On Consensus Indices of Triplex Multiagent Networks Based on Complete k-Partite Graph

Abstract

In this article, the performance indices on consensus problems for three-layered, multiagent systems are studied from the perspective of algebraic graph theory, where the indices can be used as a measurement of the system performance and refer to the network coherence and algebraic connectivity. Specifically, some operations of two graphs are applied to established the three-layered networks based on k-partite structure, and the mathematical expression of the coherence is derived by the methods of algebraic graph theory. We found that the operations of adding star-shaped copies or fan-graph copies will make the coherence increase by some scalars under the computations of limitation. Then, the indices of the three-layered systems with non-isomorphic topologies but the same number of nodes were compared and simulated; it is found that, when the number of nodes in the counterpart node classes tend to infinity, their difference in coherence are only relevant with the number of peripheral nodes in the sense of limitation.

1. Introduction

The consensus problem is a significant research field of multiagent systems; it requires that all nodes in the networked system have to reach a common physical state value based on some control protocols, the nodes are designed to cooperate effectively under the topologies of system in order such that they can reach the predetermined aims, and the consensus is expected to be distributed.

As a significant interdisciplinary field, consensus-related researches have received much attention from both engineers and scholars in the past two decades, and there are lots of applications or research works in many fields on consensus, for instance, blockchains [1], sensor networks [2], decision making, and so on. Scholars have conducted lots of important, valuable research on consensus from various aspects, such as the order of system (first- or higher-order [3,4,5,6,7,8,9,10,11,12,13,14,15,16]), ways of communication (continuous or intermittent) [6,8], control methods (adaptive control [7], intermittent control [8], optimization control [9]), and so on.

For solving consensus problems, the connecting relations among nodes are characterized by the linking topologies of the network, and the convergence index of the dynamics model, such as consensus speed [3,10] and network coherence [12,13,14,15,16], can be determined by the second largest Laplacian eigenvalue and the Laplacian spectrum. Similarly, the synchronization problem, which is also a sort of coordination problem, can also be studied through the angle of system topologies [16,17,18,19].

There exist numerous significant coordination researches related to the methods of algebraic graph theory [3,11,12,13,14,15,16,17,18,19,20,21,22]. In these enlightening works, ref [3] showed that the algebraic connectivity ${\lambda }_2$, which is the second smallest Laplacian eigenvalue of the network topology, can explain the speed of convergence process of the system. Reference [11] quantified the robustness of the systems with classic network structures by Laplacian eigenvalues.

The robustness of the systems with noise can be described by the network coherence; the concept of network coherence was proposed in [12,13], and it obtains the fact that the robustness-related index can be characterized by nonzero Laplacian eigenvalues. In reference [15], an analytical expression for the leader–follower coherence is determined depending on the number of leaders and network parameters. In [16], based on the tree construction, the authors show that the consensus in symmetric and asymmetric trees becomes better with an increasing number of leader nodes.

In recent years, the multilayer network [23] has become a hot research field because of its wide applications. Since lots of real-world networks have multilayered topologies [17,18,20,21,23,24,25], it is natural and meaningful to extend the application of Laplacian spectrum for consensus theory to multilayered topologies. In view of the aspect of application, the star-shaped structure is a sort of classic computer network graph, is an important topology in sensor networks [2], and is widely studied in many fields including the coordination-related problem [11,19,20,21]. k-partite graph is also a significant structure in heterogeneous networks [26] and in the fields of graph searching [27] and electric networks [28].

Inspired by the above facts, this work considers three-layered, multiagent systems with specific graphs that are constructed by the operations of two graphs. The obtained networks with complete k-partite substructures and star (fan) substructures can be interpreted as adhering communication edges between the counterpart nodes of different layers.

Specifically, the main novelties are listed as follows:

• Complete k-partite graph is introduced to construct the regular network structures. Several novel, three-layered networks with different sorts of topologies but the same number of nodes were composed by the graph operations.
• Theories on algebraic graph theory are applied to calculate the coherence, and novel results with the asymptotic behavior are acquired.
• It is found that, when the partition number and the number of vertices in one partition tend to infinity, the difference of the first-order robustness can be measured by a constant related only to the fixed number of nodes within each partition of the k-partite structure.

