On the Asymptotic Network Indices of Weighted Three-Layered Structures with Multi-Fan Composed Subgraphs

Abstract

In this paper, three sorts of network indices for the weighted three-layered graph are studied through the methods of graph spectra theory combined with analysis methods. The concept of union of graphs are applied to design two sorts of weighted layered multi-fan composed graphs, and the accurate mathematical expressions of the network indices are obtained through the derivations of Laplacian spectra; furthermore, the asymptotic properties are also derived. We find that when the cardinalities of the vertices on a sector-edge-link tend to infinity, the indices of FONC and EMFPT are irrelevant with the number of copies of the fan-substructure based on the considered graph framework.

1. Introduction

Algebraic graph theory and the topological indices of graphs not only have wide-ranging applications in fields such as materials physics and analytical chemistry, but also have extensive applications in the consensus problem and synchronization of complex networks.

To solve the coordination problems, the connecting relations of the networked systems are always characterized by the graph structures ([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]). There exists a lot of important articles on coordination topics related to the approach of graph theory [1,10,11,12,13,14,15,16,17,18,19,20,21,22]. In the enlightening articles, Ref. [10] described the robustness of the systems by Laplacian eigenvalues of some classic graphs. The robustness of the consensus models with disturbance is characterized by network coherence [11,12]. The articles mention that the index can be described by the expression on Laplacian eigenvalues. Ref. [13] gives solutions to the optimal leader node selection for several sorts of graphs. Reference [15] shows the relationship between the consensus index in symmetric and asymmetric trees and the number of leader nodes. In [16], the authors obtain the recursive expressions of Laplacian eigenvalues of the nested model and further derive the exact expression of the network coherence.

Similar with the network coherence, the Kirchhoff index [27,28] is a significant index of a graph, which can be used to describe the structural characteristics of molecules in chemistry. In addition, from the perspective of electrical networks, the magnitude of the Kirchhoff index not only reflects the connectivity of the network, but can also mean the resistance distance between nodes and the average electrical energy consumed by the entire network. Another topological index for the random walks which can also contain the sum of the reciprocals of the Laplacian eigenvalues is the entire mean first-passage time (EMFPT), and it is defined as the average of the first-passage time over all pairs of nodes [29]. The EMFPT is a sort of topological index to measure the efficiency of the stochastic dynamic process on complex networks. The EMFPT and the transportation efficiency on random walks have a negative correlation.

Over the past several decades, the multilayer network ([18,24,30,31]) has become a frontier topic as a result of its wide existence in real applications. A lot of networked systems have layered graphs ([17,18,19,20,21,23,26,27]); however, as a branch topic of coordination research, the related papers that incorporate the graph spectra methods on the L-spectrum for the topological indices of weighted layered networks are relatively rare. As a special multilayer structure, a three-layered structure ([19]), which is more complex than the two-layered one, has two layers that are symmetrical with respect to the middle layer.

Inspired by these enlightening papers, our research considers two sorts of networks with weighted multi-fan composed topologies that are generated by the graph operations.

The contributions of this work are the following three points:

I

Two novel non-isomorphic weighted three-layered multi-fan composed networks are constructed by graph operation.

II

Novel results of expression for the three sorts of network indices of the considered weighted graphs are given, and an analysis approach with multiple variables is utilized to calculate the asymptotic properties.

III

We find that if the cardinalities of the vertices on a sector-edge-link tend to infinity, the asymptotic properties of indices of FONC and EMFPT are irrelevant with the number of copies of the fan-substructure based on this paper’s topological framework.

Section 2 describes some basic notations in graph theory, and also interpret the relationship between the L-spectra and the indices. In Section 3, by the graph spectra approach, the graphs of the networks are designed and the mathematical expression of the indices are given; in addition, the asymptotic properties are also derived.

2. Preliminaries

Basic Notations

Let G be an undirected graph with vertex set $\mathcal{V}\left(G\right)=\{v_1,v_2,\dots ,v_N\}$, and the edge set $\mathcal{E}\left(G\right)=\left\{(v_i,v_j)\right|i,j=1,2,\dots ,N;i\ne j\}$. $\mathcal{A}\left(G\right)={\left[a_{ij}\right]}_N$ denotes the adjacency matrix, where $a_{ij}$ satisfies $a_{ij}=a_{ji}$. Set $a_{ij}=1$, if $(v_i,v_j)\in \mathcal{E}\left(G\right)$; $a_{ij}=0$, if $v_i$ and $v_j$ are disconnected. The Laplacian matrix is defined as $\mathfrak{L}\left(G\right)=\mathcal{D}\left(G\right)-\mathcal{A}\left(G\right)$, where $\mathcal{D}\left(G\right):=diag(d_1,d_2,\dots ,d_N)$, and $d_i=\sum _{j\ne i}{a}_{ij}$. The Laplacian spectrum of G is defined as $SL\left(G\right)=\left(\begin{array}{cccc}{\rho }_1\left(G\right)& {\rho }_2\left(G\right)& \dots & {\rho }_r\left(G\right)\\ l_1& l_2& \dots & l_r\end{array}\right)$, where ${\rho }_1\left(G\right)<{\rho }_2\left(G\right)<\dots <{\rho }_r\left(G\right)$ are the eigenvalues of $\mathfrak{L}\left(G\right)$, and $l_1,l_2,\dots ,l_r$ are their multiplicities ([32]).

