Complex Networked Research Articles

A machine learning approach to predicting dynamical observables from network structure

Estimating the outcome of a given dynamical process from structural features is a key unsolved challenge in network science. This goal is hampered by difficulties associated with nonlinearities, correlations and feedbacks between the structure and dynamics of complex systems. In this work, we develop an approach based on machine learning algorithms that provides an important step towards understanding the relationship between the structure and dynamics of networks. In particular, it allows us to estimate from the network structure the outbreak size of a disease starting from a single node, as well as the degree of synchronicity of a system made up of Kuramoto oscillators. We show which topological features of the network are key for this estimation and provide a ranking of the importance of network metrics with much higher accuracy than previously done. For epidemic propagation, the k-core plays a fundamental role, while for synchronization, the betweenness centrality and accessibility are the measures most related to the state of an oscillator. For all the networks, we find that random forests can predict the outbreak size or synchronization state with high accuracy, indicating that the network structure plays a fundamental role in the spreading process. Our approach is general and can be applied to almost any dynamic process running on complex networks. Also, our work is an important step towards applying machine learning methods to unravel dynamical patterns that emerge in complex networked systems.

Deep learning resilience inference for complex networked systems

Resilience, the ability to maintain fundamental functionality amidst failures and errors, is crucial for complex networked systems. Most analytical approaches rely on predefined equations for node activity dynamics and simplifying assumptions on network topology, limiting their applicability to real-world systems. Here, we propose ResInf, a deep learning framework integrating transformers and graph neural networks to infer resilience directly from observational data. ResInf learns representations of node activity dynamics and network topology without simplifying assumptions, enabling accurate resilience inference and low-dimensional visualization. Experimental results show that ResInf significantly outperforms analytical methods, with an F1-score improvement of up to 41.59% over Gao-Barzel-Barabási framework and 14.32% over spectral dimension reduction. It also generalizes to unseen topologies and dynamics and maintains robust performance despite observational disturbances. Our findings suggest that ResInf addresses an important gap in resilience inference for real-world systems, offering a fresh perspective on incorporating data-driven approaches to complex network modeling.

Mittag-Leffler function based security control for fractional-order complex network system subject to deception attacks via Observer-based AETS and its applications

The goal of this paper is to investigate the security control for uncertain fractional-order delayed complex network systems under deception attacks using the Mittag-Leffler function and observer-based adaptive event-triggered scheme (AETS) with the fractional commensurate order in q ∈ (0, 1). The adaptive event-triggering scheme is used during the data transmission process from the sensors to the observer, where the triggering threshold can be dynamically modified to reduce resource waste. We make a novel model for the estimation error system that takes into account both the effects of the adaptive event-triggered scheme and the effects of deception attacks. A sufficient condition is obtained to guarantee the stochastic mean-square stability of the augmented error system using the Mittag-Leffler (M-L) functions and the Lyapunov functional method and by using the singular value decomposition (SVD) and linear matrix inequality (LMI) techniques, the co-design problem of desired observer and controller gains is found, and it is shown that the solution ensures the stability of a closed-loop uncertain fractional-order complex networked system. At the end of this study, two numerical examples and diesel engine system model are given to show that the above findings are correct.

Evolutionary Digital Twin-Oriented Complex Networked Systems driven by node features and the mutation of feature preferences.

Accurate modelling of complex social systems, where people interact with each other and those interactions change over time, has been a research challenge for many years. This study proposes an evolutionary Digital Twin-Oriented Complex Networked System (DT-CNS) framework that considers heterogeneous node features and changeable connection preferences. We create heterogeneous preference mutation mechanisms to characterise nodes' adaptive decisions on preference mutation in response to interaction patterns and epidemic risks. In this space, we use nodes' interaction utilities to characterise the positive feedback from interactions and negative impact of epidemic risks. We also introduce social capital constraint to harness the density of social connections better. The nodes' heterogeneous preference mutation styles include the (i)inactive style that keeps initial social preferences, (ii) ignorant style that randomly mutates preferences, (iii) egocentric style that optimises individual interaction utility, (iv) cooperative style that optimises the total interaction utilities by group decisions and (v) collaborative style that further allows the cooperative nodes to transfer social capital. Our simulation experiments on evolutionary DT-CNSs reveal that heterogeneous preference mutation styles lead to various interaction and infection patterns. The results also show that (i) increasing social capital enables higher interactions but higher infection risks and uncertainty in decision-making; (ii) group decisions outperform individual decisions by eliminating the unawareness of the decisions of other nodes; (iii) the collaborative nodes under a strict social capital limit can promote interactions, reduce infection risks and achieve higher overall interaction utilities.

