具有结构洞的异结构时滞复杂网络同步 Synchronization of Time-Delay Complex Networks with Structural Hole and Different Structure

doi:10.12677/AAM.2022.114232

Advances in Applied Mathematics
Vol. 11 No. 04 ( 2022 ), Article ID: 50871 , 9 pages
10.12677/AAM.2022.114232

具有结构洞的异结构时滞复杂网络同步

王晨阳

●How to Cite this Article

河北工业大学理学院，天津

收稿日期：2022年3月26日；录用日期：2022年4月21日；发布日期：2022年4月28日

摘要

借鉴社会网络研究中的结构洞概念，利用WS和NW小世界网络建立了具有结构洞的复杂网络模型。并采用时滞的Lorenz混沌系统和时滞的Chen混沌系统作为网络节点，基于Lyapunov稳定性定理，设计合适的控制器，实现了具有结构洞的异结构时滞复杂网络同步。数值模拟证明了具有结构洞的异结构时滞网络实现了完全同步。

关键词

复杂网络，结构洞，时滞，异结构，网络同步

Synchronization of Time-Delay Complex Networks with Structural Hole and Different Structure

Chenyang Wang

College of Science, Hebei University of Technology, Tianjin

Received: Mar. 26^th, 2022; accepted: Apr. 21^st, 2022; published: Apr. 28^th, 2022

ABSTRACT

Drawing on the concept of structural holes in the study of social networks, a complex network model with a structural hole is established using WS and NW small-world networks. And the time-lagged Lorenz chaotic system and the time-lagged Chen chaotic system are used as network nodes to design suitable controllers based on Lyapunov stability theorem to realize the synchronization of time-lagged complex networks with structural hole and different structural. Numerical simulations demonstrate that the time-lagged networks with structural hole and different structure is fully synchronized.

Keywords:Complex Networks, Structural Hole, Time Lags, Different Structure, Network Synchronization

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 简介

近年来，自1998年Watts和Strogatz首次提出小世界网络(即所谓的WS小世界网络 [1] )以来，由于其在许多领域的全面应用，如万维网、计算机网络、神经网络、交通网络等 [2] - [8]，人们对复杂网络的研究给予了越来越多的关注。随后，Newman和Watts提出了一种新的小世界网络模型，即NW小世界网络 [9]。这些开创性的工作吸引了研究人员深入分析复杂网络的动力学特征，特别是复杂网络同步的研究已成为该领域中的一个热门话题。

人们在对实际网络研究过程中发现，许多网络(如万维网、通信网络等)中存在着结构洞。结构洞的概念最初由Ronald Stuart Burt所提出 [10]，用于解释社会资本差异的起源。Burt指出若具有互补资源的两个群体不存在直接联系，从整体上看它们之间的空白好像出现了洞穴，当第三者将两个分隔的群体连接起来，空白得到补充，称之为结构洞。随着人们对社会网络研究的深入，结构洞逐渐成为了一种全新的理论并得到了学者们的广泛认可，已在经济学、计算机科学和情报学等领域得到广泛应用 [11] [12] [13] [14]。

在现实世界中，许多网络都是异结构的，即网络的结构和节点动力学是不同的，这些异结构网络引起了研究者们的关注。例如，Song等人采用自适应控制策略与脉冲控制策略实现了具有不同节点的复杂网络同步 [15]。Wang等人研究了具有不同分数阶混沌节点的网络的投影同步 [16]。Yang等人采用牵制控制策略实现了具有不同节点的复杂网络同步 [17]。高艳选用三种不同的时空混沌模型作为网络节点，实现了异结构时空网络的聚类同步 [18]。

由于交换速度和信号传输的有限性，实际网络中不可避免地存在时间延迟。具有耦合延迟的复杂网络的同步在过去几年中得到了广泛的研究。例如，Li等人研究了具有耦合延迟和非线性内部耦合函数的复杂网络同步问题 [19]。Lu等人探索了由具有耦合延迟的微分方程描述的线性耦合系统的同步问题 [20]。Cheng等人研究了具有延迟耦合的离散时间复杂网络中的同步 [21]。然而，动态节点中的时间延迟由于其复杂性而在文献中涉及较少。在文献 [22] 中，Mensour等人探讨了延迟微分方程的同步问题。Zhang等人研究了具有延迟节点的复杂网络的自适应反馈同步 [23]。事实上，在许多真实网络中，其动态节点可用具有时间延迟的微分方程进行描述 [22] [23]。例如，混沌通信领域中，由于时滞的混沌系统具有更为丰富的动力学特性，将其作为传输信号可有效提高通信的安全性。在通信过程中，通常需要借助基站、卫星等第三方平台进行信息传输，这些第三方平台可以看作为结构洞。在分析这类通信问题时，可抽象为具有结构洞的时滞网络进行研究。

