Explicit Solutions to the Continuous Time Albert-Barab\'asi Scale-Free Model

TL;DR

This paper derives explicit solutions for the continuous time Albert-Barabási scale-free network model by solving differential equations recursively, providing a clear mathematical understanding of the network's evolution over time.

Contribution

The paper introduces a novel recursive method to explicitly solve the differential equations governing the continuous time Albert-Barabási model, enhancing analytical understanding.

Findings

Explicit solutions for the model at any time t

Recursive solution method for differential equations

Mathematical proof of the solution pattern

Abstract

Presented here is the explicit solution to the continuous time approximation of the Albert-Barab\'asi scale-free network model at any time t. The solution is found by recursively solving the differential equations via integrating factors, identifying a pattern for the coefficients and then proving this observed pattern using induction.