Robustness of flow networks against cascading failures under partial load redistribution

We study the robustness of flow networks against cascading failures under a partial load redistribution model. In particular, we consider a flow network of $N$ lines with initial loads $L_1, \ldots, L_N$ and free-spaces (i.e., redundant space) $S_1, \ldots, S_N$ that are independent and identically distributed with joint distribution $P_{LS}(x,y)=\mathbb{P}(L \leq x, S \leq y)$. The capacity $C_i$ is the maximum load allowed on line $i$, and is given by $C_i=L_i + S_i$. When a line fails due to overloading, it is removed from the system and $(1-\varepsilon)$-fraction of the load it was carrying (at the moment of failing) gets redistributed equally among all remaining lines in the system; hence we refer to this as the {\it partial} load redistribution model. The rest (i.e., $\varepsilon$-fraction) of the load is assumed to be lost or absorbed, e.g., due to advanced circuitry disconnecting overloaded power lines or an inter-connected network/material absorbing a fraction of the flow from overloaded lines. We analyze the robustness of this flow network against random attacks that remove a $p$-fraction of the lines. Our contributions include (i) deriving the final fraction of alive lines $n_{\infty}(p,\varepsilon)$ for all $p, \varepsilon \in (0,1)$ and confirming the results via extensive simulations; (ii) showing that partial redistribution might lead to (depending on the parameter $0<\varepsilon \leq 1$) the order of transition at the critical attack size $p^{*}$ changing from first to second-order; and (iii) proving analytically that flow networks achieve maximum robustness (quantified by the area $\int_{0}^{1} n_{\infty}(p,\varepsilon) \mathrm{d}p$) when all lines have the same free-space regardless of their initial load. The optimality of equal free-space allocation is also confirmed on real-world data from the UK National Power Grid.