To design the weighted layered graph of the network, denote the corona operation by ‘∘’ ([33,34,35]), and the Cartesian product by ‘□’ ([36,37,38]).

The following Definition and Lemma are needed during the derivation.

Definition 1

(The union of two graphs). Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be any two graphs; the union of $G_1$ and $G_2$ is denoted by $G_1\cup G_2$, and it has the vertex set $V_1\cup V_2$, where $V_1\cap V_2=\varnothing$, and it has the edge set $E_1\cup E_2$.

Lemma 1

([21]). Let G and H be two weighted graphs with m and n vertices, respectively; then, the Laplacian eigenvalues of $G\square H$ are the $mn$ numbers: $w_1{\nu }_i\left(G\right)+w_2{\nu }_j\left(H\right)(i=1,2\dots ,m;$$j=1,2,\dots ,n)$, where $w_1$ and $w_2$ are the edge weights of G and H, respectively, ${\nu }_i\left(G\right)$ and ${\nu }_j\left(H\right)$ are the Laplacian eigenvalues of the corresponding 0–1 weighted graph of G and H, respectively.

By referring to the reference [12,13,14,15,16,17], the first-order network coherence (FONC) is defined as

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

where $x_i\left(t\right)$ is the state of the nodes of noisy MAS network, and it has the Laplacian spectrum-related expression:

$$\begin{array}{c}\hfill \mathcal{H}={\displaystyle \frac{1}{2N}}\sum _{i=2}^N{\displaystyle \frac{1}{{\rho }_i}},\end{array}$$

(1)

It owns similarities with Kirchhoff index [27,28] (denoted by Kf(G)) and the entire mean first passage time (EMFPT) (denoted by F(G)) ([29]). The Kirchhoff index is defined by

$$\begin{array}{c}\hfill \mathrm{Kf}\left(G\right)=\sum _{i<j}{r}_{ij},\end{array}$$

where $r_{ij}$ is the resistance distance [27,28] between all pairs of vertices of the network, and it is proved that for a connected graph with order N,

$$\begin{array}{c}\hfill \mathrm{Kf}\left(G\right)=N\sum _{k=2}^N{\displaystyle \frac{1}{{\lambda }_k}},\end{array}$$

and the EMFPT [29] has the following expression by L-eigenvalues:

$$\begin{array}{c}\hfill \mathrm{F}\left(G\right)=\left({\displaystyle \frac{\left|\mathfrak{E}\right(G\left)\right|}{\left|\mathfrak{V}\right(G\left)\right|-1}}\right)\sum _{k=2}^N{\displaystyle \frac{1}{{\lambda }_k}}.\end{array}$$

3. Main Results

3.1. The Performance Indices for Network ${\mathfrak{M}}_1$

In this section, a sort of weighted triplex graph composed by the complete graph and multi-fan-shaped structure is considered (see Figure 1). The graph can be understood as linking the hub nodes of the fan-shaped copies; thus, the hub nodes in each layer form into a complete subgraph. The structure can be defined by $M_1(n,d_1,q_1):=K_n\circ \left({\cup }_{d_1}{P}_{q_1}\right)$, where n denotes the cardinality of the node set of the complete subgraph; $d_1$ is the number of copies of the fan-substructure; and $q_1$ denotes the number of boundary vertices in the fan structure. Thus, the triplex graph can be designed by $M_1(n,d_1,q_1)\square P_3$, which is abbreviated by ${\mathcal{M}}_1$, and the network with disturbance is denoted by ${\mathfrak{M}}_1$. Figure 2 shows an illustration of the three-layered graph.

Since $SL\left(P_3\right)=\left(\begin{array}{ccc}0& 1& 3\\ 1& 1& 1\end{array}\right)$, and $SL\left(K_n\right)=\left(\begin{array}{cc}0& n\\ 1& n-1\end{array}\right)$,

$SL\left({\cup }_{d_1}{P}_{q_1}\right)=\left(\begin{array}{cc}0& 4si{n}^2{\displaystyle \frac{k\pi }{2q_1}}\\ d_1& d_1\end{array}\right)$, where $k=1,2,\dots ,q_1-1$.