The main research aim of the article is to study the robustness of the multiagent network with stochastic disturbances, which can be measured by the network coherence. In Section 2, notations related with graph theory are introduced and the mathematical relationships between the indices and Laplacian eigenvalues are characterized. In Section 3, the graph structures of the three-layered systems are described and the main results on coherence are given. Due to the theories and lemmas on graph spectra, the simulation examples are analyzed and compared in Section 4.

2. Introduction

2.1. Graph Theory and Notations

Let ${\mathfrak{K}}_{n_1,n_2,\dots ,n_k}=(V_1,\dots ,V_k,E)$ denote a complete k-partite graph, where $V_1,\dots ,V_k$ are disjoint vertex sets, $|V_i|=n_i$, and $1\le i\le k$; each vertex in $V_i$ is adjacent to all the vertices in $V\left({\mathfrak{K}}_{n_1,\dots ,n_k}\right)\backslash V_i$. In particular, when $n_i=m$, it is called balanced complete k-partite graph. ${\mathfrak{E}}_p$ is the empty graph with p vertices. Let G be a graph with vertex set $V=\{v_1,v_2,\dots ,v_N\}$, and its edge set is defined as $\mathcal{E}=\left\{\right(i,j\left)\right|i,j=1,2,\dots ,N;i\ne j\}$. $A\left(G\right)={\left[a_{ij}\right]}_N$ denotes the adjacency matrix, where $a_{ij}$ is the edge weight on $(i,j)$. To undirected graphs, $(i,j)$ and $(j,i)$ can be viewed as the same edge in $\mathcal{E}$, i.e., $a_{ij}=a_{ji}$. All the edges in this article are 0-1 weighted—that is, $a_{ij}=\left\{\begin{array}{cc}1,\hfill & (i,j)\in \mathcal{E};\hfill \\ 0,\hfill & (i,j)\notin \mathcal{E}.\hfill \end{array}\right.$. The Laplacian matrix of G is defined as $L\left(G\right)=D\left(G\right)-A\left(G\right)$, where $D\left(G\right)$ is the diagonal degree matrix of G defined by $D\left(G\right)=diag(d_1,d_2,\dots ,d_N)$ with $d_i={\displaystyle \sum _{j\ne i}}{a}_{ij}$. The Laplacian spectrum of G is defined as $S\left(L\left(G\right)\right)=\left(\begin{array}{cccc}{\lambda }_1\left(G\right)& {\lambda }_2\left(G\right)& \dots & {\lambda }_r\left(G\right)\\ l_1& l_2& \dots & l_r\end{array}\right)$, where ${\lambda }_1\left(G\right)<{\lambda }_2\left(G\right)<\dots <{\lambda }_r\left(G\right)$ are the eigenvalues of $L\left(G\right)$ and $l_1,l_2,\dots ,l_r$ are the multiplicities of the eigenvalues [29].

To derive coherence of the three-layered networks, the definitions on graph operations are introduced as follows.

Definition 1

([30,31]). (The corona of two graphs) Let $G_1$ and $G_2$ be two graphs on disjoint sets of u and v vertices, respectively. The corona $G_1\circ G_2$ of $G_1$ and $G_2$ is defined as the graph obtained by taking one copy of $G_1$ and u copies of $G_2$, and then joining the ith vertex of $G_1$ to every vertex in the ith copy of $G_2$.

Definition 2

([32,33]). (The Cartesian product of two graphs) For two graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$, the Cartesian product graph $G=G_1\times G_2$ is the graph with vertex set $V_1\times V_2$; there is an edge from the vertex $(x_1,y_1)$ to the vertex $(x_2,y_2)$ if and only if either $x_1=x_2$ and $y_1,y_2\in E_2$ or $y_1=y_2$ and $x_1,x_2\in E_1$.

2.2. The Relations between Laplacian Spectrum and Consensus Index

Refs [13,14,15,16]. The first-order system with disturbance vector is

$$\begin{array}{c}\hfill \dot{x}\left(t\right)=-L\left(G\right)x\left(t\right)+\xi \left(t\right),\end{array}$$

(1)

where $x\in R^N$, $L\left(G\right)$ is the Laplacian matrix, and $\xi \left(t\right)\in R^N$ is a vector of uncorrelated noise.

