Asymptotic Network Robustness
Tuhin Sarkar, Mardavij Roozbehani, Munther A. Dahleh

TL;DR
This paper investigates how network performance measures, specifically energy and tail risk, scale with network size, revealing fundamental differences between directed and undirected networks and connecting control theory with economic risk analysis.
Contribution
It establishes a connection between energy metrics and tail risk in networks, providing new insights into how network topology influences performance deterioration at large scales.
Findings
Undirected networks have the most graceful energy growth rates.
Directed networks can exhibit exponentially faster energy growth.
Topology, not specific measures, governs large-scale network behavior.
Abstract
This paper examines the dependence of network performance measures on network size and considers scaling results for large networks. We connect two performance measures that are well studied, but appear to be unrelated. The first measure is concerned with energy metrics, namely the --norm of a network, which arises in control theory applications. The second measure is concerned with the notion of "tail risk" which arises in economic and financial networks. We study the question of why such performance measures may deteriorate at a faster rate than the growth rate of the network. We first focus on the energy metric and its well known connection to controllability Gramian of the underlying dynamical system. We show that undirected networks exhibit the most graceful energy growth rates as network size grows. This rate is quantified completely by the proximity of spectral radius to…
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