Convergence analysis of the time-varying discrete-time Altafini model on signed network

This paper mainly studies the convergence of time-varying discrete-time Altafini model. Firstly, we convert the bipartite consensus of the Altafini model with n nodes into consensus of the DeGroot model with $ 2n $ 2n nodes by adopting lifting approach. Secondly, based on structural balance theory and node relabelling theory, the consensus of enlarged DeGroot model can be further equivalently converted into the consensus of one simple DeGroot model with n nodes. Thirdly, by using the equivalent transformation and the ergodic coefficient theory, we change the sufficient condition from ‘repeatedly jointly strongly connected’ graphs to ‘repeatedly jointly scrambling’ graphs, and verify that the new condition also can lead to the bipartite consensus of the Altafini model. This novel criterion allows for determining the bipartite consensus of the Altafini model without the condition of ‘repeatedly jointly strongly connected’ graphs. Next, we provide a concise proof for the case of opinions converging to 0 with the condition of ‘repeatedly jointly strongly connected’ by the ergodic coefficient theory. Finally, we develop a reasonable coevolution model to illustrate the effectiveness and strong applicability of our main theoretical results.