Imagine the infrastructure systems around us, such as the power grids or communication networks, collapsing due to a minor disruption. What if a small local failure could trigger a catastrophic collapse affecting the entire system? While this might sound implausible, our recent research explores precisely this scenario and reveals its high chance to occur and discover fascinating insights that could reshape how we understand and design such critical systems.
The concept and importance of k-core percolation
In many real-world systems, maintaining tight connections among fundamental units is essential for effective operation, often exemplified by k-core structures. A k-core structure in a network is defined as a subgraph where each node is connected to at least k other nodes within the subgraph. This characteristic — ensuring that each node maintains a minimum number of connections — is crucial for the stability, resilience, and overall efficiency of various practical systems, such as transportation networks, communication infrastructures, and social and biological systems.
The concept of k-core percolation involves studying the conditions under which tightly connected structures can withstand cascading failure under random removal of nodes. By examining how the k-core structure changes as nodes are gradually removed, we can identify a critical point: above this point, the network maintains its k-core structure; below it, cascading failure occur, and the structure disintegrates. This approach helps estimate the network’s resilience to failures or attacks by identifying critical thresholds and observing critical behaviors near these points. Understanding this phase transition and the mechanism behind these phenomena is crucial for designing more robust systems and revealing the origins of the complex abrupt collapses observed in many real-world systems.
Previous works mainly focus on nonspatial random networks and reveal intriguing mixed-order transitions. Given the widespread existence of spatial characteristics in real-world systems, understanding cascading failures is particularly crucial for spatially embedded networks. Our research highlights that physical connection lengths significantly impact network robustness and lead to further complex transition behaviors, such as nucleation, which differ markedly from k-core percolation in random networks.
Key finding on k-core percolation in spatially embedded networks
Through numerous simulations of k-core percolation in spatially embedded networks, our research uncovered four distinct types of phase transitions governed by the characteristic spatial length of the links ζ:
Small ζ: When the link lengths are short, damage remains localized, and the transition is continuous and smooth near the critical point. This results into fractal structures, as shown in Fig.1(b).
Intermediate ζ: For moderate link lengths, the transition involves an abrupt system collapse  due to radial propagation of small, damaged holes, analogous to nucleation phenomenon, as illustrated in Fig. 1(d).
Large ζ: For long link lengths, the transition becomes mixed order, similar to random networks, displaying both abrupt jumps and scaling behavior near the critical point. This process is driven by random locations of cascading failures, ultimately leading to a cascading avalanche, as depicted in Fig.1(c).
Novel metastable phase: a newly discovered regime where localized microscopic damage above a certain critical size can spontaneously propagate until the system fully collapses. If the damage is below this size, the collapse does not occur. This phenomenon is highlighted in Fig. 2.
Figure1  Changes in link length ζ  lead to three distinct types of  phase transition in 5-core percolation.  (a) The relationship between the value of critical point and link length.(b)(c) Giant connected component before the collapse of the spatially embedded network with (b) a continuous phase transition at ζ=4, and (c) a mixed-order phase transition at ζ=100. (d)  Evolution of the giant connected component  with ζ=10 (first-order phase transition).
One of the most significant findings was the existence of the metastable phase, where a microscopic, localized attack anywhere in the system can cause a macroscopic network collapse. This discovery suggests that spatially embedded networks, such as those in critical infrastructure, might be significantly more vulnerable than previously thought. This novel finding is not merely academic—it has real-world implications for designing and maintaining more robust systems.Â
Figure2 Illustration of the propagation of damage initiated from localized attacks with a hole in the spatial network. The hole at the bottom is smaller than the critical size and does not propagate, thus maintaining the same size as the initial state. However, damage caused by holes larger than the critical size at the center evolves and spreads radially throughout the system over time.
Broader Implications
Our findings reveal novel characteristics and significant vulnerabilities of spatially embedded k-core systems, emphasizing the importance of considering the characteristic link length when designing robust spatial networks. The existence of a metastable phase indicates that critical infrastructure networks are far more vulnerable than previously believed. This vulnerability necessitates a reevaluation of design and maintenance strategies to enhance robustness and prevent catastrophic failures. Additionally, our understanding of the microscopic processes and their origins during mixed-order and first-order abrupt transitions in k-core networks may provide insights into the mechanisms of many systems experiencing similar transitions. These insights underscore the need for a comprehensive framework to address the vulnerabilities inherent in spatially embedded networks, protecting policies and practices in network design, urban planning, and infrastructure resilience. This research paves the way for developing more resilient and adaptive networks capable of withstanding localized attacks without cascading failures.