Percolation of attack with tunable limited knowledge

Degree distributions under general node removal: Power-law or Poisson?

Perturbations made to networked systems may result in partial structural loss, such as a blackout in a power-grid system. Investigating the resulting disturbance in network properties is quintessential to understand real networks in action. The removal of nodes is a representative disturbance, but previous studies are seemingly contrasting about its effect on arguably the most fundamental network statistic, the degree distribution. The key question is about the functional form of the degree distributions that can be altered during node removal or sampling. The functional form is decisive in the remaining subnetwork's static and dynamical properties. In this work, we clarify the situation by utilizing the relative entropies with respect to the reference distributions in the Poisson and power-law form, to quantify the distance between the subnetwork's degree distribution and either of the reference distributions. Introducing general sequential node removal processes with continuously different levels of hub protection to encompass a series of scenarios including uniform random removal and preferred or protective (i.e., biased random) removal of the hub, we classify the altered degree distributions starting from various power-law forms by comparing two relative entropy values. From the extensive investigation in various scenarios based on direct node-removal simulations and by solving the rate equation of degree distributions, we discover in the parameter space two distinct regimes, one where the degree distribution is closer to the power-law reference distribution and the other closer to the Poisson distribution.

Feature-enriched core percolation in multiplex networks

Percolation models have long served as a paradigm for unraveling the structure and resilience of complex systems comprising interconnected nodes. In many real networks, nodes are identified by not only their connections but nontopological metadata such as age and gender in social systems, geographical location in infrastructure networks, and component contents in biochemical networks. However, there is little known regarding how the nontopological features influence network structures under percolation processes. In this paper we introduce a feature-enriched core percolation framework using a generic multiplex network approach. We thereby analytically determine the corona cluster, size, and number of edges of the feature-enriched cores. We find a hybrid percolation transition combining a jump and a square root singularity at the critical points in both the network connectivity and the feature space. Integrating the degree-feature distribution with the Farlie-Gumbel-Morgenstern copula, we show the existence of continuous and discrete percolation transitions for feature-enriched cores at critical correlation levels. The inner and outer cores are found to undergo distinct phase transitions under the feature-enriched percolation, all limited by a characteristic curve of the feature distribution.

Percolation of interdependent networks with limited knowledge

Real-world networks are often not isolated and the interdependence between different networks in a complex system is as important as the topological connectivity within individual networks. We develop a theoretical framework to study the robustness of interdependent networks under attacks with limited knowledge. A node may be attacked if it is the most connected node among a given number of potential victims. This number is referred to as the attacker's knowledge level, which joins the two ends, namely, the random failure with zero knowledge and the intentional attack with full knowledge of the network. We introduce percolation models with attacks over one layer and two layers as well as mixed site-bond percolation. Along with the discontinuous phase transition, we show the existence of a critical knowledge level, which indicates a transition of network robustness under the competition between connectivity and interdependence. It is unraveled that interdependent networks can be extremely fragile to the extent that a random failure on two layers would be more deleterious than a targeted attack with full knowledge over one layer. Moreover, we find that a balanced distribution of attack knowledge on both layers tends to be most destructive if the total knowledge is a conserved quantity.

A sampling-guided unsupervised learning method to capture percolation in complex networks

The use of machine learning methods in classical and quantum systems has led to novel techniques to classify ordered and disordered phases, as well as uncover transition points in critical phenomena. Efforts to extend these methods to dynamical processes in complex networks is a field of active research. Network-percolation, a measure of resilience and robustness to structural failures, as well as a proxy for spreading processes, has numerous applications in social, technological, and infrastructural systems. A particular challenge is to identify the existence of a percolation cluster in a network in the face of noisy data. Here, we consider bond-percolation, and introduce a sampling approach that leverages the core-periphery structure of such networks at a microscopic scale, using onion decomposition, a refined version of the k-core. By selecting subsets of nodes in a particular layer of the onion spectrum that follow similar trajectories in the percolation process, percolating phases can be distinguished from non-percolating ones through an unsupervised clustering method. Accuracy in the initial step is essential for extracting samples with information-rich content, that are subsequently used to predict the critical transition point through the confusion scheme, a recently introduced learning method. The method circumvents the difficulty of missing data or noisy measurements, as it allows for sampling nodes from both the core and periphery, as well as intermediate layers. We validate the effectiveness of our sampling strategy on a spectrum of synthetic network topologies, as well as on two real-word case studies: the integration time of the US domestic airport network, and the identification of the epidemic cluster of COVID-19 outbreaks in three major US states. The method proposed here allows for identifying phase transitions in empirical time-varying networks.

Dynamical Recovery of Complex Networks under a Localised Attack

In real systems, some damaged nodes can spontaneously become active again when recovered from themselves or their active neighbours. However, the spontaneous dynamical recovery of complex networks that suffer a local failure has not yet been taken into consideration. To model this recovery process, we develop a framework to study the resilience behaviours of the network under a localised attack (LA). Since the nodes’ state within the network affects the subsequent dynamic evolution, we study the dynamic behaviours of local failure propagation and node recoveries based on this memory characteristic. It can be found that the fraction of active nodes switches back and forth between high network activity and low network activity, which leads to the spontaneous emergence of phase-flipping phenomena. These behaviours can be found in a random regular network, Erdős-Rényi network and Scale-free network, which shows that these three types of networks have the same or different resilience behaviours under an LA and random attack. These results will be helpful for studying the spontaneous recovery real systems under an LA. Our work provides insight into understanding the recovery process and a protection strategy of various complex systems from the perspective of damaged memory.

Immunization of networks with limited knowledge and temporary immunity

Modern view of network resilience and epidemic spreading has been shaped by percolation tools from statistical physics, where nodes and edges are removed or immunized randomly from a large-scale network. In this paper, we produce a theoretical framework for studying targeted immunization in networks, where only n nodes can be observed at a time with the most connected one among them being immunized and the immunity it has acquired may be lost subject to a decay probability ρ. We examine analytically the percolation properties as well as scaling laws, which uncover distinctive characters for Erdős-Rényi and power-law networks in the two dimensions of n and ρ. We study both the case of a fixed immunity loss rate as well as an asymptotic total loss scenario, paving the way to further understand temporary immunity in complex percolation processes with limited knowledge.