The cost of attack in competing networks

Non-Markovian recovery makes complex networks more resilient against large-scale failures

Non-Markovian spontaneous recovery processes with a time delay (memory) are ubiquitous in the real world. How does the non-Markovian characteristic affect failure propagation in complex networks? We consider failures due to internal causes at the nodal level and external failures due to an adverse environment, and develop a pair approximation analysis taking into account the two-node correlation. In general, a high failure stationary state can arise, corresponding to large-scale failures that can significantly compromise the functioning of the network. We uncover a striking phenomenon: memory associated with nodal recovery can counter-intuitively make the network more resilient against large-scale failures. In natural systems, the intrinsic non-Markovian characteristic of nodal recovery may thus be one reason for their resilience. In engineering design, incorporating certain non-Markovian features into the network may be beneficial to equipping it with a strong resilient capability to resist catastrophic failures.

Dynamic evolutionary metamodel analysis of the vulnerability of complex systems

Because the collapse of complex systems can have severe consequences, vulnerability is often seen as the core problem of complex systems. Multilayer networks are powerful tools to analyze complex systems, but complex networks may not be the best choice to mimic subsystems. In this work, a cellular graph (CG) model is proposed within the framework of multilayer networks to analyze the vulnerability of complex systems. Specifically, cellular automata are considered the vertices of a dynamic graph-based model at the microlevel, and their links are modeled by graph edges governed by a stochastic model at the macrolevel. A Markov chain is introduced to illustrate the evolution of the graph-based model and to obtain the details of the vulnerability evolution with low-cost inferences. This CG model is proven to describe complex systems precisely. The CG model is implemented with two actual organizational systems, which are used on behalf of the typical flat structure and the typical pyramid structure, respectively. The computational results show that the pyramid structure is initially more robust, while the flat structure eventually outperforms it when being exposed to multiple-rounds strike. Finally, the sensitivity analysis results verify and strengthen the reliability of the conclusions.

Emergence of frustration signals systemic risk

We show that the emergence of systemic risk in complex systems can be understood from the evolution of functional networks representing interactions inferred from fluctuation correlations between macroscopic observables. Specifically, we analyze the long-term collective dynamics in the New York Stock Exchange, the largest financial market in the world, for almost a century and show that periods marked by systemic crisis are associated with emergence of frustration. This is indicated by the loss of structural balance in the networks of interaction between stocks. Moreover, the mesoscopic organization of the networks during these periods exhibits prominent core-periphery organization. This suggests an increased degree of coherence in the collective dynamics of the system, which is reinforced by our observation of the transition to delocalization in the dominant eigenmodes when the systemic risk builds up. While frustration has been associated with phase transitions in physical systems such as spin glasses, its role as a signal for systemic risk buildup leading to severe crisis as shown here provides a novel perspective into the dynamical processes leading to catastrophic failures in complex systems.

Emergence of synchronization in multiplex networks of mobile Rössler oscillators

Different aspects of synchronization emerging in networks of coupled oscillators have been examined prominently in the last decades. Nevertheless, little attention has been paid on the emergence of this imperative collective phenomenon in networks displaying temporal changes in the connectivity patterns. However, there are numerous practical examples where interactions are present only at certain points of time owing to physical proximity. In this work, we concentrate on exploring the emergence of interlayer and intralayer synchronization states in a multiplex dynamical network comprising of layers having mobile nodes performing two-dimensional lattice random walk. We thoroughly illustrate the impacts of the network parameters, in particular, the vision range ϕ and the step size u together with the inter- and intralayer coupling strengths ε and k on these synchronous states arising in coupled Rössler systems. The presented numerical results are very well validated by analytically derived necessary conditions for the emergence and stability of the synchronous states. Furthermore, the robustness of the states of synchrony is studied under both structural and dynamical perturbations. We find interesting results on interlayer synchronization for a continuous removal of the interlayer links as well as for progressively created static nodes. We demonstrate that the mobility parameters responsible for intralayer movement of the nodes can retrieve interlayer synchrony under such structural perturbations. For further analysis of survivability of interlayer synchrony against dynamical perturbations, we proceed through the investigation of single-node basin stability, where again the intralayer mobility properties have noticeable impacts. We also discuss the scenarios related mainly to effects of the mobility parameters in cases of varying lattice size and percolation of the whole network.

