A SPACE–TIME FRACTIONAL REACTION–DIFFUSION MODEL FOR NON-MARKOVIAN MISINFORMATION PROPAGATION: WELL-POSEDNESS AND SPECTRAL SIMULATION

Source abstract

Classical rumour and opinion-dynamics models assume that information passes between adjacent agents and that waiting times between successive contacts are exponentially distributed. Neither assumption survives contact with empirical social-media data, where inter-event times are heavy-tailed and where a single reshare transports a message across long topological distances. We propose a space–time fractional generalization of the Maki–Thompson rumour model in which the temporal evolution is governed by a Caputo derivative of order and spatial transport by the fractional Laplacian , . The order encodes memory in re-engagement; the order encodes non-local, long-range propagation. We formulate the coupled ignorant–spreader–stifler system, establish local existence and uniqueness of mild solutions through a Mittag-Leffler operator representation combined with a Banach fixed-point argument, and extend to global solutions using conservation of total population together with a positivity principle. The model recovers the classical spatial rumour equation as and the mean-field Maki–Thompson system in the spatially homogeneous limit. A Fourier spectral scheme in space, coupled with an L1 discretization in time, is used to simulate propagation on continuous domains and on finite networks through the graph Laplacian . Numerical experiments quantify the joint effect of on peak spreader density and cascade lifetime, and confirm that the temporal order governs the timescale of the outbreak while the spatial order modulates its amplitude.