Robustness and resilience of complex networks

TL;DR

The paper surveys a unified framework for robustness and resilience in complex networks, focusing on how perturbations propagate and how to mitigate systemic collapse. It integrates static percolation analyses, optimal dismantling algorithms, cascading-failure models in multilayer networks, and strategies for prevention, early warning, and adaptive recovery. Key contributions include a synthesis of percolation-based and message-passing methods, a comparative evaluation of state-of-the-art dismantling algorithms, and guidance on applying these tools to real-world networks with diverse topologies. The work is complemented by a public code repository to facilitate replication and practical deployment of robust design and response strategies.

Abstract

Complex networks are ubiquitous: a cell, the human brain, a group of people and the Internet are all examples of interconnected many-body systems characterized by macroscopic properties that cannot be trivially deduced from those of their microscopic constituents. Such systems are exposed to both internal, localized, failures and external disturbances or perturbations. Owing to their interconnected structure, complex systems might be severely degraded, to the point of disintegration or systemic dysfunction. Examples include cascading failures, triggered by an initially localized overload in power systems, and the critical slowing downs of ecosystems which can be driven towards extinction. In recent years, this general phenomenon has been investigated by framing localized and systemic failures in terms of perturbations that can alter the function of a system. We capitalize on this mathematical framework to review theoretical and computational approaches to characterize robustness and resilience of complex networks. We discuss recent approaches to mitigate the impact of perturbations in terms of designing robustness, identifying early-warning signals and adapting responses. In terms of applications, we compare the performance of the state-of-the-art dismantling techniques, highlighting their optimal range of applicability for practical problems, and provide a repository with ready-to-use scripts, a much-needed tool set.

Figures (4)

• Figure 1: Percolation as static approach to network robustness. (a) Sketch of a network disintegration due to the removal of subsets of nodes, chosen according to a predefined protocol of removal $\phi$. Node removal predominates in this review, yet most of the concepts, metrics and techniques discussed throughout can be easily framed for link removal. The nature of $\phi$, related to the control parameter, heavily impacts on the response of the network, resulting in different sorts of phase transitions (b-d) for the size of the giant (largest) component, see the main text for details. In (e), we sketch how networks with homogeneous and heterogeneous connectivity patterns (roughly speaking, degree distributions with a second moment of similar order to the square of the mean or much larger than the mean, respectively) respond to failures and targeted attacks. Circles represent isolated components whose radius depend on their number of nodes. Each vertical column assumes the same number of nodes/links removed. Parts of the panel (e) are reproduced from Ref. artime2020effectiveness.
• Figure 2: Comparison of state-of-the-art dismantling methods. (a) The cutting-edge algorithms, described in Section \ref{['sec:network_dismantling']} are compared in terms of their ability to drive the system -- a Brazilian corruption network 10.1093/comnet/cny002 with 309 nodes -- towards disintegration, as measured by the relative size of the largest connected component (LCC). In panels (b)--(g), we display the disintegrating paths of the different curves shown in (a). The color of the nodes represents (from dark red to white) the attack order (i.e., nodes in gray are not removed), while their size represents their betweenness value. The contour color of remaining nodes represents the cluster they belong to after the attack.
• Figure 3: Evolving failures on networks. (a) Risk assessment to cascade spreading in the U.S.-South Canada power grid (from yang2017small). Cartogram with the state of power lines after the simulations of cascade spreading: lines that never underwent outage (green) vs. affected power lines (gray). In the bottom row, snapshots of the evolution of the damaged lines (in yellow) after a cascade triggered by 3 failures at a rescaled time $t=0$. Notice the nonlocal behavior, characteristic of overload cascading failures. (b) Information cascade in Twitter, showing the density of tweets (top row) about the discovery of the Higgs boson before, during, and after the Nobel announcement, as well as the corresponding network of re-shared messages (bottom row). Note the global reaches of the cascade at the peak, compared to the localized behavior of the information spreading before and after the cascading event (from de2013anatomy). (c) Sketch of the evolution of a failure sustained by dependency relations in a toy system with two coupled layers. Directed vertical arrows represent dependency relations. Functional nodes are represented in black; the node that initially fails is indicated in yellow; red and blue nodes indicate removed units because either they do not belong to the largest cluster (red) or they depend on failed nodes in the other network (blue). (d) Size of the giant component as a function of the cascade step. Grey lines correspond to individual realizations of the dynamics, the markers indicated their average. The black line is the theoretical prediction. (e) Stationary size of the giant component as a function of the fraction of initially removed nodes. The different markers correspond to different number of coupled layers, and solid lines show the theoretical predictions. Same for panel (f), but the different markers/lines correspond to the fraction of interdependent nodes in a bilayer network. Panels (c-f) are reproduced from gao2012networks. (g) Size of the giant component at the cascade stop for the threshold model introduced by watts2002simple and discussed in the main text, as a function of the threshold $R$ and the mean degree $\langle k \rangle$ of the underlying Erdos-Renyi network. The dashed line indicates the analytical prediction for the cascade emergence. (h) Size of the giant component at the cascade stop for fixed threshold $R=0.18$, as a function of the mean degree. Different markers/lines, from left to right, correspond to initial seeds of $10^{-3}$, $5 \times 10^{-3}$ and $10^{-2}$. Figures (g) and (h) are reproduced from gleeson2007seed. (i) Large load-shedding cascades can be mitigated at a nontrivial intermediate level of interconnectivity $p$ between two networks (golden curve, diamond markers). However, too many or too few interconnections induce larger cascades and become detrimental for the robustness. $T_{aa}$ ($T_{ba}$) is the cascade size unfolded on network $a$ for cascades that started at network $a$ ($b$). $T_a$ is the size of cascades in network $a$ without distinguishing where the cascade begins. Network $a$ and $b$ have $2000$ nodes. Figure from brummitt2012suppressing.
• Figure 4: Preventing and reacting to network collapse. (a) Early-warning signal measured by $\Omega$ (see text) for three distinct network infrastructures (shown in the top row) repeatedly attacked using a degree-based protocol, and size of the largest and second-largest connected components (bottom row) (from grassia2021machine). (b-d) Dynamical changes in a molecular network from lung tissue of mouse, with a critical transition at 8 hours (top) and node color encoding the fluctuation strength in gene expressions which correlates with a dynamical network marker (DNM) introduced in liu2015identifying; DNM has been used for capturing early-warning signals in other systems, such as eutrophic lake states (left) and daily prices of interest-rate swaps in the USD and EUR currency (right). Figure readapted from liu2015identifying. (e-f) Regular network ($k=10$, $m=4$, $N=100$) with a fraction $z$ of active nodes flipping between two collective modes over time (top) and trajectory of the system in the phase diagram, from $t=0$ to the moment of the first transition (bottom), for the marked states. (g-h) An optimal repair strategy Majdandzic2016optimal for a system with two networks, characterized by a fraction $p_{A}^{\star}$ and $p_{B}^{\star}$ of internally failed nodes. Given the initial state $S_i$ of a collapsed system, the repairing corresponds to minimize the distance between $S_i$ and the nearest border of the green region, where the systems goes back to a fully functional state. Arrows indicate the followed trajectories, while $R_{1}$ and $R_{2}$ are triple points. Collective states, identified by marked numbers, for two synthetic (left) and empirical (right) coupled networks are show, in the bottom. (i-k) Restructuring and reigniting: a 2-steps procedure that drives a perturbed yeast protein interaction network from a collapsed phase (red) into the recoverable phase (blue). Figure from sanhedrai2022reviving.