Solitary states in multiplex networks owing to competing interactions

Recent researches in network science demonstrate the coexistence of different types of interactions among the individuals within the same system. A wide range of situations appear in ecological and neuronal systems that incorporate positive and negative interactions. Also, there are numerous examples of systems that are best represented by the multiplex configuration. The present article investigates a possible scenario for the emergence of a newly observed remarkable phenomenon named as solitary state in coupled dynamical units in which one or a few units split off and behave differently from the other units. For this, we consider dynamical systems connected through a multiplex architecture in the presence of both positive and negative couplings. We explore our findings through analysis of the paradigmatic FitzHugh-Nagumo system in both equilibrium and periodic regimes on the top of a multiplex network having positive inter-layer and negative intra-layer interactions. We further substantiate our proposition using a periodic Lorenz system with the same scheme and show that an opposite scheme of competitive interactions may also work for the Lorenz system in the chaotic regime.

Self-organization of dragon king failures

The mechanisms underlying cascading failures are often modeled via the paradigm of self-organized criticality. Here we introduce a simple network model where nodes self-organize to be either weakly or strongly protected against failure in a manner that captures the trade-off between degradation and reinforcement of nodes inherent in many network systems. If strong nodes cannot fail, any failure is contained to a single, isolated cluster of weak nodes and the model produces power-law distributions of failure sizes. We classify the large, rare events that involve the failure of only a single cluster as "black swans." In contrast, if strong nodes fail once a sufficient fraction of their neighbors fail, then failure can cascade across multiple clusters of weak nodes. If over 99.9% of the nodes fail due to this cluster hopping mechanism, we classify this as a "dragon king," which are massive failures caused by mechanisms distinct from smaller failures. The dragon kings observed are self-organized, existing over a wide range of reinforcement rates and system sizes. We find that once an initial cluster of failing weak nodes is above a critical size, the dragon king mechanism kicks in, leading to piggybacking system-wide failures. We demonstrate that the size of the initial failed weak cluster predicts the likelihood of a dragon king event with high accuracy and we develop a simple control strategy that can dramatically reduce dragon kings and other large failures.

Chimera states in neuronal networks: A review

Neuronal networks, similar to many other complex systems, self-organize into fascinating emergent states that are not only visually compelling, but also vital for the proper functioning of the brain. Synchronous spatiotemporal patterns, for example, play an important role in neuronal communication and plasticity, and in various cognitive processes. Recent research has shown that the coexistence of coherent and incoherent states, known as chimera states or simply chimeras, is particularly important and characteristic for neuronal systems. Chimeras have also been linked to the Parkinson's disease, epileptic seizures, and even to schizophrenia. The emergence of this unique collective behavior is due to diverse factors that characterize neuronal dynamics and the functioning of the brain in general, including neural bumps and unihemispheric slow-wave sleep in some aquatic mammals. Since their discovery, chimera states have attracted ample attention of researchers that work at the interface of physics and life sciences. We here review contemporary research dedicated to chimeras in neuronal networks, focusing on the relevance of different synaptic connections, and on the effects of different network structures and coupling setups. We also cover the emergence of different types of chimera states, we highlight their relevance in other related physical and biological systems, and we outline promising research directions for the future, including possibilities for experimental verification.

Topological properties of the limited penetrable horizontal visibility graph family

The limited penetrable horizontal visibility graph algorithm was recently introduced to map time series in complex networks. In this work, we extend this algorithm to create a directed-limited penetrable horizontal visibility graph and an image-limited penetrable horizontal visibility graph. We define two algorithms and provide theoretical results on the topological properties of these graphs associated with different types of real-value series. We perform several numerical simulations to check the accuracy of our theoretical results. Finally, we present an application of the directed-limited penetrable horizontal visibility graph to measure real-value time series irreversibility and an application of the image-limited penetrable horizontal visibility graph that discriminates noise from chaos. We also propose a method to measure the systematic risk using the image-limited penetrable horizontal visibility graph, and the empirical results show the effectiveness of our proposed algorithms.

A dynamic approach merging network theory and credit risk techniques to assess systemic risk in financial networks

The interconnectedness of financial institutions affects instability and credit crises. To quantify systemic risk we introduce here the PD model, a dynamic model that combines credit risk techniques with a contagion mechanism on the network of exposures among banks. A potential loss distribution is obtained through a multi-period Monte Carlo simulation that considers the probability of default (PD) of the banks and their tendency of defaulting in the same time interval. A contagion process increases the PD of banks exposed toward distressed counterparties. The systemic risk is measured by statistics of the loss distribution, while the contribution of each node is quantified by the new measures PDRank and PDImpact. We illustrate how the model works on the network of the European Global Systemically Important Banks. For a certain range of the banks' capital and of their assets volatility, our results reveal the emergence of a strong contagion regime where lower default correlation between banks corresponds to higher losses. This is the opposite of the diversification benefits postulated by standard credit risk models used by banks and regulators who could therefore underestimate the capital needed to overcome a period of crisis, thereby contributing to the financial system instability.