Definition 3

([12,13]). The first-order network coherence is defined as the mean steady-state variance of the deviation from the average of all node states:

$$\begin{array}{c}\hfill H_f=\underset{t\to \infty }{lim}\frac{1}{N}\sum _{i=1}^N\mathrm{Var}\left\{x_i\left(t\right)-\frac{1}{N}\sum _{j=1}^N{x}_j\left(t\right)\right\}.\end{array}$$

The first-order coherence $H_f$ can be characterized by the L spectrum and has the following form:

$$\begin{array}{c}\hfill H_f=\frac{1}{2N}\sum _{i=2}^N\frac{1}{{\lambda }_i},\end{array}$$

(2)

The characterization of the consensus index has some similarity with the Kirhoff index [34] or other graphical indices [35] related to graph structures.

3. Main Results

The three-layered network structures in this article can be viewed as a sort of structure formed by connecting the hub nodes of the star copies or fan-shaped substructures. In real-world application, the structure may contain vertices of small degree that only connect to vertices in the same layer. For instance, the leaf vertices in each layer are designed to be disconnected from the other layers (see Figure 1).

The graphs considered in this article have well symmetry (see Figure 1 and Figure 2)—that is, the central layer is symmetric about the two side layers, the balanced complete k-partite graph in each layer is regular, and the leaf nodes in Figure 1 and the lower degree nodes colored orange in Figure 2 are called peripheral nodes in the remaining context. The three-layered network topologies are described in the following subsections.

3.1. The Performance Index for $G\left(\mathfrak{A}\right)$ and $G\left({\mathfrak{A}}^{*}\right)$

In this subsection, a class of triplex structure and a sort of three-layered, star-composed network based on the balanced complete k-partite graph are considered.

As shown in Figure 1, the black nodes are designed to form into the k-partite graph inner one layer, which is a sort of regular graph—that is, $\mathcal{K}(a,m)$, where a denotes the number of parts of the graph and m is the number of nodes inside each part. The copies of star topology consist of the blue nodes. Then, the triplex network structure can be denoted by $G\left(\mathfrak{A}\right):=K(a,m)\times P_3$. Let each vertex in $G\left(\mathfrak{A}\right)$ be the hub nodes adhered to a star-shaped substructure with p leaves; they are designed to disconnect with the other layers. Then, the graph of the acquired network ${\mathfrak{A}}^{*}$ can be defined by $G\left({\mathfrak{A}}^{*}\right):=G\left(\mathfrak{A}\right)\circ {\mathfrak{E}}_p$.

By the Laplacian characteristic polynomial, it can be derived that $SL\left(\mathcal{K}(a,m)\right)=\left(\begin{array}{ccc}0& am& (a-1)m\\ 1& a-1& am-a\end{array}\right)$, and since $SL\left(P_3\right)=\left(\begin{array}{ccc}0& 1& 3\\ 1& 1& 1\end{array}\right)$

We have $SL\left[G\right(\mathfrak{A}\left)\right]$

$$=\left(\begin{array}{ccccccccc}0& am& (a-1)m& 1& 1+am& 1+(a-1)m& 3& 3+am& 3+(a-1)m\\ 1& a-1& am-a& 1& a-1& am-a& 1& a-1& am-a\end{array}\right)$$

Hence, ${\lambda }_2\left(\mathfrak{A}\right)=1$, and the coherence $H^{\left(1\right)}\left(\mathfrak{A}\right)=\frac{1}{6am}(\frac{a-1}{am}+\frac{am-a}{(a-1)m}+1+\frac{a-1}{1+am}+\frac{am-a}{1+(a-1)m}+\frac{1}{3}+\frac{a-1}{3+am}+\frac{am-a}{3+(a-1)m})$.

By the theorem on corona graph [30,31], $SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ can be described as follows:

(1).

0, $p+1\in SL\left(G\left({\mathfrak{A}}^{*}\right)\right)$ with multiplicity one;

(2).

