimization of the Frobenius Norm Symmetric Weights. □

Scheme 3.

The Minimization of the Frobenius Norm Random Walk Weights: In the consensus algorithm with matrix communication weights in the form $X\left(k+1\right)=\left(I-H\odot L\right)X\left(k\right)$, when $H=H^T$ is not constrained, the optimal solution for $H$ to minimize the Frobenius norm of matrix $P$ is denoted as $H_{rw}$.

$$H_{rw}=\left\{\frac{1}{1+d_i}\right.; for all elements in the ith row, i\in V$$

(24)

Proof. 

Using the Frobenius norm of matrix $P$ as the optimization objective, the optimization model is established as follows:

$$\begin{gathered} \underset{H}{min}{\parallel I-H\odot L\parallel }_F^2 \\ s.t P1=1 \end{gathered}$$

(25)

In this case, $H$ is not necessarily symmetric, so ${\parallel I-H\odot L\parallel }_F^2\ge {\displaystyle \sum }_{r=1}^N{\lambda }_{P,r}^2$. However, this study still opts to minimize the Frobenius norm of the iteration matrix $P$ to minimize ${\displaystyle \sum }_{r=1}^N{\lambda }_{P,r}^2$, despite a possible gap between these two objectives. This approach is similar to those in existing research aiming to enhance convergence speed by minimizing the spectral radius of the matrix $P-\frac{{11}^{\mathit{T}}}{N}$ to bring all eigenvalues closer to 0.

Similarly, simplifying the objective function and differentiating with respect to $h_{ij}$ yields

$$\frac{\partial {\parallel I-H\odot L\parallel }_F^2}{\partial h_{ij}}=-2+2h_{ij}+2{\displaystyle \sum }_{j\in N_i}{h}_{ij}$$

(26)

Clearly, the objective function is convex. With Equation (26) equal to zero,

$$h_{ij}=1-{\displaystyle \sum }_{j\in N_i}{h}_{ij}$$

(27)

Substituting the constraint $P1=1$ and summing over $j\in N_i$ on both sides yields

$$h_{ii}{d}_i=d_i-h_{ii}{d}_i^2\to h_{ii}=\frac{1}{1+d_i}$$

(28)

Substituting Equation (28) into Equation (27) yields

$$h_{ij}=\frac{1}{1+d_i}$$

(29)

Thus, Scheme 3 is established. The obtained result is similar in form to the Laplacian Random Walk Normalized matrix, where each row of $H$ has the same element $\frac{1}{1+d_i}$. It is termed as the Minimization of the Frobenius Norm Random Walk Weights. □

Among the three schemes, $P$ is not constrained to be a non-negative matrix. In Scheme 3, only one eigenvalue of $P$ is constrained to be 1, corresponding to eigenvector $1$. Furthermore, the optimal value ${ϵ}_c^{*}$ in Scheme 1 is a special case where all elements in $H$ are equal. This case implies that $\underset{H}{min}{\parallel I-H\odot L\parallel }_F^2$ always has a solution, ensuring the convergence of the consensus algorithm to the weighted average of all node initial states. In the proposed schemes and the comparison schemes in this study, when the iteration matrix $P$ is symmetric, the state values of all nodes converge to $\frac{1}{N}{\displaystyle \sum }_{i=1}^N{x}_i\left(0\right)$. When the iteration matrix $P$ is asymmetric, the state values of all nodes converge to ${\displaystyle \sum }_{i=1}^N\frac{d_i+1}{{{\displaystyle \sum }}_{j=1}^N\left(d_j+1\right)}{x}_i\left(0\right)$. The iteration matrix $P$ in the other schemes is symmetric except for Scheme 3.

In Scheme 3, the weighted Laplacian matrix $H_{rw}\odot L$ is equivalent to ${(I+D)}^{-1}L$. Although this correspondence with findings from prior research may be unexpected, previous studies have primarily examined its convergence and convergence conditions from a different perspective, not analyzing convergence speed [33,37,65]. In contrast, this study aims to enhance the convergence speed by minimizing the Frobenius norm of the iteration matrix $P$.

