General Connectivity Distribution Functions for Growing Networks with Preferential Attachment of Fractional Power

TL;DR

This paper introduces a unified framework for analyzing connectivity distributions in growing networks with fractional power preferential attachment, utilizing a novel transidental function called upsilon function.

Contribution

It develops a new fractional differential equation approach and defines the upsilon function to solve it, unifying previous results in network growth models.

Findings

Derived a fractional differential equation for fractional power preferential attachment.

Introduced the upsilon function to solve the fractional differential equation.

Unified previous results within a new theoretical framework.

Abstract

We study the general connectivity distribution functions for growing networks with preferential attachment of fractional power, $\Pi_{i} \propto k^{\alpha}$, using the Simon's method. We first show that the heart of the previously known methods of the rate equations for the connectivity distribution functions is nothing but the Simon's method for word problem. Secondly, we show that the case of fractional $\alpha$ the $Z$-transformation of the rate equation provides a fractional differential equation of new type, which coincides with that for PA with linear power, when $\alpha = 1$. We show that to solve such a fractional differential equation we need define a transidental function $\Upsilon (a,s,c;z)$ that we call {\it upsilon function}. Most of all previously known results are obtained consistently in the frame work of a unified theory.