The nonlinear heat equation on W-random graphs

For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in [9] justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs

Small-world networks of Kuramoto oscillators

The Kuramoto model of coupled phase oscillators on small-world (SW) graphs is analyzed in this work. When the number of oscillators in the network goes to infinity, the model acquires a family of steady state solutions of degree q, called q-twisted states. We show that this class of solutions plays an important role in the formation of spatial patterns in the Kuramoto model on SW graphs. In particular, the analysis of q-twisted elucidates the role of long-range random connections in shaping the attractors in this model. We develop two complementary approaches for studying q-twisted states in the coupled oscillator model on SW graphs: the linear stability analysis and the numerical continuation. The former approach shows that long-range random connections in the SW graphs promote synchronization and yields the estimate of the synchronization rate as a function of the SW randomization parameter. The continuation shows that the increase of the long-range connections results in patterns consisting of one or several plateaus separated by sharp interfaces. These results elucidate the pattern formation mechanisms in nonlocally coupled dynamical systems on random graphs

Parabolic theory of the discrete p-Laplace operator

We study the discrete version of the $p$-Laplacian. Based on its variational properties we discuss some features of the associated parabolic problem. Our approach allows us in turn to obtain interesting information about positivity and comparison principles as well as compatibility with the symmetries of the graph. We conclude briefly discussing the variational properties of a handful of nonlinear generalized Laplacians appearing in different parabolic equations.Comment: 35 pages several corrections and enhancements in comparison to the v