Inspired by the form of ${(I+D)}^{-1}L$, this study investigates the enhancement of convergence speed on the basis of local information from the perspectives of non-negative matrices.

Scheme 4.

The Minimization of the Frobenius Norm Random Walk Weights: In the consensus algorithm with matrix communication weights in the form $X\left(k+1\right)=\left(I-{(\alpha +D)}^{-1}L\right)X\left(k\right)$, where $\alpha \in R^{N\times N}$ is a diagonal matrix, the optimal solution for minimizing the Frobenius norm of the matrix  $P$ is ${\alpha }^{*}=I$.

Proof. 

Using the Frobenius norm of matrix $P$ as the optimization objective, the optimization model is developed with the diagonal matrix $\alpha \in R^{N\times N}$ as the optimization variable.

$$\begin{gathered} \underset{\alpha }{min}{\parallel I-{(\alpha +D)}^{-1}L\parallel }_F^2 \\ s.t P1=1;P\ge 0 \end{gathered}$$

(30)

where $P\ge 0$, with $P$ being a non-negative matrix, i.e., ${\alpha }_i>-d_i$.

The objective function is convex. Simplifying and taking the derivative with respect to ${\alpha }_i$ yield

$$\frac{\partial {\parallel I-\alpha L\parallel }_F^2}{\partial {\alpha }_i}=2d_i{({\alpha }_i+d_i)}^{-2}-2d_i^2{({\alpha }_i+d_i)}^{-3}-2d_i{({\alpha }_i+d_i)}^{-3}$$

(31)

With Equation (31) equal to zero, ${\alpha }_i=1,\forall i\to {\alpha }^{*}=I$. Thus, Scheme 4 is formed. □

In Scheme 4, the convergence of the iteration matrix $P=I-{(I+D)}^{-1}L$ can be directly inferred from Lemma 6. This conclusion aligns with the one obtained in Scheme 3. Moreover, owing to the asymmetry of $P$, a gap may exist between the Frobenius norm of $P$ and ${\displaystyle \sum }_{r=1}^N{\lambda }_{P,r}^2$.

The proposed approaches leverage the Frobenius norms of matrices to reformulate the problem of accelerating the convergence rates of consensus algorithms into a convex problem. Through straightforward computations, analytical solutions can be obtained, eliminating the necessity of solving semidefinite programming problems and computing matrix eigenvalues.

4. Performance Evaluation

In this section, the proposed approaches are compared with existing schemes in terms of convergence speed under different scenarios. The limitations of existing metrics for evaluating the convergence speeds of distributed consensus algorithms are reviewed.

4.1. Global Information Acceleration Scheme Comparison

Different acceleration scheme designs require varying levels of network topology information. To compare the convergence speeds of each scheme comprehensively, this section assumes that nodes have access to the entire network communication topology information, corresponding to static or pre-planned network topologies.

Four simple connected graphs are depicted in Figure 1. Subsequently, a communication topology network consisting of 30 nodes was randomly generated to assess the performance of the proposed approach using relatively complex networks (see Figure 2). The points represent individual nodes and lines represent bidirectional communication between two nodes. The initial state of the nodes was set as $X\left(0\right)={(1,2,\dots ,N)}^T$.

4.1.1. Noise-Free Transmission Scenario

Eleven acceleration schemes for consensus algorithms were compared without introducing transmission noise. The simulation results depicted in Figure 3 and Figure 4 correspond to Figure 1 and Figure 2, respectively. The vertical axis represents the standard deviation $\zeta \left(k\right)$ of $X\left(k\right)$ obtained in the $k$th iteration, and the horizontal axis represents the iteration count $k$.

Except for Figure 1a, the proposed Scheme 3 achieves the fastest convergence speed compared with the other 10 schemes. It still demonstrates an excellent convergence speed in Figure 1a.

4.1.2. Scenario with Transmission Noise

In practice, communication between any two nodes always involves noise, so errors in the information accepted by the receiving node are inevitable. Research on noise suppression issues goes beyond the scope of this study and requires discussion of the characteristics of noise for different state variables in specific scenarios. The state values of each node involved in the consensus algorithm should ideally undergo corresponding filtering schemes to improve their accuracy. Depending solely on consensus algorithms cannot selectively suppress noise.

