Deduction of the Bromilow's time-cost model from the fractal nature of activity networks

In 1969 Bromilow observed that the time $T$ to execute a construction project follows a power law scaling with the project cost $C$, $T\sim C^B$ [Bromilow 1969]. While the Bromilow's time-cost model has been extensively tested using data for different countries and project types, there is no theoretical explanation for the algebraic scaling. Here I mathematically deduce the Bromilow's time-cost model from the fractal nature of activity networks. The Bromislow's exponent is $B=1-\alpha$, where $1-\alpha$ is the scaling exponent between the number of activities in the critical path $L$ and the number of activities $N$, $L\sim N^{1-\alpha}$ with $0\leq\alpha<1$ [Vazquez et al 2023]. I provide empirical data showing that projects with low serial/parallel (SP)% have lower $B$ values than those with higher SP%. I conclude that the Bromilow's time-cost model is a law of activity networks, the Bromilow's exponent is a network property and forecasting project duration from cost should be limited to projects with high SP%.