Therefore, by the theorem on the Laplacian spectrum of the corona graph [33,35], $SL\left({\mathcal{M}}_1\right)$ has the following description:

(1). $0,w_2,3w_2\in SL\left({\mathcal{M}}_1\right)$ with multiplicity 1;

(2). $w_1{d}_1{q}_1+w_1,w_1{d}_1{q}_1+w_1+w_2,w_1{d}_1{q}_1+w_1+3w_2\in SL\left({\mathcal{M}}_1\right)$ once;

(3). ${\displaystyle \frac{w_1(n+d_1{q}_1+1)\pm w_1\sqrt{{(n+d_1{q}_1+1)}^2-4n}}{2}}\in SL\left({\mathcal{M}}_1\right)$ repeated $(n-1)$ times;

${\displaystyle \frac{w_1(n+d_1{q}_1+1)\pm w_1\sqrt{{(n+d_1{q}_1+1)}^2-4n}}{2}}+w_2\in SL\left({\mathcal{M}}_1\right)$ repeated $(n-1)$ times;

${\displaystyle \frac{w_1(n+d_1{q}_1+1)\pm w_1\sqrt{{(n+d_1{q}_1+1)}^2-4n}}{2}}+3w_2\in SL\left({\mathcal{M}}_1\right)$ repeated $(n-1)$ times;

(4). $w_1,w_1+w_2,w_1+3w_2\in SL\left({\mathcal{M}}_1\right)$ with $(d_1-1)n$;

(5). $w_1(4si{n}^2\left({\displaystyle \frac{k\pi }{2q_1}}\right)+1),w_1(4si{n}^2\left({\displaystyle \frac{k\pi }{2q_1}}\right)+1)+w_2,w_1(4si{n}^2\left({\displaystyle \frac{k\pi }{2q_1}}\right)+1)+3w_2\in SL\left({\mathcal{M}}_1\right)$ repeated $d_{1}n$ times, where $k=1,2,\dots ,q_1-1$.

Hence, the coherence index of ${\mathfrak{M}}_1$ is

$$\begin{array}{cc}\hfill H\left({\mathfrak{M}}_1\right)=& {\displaystyle \frac{1}{2N}}\sum _{i=2}^N{\displaystyle \frac{1}{{\rho }_i}}\hfill \\ \hfill =& {\displaystyle \frac{1}{6(n+n{d}_1{q}_1)}}({\displaystyle \frac{1}{w_1{d}_1{q}_1+w_1}}+{\displaystyle \frac{1}{w_1{d}_1{q}_1+w_1+w_2}}+{\displaystyle \frac{1}{w_1{d}_1{q}_1+w_1+3w_2}}\hfill \\ \hfill +& {\displaystyle \frac{1}{w_2}}+{\displaystyle \frac{1}{3w_2}}+\left({\displaystyle \frac{1}{w_1}}+{\displaystyle \frac{1}{w_1+w_2}}+{\displaystyle \frac{1}{w_1+3w_2}}\right)(d_1-1)n_1+{\displaystyle \frac{(n_1+d_1{q}_1+1)(n_1-1)}{w_1{n}_1}}\hfill \\ \hfill +& {\displaystyle \frac{(w_1(n_1+d_1{q}_1+1)+2w_2)(n_1-1)}{w_1{w}_2(n_1+d_1{q}_1+1)+w_2^2+w_1^2{n}_1}}+{\displaystyle \frac{(w_1(n_1+d_1{q}_1+1)+6w_2)(n_1-1)}{3w_1{w}_2(n_1+d_1{q}_1+1)+9w_2^2+w_1^2{n}_1}}\hfill \\ \hfill +& d_1{n}_1\left(\sum _{k=1}^{q_1-1}{\displaystyle \frac{1}{w_1(4si{n}^2\frac{k\pi }{2q_1}+1)}}+{\displaystyle \frac{1}{w_1(4si{n}^2\frac{k\pi }{2q_1}+1)+w_2}}+{\displaystyle \frac{1}{w_1(4si{n}^2\frac{k\pi }{2q_1}+1)+3w_2}}\right)).\hfill \end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill \underset{q_1\to \infty }{lim}H\left({\mathfrak{M}}_1\right)={\displaystyle \frac{1}{6}}({\displaystyle \frac{\sqrt{5}}{5w_1}}+{\displaystyle \frac{1}{\sqrt{(5w_1+w_2)(w_1+w_2)}}}+{\displaystyle \frac{1}{\sqrt{(5w_1+3w_2)(w_1+3w_2)}}}+{\displaystyle \frac{n_1-1}{w_1{n}_1^2}}),\end{array}$$

from Equations (1) and (2), one has $K\left(M_1\right)\sim 3n^2{(1+d_1{q}_1)}^2({\displaystyle \frac{\sqrt{5}}{5w_1}}+{\displaystyle \frac{1}{\sqrt{(5w_1+w_2)(w_1+w_2)}}}+{\displaystyle \frac{1}{\sqrt{(5w_1+3w_2)(w_1+3w_2)}}}+{\displaystyle \frac{n_1-1}{w_1{n}_1^2}})$, as $q_1\to \infty$.

And since ${lim}_{q_1\to \infty }\left({\displaystyle \frac{\left|E\right(M_1\left)\right|}{\left|V\right(M_1\left)\right|-1}}\right)/q_1=4$, we have $F\left(M_1\right)\sim 4q_1({\displaystyle \frac{\sqrt{5}}{5w_1}}+{\displaystyle \frac{1}{\sqrt{(5w_1+w_2)(w_1+w_2)}}}$$+{\displaystyle \frac{1}{\sqrt{(5w_1+3w_2)(w_1+3w_2)}}}+{\displaystyle \frac{n_1-1}{w_1{n}_1^2}})$.

