Mathematical Analysis of Nonlocal PDEs for Network Generation

TL;DR

This paper rigorously analyzes a class of nonlocal PDEs modeling network generation, transforming them into local PDEs, proving solution properties, and exploring steady states with numerical validation.

Contribution

It introduces a novel method to convert nonlocal PDEs into local PDEs for network generation, providing new insights into solution regularity and steady state stability.

Findings

Proved solvability and regularity of the PDE solutions.

Reduced steady state analysis to an implicit ODE.

Numerical simulations confirm stability of steady states.

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.