In this section, simple Gaussian white noise with a mean of $0$ and a standard deviation ${\sigma }_{noise}$ of $1\times {10}^{-10}$ is introduced into the bidirectional communication between nodes to achieve high-precision synchronization with various consensus algorithm acceleration schemes in a noisy environment. Simulations were conducted for the scenarios depicted in Figure 1 and Figure 2, and the results are presented in Figure 5 and Figure 6, respectively.

4.1.3. Comparison of Synchronization Accuracy

Figure 3 was compared with Figure 5, and Figure 4 was compared with Figure 6. The findings show that noise primarily affects the final synchronization accuracy of the consensus algorithm, and the impact on the convergence speed of the algorithm within reachable precision levels is minimal.

Table 3. Synchronization accuracy of each scheme.

Schemes	Figure 1a	Figure 1b	Figure 1c	Figure 1d	Figure 2
Scheme 1	$3.85\times {10}^{-11}$	$3.23\times {10}^{-11}$	$3.27\times {10}^{-11}$	$2.82\times {10}^{-11}$	$1.71\times {10}^{-11}$
Maximum-degree weight	$3.19\times {10}^{-11}$	$2.68\times {10}^{-11}$	$2.83\times {10}^{-11}$	$2.48\times {10}^{-11}$	$1.35\times {10}^{-11}$
Best constant edge weight	$4.94\times {10}^{-11}$	$4.50\times {10}^{-11}$	$3.17\times {10}^{-11}$	$3.09\times {10}^{-11}$	$2.66\times {10}^{-11}$
Scheme 2	$3.91\times {10}^{-11}$	$3.28\times {10}^{-11}$	$3.24\times {10}^{-11}$	$2.83\times {10}^{-11}$	$2.20\times {10}^{-11}$
Fastest mixing chain	$1.59\times {10}^{-10}$	$3.35\times {10}^{-10}$	$4.69\times {10}^{-10}$	$3.37\times {10}^{-11}$	$2.41\times {10}^{-10}$
Optimal symmetric edge weights	$5.29\times {10}^{-11}$	$3.97\times {10}^{-11}$	$8.87\times {10}^{-10}$	$4.10\times {10}^{-10}$	$7.26\times {10}^{-10}$
Least-mean-square weights	$4.36\times {10}^{-11}$	$3.63\times {10}^{-11}$	$3.25\times {10}^{-11}$	$3.04\times {10}^{-11}$	$2.44\times {10}^{-11}$
Effective resistance minimization weights	$5.32\times {10}^{-11}$	$5.50\times {10}^{-11}$	$3.92\times {10}^{-11}$	$4.29\times {10}^{-11}$	$2.91\times {10}^{-11}$
Metropolis–Hastings weights	$3.22\times {10}^{-11}$	$2.73\times {10}^{-11}$	$2.79\times {10}^{-11}$	$2.79\times {10}^{-11}$	$1.79\times {10}^{-11}$
Lazy Metropolis weights	$1.95\times {10}^{-11}$	$1.69\times {10}^{-11}$	$1.79\times {10}^{-11}$	$1.60\times {10}^{-11}$	$1.05\times {10}^{-11}$
Scheme 3	$3.90\times {10}^{-11}$	$3.42\times {10}^{-11}$	$3.33\times {10}^{-11}$	$2.94\times {10}^{-11}$	$2.11\times {10}^{-11}$

presents the synchronization accuracy maintained by each scheme when reaching a steady state in the presence of noise.

The final synchronization accuracy of “Fastest mixing chain” and “Optimal symmetric edge weights” are lower compared with that of other acceleration schemes. Combining this observation with Figure 3 and Figure 4, this discrepancy is not due to the introduction of minimal communication noise but rather the inherent properties of different communication topology structures and acceleration schemes. These properties limit the applicability of these two schemes in scenarios requiring high synchronization accuracy. Among these 11 schemes, the synchronization accuracy of the proposed schemes in this study is within the normal range. Additionally, it is worth noting that Lazy Metropolis weights achieves the highest synchronization accuracy among the 11 schemes.