On the other hand, if the edge weights are 0–1 weighted, and the graph structure remains, then by the theorem on the L-spectrum of 0–1 weighted composite graph [34], $H\left({\mathfrak{M}}_1\right)$ has the form

$$\begin{array}{cc}\hfill H\left({\mathfrak{M}}_1\right)=& {\displaystyle \frac{1}{2N}}\sum _{i=2}^N{\displaystyle \frac{1}{{\rho }_i}}\hfill \\ \hfill =& {\displaystyle \frac{1}{6(n+n{d}_1{q}_1)}}({\displaystyle \frac{1}{d_1{q}_1+1}}+{\displaystyle \frac{1}{d_1{q}_1+2}}+{\displaystyle \frac{1}{d_1{q}_1+4}}+(d_{1}n-n)+{\displaystyle \frac{4}{3}}+(n-1){\displaystyle \frac{n+d_1{q}_1+1}{n}}\hfill \\ \hfill & +(n-1){\displaystyle \frac{n+d_1{q}_1+3}{2n+d_1{q}_1+2}}+(n-1){\displaystyle \frac{n+d_1{q}_1+7}{4n+3d_1{q}_1+12}}+{\displaystyle \frac{(d_1-1)n}{2}}+{\displaystyle \frac{(d_1-1)n}{4}}\hfill \\ \hfill & +d_{1}n\sum _{k=1}^{q_1-1}{\displaystyle \frac{1}{4si{n}^2\frac{k\pi }{2q_1}+1}}+d_{1}n\sum _{k=1}^{q_1-1}{\displaystyle \frac{1}{4si{n}^2\frac{k\pi }{2q_1}+2}}+d_{1}n\sum _{k=1}^{q_1-1}{\displaystyle \frac{1}{4si{n}^2\frac{k\pi }{2q_1}+4}}).\hfill \end{array}$$

Therefore, we have the following:

(i). When $q_1\to \infty$, one has $H\left({\mathfrak{M}}_1\right)\to {\displaystyle \frac{n-1}{6n^2}}+{\displaystyle \frac{\sqrt{5}}{30}}+{\displaystyle \frac{\sqrt{3}}{36}}+{\displaystyle \frac{\sqrt{2}}{48}}$; it can be seen that the 0–1 weighted result coincides with the above weighted one, that is, when $w_1=w_2=1$;

(ii). When $d_1\to \infty$, $H\left({\mathfrak{M}}_1\right)\to {\displaystyle \frac{7}{24q_1}}+{\displaystyle \frac{n-1}{6n^2}}+{\displaystyle \frac{1}{6q_1}}·{\eta }_1$;

(iii). When $n\to \infty$, $H\left({\mathfrak{M}}_1\right)\to {\displaystyle \frac{7d_1}{24(1+d_1{q}_1)}}+{\displaystyle \frac{d_1}{6(1+d_1{q}_1)}}·{\eta }_1$;

where ${\eta }_1={\sum }_{k=1}^{q_1-1}{\displaystyle \frac{1}{4si{n}^2\frac{k\pi }{2q_1}+1}}+{\sum }_{k=1}^{q_1-1}{\displaystyle \frac{1}{4si{n}^2\frac{k\pi }{2q_1}+2}}+{\sum }_{k=1}^{q_1-1}{\displaystyle \frac{1}{4si{n}^2\frac{k\pi }{2q_1}+4}}$, and the term is generated by the Cartesian product of the fan-shaped structures. Similarly, the parameter ${\eta }_2$ is presented in the results of Section 3.2.

It can be obtained that if $q_1\to \infty$, the asymptotic behaviour of FONC is only relevant with n, $w_1$, and $w_2$, that is, the cardinalities of vertices in the complete subgraph, and the edge weights. The value is irrelevant with $d_1$, i.e., the number of copies that stick to each node of the complete subgraph, and the EMFPT $F\left(M_1\right)$ is also irrelevant with $d_1$.

In view of (ii) and (iii), when $n,d_1$ are large enough, $H\left({\mathfrak{M}}_1\right)\to {\displaystyle \frac{7}{24q_1}}+{\displaystyle \frac{1}{6q_1}}·{\eta }_1$.

From Equations (1) and (2), one has $K\left(M_1\right)\sim 3n^2{(1+d_1{q}_1)}^2({\displaystyle \frac{\sqrt{5}}{5w_1}}+{\displaystyle \frac{1}{\sqrt{(5w_1+w_2)(w_1+w_2)}}}$$+{\displaystyle \frac{1}{\sqrt{(5w_1+3w_2)(w_1+3w_2)}}}+{\displaystyle \frac{n_1-1}{w_1{n}_1^2}})$, as $q_1\to \infty$. One can see that the Kirhoff index of this weighted graph is relevant with $d_1$; however, the corresponding coherence of the same weighted structure is irrelevant with $d_1$, that is, the number of copy of the fan subgraphs.

