Consensus seeking in diffusive multidimensional networks with a repeated interaction pattern and time-delays

In recent years, the demand for understanding complex and large-scale structures such as social, traffic, material, or brain networks has raised increasing attention on modeling, analyzing, and synthesizing multilayer networks [1]. A multilayer network consists of multiple agents (or subsystems) interacting via $d$-dimensional single-layer networks $(d\geq 2)$. The dynamic of each agent is captured by a set of state variables corresponding to $d$ layers, and the agent-to-agent influences govern the entire network’s behavior. If the network contains only intra-layer interactions (interactions of state variables belonging to the same layer) and inter-layer interactions (couplings of state variables from different layers of the same agent), we refer to the network as a multiplex. A multilayer network contains cross-layer interactions - the couplings of the state variables lying in different layers and from distinct agents [2]. In a diffusive network, the variation of each agent’s state variables depends on the differences between the agent’s state and its neighboring agents. Figure 1 illustrates a two-layer network of four agents.

Consensus algorithms, because of their simplicity and generality, were extensively used to study the dynamics of single-layer networks (monoplexes) [3]. For multilayer networks, [4, 5, 6] considered the consensus and synchronization on multiplexes. The authors in [7, 8, 9] proposed matrix-weighted consensus in which the state variable in a layer of an agent is updated based on a weighted sum of relative states from every layer, taken for all neighboring agents. Time delay is a source of uncertainty that usually affects the performance of communication networks [10]. As a result, many studies have focused on delayed single-layer consensus networks [3, 11]. On the other hand, there is not much research on delays in multilayer consensus systems. The consensus and synchronization on multiplexes with time delays were studied in [12].

This paper attempts to derive consensus conditions for the matrix-weighted consensus model [8] with heterogeneous time delays. We assume that all agent-to-agent interactions have the same graph pattern and time delays may exist in both intra- and cross-layer interactions. We first prove that if there is no intra-layer time delay and the maximum magnitude of all eigenvalues of the adjacency matrix corresponding to cross-layer interactions is smaller than unity, the network globally asymptotically achieves a consensus. The result reveals that significant delays from weak cross-layer interactions cannot destabilize a multilayer consensus network. Second, the network is considered with both intra- and cross-layer time delays. The stability of an equation involving two different constant time delays was considered in [13], where the authors derived a necessary and sufficient condition based on the root of a transcendental equation. In this paper, due to the interweaving of the (complex) eigenvalues of the overall network and the local interaction graph, the method developed in [13] cannot be applied. Instead, we examine the network when two-time delays are equal and determine a corresponding delay margin. Then, we show that when intra- and cross-layer time delays do not exceed a calibrated delay margin, the network asymptotically achieves a consensus. The delay margin is tight in the sense that once one of the time delays exceeds the margin and the other is equal to the delay margin, the consensus network becomes unstable. Finally, the delayed two-layer network is considered and several detailed consensus conditions are determined by the graph’s parameters.

The remainder of this paper is organized as follows. Section II contains theoretical background and problem formulation. Section III provides consensus conditions and the corresponding analysis for networks with a general interaction pattern matrix. Two-layer networks are considered and simulated in Section IV. Lastly, Section V concludes the paper.

Consider a diffusive multi-layer network of $n\geq 2$ agents. Each agent $i\in\{1,\ldots,n\}$ is a $d$-dimensional subsystem with the state vector $\mathbf{x}_{i}=[x_{i1},\ldots,x_{id}]^{\top}\in\mathbb{R}^{d}$ $(d\geq 2)$. The interaction between two agents $i$ and $j$ in the network is modeled by a matrix weight $\mathbf{A}_{ij}=\mathbf{A}_{ji}\in\mathbb{R}^{d\times d}$,

where $i,j\in\{1,\ldots,n\}$, $i\neq j$, and $p,q\in\{1,\ldots,d\}$. Thus, each element $a_{ij}^{pq}=a_{ji}^{pq}\in\mathbb{R}$ captures the influence from the $p$-th layer to the $q$-th layer in the network. The elements $a_{ij}^{pp},~{}p=1,\ldots,d$ represent intra-layer interactions (same layer), while the $a_{ij}^{pq},~{}p\neq q$ stand for cross-layer interactions between two agents $i,~{}j$. We use an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ to describe the interaction topology between agents in the network, i.e., each agent $i$ is represented by a vertex $i$ in the vertex set $\mathcal{V}=\{1,\ldots,n\}$, each edge $(i,j)\in\mathcal{E}$ exists if the matrix weight $\mathbf{A}_{ij}\neq\mathbf{0}_{d\times d}$. Assume that the graph $\mathcal{G}$ does not exist any self-loop, i.e., edges connecting a vertex with itself. The undirectedness assumption implies that if $(i,j)\in\mathcal{E}$, then $(j,i)\in\mathcal{E}$. Let $\mathcal{N}_{i}=\{j\in\mathcal{V}|~{}(i,j)\in\mathcal{E}\}$ denote the neighbor set of vertex $i$. A path $i_{1}i_{2}\ldots i_{k}$ in $\mathcal{G}$ is a sequence of vertices $i_{k}\in\mathcal{V},~{}k=1,\ldots,k,$ joining the starting vertex $i_{1}$ to the end vertices $i_{k}$ by $k-1$ edges $(i_{r},i_{r+1})\in\mathcal{E}$, $r=1,\ldots,k-1$. The graph $\mathcal{G}$ is connected if and only if for any pairs of vertices in $\mathcal{V}$, there exists a path connecting them.

