The asymptotic behavior of the Altafini model on signed graphons

The Altafini model, also known as opposing dynamics, is an opinion dynamics model on signed graphs. In this paper, we study an extension of this model to signed graphons, mathematical objects that approximate large undirected graphs. After defining natural extensions of the notions of connectivity and structural balance to signed graphons, we analyze the asymptotic behavior of the dynamics, through the use of semigroup theory and the study of the properties of the opposing Laplacian operator. Our results are consistent with known facts about the classical Altafini model on signed graphs: on a connected signed graphon with degree essentially bounded away from zero, the dynamics converges to a bipartite consensus if the signed graphon is structurally balanced, and converges to zero if it is not.