3.2. The Performance Indices for ${\mathfrak{M}}_2$

The weighted graph in this section is designed and interpreted in Figure 3 and Figure 4. The counterpart leaf nodes of different layers in the previous section is designed to connect with each other to form a path. The graph can be characterized by $M_2(q_2,d_2,r):=[E_1▿\left({\cup }_{d_2}{P}_{q_2}\right)\circ E_r]\square P_3$; it is abbreviated as $M_2$, where the parameters $q_2,d_2,r$ are the same as those in Section 3.2, and the related network is set by ${\mathfrak{M}}_2$. For an explanation of the triplex graph, Figure 4 is given to show the duplex one.

The L-spectrum of $M_2$ can be described as follows:

(1). $w_1(r+1),w_1(r+1)+w_2,w_1(r+1)+3w_2\in SL\left(M_2\right)$ with multiplicity 1, respectively;

(2). $0,w_2,3w_2\in SL\left(M_2\right)$ with multiplicity 1;

(3). ${\displaystyle \frac{w_1(2+d_2{q}_2+r)\pm w_1\sqrt{{(2+d_2{q}_2+r)}^2-4(1+d_2{q}_2)}}{2}},{\displaystyle \frac{w_1(2+d_2{q}_2+r)\pm w_1\sqrt{{(2+d_2{q}_2+r)}^2-4(1+d_2{q}_2)}}{2}}+w_2$, ${\displaystyle \frac{w_1(2+d_2{q}_2+r)\pm w_1\sqrt{{(2+d_2{q}_2+r)}^2-4(1+d_2{q}_2)}}{2}}+3w_2\in SL\left(M_2\right)$ with multiplicity 1;

(4). ${\displaystyle \frac{w_1(2+r)\pm w_1\sqrt{{(2+r)}^2-4}}{2}},{\displaystyle \frac{w_1(2+r)\pm w_1\sqrt{{(2+r)}^2-4}}{2}}+w_2,$ ${\displaystyle \frac{w_1(2+r)\pm w_1\sqrt{{(2+r)}^2-4}}{2}}+3w_2\in SL\left(M_2\right)$ repeated $d_2-1$ times;

(5). ${\displaystyle \frac{w_1(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)\pm w_1\sqrt{{(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)}^2-4(1+4si{n}^2\left(\frac{k\pi }{2q_2}\right))}}{2}}$,

${\displaystyle \frac{w_1(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)\pm w_1\sqrt{{(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)}^2-4(1+4si{n}^2\left(\frac{k\pi }{2q_2}\right))}}{2}}+w_2$,

${\displaystyle \frac{w_1(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)\pm w_1\sqrt{{(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)}^2-4(1+4si{n}^2\left(\frac{k\pi }{2q_2}\right))}}{2}}+3w_2\in SL\left(M_2\right)$ repeated $d_2$ times, where $k=1,2,\dots ,q_2-1$;

(6). $w_1,w_1+w_2,w_1+3w_2\in SL\left(M_2\right)$ repeated $(1+d_2{q}_2)(r-1)$ times.

Therefore, we have

$$\begin{array}{cc}\hfill H\left({\mathfrak{M}}_2\right)=& {\displaystyle \frac{1}{6(d_2{q}_2+1)(1+r)}}({\displaystyle \frac{1}{w_1(r+1)}}+{\displaystyle \frac{1}{w_1(r+1)+w_2}}+{\displaystyle \frac{1}{w_1(r+1)+3w_2}}+{\displaystyle \frac{4}{3w_2}}\hfill \\ \hfill & +{\displaystyle \frac{2+d_2{q}_2+r}{w_1(1+d_2{q}_2)}}+{\displaystyle \frac{w_1(2+d_2{q}_2+r)+2w_2}{w_1{w}_2(2+d_2{q}_2+r)+w_2^2+w_1^2(1+d_2{q}_2)}}\hfill \\ \hfill & +{\displaystyle \frac{w_1(2+d_2{q}_2+r)+6w_2}{3w_1{w}_2(2+d_2{q}_2+r)+9w_2^2+w_1^2(1+d_2{q}_2)}}+{\displaystyle \frac{2+r}{w_1}}(d_2-1)\hfill \\ \hfill & +{\displaystyle \frac{w_1(2+r)+2w_2}{w_1{w}_2(2+r)+w_2^2+w_1^2}}(d_2-1)+{\displaystyle \frac{w_1(2+r)+6w_2}{3w_1{w}_2(2+r)+9w_2^2+w_1^2}}(d_2-1)\hfill \\ \hfill & +\sum _{k=1}^{q_2-1}{\displaystyle \frac{(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)}{w_1(1+4si{n}^2\left(\frac{k\pi }{2q_2}\right))}}{d}_2+\sum _{k=1}^{q_2-1}{\displaystyle \frac{w_1(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)+2w_2}{w_1{w}_2(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)+w_2^2+w_1^2(1+4si{n}^2\left(\frac{k\pi }{2q_2}\right))}}{d}_2\hfill \\ \hfill & +\sum _{k=1}^{q_2-1}{\displaystyle \frac{w_1(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)+6w_2}{3w_1{w}_2(2+4si{n}^2\left(\frac{k\pi }{2q_2}\right)+r)+9w_2^2+w_1^2(1+4si{n}^2\left(\frac{k\pi }{2q_2}\right))}}{d}_2\hfill \\ \hfill & +{\displaystyle \frac{(1+d_2{q}_2)(r-1)}{w_1}}+{\displaystyle \frac{(1+d_2{q}_2)(r-1)}{w_1+w_2}}+{\displaystyle \frac{(1+d_2{q}_2)(r-1)}{w_1+3w_2}}).\hfill \end{array}$$

