# The dynamics of power laws: Fitness and aging in preferential attachment trees

## TL;DR

This paper models the evolution of networks using continuous-time branching processes incorporating fitness and aging, revealing how these factors influence the degree distribution's tail behavior, with implications for understanding citation networks.

## Contribution

It introduces a new model combining fitness and aging in preferential attachment trees, analyzing how these factors affect the degree distribution's tail behavior.

## Key findings

- Power-law tails are preserved with unbounded fitness and aging.
- Exponential tails occur with bounded fitness and aging.
- Processes without aging can become explosive.

## Abstract

Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive.