Mathematical analysis of nonlocal PDEs for network generation

Abstract

In this paper, we study a certain class of nonlocal partial differential equations (PDEs). The equations arise from a key problem in network science, i.e., network generation from local interaction rules, which result in a change of the degree distribution as time progresses. The evolution of the generating function of this degree distribution can be described by a nonlocal PDE. To address this equation we will rigorously convert it into a local first order PDE. Then, we use theory of characteristics to prove solvability and regularity of the solution. Next, we investigate the existence of steady states of the PDE. We show that this problem reduces to an implicit ODE, which we subsequently analyze. Finally, we perform numerical simulations, which show stability of the steady states.

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

“…Among others, such models as nonlocal Fisher-KPP (Kolmogorov-Petrovskii-Piskunov) and Lotka-Volterra are now firmly established as indispensable tool for modelling in these areas [24,25]. Additionally, other sources of nonlocal models are coming from convection-diffusion and reaction-diffusion systems of various types and their considerable range of applications in sciences and engineering [26], as well as from new applications in complex systems and dynamic networks [27][28][29]. This popularity is largely due to the fact that nonlocal models can handle complex systems involving nonsmooth and possible (discontinuous) singular solutions.…”

“…It follows the main idea of the numerical technique from [203]. Specifically, in order to regularize problem (1), (28), we perturb the quasi-reversibility condition with a term εψ(0) which, then, transforms to…”