Therefore, we have

$$\begin{array}{cc}\hfill \underset{q_2\to \infty }{lim}H\left({\mathfrak{M}}_2\right)=& {\displaystyle \frac{1}{6(r+1)}}\left({\displaystyle \frac{1}{w_1}}+{\displaystyle \frac{1}{w_1+w_2}}+{\displaystyle \frac{1}{w_1+3w_2}}\right)+{\displaystyle \frac{r-1}{6(r+1)}}({\displaystyle \frac{1}{w_1+w_2}}+{\displaystyle \frac{1}{w_1}}\hfill \\ \hfill & +{\displaystyle \frac{1}{w_1+3w_2}})+{\displaystyle \frac{1}{6(r+1)}}[(1+r){\displaystyle \frac{\sqrt{5}}{5}}\hfill \\ \hfill & +{\int }_0^1{\displaystyle \frac{w_1(2+4si{n}^2\left(\frac{\pi x}{2}\right)+r)+2w_2}{w_1{w}_2(2+4si{n}^2\left(\frac{\pi x}{2}\right)+r)+w_2^2+w_1^2(1+4si{n}^2\left(\frac{\pi x}{2}\right))}}dx\hfill \\ \hfill & +{\int }_0^1{\displaystyle \frac{w_1(2+4si{n}^2\left(\frac{\pi x}{2}\right)+r)+6w_2}{3w_1{w}_2(2+4si{n}^2\left(\frac{\pi x}{2}\right)+r)+9w_2^2+w_1^2(1+4si{n}^2\left(\frac{\pi x}{2}\right))}}dx]\hfill \\ \hfill =& {\displaystyle \frac{1}{6(r+1)}}\left({\displaystyle \frac{1}{w_1}}+{\displaystyle \frac{1}{w_1+w_2}}+{\displaystyle \frac{1}{w_1+3w_2}}\right)+{\displaystyle \frac{r-1}{6(r+1)}}({\displaystyle \frac{1}{w_1+w_2}}+{\displaystyle \frac{1}{w_1}}\hfill \\ \hfill & +{\displaystyle \frac{1}{w_1+3w_2}})+{\displaystyle \frac{1}{6(r+1)}}[(1+r){\displaystyle \frac{\sqrt{5}}{5}}\hfill \\ \hfill & +{\displaystyle \frac{(2w_1{w}_2+w_1^{2}r+w_1^2+w_2^2)}{(w_1+w_2)\sqrt{(2w_1{w}_2+w_1{w}_{2}r+w_1^2+w_2^2)(6w_1{w}_2+5w_1^2+w_1{w}_{2}r+w_2^2)}}}\hfill \\ & +{\displaystyle \frac{(6w_1{w}_2+w_1^{2}r+w_1^2+9w_2^2)}{(w_1+3w_2)\sqrt{(18w_1{w}_2+3w_1{w}_{2}r+5w_1^2+9w_2^2)(6w_1{w}_2+w_1^2+3w_1{w}_{2}r+9w_2^2)}}}].\hfill \end{array}$$

Denote the 0–1 weighted graph of the corresponding weighted graph $M_2$ by ${\mathcal{M}}_2$, and denote the corresponding network by ${\overline{M}}_2$. Then, by the lemma on the L-spectrum of the Cartesian product of 0–1 weighted graphs [32,34], the L-spectrum of ${\mathcal{M}}_2$ can be described as follows:

(1). $0,1,3\in SL\left({\mathcal{M}}_2\right)$ with multiplicity 1, respectively;

(2). $r+1,r+2,r+4\in SL\left({\mathcal{M}}_2\right)$ with multiplicity 1, respectively;

(3). ${\displaystyle \frac{(d_2{q}_2+2+r)\pm \sqrt{{(d_2{q}_2+2+r)}^2-4(d_2{q}_2+1)}}{2}}\in SL\left({\mathcal{M}}_2\right)$ with multiplicity 1;

(4). ${\displaystyle \frac{(d_2{q}_2+4+r)\pm \sqrt{{(d_2{q}_2+2+r)}^2-4(d_2{q}_2+1)}}{2}}\in SL\left({\mathcal{M}}_2\right)$ with multiplicity 1;

(5). ${\displaystyle \frac{(d_2{q}_2+8+r)\pm \sqrt{{(d_2{q}_2+2+r)}^2-4(d_2{q}_2+1)}}{2}}\in SL\left({\mathcal{M}}_2\right)$ with multiplicity 1;

(6). ${\displaystyle \frac{(2+r)\pm \sqrt{{(2+r)}^2-4}}{2}}\in SL\left({\mathcal{M}}_2\right)$ repeated $d_2-1$ times;

