The paper makes three main contributions:
It proves the existence and uniqueness of solutions to the repelling and opposing opinion dynamics models on signed graphons (Theorem 1).
It provides sufficient conditions for the solutions of the opinion dynamics on graphs of size n to converge, as n goes to infinity, to the solutions of the graphon dynamics (Theorem 2). This convergence result applies when the sequence of graphs converges to a graphon.
It shows that the convergence conditions in Theorem 2 apply to large random graphs sampled from signed graphons, as long as the graphon is piece-wise Lipschitz continuous (Theorem 3).
The paper first recalls the repelling and opposing opinion dynamics models on signed graphs, and then defines their counterparts on signed graphons. It then proves the existence and uniqueness of solutions to the graphon dynamics (Theorem 1).
Next, the paper establishes a convergence result (Theorem 2), showing that the solutions on graphs converge to the solutions on graphons, provided that the initial conditions and the graphon approximation error converge appropriately.
Finally, the paper demonstrates that the convergence conditions in Theorem 2 are satisfied when the graphs are sampled from a piece-wise Lipschitz signed graphon (Theorem 3), using results on the convergence of sampled graphs.
The numerical example illustrates the differences between the solutions on graphs and graphons for the repelling and opposing dynamics, and confirms the convergence of the graph solutions to the graphon solutions as the graph size increases.
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by Paolo Frasca... at arxiv.org 04-15-2024
https://arxiv.org/pdf/2404.08372.pdfDeeper Inquiries