本文中，我们利用WS小世界网络与NW小世界网络构建了具有结构洞的网络，并采用时滞的Chen混沌系统和时滞的Lorenz混沌系统作为网络的节点，基于李雅普诺夫稳定性定理设计了控制器，实现了网络的完全同步。

2. 具有结构洞网络模型的构建

本文中建立的具有结构洞的复杂网络模型包括两侧的两个小世界网络和位于两个小世界网络中间的核心节点部分，为了简化模型，在网络的结构洞位置仅使用了一个核心节点。首先生成有着12个节点的WS小世界网络和有着14个节点的NW小世界网络，再分别从这两个小世界网络中各选出1个节点与核心节点相连。这样就构成了一个节点数为27的具有结构洞的复杂网络。如图1所示：

Figure 1. Network model with structural hole

图1. 具有结构洞的网络模型

3. 网络同步理论分析

考虑一个由两类不同结构的时滞混沌系统作为节点的复杂网络，其状态方程可以描述为：

${\begin{array}{l} {\dot{x}}_{i} (t) = f (x_{i} (t), x_{i} (t - τ_{1})) + ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{i} (t)), i = 1, 2, \dots, l \\ {\dot{x}}_{i} (t) = g (x_{i} (t), x_{i} (t - τ_{2})) + ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{i} (t)), i = l + 1, l + 2, \dots, N \end{array}$ (1.1)

其中 $x_{i} = {(x_{1}^{i}, x_{2}^{i}, \dots, x_{n}^{i})}^{T} \in R^{n}$ 是对应第i个节点的状态变量， $f, g = R^{n} \to R^{n}$ 表示节点的系统方程，ε是节点间的耦合强度， $C = {(C_{i j})}_{N \times N} \in R^{N \times N}$ 为节点的耦合矩阵。 $τ_{1}, τ_{2}$ 均为大于零的常值时滞。将网络中前l个节点上放置时滞的Chen混沌系统 [24]，以 $x_{1}^{i} (t), y_{1}^{i} (t), z_{1}^{i} (t)$ 表示它的状态变量，其状态方程如下：

${\begin{cases} {\dot{x}}_{1}^{i} (t) = - a x_{1}^{i} (t) + a y_{1}^{i} (t) + ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{1}^{i} (t)) \\ {\dot{y}}_{1}^{i} (t) = (c - a) x_{1}^{i} (t) - x_{1}^{i} (t) z_{1}^{i} (t) + (c - d_{1}) y_{1}^{i} (t) + d_{1} y_{1}^{i} (t - τ_{1}) + ε \sum_{j = 1}^{N} C_{i j} (y_{j} (t) - y_{1}^{i} (t)) \\ {\dot{z}}_{1}^{i} (t) = x_{1}^{i} (t) y_{1}^{i} (t) - b z_{1}^{i} (t) + ε \sum_{j = 1}^{N} C_{i j} (z_{j} (t) - z_{1}^{i} (t)) \end{cases}$ (1.2)

在第l + 1到第N个节点上放置时滞的Lorenz混沌系统 [25]，用 $x_{2}^{i} (t), y_{2}^{i} (t), z_{2}^{i} (t)$ 表示它的状态变量，其状态方程如下：

${\begin{cases} {\dot{x}}_{2}^{i} (t) = - σ x_{2}^{i} (t) + σ y_{2}^{i} (t) + ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{2}^{i} (t)) \\ {\dot{y}}_{2}^{i} (t) = γ x_{2}^{i} (t) - x_{2}^{i} (t) z_{2}^{i} (t) + (d_{2} - 1) y_{2}^{i} (t) + d_{2} y_{2}^{i} (t - τ_{2}) + ε \sum_{j = 1}^{N} C_{i j} (y_{j} (t) - y_{2}^{i} (t)) \\ {\dot{z}}_{2}^{i} (t) = x_{2}^{i} (t) y_{2}^{i} (t) - β z_{2}^{i} (t) + ε \sum_{j = 1}^{N} C_{i j} (z_{j} (t) - z_{2}^{i} (t)) \end{cases}$ (1.3)

