Cascading failures in scale-free interdependent networks

Motif-based community detection in heterogeneous multilayer networks

Multilayer networks composed of intralayer edges and interlayer edges are an important type of complex networks. Considering the heterogeneity of nodes and edges, it is necessary to design more reasonable and diverse community detection methods for multilayer networks. Existing research on community detection in multilayer networks mainly focuses on multiplexing networks (where the nodes are homogeneous and the edges are heterogeneous), but few studies have focused on heterogeneous multilayer networks where both nodes and edges represent different semantics. In this paper, we studied community detection on heterogeneous multilayer networks and proposed a motif-based detection algorithm. First, the communities and motifs of multilayer networks are defined, especially the interlayer motifs. Then, the modularity of multilayer networks based on these motifs is designed, and the community structure of the multilayer network is detected by maximizing the modularity of multilayer networks. Finally, we verify the effectiveness of the detection algorithm on synthetic networks. In the experiments on synthetic networks, comparing with the classical community detection algorithms (without considering interlayer heterogeneity), the motif-based modularity community detection algorithm can obtain better results under different evaluation indexes, and we found that there exists a certain relationship between motifs and communities. In addition, the proposed algorithm is applied in the empirical network, which shows its practicability in the real world. This study provides a solution for the investigation of heterogeneous information in multilayer networks.

Catastrophic cascade of failures in interdependent hypergraphs

The failures of individual agents can significantly impact the functionality of associated groups in interconnected systems. To reveal these impacts, we develop a threshold model to investigate cascading failures in double-layer hypergraphs with interlayer interdependence. We hypothesize that a hyperedge disintegrates when the proportion of failed nodes within it exceeds a threshold. Due to the interdependence between a node and its replica in the other layer, the disintegrations of these hyperedges could trigger a cascade of events, leading to an iterative collapse across these two layers. We find that double-layer hypergraphs undergo abrupt, discontinuous first-order phase transitions during systemic collapse regardless of the specific threshold value. Additionally, the connectivity measured by average cardinality and hyperdegree plays a crucial role in shaping system robustness. A higher average hyperdegree always strengthens system robustness. However, the relationship between system robustness and average cardinality exhibits non-monotonic behaviors. Specifically, both excessively small and large average cardinalities undermine system robustness. Furthermore, a higher threshold value can boost the system's robustness. In summary, our study provides valuable insights into cascading failure dynamics in double-layer hypergraphs and has practical implications for enhancing the robustness of complex interdependent systems across domains.

Robustness of coupled networks with multiple support from functional components at different scales

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.

Sandpile cascades on oscillator networks: The BTW model meets Kuramoto

Cascading failures abound in complex systems and the Bak-Tang-Weisenfeld (BTW) sandpile model provides a theoretical underpinning for their analysis. Yet, it does not account for the possibility of nodes having oscillatory dynamics, such as in power grids and brain networks. Here, we consider a network of Kuramoto oscillators upon which the BTW model is unfolding, enabling us to study how the feedback between the oscillatory and cascading dynamics can lead to new emergent behaviors. We assume that the more out-of-sync a node is with its neighbors, the more vulnerable it is and lower its load-carrying capacity accordingly. Also, when a node topples and sheds load, its oscillatory phase is reset at random. This leads to novel cyclic behavior at an emergent, long timescale. The system spends the bulk of its time in a synchronized state where load builds up with minimal cascades. Yet, eventually, the system reaches a tipping point where a large cascade triggers a "cascade of larger cascades," which can be classified as a dragon king event. The system then undergoes a short transient back to the synchronous, buildup phase. The coupling between capacity and synchronization gives rise to endogenous cascade seeds in addition to the standard exogenous ones, and we show their respective roles. We establish the phenomena from numerical studies and develop the accompanying mean-field theory to locate the tipping point, calculate the load in the system, determine the frequency of the long-time oscillations, and find the distribution of cascade sizes during the buildup phase.

