Manageable to unmanageable transition in a fractal model of project networks

TL;DR

This work analyzes delay propagation in project networks through a percolation lens and introduces a duplication-split growth model to generate fractal directed networks. It establishes fractal scaling in these networks: $\langle d\rangle \sim N^{\beta}$ with $0<\beta<1$ and $\langle N\rangle_d \sim d^{D_f}$ where $D_f=1/\beta>1$, showing that the structure is fractal rather than small-world. A key result is the existence of a critical duplication rate $q_c=1/2$ separating regimes: for $q<q_c$, the percolation threshold is $p_c=1$ (manageable), while for $q>q_c$ there is a finite $0<p_c<1$ (unmanageable) despite diverging $\langle k_{out}\rangle$. This contrast with random directed networks highlights the impact of fractal structure on intervention strategies and provides guidance for designing projects to minimize delay cascades.

Abstract

Project networks are characterized by power law degree distributions, a property that is known to promote spreading. In contrast, the longest path length of project networks scales algebraically with the network size, which improves the impact of random interventions. Using the duplication-split model of project networks, I provide convincing evidence that project networks are fractal networks. The average distance between nodes scales as $\langle d\rangle \sim N^β$ with $0<β<1$. The average number of nodes $\langle N\rangle_d$ within a distance $d$ scales as $\langle N\rangle_d\sim d^{D_f}$, with a fractal dimension $D_f=1/β>1$. Furthermore, I demonstrate that the duplication-split networks are fragile for duplication rates $q<q_c=1/2$: The size of the giant out-component decreases with increasing the network size for any site occupancy probability less than 1. In contrast, they exhibit a non trivial percolation threshold $0<p_c<1$ for $q>q_c$, in spite the mean out-degree diverges with increasing the network size. I conclude the project networks generated by the duplication-split model are manageable for $q<q_c$ and unmanageable otherwise.

Figures (5)

• Figure 1: Activity networks generated by the duplication-split model with 100 nodes and A) $q=0.1$, B) $q=0.3$ and C) $q=0.6$. The nodes layout was generated with graphviz.
• Figure 2: Scaling of the average distance $\langle d\rangle$ from root nodes to reachable nodes with the network size $N$. The symbols represent an average over all roots in the network and 100 network realizations. The lines are a fit to a power law scalings. The error bars represent the standard deviation, including variations between nodes in the same network and across network realizations.
• Figure 3: Average number of nodes scaling with the distance in duplication-split networks. A) $q=0.3$ and average over 10 networks. B) $q=0.6$ and one network.
• Figure 4: The value of $\beta$ for different values of $q$, as estimated from the slope of $\log \langle d\rangle$ vs $\log N$. The two extreme points at $q=0$ and $1$ are the expectations for a linear chain and a network where the distance does not change with increasing the network size, respectively. For $q=0.8$ and 0.9 we could not finish the analysis for the network size $N=10^6$. This large or larger networks sizes are required due to the very slow increase of $\langle d\rangle$ with $N$ when $q$ approaches 1.
• Figure 5: Fraction of nodes in the giant out-component $G$ and its size-dependent slope $dG/dN$ as a function of the node occupancy $p$, for duplication-split networks with different duplication rates $q$ and network size $N$. All data points represent averages over 10 networks and 10 sets of node occupancies.