由于网络中的节点由不同的混沌系统组成，在网络的耦合作用下很难实现同步，所以对网络节点添加控制器从而使网络实现同步， $U_{1}^{i}^{'}, U_{2}^{i}^{'}, U_{3}^{i}^{'}; U_{1}^{i}, U_{2}^{i}, U_{3}^{i}$ 分别表示 $x_{1}^{i} (t), y_{1}^{i} (t), z_{1}^{i} (t); x_{2}^{i} (t), y_{2}^{i} (t), z_{2}^{i} (t)$ 上的同步控制器，此时在第1到第l个节点的状态方程变为：

${\begin{cases} {\dot{x}}_{1}^{i} (t) = - a x_{1}^{i} (t) + a y_{1}^{i} (t) + ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{1}^{i} (t)) + U_{1}^{i}^{'} \\ {\dot{y}}_{1}^{i} (t) = (c - a) x_{1}^{i} (t) - x_{1}^{i} (t) z_{1}^{i} (t) + (c - d_{1}) y_{1}^{i} (t) + d_{1} y_{1}^{i} (t - τ_{1}) + ε \sum_{j = 1}^{N} C_{i j} (y_{j} (t) - y_{1}^{i} (t)) + U_{2}^{i}^{'} \\ {\dot{z}}_{1}^{i} (t) = x_{1}^{i} (t) y_{1}^{i} (t) - b z_{1}^{i} (t) + ε \sum_{j = 1}^{N} C_{i j} (z_{j} (t) - z_{1}^{i} (t)) + U_{3}^{i}^{'} \end{cases}$ (1.4)

在第l + 1到第N个节点的状态方程变为：

${\begin{cases} {\dot{x}}_{2}^{i} (t) = - σ x_{2}^{i} (t) + σ y_{2}^{i} (t) + ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{2}^{i} (t)) + U_{1}^{i} \\ {\dot{y}}_{2}^{i} (t) = γ x_{2}^{i} (t) - x_{2}^{i} (t) z_{2}^{i} (t) + (d_{2} - 1) y_{2}^{i} (t) + d_{2} y_{2}^{i} (t - τ_{2}) + ε \sum_{j = 1}^{N} C_{i j} (y_{j} (t) - y_{2}^{i} (t)) + U_{2}^{i} \\ {\dot{z}}_{2}^{i} (t) = x_{2}^{i} (t) y_{2}^{i} (t) - β z_{2}^{i} (t) + ε \sum_{j = 1}^{N} C_{i j} (z_{j} (t) - z_{2}^{i} (t)) + U_{3}^{i} \end{cases}$ (1.5)

我们选取具有结构洞的复杂网络中1号节点上放置时滞的Chen混沌系统，并将其作为目标节点，网络中其他节点放置时滞的Lorenz混沌系统，则根据(1.4)式与(1.5)式可以得：

${\begin{cases} {\dot{x}}_{1}^{1} (t) = - a x_{1}^{1} (t) + a y_{1}^{1} (t) + ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{1}^{1} (t)) \\ {\dot{y}}_{1}^{1} (t) = (c - a) x_{1}^{1} (t) - x_{1}^{1} (t) z_{1}^{1} (t) + (c - d_{1}) y_{1}^{1} (t) + d_{1} y_{1}^{1} (t - τ_{1}) + ε \sum_{j = 1}^{N} C_{i j} (y_{j} (t) - y_{1}^{1} (t)) \\ {\dot{z}}_{1}^{1} (t) = x_{1}^{1} (t) y_{1}^{1} (t) - b z_{1}^{1} (t) + ε \sum_{j = 1}^{N} C_{i j} (z_{j} (t) - z_{1}^{1} (t)) \end{cases}$ (1.6)

网络系统状态变量间的误差为：

${\begin{array}{l} e_{1}^{i} (t) = x_{2}^{i} (t) - x_{1}^{1} (t) \\ e_{2}^{i} (t) = y_{2}^{i} (t) - y_{1}^{1} (t) \\ e_{3}^{i} (t) = z_{2}^{i} (t) - z_{1}^{1} (t) \end{array}$ (1.8)

其中 $i = 2, 3, \dots, N$ 。

为了计算方便，我们定义：

${\begin{array}{l} Δ Φ_{1}^{i} = ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{2}^{i} (t)) - ε \sum_{j = 1}^{N} C_{i j} (x_{j} (t) - x_{1}^{1} (t)) \\ Δ Φ_{2}^{i} = ε \sum_{j = 1}^{N} C_{i j} (y_{j} (t) - y_{2}^{i} (t)) - ε \sum_{j = 1}^{N} C_{i j} (y_{j} (t) - y_{1}^{1} (t)) \\ Δ Φ_{3}^{i} = ε \sum_{j = 1}^{N} C_{i j} (z_{j} (t) - z_{2}^{i} (t)) - ε \sum_{j = 1}^{N} C_{i j} (z_{j} (t) - z_{1}^{1} (t)) \end{array}$ (1.9)

