Mechanisms of pattern formation-of which the Turing instability is an archetype-constitute an important class of dynamical processes occurring in biological, ecological and chemical systems. Recently, it has been shown that the Turing instability can induce pattern formation in discrete media such as complex networks, opening up the intriguing possibility of exploring it as a generative mechanism in a plethora of socioeconomic contexts. Yet, much remains to be understood in terms of the precise connection between network topology and its role in inducing the patterns. Here, we present a general mathematical description of a two-species reaction-diffusion process occurring on different flavors of network topology. The dynamical equations are of the predator-prey class, that while traditionally used to model species population, has also been used to model competition between antagonistic ideas in social systems. We demonstrate that the Turing instability can be induced in any network topology, by tuning the diffusion of the competing species, or by altering network connectivity. The extent to which the emergent patterns reflect topological properties is determined by a complex interplay between the diffusion coefficients and the localization properties of the eigenvectors of the graph Laplacian. We find that networks with large degree fluctuations tend to have stable patterns over the space of initial perturbations, whereas patterns in more homogenous networks are purely stochastic. arXiv:1903.03845v1 [cond-mat.dis-nn]
Sociolinguistic phenomena often involve interactions across different scales and result in social and linguistic changes that can be tracked over time. Here, we focus on the dynamics of language shift in Galicia, a bilingual community in northwest Spain. Using historical data on Galician and Spanish speakers, we show that the rate at which shift dynamics unfold correlates inversely with the internal complexity of a region (approximated by the proportion of urban area). Less complex areas converge faster to steady states, while more complex ones sustain transitory dynamics longer. We further explore the contextual relevance of each region within the network of regions that constitute Galicia. The network is observed to sustain or reverse the dynamic rates. This model can introduce a competition between the internal complexity of a region and its contextual relevance in the network. Harnessing these sociodynamic features may prove useful in policy making to limit conflicts.
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