Mathematical analysis of nonlocal PDEs for network generation

Technical University of Munich, Faculty of Mathematics, Boltzmannstr. 3, 85748 Garching b. München, Germany.

^* Corresponding author: ckuehn@ma.tum.de

Received: 28 December 2018
Accepted: 27 November 2019

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.