则基于(1.8)式可得到以下表达式：

$\begin{matrix} {\dot{e}}_{1}^{i} (t) = {\dot{x}}_{2}^{i} (t) - {\dot{x}}_{1}^{1} (t) \\ = - σ x_{2}^{i} + σ y_{2}^{i} + a x_{1}^{1} - a y_{1}^{1} + Δ Φ_{1}^{i} + U_{1}^{i} \end{matrix}$ (1.10)

$\begin{matrix} {\dot{e}}_{2}^{i} (t) = {\dot{y}}_{2}^{i} (t) - {\dot{y}}_{1}^{1} (t) \\ = γ x_{2}^{i} - x_{2}^{i} z_{2}^{i} + (d_{2} - 1) y_{2}^{i} + d_{2} y_{2}^{i} (t - τ_{2}) - (c - a) x_{1}^{1} \\ + x_{1}^{1} z_{1}^{1} - (c - d_{1}) y_{1}^{1} - d_{1} y_{1}^{1} (t - τ_{1}) + Δ Φ_{2}^{i} + U_{2}^{i} \end{matrix}$ (1.11)

$\begin{matrix} {\dot{e}}_{3}^{i} (t) = {\dot{z}}_{2}^{i} (t) - {\dot{z}}_{1}^{1} (t) \\ = x_{2}^{i} y_{2}^{i} - β z_{2}^{i} - x_{1}^{1} y_{1}^{1} + b z_{1}^{1} + Δ Φ_{3}^{i} + U_{3}^{i} \end{matrix}$ (1.12)

所设计的控制器 $U_{1}^{i}, U_{2}^{i}, U_{3}^{i}$ 为：

${\begin{array}{l} U_{1}^{i} = σ x_{2}^{i} - σ y_{2}^{i} - a x_{1}^{1} + a y_{1}^{1} - k_{1} e_{1}^{i} - Δ Φ_{1}^{i} \\ \begin{matrix} U_{2}^{i} = - γ x_{2}^{i} + x_{2}^{i} z_{2}^{i} - (d_{2} - 1) y_{2}^{i} - d_{2} y_{2}^{i} (t - τ_{2}) + (c - a) x_{1}^{1} \\ - x_{1}^{1} z_{1}^{1} + (c - d_{1}) y_{1}^{1} + d_{1} y_{1}^{1} (t - τ_{1}) - k_{2} e_{2}^{i} - Δ Φ_{2}^{i} \end{matrix} \\ U_{3}^{i} = - x_{2}^{i} y_{2}^{i} + β z_{2}^{i} + x_{1}^{1} y_{1}^{1} - b z_{1}^{1} - k_{3} e_{3}^{i} - Δ Φ_{3}^{i} \end{array}$ (1.13)

其中k₁、k₂、k₃均为大于零的常数。

证明：我们构建Lyapunov函数为：

$v = \frac{1}{2} \sum_{k = 1}^{3} {(e_{k}^{i})}^{2} = \frac{1}{2} ({(e_{1}^{i})}^{2} + {(e_{2}^{i})}^{2} + {(e_{3}^{i})}^{2})$ (1.14)

对上式进行求导，将(1.8)、(1.9)、(1.10)、(1.11)、(1.12)代入可得：

$\begin{matrix} \dot{v} = e_{1}^{i} {\dot{e}}_{1}^{i} + e_{2}^{i} {\dot{e}}_{2}^{i} + e_{3}^{i} {\dot{e}}_{3}^{i} \\ = (x_{2}^{i} - x_{1}^{1}) ({\dot{x}}_{2}^{i} - {\dot{x}}_{1}^{1}) + (y_{2}^{i} - y_{1}^{1}) ({\dot{y}}_{2}^{i} - {\dot{y}}_{1}^{1}) + (z_{2}^{i} - z_{1}^{1}) ({\dot{z}}_{2}^{i} - {\dot{z}}_{1}^{1}) \\ = e_{1}^{i} (- σ x_{2}^{i} + σ y_{2}^{i} + a x_{1}^{1} - a y_{1}^{1} + Δ Φ_{1}^{i} + U_{1}^{i}) \\ + e_{2}^{i} [γ x_{2}^{i} - x_{2}^{i} z_{2}^{i} + (d_{2} - 1) y_{2}^{i} + d_{2} y_{2}^{i} (t - τ_{2}) - (c - a) x_{1}^{1} \end{matrix}$