Greedy control of cascading failures in interdependent networks

Complex systems are challenging to control because the system responds to the controller in a nonlinear fashion, often incorporating feedback mechanisms. Interdependence of systems poses additional difficulties, as cross-system connections enable malicious activity to spread between layers, increasing systemic risk. In this paper we explore the conditions for an optimal control of cascading failures in a system of interdependent networks. Specifically, we study the Bak-Tang-Wiesenfeld sandpile model incorporating a control mechanism, which affects the frequency of cascades occurring in individual layers. This modification allows us to explore sandpile-like dynamics near the critical state, with supercritical region corresponding to infrequent large cascades and subcritical zone being characterized by frequent small avalanches. Topological coupling between networks introduces dependence of control settings adopted in respective layers, causing the control strategy of a given layer to be influenced by choices made in other connected networks. We find that the optimal control strategy for a layer operating in a supercritical regime is to be coupled to a layer operating in a subcritical zone, since such condition corresponds to reduced probability of inflicted avalanches. However this condition describes a parasitic relation, in which only one layer benefits. Second optimal configuration is a mutualistic one, where both layers adopt the same control strategy. Our results provide valuable insights into dynamics of cascading failures and and its control in interdependent complex systems.

Mutualistic networks emerging from adaptive niche-based interactions

Mutualistic networks are vital ecological and social systems shaped by adaptation and evolution. They involve bipartite cooperation via the exchange of goods or services between actors of different types. Empirical observations of mutualistic networks across genres and geographic conditions reveal correlated nested and modular patterns. Yet, the underlying mechanism for the network assembly remains unclear. We propose a niche-based adaptive mechanism where both nestedness and modularity emerge simultaneously as complementary facets of an optimal niche structure. Key dynamical properties are revealed at different timescales. Foremost, mutualism can either enhance or reduce the network stability, depending on competition intensity. Moreover, structural adaptations are asymmetric, exhibiting strong hysteresis in response to environmental change. Finally, at the evolutionary timescale we show that the adaptive mechanism plays a crucial role in preserving the distinctive patterns of mutualism under species invasions and extinctions.

Nested and modular patterns are vastly observed in mutualistic networks across genres and geographic conditions. Here, the authors show a unified mechanism that underlies the assembly and evolution of such networks, based on adaptive niche interactions of the participants.

The transsortative structure of networks

Network topologies can be highly non-trivial, due to the complex underlying behaviours that form them. While past research has shown that some processes on networks may be characterized by local statistics describing nodes and their neighbours, such as degree assortativity, these quantities fail to capture important sources of variation in network structure. We define a property called transsortativity that describes correlations among a node's neighbours. Transsortativity can be systematically varied, independently of the network's degree distribution and assortativity. Moreover, it can significantly impact the spread of contagions as well as the perceptions of neighbours, known as the majority illusion. Our work improves our ability to create and analyse more realistic models of complex networks.

How motifs condition critical thresholds for tipping cascades in complex networks: Linking micro- to macro-scales

In this study, we investigate how specific micro-interaction structures (motifs) affect the occurrence of tipping cascades on networks of stylized tipping elements. We compare the properties of cascades in Erdős-Rényi networks and an exemplary moisture recycling network of the Amazon rainforest. Within these networks, decisive small-scale motifs are the feed forward loop, the secondary feed forward loop, the zero loop, and the neighboring loop. Of all motifs, the feed forward loop motif stands out in tipping cascades since it decreases the critical coupling strength necessary to initiate a cascade more than the other motifs. We find that for this motif, the reduction of critical coupling strength is 11% less than the critical coupling of a pair of tipping elements. For highly connected networks, our analysis reveals that coupled feed forward loops coincide with a strong 90% decrease in the critical coupling strength. For the highly clustered moisture recycling network in the Amazon, we observe regions of a very high motif occurrence for each of the four investigated motifs, suggesting that these regions are more vulnerable. The occurrence of motifs is found to be one order of magnitude higher than in a random Erdős-Rényi network. This emphasizes the importance of local interaction structures for the emergence of global cascades and the stability of the network as a whole.

Crash dynamics of interdependent networks

The emergence and evolution of real-world systems have been extensively studied in the last few years. However, equally important phenomena are related to the dynamics of systems' collapse, which has been less explored, especially when they can be cast into interdependent systems. In this paper, we develop a dynamical model that allows scrutinizing the collapse of systems composed of two interdependent networks. Specifically, we explore the dynamics of the system's collapse under two scenarios: in the first one, the condition for failure should be satisfied for the focal node as well as for its corresponding node in the other network; while in the second one, it is enough that failure of one of the nodes occurs in either of the two networks. We report extensive numerical simulations of the dynamics performed in different setups of interdependent networks, and analyze how the system behavior depends on the previous scenarios as well as on the topology of the interdependent system. Our results can provide valuable insights into the crashing dynamics and evolutionary properties of interdependent complex systems.