4.1.4. Comparison of Convergence Speed

For an intuitive and comprehensive demonstration of the convergence speed of each scheme, the convergence time $k$ is quantitatively evaluated using $k=T_{P\left(k\right)}\left(\zeta \left(k\right),\sigma \right)$. Given the differences in final synchronization accuracy among schemes in the presence of noise, $\sigma =1\times {10}^{-9}$ is set as the synchronization accuracy achievable by all schemes.

Table 4. Convergence time $k$ of each scheme.

Schemes	Figure 1a	Figure 1b	Figure 1c	Figure 1d	Figure 2
Scheme 1	86	58	26	19	218
Maximum-degree weight	98	74	31	24	306
Best constant edge weight	62	42	26	17	177
Scheme 2	96	56	26	17	90
Fastest mixing chain	62	47	26	17	65
Optimal symmetric edge weights	62	40	26	17	60
Least-mean-square weights	75	48	26	17	70
Effective resistance minimization weights	63	66	52	29	67
Metropolis–Hastings weights	98	74	31	19	153
Lazy Metropolis weights	206	158	74	50	315
Scheme 3	68	38	25	17	51

presents the convergence time $k$ obtained by each scheme.

Table 3 and Table 4 reveal that among the three constant weight schemes, Scheme 1 converges faster than the maximum-degree weight in the aforementioned five scenarios. Scheme 1 converges slower than the best constant edge weight but has a simpler solution approach and requires less information. Among the 11 schemes, except for Figure 1a, Scheme 3 achieves the fastest convergence speed while ensuring synchronization accuracy. Additionally, optimal symmetric edge weights also demonstrate a noticeable improvement in convergence speed, with the fastest convergence observed in the scenario depicted in Figure 1a, being second only to Scheme 3 in other scenarios.

4.1.5. Evaluation Metric Analysis

As the convergence of the consensus algorithm is related to the sum of squares of all eigenvalues of the iteration matrix $P$,

Table 5. The Frobenius norm of $P$ for each scheme.

Schemes	Figure 1a	Figure 1b	Figure 1c	Figure 1d	Figure 2
Scheme 1	1.32	1.24	1.25	1.11	2.41
Maximum-degree weight	1.33	1.27	1.27	1.15	2.78
Best constant edge weight	1.41	1.33	1.25	1.12	2.96
Scheme 2	1.31	1.22	1.25	1.10	1.74
Fastest mixing chain	1.41	1.24	1.25	1.12	2.75
Optimal symmetric edge weights	1.41	1.29	1.25	1.12	3.09
Least-mean-square weights	1.34	1.24	1.25	1.10	1.80
Effective resistance minimization weights	1.42	1.44	1.29	1.27	2.01
Metropolis–Hastings weights	1.33	1.25	1.27	1.10	2.05
Lazy Metropolis weights	1.56	1.52	1.63	1.50	3.49
Scheme 3	1.29	1.19	1.22	1.10	1.69

displays the Frobenius norm of $P$ for different schemes. Additionally, the algebraic connectivity ${\lambda }_2$ or eigenratio $\left|\frac{{\lambda }_2}{{\lambda }_n}\right|$ of the Laplacian matrix $L$ or weighted Laplacian matrix $\tilde{L}$ is used as a metric to measure the convergence performances of consensus algorithms.

Table 6. Algebraic connectivity of the weighted Laplacian matrix $\tilde{L}$ for each scheme.

Schemes	Figure 1a	Figure 1b	Figure 1c	Figure 1d	Figure 2
Scheme 1	0.2197	0.3077	0.5714	0.7059	0.0842
Maximum-degree weight	0.1953	0.2500	0.5000	0.6000	0.0605
Best constant edge weight	0.2929	0.4000	0.5714	0.7500	0.1136
Scheme 2	0.1992	0.3170	0.5714	0.7368	0.1874
Fastest mixing chain	0.2929	0.3636	0.5714	0.7500	0.2905
Optimal symmetric edge weights	0.2929	0.4226	0.5714	0.7500	0.3222
Least-mean-square weights	0.2501	0.3648	0.5676	0.7373	0.2299
Effective resistance minimization weights	0.2893	0.3632	0.6667	1.0000	0.2536
Metropolis–Hastings weights	0.1953	0.2500	0.5000	0.7000	0.1157
Lazy Metropolis weights	0.0976	0.1250	0.2500	0.3500	0.0578
Scheme 3	0.2713	0.4357	0.6667	0.7500	0.2955

,

Table 7. The eigenratio of the weighted Laplacian matrix $\tilde{L}$ for each scheme.