Biological conservation law as an emerging functionality in dynamical neuronal networks

Scientists strive to understand how functionalities, such as conservation laws, emerge in complex systems. Living complex systems in particular create high-ordered functionalities by pairing up low-ordered complementary processes, e.g., one process to build and the other to correct. We propose a network mechanism that demonstrates how collective statistical laws can emerge at a macro (i.e., whole-network) level even when they do not exist at a unit (i.e., network-node) level. Drawing inspiration from neuroscience, we model a highly stylized dynamical neuronal network in which neurons fire either randomly or in response to the firing of neighboring neurons. A synapse connecting two neighboring neurons strengthens when both of these neurons are excited and weakens otherwise. We demonstrate that during this interplay between the synaptic and neuronal dynamics, when the network is near a critical point, both recurrent spontaneous and stimulated phase transitions enable the phase-dependent processes to replace each other and spontaneously generate a statistical conservation law-the conservation of synaptic strength. This conservation law is an emerging functionality selected by evolution and is thus a form of biological self-organized criticality in which the key dynamical modes are collective.

Chimera states in a multilayer network of coupled and uncoupled neurons

We study the emergence of chimera states in a multilayer neuronal network, where one layer is composed of coupled and the other layer of uncoupled neurons. Through the multilayer structure, the layer with coupled neurons acts as the medium by means of which neurons in the uncoupled layer share information in spite of the absence of physical connections among them. Neurons in the coupled layer are connected with electrical synapses, while across the two layers, neurons are connected through chemical synapses. In both layers, the dynamics of each neuron is described by the Hindmarsh-Rose square wave bursting dynamics. We show that the presence of two different types of connecting synapses within and between the two layers, together with the multilayer network structure, plays a key role in the emergence of between-layer synchronous chimera states and patterns of synchronous clusters. In particular, we find that these chimera states can emerge in the coupled layer regardless of the range of electrical synapses. Even in all-to-all and nearest-neighbor coupling within the coupled layer, we observe qualitatively identical between-layer chimera states. Moreover, we show that the role of information transmission delay between the two layers must not be neglected, and we obtain precise parameter bounds at which chimera states can be observed. The expansion of the chimera region and annihilation of cluster and fully coherent states in the parameter plane for increasing values of inter-layer chemical synaptic time delay are illustrated using effective range measurements. These results are discussed in the light of neuronal evolution, where the coexistence of coherent and incoherent dynamics during the developmental stage is particularly likely.

The promotion of cooperation by the poor in dynamic chicken games

The evolution of cooperative behavior is one of the most important issues in game theory. Previous studies have shown that cooperation can evolve only under highly limited conditions, and various modifications have been introduced to games to explain the evolution of cooperation. Recently, a utility function basic to game theory was shown to be dependent on current wealth as a conditional (state) variable in a dynamic version of utility theory. Here, we introduce this dynamic utility function to several games. Under certain conditions, poor players exhibit cooperative behavior in two types of chicken games (the hawk-dove game and the snowdrift game) but not in the prisoner’s dilemma game and the stag hunt game. This result indicates that cooperation can be exhibited by the poor in some chicken games. Thus, the evolution of cooperation may not be as limited as has been suggested in previous studies.

Chimera states in uncoupled neurons induced by a multilayer structure

Spatial coexistence of coherent and incoherent dynamics in network of coupled oscillators is called a chimera state. We study such chimera states in a network of neurons without any direct interactions but connected through another medium of neurons, forming a multilayer structure. The upper layer is thus made up of uncoupled neurons and the lower layer plays the role of a medium through which the neurons in the upper layer share information among each other. Hindmarsh-Rose neurons with square wave bursting dynamics are considered as nodes in both layers. In addition, we also discuss the existence of chimera states in presence of inter layer heterogeneity. The neurons in the bottom layer are globally connected through electrical synapses, while across the two layers chemical synapses are formed. According to our research, the competing effects of these two types of synapses can lead to chimera states in the upper layer of uncoupled neurons. Remarkably, we find a density-dependent threshold for the emergence of chimera states in uncoupled neurons, similar to the quorum sensing transition to a synchronized state. Finally, we examine the impact of both homogeneous and heterogeneous inter-layer information transmission delays on the observed chimera states over a wide parameter space.

System crash as dynamics of complex networks

Complex systems, from animal herds to human nations, sometimes crash drastically. Although the growth and evolution of systems have been extensively studied, our understanding of how systems crash is still limited. It remains rather puzzling why some systems, appearing to be doomed to fail, manage to survive for a long time whereas some other systems, which seem to be too big or too strong to fail, crash rapidly. In this contribution, we propose a network-based system dynamics model, where individual actions based on the local information accessible in their respective system structures may lead to the "peculiar" dynamics of system crash mentioned above. Extensive simulations are carried out on synthetic and real-life networks, which further reveal the interesting system evolution leading to the final crash. Applications and possible extensions of the proposed model are discussed.