# Limits of Predictability of Cascading Overload Failures in Spatially-Embedded Networks with Distributed Flows

## TL;DR

This paper investigates load-based cascading failures in spatially-embedded networks, proposing resource distribution strategies and analyzing their effectiveness against various failure scenarios, revealing phase transitions and the limited role of topology.

## Contribution

It introduces a stochastic method for optimal heterogeneous resource distribution and compares network stability under different capacity allocation strategies.

## Key findings

- Increased node protection effectively mitigates single node failures.
- Multiple node failures are harder to prevent due to combinatorial complexity.
- A critical tolerance level causes a phase transition in network robustness.

## Abstract

Cascading failures are a critical vulnerability of complex information or infrastructure networks. Here we investigate the properties of load-based cascading failures in real and synthetic spatially-embedded network structures, and propose mitigation strategies to reduce the severity of damages caused by such failures. We introduce a stochastic method for optimal heterogeneous distribution of resources (node capacities) subject to a fixed total cost. Additionally, we design and compare the performance of networks with N-stable and (N-1)-stable network-capacity allocations by triggering cascades using various real-world node-attack and node-failure scenarios. We show that failure mitigation through increased node protection can be effectively achieved against single node failures. However, mitigating against multiple node failures is much more difficult due to the combinatorial increase in possible failures. We analyze the robustness of the system with increasing protection, and find that a critical tolerance exists at which the system undergoes a phase transition, and above which the network almost completely survives an attack. Moreover, we show that cascade-size distributions measured in this region exhibit a power-law decay. Finally, we find a strong correlation between cascade sizes induced by individual nodes and sets of nodes. We also show that network topology alone is a weak factor in determining the progression of cascading failures.