Initial State Causes the Structural Balance of Complex Networks With Dynamical Models

Abstract

What initial conditions may lead the dynamic complex network into structural balance? In the sense of social network, the solution to this problem explains that the initial connection relations between individuals are one of the reasons to achieve social stability. In order to obtain the solution, the special Riccati differential equation is chosen as the mathematic model of the dynamical complex network in this paper. Different from the existing results, we focus on the exact solution to the Riccati differential equation and draw the mathematic initial conditions from the exact solution. By using the existing results about the real logarithm of the matrix, the unique exact solution is obtained under the given initial conditions. This solution is a real continuous and differentiable symmetric matrix, and it is seen from which that how the initial state can lead to approximate asymptotically the structural balance more clearly. It turns out that the plus or minus sign of eigenvalues of the initial link matrix affects the asymptotic behavior of the solution, and the maximum positive eigenvalue and its eigenvector with nonzero entries play a key role in approximating asymptotically the structural balance. Finally, the numerical simulations show the validity of methods in this paper.