$\frac{am+p+1\pm \sqrt{{(am+p+1)}^2-4am}}{2}\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ repeated $2(a-1)$ times;

(3).

$\frac{(a-1)m+p+1\pm \sqrt{{((a-1)m+p+1)}^2-4(a-1)m}}{2}\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ repeated $2a(m-1)$ times;

(4).

$\frac{2+p\pm \sqrt{{(2+p)}^2-4}}{2}\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ with multiplicity one;

(5).

$\frac{am+p+2\pm \sqrt{{(am+p+2)}^2-4(1+am)}}{2}\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ repeated $2(a-1)$ times;

(6).

$\frac{(a-1)m+p+2\pm \sqrt{{((a-1)m+p+2)}^2-4(1+(a-1)m)}}{2}\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ repeated $2a(m-1)$ times;

(7).

$\frac{(4+p)\pm \sqrt{{(4+p)}^2-12}}{2}\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ with multiplicity one;

(8).

$\frac{am+p+4\pm \sqrt{{(am+p+4)}^2-4(3+am)}}{2}\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ repeated $2(a-1)$ times;

(9).

$\frac{(a-1)m+p+4\pm \sqrt{{((a-1)m+p+4)}^2-4(3+(a-1)m)}}{2}\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ repeated $2a(m-1)$ times;

(10).

$1\in SL\left[G\left({\mathfrak{A}}^{*}\right)\right]$ repeated $3am(p-1)$ times.

Therefore, ${\lambda }_2\left({\mathfrak{A}}^{*}\right)=\frac{2+p-\sqrt{{(2+p)}^2-4}}{2}$, and the coherence of ${\mathfrak{A}}^{*}$ is as follows:

$$\begin{array}{cc}\hfill H\left(1\right)=& \frac{1}{2N_a}\sum _{i=2}^N\frac{1}{{\lambda }_i}\hfill \\ \hfill =& \frac{1}{6am(p+1)}(\frac{1}{p+1}+(a-1)\frac{am+p+1}{am}+a(m-1)\frac{(a-1)m+p+1}{(a-1)m}\hfill \\ \hfill & +(2+p)+(a-1)\frac{am+p+2}{1+am}+a(m-1)\frac{(a-1)m+p+2}{(a-1)m+1}+\frac{4+p}{3}\hfill \\ \hfill & +(a-1)\frac{4+am+p}{3+am}+a(m-1)\frac{4+(a-1)m+p}{3+(a-1)m}+3am(p-1)).\hfill \end{array}$$

Therefore, we have

(i).

when $a,m$ are fixed, $p\to \infty$, one has, $H^{\left(1\right)}\to \frac{a-1}{6a^{2}m}+\frac{m-1}{6m^2(a-1)}+\frac{2}{9am}+\frac{a-1}{6am(1+am)}+\frac{m-1}{6m(1+(a-1\left)m\right)}+\frac{a-1}{6am(3+am)}+\frac{m-1}{6m(3+(a-1\left)m\right)}+\frac{1}{2}$;

(ii).

when $a,p$ are fixed, $m\to \infty$, $H^{\left(1\right)}\to \frac{p}{2(p+1)}<\frac{1}{2}$;

(iii).

when $m,p$ are fixed, $a\to \infty$, $H^{\left(1\right)}\to \frac{1}{2m(1+p)}+\frac{p}{2(p+1)}$.

One can see that when $m\to \infty$, the limitation of $H^{\left(1\right)}$ is only relevant with p—that is, the number of leaves, and when the partition number $a\to \infty$, the limitation of $H^{\left(1\right)}$ is not only relevant with the number of partitions but also the number of nodes inside one partition. In addition, the robustness of case (ii) is $\frac{1}{2m(1+p)}$ better than case (iii) in numerical terms, and case (ii) also has the best robustness of the three cases.

3.2. The Performance Index for $G\left(\overline{\mathfrak{A}}\right)$

Due to the notions that the leaves might have connections with each other in one layer, fan-shaped copies are designed to adhere to the original triplex k-partite graph, and the obtained structure can be denoted by $G\left(\overline{\mathfrak{A}}\right):=G\left(\mathfrak{A}\right)\circ P_n$. In Figure 2, the black nodes are defined the same as those in Section 3.1, and the edges of the fan-shaped substructure are composed by the orange vertices. The orange vertices in each layer are designed to disconnect with the others; thus, a sort of triplex fan-composed structure with complete k-partite graph based on $\overline{\mathfrak{A}}$ is analyzed as follows.