Schemes	Figure 1a	Figure 1b	Figure 1c	Figure 1d	Figure 2
Scheme 1	0.1716	0.2500	0.4000	0.6000	0.0602
Maximum-degree weight	0.1716	0.2500	0.4000	0.6000	0.0602
Best constant edge weight	0.1716	0.2500	0.4000	0.6000	0.0602
Scheme 2	0.1620	0.2679	0.4000	0.5833	0.1443
Fastest mixing chain	0.1716	0.2667	0.4000	0.6000	0.1700
Optimal symmetric edge weights	0.1716	0.2679	0.4000	0.6000	0.1920
Least-mean-square weights	0.1713	0.2679	0.4000	0.5840	0.1607
Effective resistance minimization weights	0.1691	0.2093	0.4000	0.5714	0.1480
Metropolis–Hastings weights	0.1716	0.2500	0.4000	0.5833	0.1082
Lazy Metropolis weights	0.1716	0.2500	0.4000	0.5833	0.1082
Scheme 3	0.2208	0.3796	0.4706	0.6000	0.2365

and

Table 8. Algebraic connectivity and eigenratio of the Laplacian matrix $L$.

	Figure 1a	Figure 1b	Figure 1c	Figure 1d	Figure 2
Algebraic connectivity of Laplacian matrix $L$	0.5858	1.0000	2.0000	3.0000	1.3310
Eigenratio of Laplacian matrix $L$	0.1716	0.2500	0.4000	0.6000	0.0602

present the various metrics corresponding to the schemes in Figure 1 and Figure 2.

The Frobenius norm corresponding to Scheme 3 is always the smallest, as supported by the theoretical analysis conducted in Section 3.2.

Table 8 indicates that the algebraic connectivity and eigenratio of $L$ cannot uniformly measure the convergence speed of different acceleration schemes for consensus algorithms once the communication topology network structure is determined. Moreover, the Frobenius norm of the iteration matrix $P$, algebraic connectivity, and eigenratio of the weighted Laplacian matrix $\tilde{L}$ cannot accurately measure the convergence speeds of acceleration schemes for consensus algorithms, as shown in Table 4, Table 5, Table 6 and Table 7.

After selecting an acceleration scheme for consensus algorithms, the algebraic connectivity and eigenratio of the weighted Laplacian matrix $\tilde{L}$ can be employed to measure the differences in the convergence speeds of consensus algorithms in different scenarios. However, applying these metrics to Scheme 2 and least-mean-square weights still has limitations because the decrements in the algebraic connectivity and eigenratio of the weighted Laplacian matrix $\tilde{L}$ do not fully align with the increase in convergence time $k$.

At present, no theoretical metric can comprehensively and accurately describe the convergence speeds of consensus algorithms in different acceleration schemes and scenarios. The convergence time $k=T_{P\left(k\right)}\left(\zeta \left(k\right),\sigma \right)$ required to achieve a certain steady-state accuracy with consensus algorithms can only be used for demonstration. This metric cannot directly guide the research on acceleration schemes for consensus algorithms.

4.2. Local Information Acceleration Scheme Comparison

Given the practical communication capabilities and dynamic nature of UAVs in a UANET, nodes typically cannot obtain real-time information about the communication topology of the entire network. Therefore, among the 11 schemes, only maximum-degree weights, Metropolis–Hastings weights, lazy Metropolis weights, and Scheme 3 can be applied to accelerate the convergence of consensus algorithms in real-time dynamic scenarios. The maximum-degree weight scheme can be directly set on the basis of the maximum number of communication nodes for a node or the total number of nodes in the entire network. It can preset communication weights without relying on information about the communication topology structure. The remaining three schemes only require nodes to adjust communication weights on the basis of local information to enhance convergence speed.