(7). ${\displaystyle \frac{(4+r)\pm \sqrt{{(2+r)}^2-4}}{2}}\in SL\left({\mathcal{M}}_2\right)$ repeated $d_2-1$ times;

(8). ${\displaystyle \frac{(8+r)\pm \sqrt{{(2+r)}^2-4}}{2}}\in SL\left({\mathcal{M}}_2\right)$ repeated $d_2-1$ times;

(9). ${\displaystyle \frac{2+4si{n}^2\frac{k\pi }{2q_2}+r\pm \sqrt{{(2+4si{n}^2\frac{k\pi }{2q_2}+r)}^2-4(1+4si{n}^2\frac{k\pi }{2q_2})}}{2}}\in SL\left({\mathcal{M}}_2\right)$ with multiplicity $d_2$, where $k=1,2,\dots ,q_2-1$;

(10). ${\displaystyle \frac{4+4si{n}^2\frac{k\pi }{2q_2}+r\pm \sqrt{{(2+4si{n}^2\frac{k\pi }{2q_2}+r)}^2-4(1+4si{n}^2\frac{k\pi }{2q_2})}}{2}}\in SL\left({\mathcal{M}}_2\right)$ with multiplicity $d_2$, where $k=1,2,\dots ,q_2-1$;

(11). ${\displaystyle \frac{8+4si{n}^2\frac{k\pi }{2q_2}+r\pm \sqrt{{(2+4si{n}^2\frac{k\pi }{2q_2}+r)}^2-4(1+4si{n}^2\frac{k\pi }{2q_2})}}{2}}\in SL\left({\mathcal{M}}_2\right)$ repeated $d_2$ times, where $k=1,2,\dots ,q_2-1$;

(12). 1,2 and 4 $\in SL\left({\mathcal{M}}_2\right)$ repeated $(d_2{q}_2+1)(r-1)$ times, respectively.

Hence, the coherence for the corresponding network with disturbance can be derived as

$$\begin{array}{cc}\hfill H\left({\overline{M}}_2\right)=& {\displaystyle \frac{1}{6(d_2{q}_2+1)(1+r)}}({\displaystyle \frac{1}{r+1}}+{\displaystyle \frac{1}{r+2}}+{\displaystyle \frac{1}{r+4}}+{\displaystyle \frac{4}{3}}+{\displaystyle \frac{d_2{q}_2+2+r}{d_2{q}_2+1}}+{\displaystyle \frac{d_2{q}_2+4+r}{2d_2{q}_2+4+r}}\hfill \\ \hfill & +{\displaystyle \frac{d_2{q}_2+8+r}{4d_2{q}_2+16+3r}}+(d_2-1)(2+r)+{\displaystyle \frac{4+r}{6+r}}(d_2-1)+{\displaystyle \frac{8+r}{3r+16}}(d_2-1)\hfill \\ \hfill & +d_2\sum _{k=1}^{q_2-1}{\displaystyle \frac{2+4si{n}^2\frac{k\pi }{2q_2}+r}{1+4si{n}^2\frac{k\pi }{2q_2}}}+d_2\sum _{k=1}^{q_2-1}{\displaystyle \frac{4+4si{n}^2\frac{k\pi }{2q_2}+r}{4+8si{n}^2\frac{k\pi }{2q_2}+r}}+d_2\sum _{k=1}^{q_2-1}{\displaystyle \frac{8+4si{n}^2\frac{k\pi }{2q_2}+r}{16+16si{n}^2\frac{k\pi }{2q_2}+3r}}\hfill \\ \hfill & +(d_2{q}_2+1)(r-1)+{\displaystyle \frac{1}{2}}(d_2{q}_2+1)(r-1)+{\displaystyle \frac{1}{4}}(d_2{q}_2+1)(r-1)).\hfill \end{array}$$

Therefore, when

(i). $q_2\to \infty$, we have

$$\begin{array}{cc}\hfill \underset{q_2\to \infty }{lim}H\left({\overline{M}}_2\right)=& {\displaystyle \frac{1}{6(1+r)}}({\int }_0^1{\displaystyle \frac{2+4si{n}^2\frac{\pi x}{2}+r}{1+4si{n}^2\frac{\pi x}{2}}}dx+{\int }_0^1{\displaystyle \frac{4+4si{n}^2\frac{\pi x}{2}+r}{4+8si{n}^2\frac{\pi x}{2}+r}}dx\hfill \\ \hfill & +{\int }_0^1{\displaystyle \frac{8+4si{n}^2\frac{\pi x}{2}+r}{16+16si{n}^2\frac{\pi x}{2}+3r}}dx\hfill \\ \hfill =& {\displaystyle \frac{1}{6(1+r)}}({\displaystyle \frac{5}{2}}+{\displaystyle \frac{16+r}{\sqrt{(32+3r)(16+3r)}}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{4+r}{12+r}}}+{\displaystyle \frac{\sqrt{5}}{5}}(1+r)\hfill \\ \hfill & +{\displaystyle \frac{7}{4}}(r-1));\hfill \end{array}$$

it can be seen that the asymptotic result coincides with that of $H\left({\mathfrak{M}}_2\right)$ on page 9.