Since $SL\left(P_n\right)=(0,4si{n}^2\left(\frac{q\pi }{2n}\right)),q=1,2,\dots ,n-1$, $SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ can be characterized as follows:

(1).

0 and $(n+1)\in SL\left(\overline{\mathfrak{A}}\right)$ with multiplicity one;

(2).

$\frac{am+n+1\pm \sqrt{{(am+n+1)}^2-4am}}{2}\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ repeated $2(a-1)$ times;

(3).

$\frac{(a-1)m+n+1\pm \sqrt{{((a-1)m+n+1)}^2-4(a-1)m}}{2}\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ repeated $2a(m-1)$ times;

(4).

$\frac{2+n\pm \sqrt{{(2+n)}^2-4}}{2}\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ with multiplicity one;

(5).

$\frac{am+n+2\pm \sqrt{{(am+n+2)}^2-4(1+am)}}{2}\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ repeated $2(a-1)$ times;

(6).

$\frac{(a-1)m+n+2\pm \sqrt{{((a-1)m+n+2)}^2-4(1+(a-1)m)}}{2}\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ repeated $2a(m-1)$ times;

(7).

$\frac{(4+n)\pm \sqrt{{(4+n)}^2-12}}{2}\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ with multiplicity one;

(8).

$\frac{am+n+4\pm \sqrt{{(am+n+4)}^2-4(3+am)}}{2}\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ repeated $2(a-1)$ times;

(9).

$\frac{(a-1)m+n+4\pm \sqrt{{((a-1)m+n+4)}^2-4(3+(a-1)m)}}{2}\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ repeated $2a(m-1)$ times;

(10).

$4si{n}^2\left(\frac{q\pi }{2n}\right)+1\in SL\left[G\left(\overline{\mathfrak{A}}\right)\right]$ repeated $3am$ times, $(q=1,2,\dots ,n-1)$.

Therefore, ${\lambda }_2\left(\overline{\mathfrak{A}}\right)=\frac{2+n-\sqrt{{(2+n)}^2-4}}{2}$, and the first-order coherence of $\overline{\mathfrak{A}}$ can be obtained by

$$\begin{array}{cc}\hfill H^{\left(1\right)}=& \frac{1}{2N_a}\sum _{i=2}^N\frac{1}{{\lambda }_i}=\frac{1}{6am(n+1)}[\frac{1}{(n+1)}+(a-1)\frac{am+n+1}{am}\hfill \\ \hfill & +(am-a)\frac{(a-1)m+n+1}{(a-1)m}+(2+n)+(a-1)\frac{2+am+n}{1+am}\hfill \\ \hfill & +a(m-1)\frac{2+(a-1)m+n}{1+(a-1)m}+\frac{4+n}{3}+(a-1)\frac{4+am+n}{3+am}\hfill \\ \hfill & +a(m-1)\frac{4+(a-1)m+n}{3+(a-1)m}+3am\sum _{q=1}^{n-1}\frac{1}{4si{n}^2\left(\frac{q\pi }{2n}\right)+1}].\hfill \end{array}$$

Hence, one has

(i).

when the values of a and m are fixed, if $n\to \infty$, we have $H^{\left(1\right)}\to \frac{a-1}{6a^{2}m}+\frac{m-1}{6m^2(a-1)}+\frac{1}{6am}+$$\frac{a-1}{6am(1+am)}+\frac{m-1}{6m(1+(a-1\left)m\right)}+\frac{1}{18am}+\frac{a-1}{6am(3+am)}+\frac{m-1}{6m(3+(a-1\left)m\right)}+\frac{\sqrt{5}}{10}>\frac{\sqrt{5}}{10}$;

(ii).

when a and n are fixed, let $m\to \infty$, then, $H^{\left(1\right)}\to \frac{1}{2(1+n)}(1+{\sum }_{q=1}^{n-1}\frac{1}{1+4si{n}^2\left(\frac{q\pi }{2n}\right)})$;

(iii).

when m and n are fixed, let $a\to \infty$, then, $H^{\left(1\right)}\to \frac{1}{2m(1+n)}+\frac{1}{2(1+n)}+$$\frac{1}{2(1+n)}{\sum }_{q=1}^{n-1}\frac{1}{1+4si{n}^2\left(\frac{q\pi }{2n}\right)}$.

