Coupling between networks is widely prevalent in real systems and has dramatic effects on their resilience and functional properties. However, current theoretical models tend to assume homogeneous coupling where all the various subcomponents interact with one another, whereas real-world systems tend to have various different coupling patterns. We develop two frameworks to explore the resilience of such modular networks, including specific deterministic coupling patterns and coupling patterns where specific subnetworks are connected randomly. We find both analytically and numerically that the location of the percolation phase transition varies nonmonotonically with the fraction of interconnected nodes when the total number of interconnecting links remains fixed. Furthermore, there exists an optimal fraction $r^{*}$ of interconnected nodes where the system becomes optimally resilient and is able to withstand more damage. Our results suggest that, although the exact location of the optimal $r^{*}$ varies based on the coupling patterns, for all coupling patterns, there exists such an optimal point. Our findings provide a deeper understanding of network resilience and show how networks can be optimized based on their specific coupling patterns. Significance In real-world scenarios, many subnetworks interact with other subnetworks to form a modular interacting network system. In previous models, it was assumed that all subnetworks may be linked pairwise; however, in real systems, only some pairs of subnetworks are connected. Understanding how to develop theoretical frameworks and study system resilience of general coupling patterns of the subnetworks is of significant importance for designing and optimizing economic, social, and infrastructure networks. Here we present two frameworks for such systems having specific coupling patterns to investigate their resilience. Specifically, we find an optimal level of interaction between subnetworks, which maximizes the system’s resilience to failures. Our findings highlight the need to consider real-world coupling patterns and possible optimizations for designing resilient systems.

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.

Robustness is an essential component of modern network science. Here, we investigate the robustness of coupled networks where the functionality of a node depends not only on its connectivity, here measured by the size of its connected component in its own network, but also the support provided by at least M links from another network. We here develop a theoretical framework and investigate analytically and numerically the cascading failure process when the system is under attack, deriving expressions for the proportion of functional nodes in the stable state, and the critical threshold when the system collapses. Significantly, our results show an abrupt phase transition and we derive the minimum inner and inter-connectivity density necessary for the system to remain active. We also observe that the system necessitates an increased density of links inside and across networks to prevent collapse, especially when conditions on the coupling between the networks are more stringent. Finally, we discuss the importance of our results in real-world settings and their potential use to aid decision-makers design more resilient infrastructure systems.

In recent years, implicit learning theory has been one of the focuses of cognitive psychology research, and foreign language learning has important enlightening significance. This article aims to introduce implicit learning and its characteristics, and on this basis, discuss the application of implicit learning in foreign language learning.

Targeted immunization or attacks of large-scale networks has attracted significant attention by the scientific community. However, in real-world scenarios, knowledge and observations of the network may be limited thereby precluding a full assessment of the optimal nodes to immunize (or remove) in order to avoid epidemic spreading such as that of current COVID-19 epidemic. Here, we study a novel immunization strategy where only n nodes are observed at a time and the most central between these n nodes is immunized (or attacked). This process is continued repeatedly until 1 − p fraction of nodes are immunized (or attacked). We develop an analytical framework for this approach and determine the critical percolation threshold p _c and the size of the giant component P _∞; for networks with arbitrary degree distributions P ( k ). In the limit of n → ∞ we recover prior work on targeted attack, whereas for n = 1 we recover the known case of random failure. Between these two extremes, we observe that as n increases, p _c increases quickly towards its optimal value under targeted immunization (attack) with complete information. In particular, we find a new scaling relationship between | p _c (∞) − p _c ( n ) | and n as | p _c (∞) − p _c ( n )| ~ n ⁻¹ exp(− αn ). For Scale-free (SF) networks, where P ( k ) ~ k ^{− γ} , 2 < γ < 3, we find that p _c has a transition from zero to non-zero when n increases from n = 1 to order of log N ( N is the size of network). Thus, for SF networks, knowledge of order of log N nodes and immunizing them can reduce dramatically an epidemics.