# Longest Path in the Price Model

## TL;DR

This paper analyzes the longest paths in directed acyclic graphs generated by the Price model, showing they scale logarithmically with network size and depend on attachment methods.

## Contribution

It introduces a reverse greedy path concept and provides analytical and numerical evidence for the logarithmic scaling of longest paths in Price model variants.

## Key findings

- Reverse greedy path scales with log of network size
- Longest path length depends on attachment method
- Numerical results confirm analytical predictions

## Abstract

The Price model, the directed version of the Barab\'{a}si-Albert model, produces a growing directed acyclic graph. We look at variants of the model in which directed edges are added to the new vertex in one of two ways: using cumulative advantage (preferential attachment) choosing vertices in proportion to their degree, or with random attachment in which vertices are chosen uniformly at random. In such networks, the longest path is well defined and in some cases is known to be a better approximation to geodesics than the shortest path. We define a reverse greedy path and show both analytically and numerically that this scales with the logarithm of the size of the network with a coefficient given by the number of edges added using random attachment. This is a lower bound on the length of the longest path to any given vertex and we show numerically that the longest path also scales with the logarithm of the size of the network but with a larger coefficient that has some weak dependence on the parameters of the model.