Define the network adjacency matrix $\mathbf{W}=[w_{ij}]\in\mathbb{R}^{n\times n}$ of $\mathcal{G}$ with elements $w_{ij}=1$ if $(i,j)\in\mathcal{E}$ and $w_{ij}=0$, otherwise. Then, the Laplacian matrix of $\mathcal{G}$ is defined as $\mathbf{L}=[l_{ij}]=\text{diag}(\mathbf{W}\mathbf{1}_{n})-\mathbf{W}\in\mathbb{R}^{n\times n}$.

We consider the consensus algorithm in the multi-layer network, where each agent updates its state variables according to a weighted sum of both intra-layer and cross-layer state differences. Moreover, it is supposed that the intra-layer interactions and cross-layer interactions are having constant time delays $\tau_{1},\tau_{2}\geq 0$. As a result, the equation governing the network dynamics under the matrix-weighted consensus algorithm is given as

for all $i\in\mathcal{V}$ and $q\in\{1,\ldots,d\}$.

We assume that the multilayer network is representable by a pair of graphs: the network graph $\mathcal{G}$ (the global graph), and an agent-to-agent interaction graph $\mathcal{H}$ (the local graph). The graph $\mathcal{H}$ encodes the mutual influences between the relative states of any two adjacent agents in the network. The multi-layer networks with a repeated pattern are described in the following assumptions.

Note that $\mathbf{A}_{ij}=\mathbf{A}_{ji}=\mathbf{A}=[a_{ij}]_{d\times d}$ and $\mathcal{H}$ is a graph of $d$ vertices with self-loops. Moreover, as all agent-to-agent interactions are represented by the same graph $\mathcal{H}$, we will refer to $\mathcal{H}$ as the interaction pattern of the network. Figure 1 illustrates a four-agent network consisting of the network graph $\mathcal{G}$ and the interaction pattern $\mathcal{H}$.

The problem studied in this paper can be stated as follows.

In this section, we consider the delayed consensus system (II) and derive some consensus conditions. We next examine specific networks of two- or three-layers and provide stability conditions related to the cross-layer interactions.

This section provides specific consensus conditions for two-layer matrix-weighted networks with time delays and illustrates the theoretical results by simulations.

The corresponding matrix capturing the agent-to-agent interaction pattern is $\mathbf{A}=\begin{bmatrix}1&a^{12}\\ a^{21}&1\end{bmatrix}$. Thus $\mathbf{A}_{\rm cross}=-\mathbf{I}_{2}+\mathbf{A}=\begin{bmatrix}0&a^{12}\\ a^{21}&0\end{bmatrix}$ has a pair of pure imaginary (real) eigenvalues when $a^{12}a^{21}<0$ (resp., $a^{12}a^{21}>0$). We can state the following result, which can be considered as a corollary of Lemma 4, Theorem 3 and Theorem 4.

Next, consider a network of 4 agents having the interaction graph as depicted in Fig. 1. We have

Simulations for intra-layer delay-free two-layer networks ($\tau_{1}=0$):

We conduct several simulations of the consensus algorithm with $\mathbf{A}_{1}=\begin{bmatrix}1&1\\ 0.5&1\end{bmatrix}$. In this case, $\mathbf{A}$ has eigenvalues satisfying $|\mu_{k}|<1,~{}k=1,2$. The simulation results for $\tau_{2}=2,5,10$ are shown in Fig. 4. Clearly, $x_{i}^{k},~{}k=1,2,$ consent on two values (consensus states) in all three cases.

Next, consider $\mathbf{A}_{2}=\begin{bmatrix}1&2\\ 0.5&1\end{bmatrix}$. Correspondingly, $|\mu_{k}|=1$. Simulation results in Fig. 5 show that cross-layer time delays do not destabilize the system but perturb the system from the consensus set.

Finally, we consider $\mathbf{A}_{2}=\begin{bmatrix}1&2\\ 1&1\end{bmatrix}$. In this case, $|\mu_{k}|>1$. Simulation results in Fig. 6 show that cross-layer time delays destabilize the consensus system.

Simulations for two time-delay networks ($\tau_{1},\tau_{2}\geq 0$):

We simulate the two-layer network under the presence of two time-delays $\tau_{1},\tau_{2}\geq 0$ with matrix $\mathbf{A}_{1}$. Corresponding to Corollary 1, we have $\tau_{\max}=0.23$. The simulation result in Fig. 7 shows that the states $x_{i}^{k},~{}k=1,2,$ approach to two sinusoidal trajectories for $\tau_{1}=\tau_{2}=\tau_{\max}$ (i.e., the system has a pair of purely imaginary eigenvalues), two common constant values (or the consensus state) for $0<\tau_{1}\leq\tau_{2}\leq\tau_{\max}$, and is unstable in case $\tau_{1}=\tau_{\max}<\tau_{2}$ (Fig. 7(d,e)), or $\tau_{2}=\tau_{\max}<\tau_{1}$ (Fig. 7(f)).

Thus, the simulation results are consistent with the theoretical analysis.

In this paper, we have derived several consensus conditions for a multilayer network with a repeated agent-to-agent interaction pattern and two different time delays in both intra- and cross-layer interactions. Some specific conditions are given for two-layer networks. For further studies, it will be interesting to consider time delays for networks with different inter-agent patterns.