Graph-Theoretic Method on Topology Identification of Stochastic Multi-weighted Complex Networks with Time-Varying Delayed Coupling Based on Adaptive Synchronization

Abstract

This paper is concerned with topology identification of stochastic multi-weighted complex networks based on adaptive synchronization. Different from previous work, time-varying delayed coupling is taken into account. By using Lyapunov method, graph theory and LaSalle-type invariance principle for stochastic differential equations, the whole topological structures and partial topological structures are successfully identified under adaptive control and adaptive pinning control respectively. Based on drive-response concept, response network can reach synchronization with drive network. In the end, two numerical examples are provided to demonstrate the effectiveness and correctness of our theoretical results, where typical Lü chaotic system is used to describe the vertex’s dynamical behavior.

Appendix

The initial conditions $\phi _{i}$ and $\psi _{i}$ for drive system (26) and response system (27) are shown as follows.

$$\begin{aligned}&\psi _{1}(s)=(-0.5\cos s+10,\psi _{12}(s),\psi _{13}(s))^{\mathrm {T}},\\&\phi _{1}(s)=(0.4\cos s+6,0.3\cos s+5,0.2\cos s+7)^{\mathrm {T}},\\&\psi _{2}(s)=(-0.4\cos s+10,\psi _{22}(s),\psi _{23}(s))^{\mathrm {T}},\\&\phi _{2}(s)=(0.4\cos s+6.5,0.3\cos s+5.5,0.2\cos s+7.5)^{\mathrm {T}},\\&\psi _{3}(s)=(-0.5\cos s+10,\psi _{32}(s),\psi _{33}(s))^{\mathrm {T}},\\&\phi _{3}(s)=(0.5\cos s+7,0.3\cos s+6,0.2\cos s+7.5)^{\mathrm {T}},\\&\psi _{4}(s)=(-0.2\cos s+10,\psi _{42}(s),\psi _{43}(s))^{\mathrm {T}},\\&\phi _{4}(s)=(0.5\cos s+7.5,0.3\cos s+6.5,0.2\cos s+8.5)^{\mathrm {T}},\\&\psi _{5}(s)=(-0.6\cos s+10,\psi _{52}(s),\psi _{53}(s))^{\mathrm {T}},\\&\phi _{5}(s)=(0.6\cos s+8,0.3\cos s+7,0.2\cos s+9)^{\mathrm {T}},\\&\psi _{6}(s)=(-0.6\cos s+10,\psi _{62}(s),\psi _{63}(s))^{\mathrm {T}},\\&\phi _{6}(s)=(0.6\cos s+8.5,0.3\cos s+7.5,0.2\cos s+9.5)^{\mathrm {T}},\\&\psi _{7}(s)=(-0.7\cos s+10,\psi _{72}(s),\psi _{73}(s))^{\mathrm {T}},\\&\phi _{7}(s)=(0.7\cos s+9,0.3\cos s+8,0.2\cos s+10)^{\mathrm {T}},\\&\psi _{8}(s)=(-0.7\cos s+10,\psi _{82}(s),\psi _{83}(s))^{\mathrm {T}},\\&\phi _{8}(s)=(0.7\cos s+9.5,0.3\cos s+8.5,0.2\cos s+10.5)^{\mathrm {T}},\\&\psi _{9}(s)=(-0.8\cos s+10,\psi _{92}(s),\psi _{93}(s))^{\mathrm {T}},\\&\phi _{9}(s)=(0.8\cos s+10,0.5\cos s+9,0.2\cos s+11)^{\mathrm {T}},\\&\psi _{10}(s)=(-0.8\cos s+10,\psi _{102}(s),\psi _{103}(s))^{\mathrm {T}},\\&\phi _{10}(s)=(0.8\cos s+10.5,0.3\cos s+9.5,0.2\cos s+11.5)^{\mathrm {T}},\\ \end{aligned}$$

in which $\psi _{i2}(s)=-0.3\cos s+10$, $\psi _{i3}(s)=-0.2\cos s+11$ for $i=1,2,\ldots ,10$.