Remark 1.

One can see that the convergence speed of cases Figure 1 and Figure 2 are the same if the number of peripheral vertices $p=n$—that is, the increase in the number of edges between the leaf nodes in case Figure 1—does not increase the convergence speed, and the convergence speed is only relevant with the number of peripheral node.

It is interesting to note that the coherence of case (iii) is $\frac{1}{2m(1+n)}$ larger than that of case (ii), where m is the number of vertices inside each partition in case (ii), which means, when n is fixed, the partition number a inside one layer has less influence than the number of vertices m in one partition. In addition, in case (iii) of both Section 3.1 and Section 3.2, when m is large enough, the coherence is approximately the same as the case of (ii), and one can see that in case (ii), the limitation that $H^{\left(1\right)}$ tend to is irrelevant with the value of a, i.e, the number of partitions of the k-partite graph.

We obtain the result that when the number of peripheral nodes p (or n) of ${\mathfrak{A}}^{*}$ (or $\overline{\mathfrak{A}}$) remains fixed, the robustness of case $m\to \infty$ is better than that of the case $a\to \infty$.

Remark 2.

For the networks ${\mathfrak{A}}^{*}$ and $\overline{\mathfrak{A}}$, if the corresponding pairs of the peripheral vertices of different layers are required to have links, the robustness of the obtained triplex systems are better than those that do not have corresponding connections. For example, for ${\mathfrak{A}}^{*}$, when the topology $(K(a,m)\times P_3)\circ {\mathfrak{E}}_p$ is changed to $(K(a,m)\circ {\mathfrak{E}}_p)\times P_3$, one can infer that the coherence behaves better than previous one. Similarly, by the interlace theorem of edge version [29], the first-order robustness of $\overline{\mathfrak{A}}$ is obviously better than that of ${\mathfrak{A}}^{*}$; however, in real systems, for saving the cost, we may give up robustness to some extent, but to what extent exactly is the meaning of accurate calculation.

4. Simulation

The performance indices for the three-layered systems are compared in this section. If the number of peripheral nodes are the same, i.e., $p=n$, then the convergence speed of the k-partite graph-based networks has the relation, $1>{\lambda }_2\left(\mathfrak{A}\right)={\lambda }_2\left({\mathfrak{A}}^{*}\right)={\lambda }_2\left(\overline{\mathfrak{A}}\right)=((p+2)-\sqrt{{(p+2)}^2-4})/2$, where $p\ge 2$. Furthermore, the largest convergence speed has the form ${\lambda }_{2max}\left({\mathfrak{A}}^{*}\right)={\lambda }_{2max}\left(\overline{\mathfrak{A}}\right)=2-\sqrt{3}$. We can acquire that the speed of the three-layered consensus models is not relevant with both the partition number a and the number of nodes inside each partition m.

For the first-order coherence of the three-layered networks—that is, $\mathfrak{A}$, ${\mathfrak{A}}^{*}$, and $\overline{\mathfrak{A}}$—the following asymptotic properties can be obtained:

In Figure 3, for $\mathfrak{A}$ and ${\mathfrak{A}}^{*}$, when one of the variables a and m tends to infinity, one has, $H^{\left(1\right)}\left(\mathfrak{A}\right)\to 0$; then, ${lim}_{m\to \infty }{H}^{\left(1\right)}\left({\mathfrak{A}}^{*}\right)={lim}_{m\to \infty }{H}^{\left(1\right)}\left(\mathfrak{A}\right)+\frac{p}{2(p+1)}$, which means the change in the robustness for the adhering star copies operation is only relevant with the number of leaf nodes p, and we also have ${lim}_{a\to \infty }{H}^{\left(1\right)}\left({\mathfrak{A}}^{*}\right)={lim}_{a\to \infty }{H}^{\left(1\right)}\left(\mathfrak{A}\right)+\frac{p}{2(p+1)}+\frac{1}{2m(1+p)}$.