(ii). If $d_2\to \infty$, $H\left({\overline{M}}_2\right)\to {\displaystyle \frac{2+r}{6q_2(1+r)}}+{\displaystyle \frac{4+r}{6q_2(1+r)(6+r)}}+{\displaystyle \frac{8+r}{6q_2(1+r)(3r+16)}}+{\displaystyle \frac{1}{6q_2(1+r)}}({\sum }_{k=1}^{q_2-1}$${\displaystyle \frac{2+4si{n}^2\frac{k\pi }{2q_2}+r}{1+4si{n}^2\frac{k\pi }{2q_2}}}+{\sum }_{k=1}^{q_2-1}{\displaystyle \frac{4+4si{n}^2\frac{k\pi }{2q_2}+r}{4+8si{n}^2\frac{k\pi }{2q_2}+r}}+{\sum }_{k=1}^{q_2-1}{\displaystyle \frac{8+4si{n}^2\frac{k\pi }{2q_2}+r}{16+16si{n}^2\frac{k\pi }{2q_2}+3r}})+{\displaystyle \frac{7r-7}{24(1+r)}}$;

(iii). If $r\to \infty$, $H\left({\overline{M}}_2\right)\to {\displaystyle \frac{1}{6{(d_2{q}_2+1)}^2}}+{\displaystyle \frac{d_2-1}{6(d_2{q}_2+1)}}+{\sum }_{k=1}^{q_2-1}{\displaystyle \frac{d_2}{6(d_2{q}_2+1)(1+4si{n}^2\frac{k\pi }{2q_2})}}+{\displaystyle \frac{7}{24}}$.

Remark 1.

One can see that when $q_2\to \infty$, the limitation result is only relevant with r, and it is irrelevant to the parameter $d_2$. By the similar derivation from above, the Kirhoff index $K\left({\mathcal{M}}_2\right)$ based on ${\mathcal{M}}_2$ satisfies the asymptotic result: $K\left({\mathcal{M}}_2\right)\sim 3{(d_2{q}_2+1)}^2(1+r)\left({\displaystyle \frac{5}{2}}+{\displaystyle \frac{16+r}{\sqrt{(32+3r)(16+3r)}}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{4+r}{12+r}}}+{\displaystyle \frac{\sqrt{5}}{5}}(1+r)+{\displaystyle \frac{7}{4}}(r-1)\right)$, and we also have $F\left(M_2\right)\sim \left({\displaystyle \frac{8+3r}{3+3r}}\right)·\left({\displaystyle \frac{5}{2}}+{\displaystyle \frac{16+r}{\sqrt{(32+3r)(16+3r)}}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{4+r}{12+r}}}+{\displaystyle \frac{\sqrt{5}}{5}}(1+r)+{\displaystyle \frac{7}{4}}(r-1)\right)$, and the asymptotic result is only relevant with the parameter r. A similar equivalence relationship can be obtained for the indices when the other variables tend to infinity—they are omitted here.

Remark 2.

The asymptotic properties can be used to analyse and improve the topological performance. The weighted three-layered multi-fan composed graph in this paper might be extended to multilayer graphs in future research. From Figure 1 and Figure 3, it seems that each layer of ${\mathcal{M}}_1$ has a more distributed graphic structure from the middle to the margin nodes, but to the layer of ${\mathcal{M}}_2$, the density of links seems to gradually become sparse. However, one can adjust the node-related parameters to change the difference between their FONC using the acquired results in this paper. In addition, this paper might enlighten the graph structure design of multilayer networks of one’s future research.

Remark 3.

The mathematical expression of coherence can be a reference for deriving the robustness of layered MAS, and the asymptotic results can be used to analyze and improve the similar topological indices [39] of networked systems. The networked MAS has many relations or applications with other fields. The design of the layered system structure and the analysis of the network performance of this paper might be a reference for other network problems, such as network traffic detection [40,41], networked knowledge-based systems [42], and neural networks [43]. MASs can be applied to optimize the training process of neural networks; both MASs and neural networks consist of distributed computing and communication. The distributed characteristics of MASs and the mathematical description of network structures in our paper might provide a more flexible, efficient, and intelligent analysis scheme for network traffic detection or other distributed networked systems [44,45].

4. Conclusions

This research mainly discussed the index of the FONC of three-layered weighted networks with multi-fan composed subgraphs. The graph spectra theory and analysis approach are utilized to analyse the weighted multi-fan composed graph, then the expressions for the indices are derived, and novel asymptotic properties on the FONC, Kirhoff index, and FMPT are acquired. We find that if $q_1\to \infty$, the asymptotic behaviour of FONC is only relevant to n, $w_1$, and $w_2$, that is, the cardinalities of vertices in the complete subgraph, and the edge weights. The value is irrelevant with $d_1$, i.e., the number of copies that stick to each node of the complete subgraph. For ${\mathcal{M}}_2$, one can see that when $q_2\to \infty$, the result is irrelevant to $d_2$, and specifically, the 0–1 weighted cases for the FONC and EMFPT are only relevant with r.