# Structural instability of large-scale functional networks

## TL;DR

This paper investigates the conditions under which large-scale functional networks remain stable against cascading overload failures, identifying key parameters that influence maximum stable size and differences between network types.

## Contribution

It introduces a model for cascading failures with fluctuating loads, quantifies maximum stable network size as a function of load reduction rate, and compares stability across network types.

## Key findings

- Fast load reduction ($r \,\ge\, r_c$) allows infinite network growth.
- Maximum stable size $n_{max}$ increases with load reduction rate $r$.
- Scale-free networks have larger $n_{max}$ than exponential networks at the same average degree.

## Abstract

We study how large functional networks can grow stably under possible cascading overload failures and evaluated the maximum stable network size above which even a small-scale failure would cause a fatal breakdown of the network. Employing a model of cascading failures induced by temporally fluctuating loads, the maximum stable size $n_{\text{max}}$ has been calculated as a function of the load reduction parameter $r$ that characterizes how quickly the total load is reduced during the cascade. If we reduce the total load sufficiently fast ($r\ge r_{\text{c}}$), the network can grow infinitely. Otherwise, $n_{\text{max}}$ is finite and increases with $r$. For a fixed $r\,(<r_{\text{c}})$, $n_{\text{max}}$ for a scale-free network is larger than that for an exponential network with the same average degree. We also discuss how one detects and avoids the crisis of a fatal breakdown of the network from the relation between the sizes of the initial network and the largest component after an ordinarily occurring cascading failure.