Similarly, for $\mathfrak{A}$ and ${\mathfrak{A}}^{-}$,

• ${lim}_{m\to \infty }{H}^{\left(1\right)}\left(\overline{\mathfrak{A}}\right)={lim}_{m\to \infty }{H}^{\left(1\right)}\left(\mathfrak{A}\right)+\frac{1}{2(1+n)}(1+{\sum }_{q=1}^{n-1}\frac{1}{1+4si{n}^2\left(\frac{q\pi }{2n}\right)})$ holds, and
• ${lim}_{a\to \infty }{H}^{\left(1\right)}\left(\overline{\mathfrak{A}}\right)={lim}_{a\to \infty }{H}^{\left(1\right)}\left(\mathfrak{A}\right)+\frac{1}{2m(1+n)}+\frac{1}{2(1+n)}+\frac{1}{2(1+n)}{\sum }_{q=1}^{n-1}\frac{1}{1+4si{n}^2\left(\frac{q\pi }{2n}\right)})$.

In Section 3.1 and Section 3.2, it can be found that the limitation of coherence in (ii) is $\frac{1}{2m(1+n)}$ better than the case in (iii).

For ${\mathfrak{A}}^{*}$ and ${\mathfrak{A}}^{-}$, in case (i), ${lim}_{p\to \infty }{H}^{\left(1\right)}\left({\mathfrak{A}}^{*}\right)={lim}_{n\to \infty }{H}^{\left(1\right)}\left(\overline{\mathfrak{A}}\right)+0.2764$. The result is consistent with Figure 4.

In both cases (ii) and (iii), when $p=n$ is large enough, the limitation of coherence of $\overline{\mathfrak{A}}$ is also 0.2764 better than that of ${\mathfrak{A}}^{*}$.

It can be derived that the adhering fan-shaped substructure operation to $\mathfrak{A}$ has more influence on the coherence than the operation of adhering star topologies.

In view of Figure 5 and Figure 6, numerically, it can be seen that the coherence of $\overline{\mathfrak{A}}$ is 0.1562 better than that of ${\mathfrak{A}}^{*}$ (see Figure 5 and Figure 6, when $m=98$ and $a=98$, respectively); due to the choice of the value $p=n=3$, it verifies the above results on limitation well, and when the choices of p and n in case (ii) and (iii) are large enough, similar curves can also verify the result well; at this time, the greater the value of p is, the more the robustness of $\overline{\mathfrak{A}}$ is better than that of ${\mathfrak{A}}^{*}$.

The change in coherence with the parameters a and m for the triplex network $\mathfrak{A}$ is shown in Figure 3 ($a,m\in [2,100]$), and the variance of first-order coherence for the three-layered networks ${\mathfrak{A}}^{*}$ and $\overline{\mathfrak{A}}$ are shown in Figure 4, Figure 5 and Figure 6; the simulation examples are in good agreement with the results. One can also find that $\overline{\mathfrak{A}}$ always has the better first-order robustness of all the three cases of limit process; this is consistent with the result that edge interlace theorem on Laplacian eigenvalues derives.

5. Conclusions

In this paper, we mainly investigated the network coherence of three-layered systems. Specifically, the approach of algebraic graph theory was applied to analyze the complete k-partite graph-based structure; then, the related network coherence was derived and novel asymptotic behaviors for the performance index were found. When the numbers of peripheral vertices—i.e., p or n—were large enough, we found that the adhering star copies operations or fan-graph copies will make the first-order coherence increase by some certain scalars under the limit computations. Then, the performance indices of networks with non-isomorphic graphs were analyzed and simulated. We found that, when the number of nodes in counterpart node classes tend to infinity, their difference in coherence are only relevant with the number of peripheral nodes in the sense of limitation. Further, it was found that, compared with ${\mathfrak{A}}^{*}$, $\overline{\mathfrak{A}}$ always has the better first-order robustness; the convergence speed does not increase because of the linking relation between the leaf nodes of ${\mathfrak{A}}^{*}$.