Table 4 reveals that in static or pre-planned scenarios, the convergence speed of Scheme 3 is always superior to those of the other three schemes when using local information. In dynamic scenarios, the convergence is equivalent to the sequential multiplication of the iteration matrix $P$ corresponding to a series of time-varying static scenarios over time.

Table 9. Simulation settings.

Parameters	Value or Description
Number of UAVs	30
Spatial Area	160 × 130 × 2 ${\mathrm{km}}^3$
Mission Details	Every three UAVs took off from a different airport and sequentially passed through four target places. After hovering for five circles above each target place, the UAVs returned to their starting point.
Communication Range	10 km
Maximum Flying Speed	340 m/s
Maximum Nodes for Communication	29
Synchronization Signal Transmission Period	1s
Simulation Software	STK and MATLAB

lists the main simulation settings, with further detailed simulation settings being consistent with our previous work, as shown in [16].

Figure 7 and Figure 8 depict the trajectories of 30 UAVs and the simulation results, respectively.

Figure 8 shows that Scheme 3 reduces the convergence time by approximately 16%, 31%, and 29% compared with the Metropolis–Hastings weights, lazy Metropolis weights, and maximum-degree weights, respectively. The final synchronization accuracies of all four schemes are within the order of ${10}^{-10}$.

Furthermore, changing the maximum communication range of each node means changes in the communication topology network. Keeping other settings unchanged, the simulation results when the communication range of each node is changed to 15 km are shown in Figure 9.

Figure 9 illustrates that Scheme 3 considerably improves the convergence speed. It reduces the convergence time by approximately 38%, 44%, and 53% compared with Metropolis–Hastings weights, lazy Metropolis weights, and maximum-degree weights. The final synchronization accuracy of all four schemes is within the order of ${10}^{-10}$.

5. Conclusions

This study investigates how to accelerate the convergence of distributed consensus algorithms in UANETs. A new upper bound for the constant weight of communication links is first introduced. Convex models are then established to approximate the original problem. Then, three acceleration schemes are proposed by minimizing the Frobenius norm of the iteration matrix $P$ to optimize the weights of communication topology links. For validation, these schemes are compared with existing schemes. The simulation results indicate that in scenarios where the topology structure is known, Scheme 1 requires moderate topology structure information and exhibits moderate convergence speed among the three constant weight schemes. In contrast, Scheme 2 features a simple solution and faster convergence speed. Scheme 3 requires the least topology structure information among the 11 schemes compared in scenarios where the topology structure is time-varying and unknown. Applicable to dynamic scenarios as well, Scheme 3 achieves the fastest convergence speed while ensuring synchronization accuracy.

This paper also discusses the existing metrics for assessing the convergence speeds of consensus algorithms. The algebraic connectivity and eigenratio of the weighted Laplacian matrix $\tilde{L}$ can be used to measure the differences in the convergence speeds of consensus algorithms in different scenarios in most cases. Nonetheless, a uniform measurement of the convergence speeds of various acceleration schemes remains an interesting and unresolved problem.

6. Future Directions

Enhancing the convergence speeds of consensus algorithms in real-time dynamic scenarios warrants further investigation. The following four aspects are suggested for future research directions.

(1)

A theoretical index that correctly and uniformly measures the convergence speeds of various consensus algorithms must be created to provide guidance for subsequent research.

(2)

Whether convergence speed is simultaneously related to the network topology structure, the distribution of current state values of nodes, and the current number of iterations can be explored from a projection perspective.

(3)

The intrinsic connection between network communication topology structure and acceleration schemes for consensus algorithms should be looked into in future work. This attempt can lead to the classification of the topology structure and proposal of optimal acceleration schemes for specific structures.

(4)

The differences between the sum of squares of eigenvalues of asymmetric matrices and the squares of the Frobenius norm should be determined as a basis for identifying faster acceleration schemes for distributed consensus algorithms.

(5)

Combined with edge-based federated learning, investigating how to train specialized models for each node in dynamic scenarios to achieve time synchronization would be an intriguing research direction [66,67].