$\begin{matrix} + x_{1}^{1} z_{1}^{1} - (c - d_{1}) y_{1}^{1} - d_{1} y_{1}^{1} (t - τ_{1}) + Δ Φ_{2}^{i} + U_{2}^{i}] \\ + e_{3}^{i} (x_{2}^{i} y_{2}^{i} - β z_{2}^{i} - x_{1}^{1} y_{1}^{1} + b z_{1}^{1} + Δ Φ_{3}^{i} + U_{3}^{i}) \\ = - k_{1} {(e_{1}^{i})}^{2} - k_{2} {(e_{2}^{i})}^{2} - k_{3} {(e_{3}^{i})}^{3} \\ \leq 0 \end{matrix}$ (1.15)

由上式可知，Lyapunov函数(1.14)的解是渐进稳定的，也就是说，网络可以实现同步。

4. 数值模拟

我们采用上述工作中由具有12个节点的WS小世界和具有14个节点的NW小世界构建成的具有结构洞的复杂网络进行数值模拟。下面给出所构建的具有结构洞的复杂网络的耦合矩阵：

$C = [\begin{matrix} A & \begin{matrix} 0 \\ 1 \\ ⋮ \\ 0 \end{matrix} & \begin{matrix} 0 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & 0 \end{matrix} \\ \begin{matrix} 0 & 1 & \dots & 0 \end{matrix} & - 2 & \begin{matrix} 1 & 0 & \dots & 0 \end{matrix} \\ \begin{matrix} 0 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 \end{matrix} & \begin{matrix} 1 \\ 0 \\ ⋮ \\ 0 \end{matrix} & B \end{matrix}]$ (1.16)

其中A、B分别为：

$A = [\begin{matrix} - 3 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & - 6 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & - 3 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & - 5 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 3 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & - 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & - 5 & 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & - 4 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & - 4 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & - 5 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & - 4 & 1 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & - 5 \end{matrix}]$ (1.17)

$B = [\begin{matrix} - 7 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\ 1 & - 5 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & - 5 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & - 5 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & - 4 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & - 4 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & - 4 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & - 5 & 1 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & - 5 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & - 6 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & - 5 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & - 4 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & - 4 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & - 4 \end{matrix}]$ (1.18)

当时滞Chen混沌系统参数设为 $a = 35$ ， $b = 3$ ， $c = 28$ ， $d_{1} = 3$ ， $τ_{1} = 0.4$ ，所有时滞Lorenz混沌系统参数设为 $σ = 10$ ， $β = 8 / 3$ ， $γ = 28$ ， $d_{2} = 6.5$ ， $τ_{2} = 0.6$ ，节点初值的取值为0~5的随机数，网络的耦合强度 $ε = 0.1$ ，设置控制器中的参数 $k_{1} = k_{2} = k_{3} = 0.3$ 。用总误差 $E (t) = ‖ e (t) ‖$ 来度量具有结构洞的异结构时滞复杂网络的总同步误差演化过程。

通过图2可以看出，在施加控制后，网络中各节点的运动状态逐渐与目标节点达成一致。从图3可知在施加控制后，网络中各节点与目标节点的同步误差经过一段时间后趋于零，从图4中我们可以知道，网络的总同步误差逐渐趋于零。那么也就说明了网络达到了完全同步。

Figure 2. Evolution of the state variables of each node in the network

图2. 网络中各节点的状态变量演化图

Figure 3. Evolution of the synchronization error variables of the network

图3. 网络的同步误差变量演化图

Figure 4. The total synchronization error E(t) of the network evolves with time

图4. 网络的总同步误差E(t)随时间演化图

5. 结论

本文研究了具有结构洞的异结构时滞复杂网络的同步问题。借鉴社会网络研究中的结构洞概念，以WS小世界与NW小世界网络为基础构建了一个具有结构洞的复杂网络，在网络的1号节点上放置了时滞的Chen混沌系统，网络中的其他节点上放置时滞的Lorenz混沌系统，根据Lyapunov稳定性定理上设计了合适的控制器，通过对网络节点施加控制实现了网络的完全同步。最后进行数值模拟，发现数值模拟结果与理论分析相吻合，证明了该同步的可行性。

文章引用

王晨阳. 具有结构洞的异结构时滞复杂网络同步
Synchronization of Time-Delay Complex Networks with Structural Hole and Different Structure[J]. 应用数学进展, 2022, 11(04): 2156-2164. https://doi.org/10.12677/AAM.2022.114232

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