Reconstructing the evolution history of networked complex systems

The evolution processes of complex systems carry key information in the systems’ functional properties. Applying machine learning algorithms, we demonstrate that the historical formation process of various networked complex systems can be extracted, including protein-protein interaction, ecology, and social network systems. The recovered evolution process has demonstrations of immense scientific values, such as interpreting the evolution of protein-protein interaction network, facilitating structure prediction, and particularly revealing the key co-evolution features of network structures such as preferential attachment, community structure, local clustering, degree-degree correlation that could not be explained collectively by previous theories. Intriguingly, we discover that for large networks, if the performance of the machine learning model is slightly better than a random guess on the pairwise order of links, reliable restoration of the overall network formation process can be achieved. This suggests that evolution history restoration is generally highly feasible on empirical networks.

The Identification of Influential Nodes Based on Neighborhood Information in Asymmetric Networks

Identifying influential nodes, with pivotal roles in practical domains like epidemic management, social information dissemination optimization, and transportation network security enhancement, is a critical research focus in complex network analysis. Researchers have long strived for rapid and precise identification approaches for these influential nodes that are significantly shaping network structures and functions. The recently developed SPON (sum of proportion of neighbors) method integrates information from the three-hop neighborhood of each node, proving more efficient and accurate in identifying influential nodes than traditional methods. However, SPON overlooks the heterogeneity of neighbor information, derived from the asymmetry properties of natural networks, leading to its lower accuracy in identifying essential nodes. To sustain the efficiency of the SPON method pertaining to the local method, as opposed to global approaches, we propose an improved local approach, called the SSPN (sum of the structural proportion of neighbors), adapted from the SPON method. The SSPN method classifies neighbors based on the h-index values of nodes, emphasizing the diversity of asymmetric neighbor structure information by considering the local clustering coefficient and addressing the accuracy limitations of the SPON method. To test the performance of the SSPN, we conducted simulation experiments on six real networks using the Susceptible–Infected–Removed (SIR) model. Our method demonstrates superior monotonicity, ranking accuracy, and robustness compared to seven benchmarks. These findings are valuable for developing effective methods to discover and safeguard influential nodes within complex networked systems.

Towards Digital Twin-Oriented Complex Networked Systems: Introducing heterogeneous node features and interaction rules.

This study proposes an extendable modelling framework for Digital Twin-Oriented Complex Networked Systems (DT-CNSs) with a goal of generating networks that faithfully represent real-world social networked systems. Modelling process focuses on (i) features of nodes and (ii) interaction rules for creating connections that are built based on individual node's preferences. We conduct experiments on simulation-based DT-CNSs that incorporate various features and rules about network growth and different transmissibilities related to an epidemic spread on these networks. We present a case study on disaster resilience of social networks given an epidemic outbreak by investigating the infection occurrence within specific time and social distance. The experimental results show how different levels of the structural and dynamics complexities, concerned with feature diversity and flexibility of interaction rules respectively, influence network growth and epidemic spread. The analysis revealed that, to achieve maximum disaster resilience, mitigation policies should be targeted at nodes with preferred features as they have higher infection risks and should be the focus of the epidemic control.

Random node reinforcement and K-core structure of complex networks

Random node reinforcement and K-core structure of complex networks

Conserved Control Path in Multilayer Networks.

The determination of directed control paths in complex networks is important because control paths indicate the structure of the propagation of control signals through edges. A challenging problem is to identify them in complex networked systems characterized by different types of interactions that form multilayer networks. In this study, we describe a graph pattern called the conserved control path, which allows us to model a common control structure among different types of relations. We present a practical conserved control path detection method (CoPath), which is based on a maximum-weighted matching, to determine the paths that play the most consistent roles in controlling signal transmission in multilayer networks. As a pragmatic application, we demonstrate that the control paths detected in a multilayered pan-cancer network are statistically more consistent. Additionally, they lead to the effective identification of drug targets, thereby demonstrating their power in predicting key pathways that influence multiple cancers.

Layered complex networks as fluctuation amplifiers

In complex networked systems theory, an important question is how to evaluate the system robustness to external perturbations. With this task in mind, I investigate the propagation of noise in multi-layer networked systems. I find that, for a two layer network, noise originally injected in one layer can be strongly amplified in the other layer, depending on how well-connected are the complex networks in each layer and on how much the eigenmodes of their Laplacian matrices overlap. These results allow to predict potentially harmful conditions for the system and its sub-networks, where the level of fluctuations is important, and how to avoid them. The analytical results are illustrated numerically on various synthetic networks.

Sampled-Data-Based $\mathcal {H}_{\infty }$ Fuzzy Pinning Synchronization of Complex Networked Systems With Adaptive Event-Triggered Communications

In this article, the fuzzy pinning adaptive event-triggered control (FPAETC) problem related to complex networked systems is investigated. In light of the fact that the underlying controllers are fuzzy pinning, and the threshold parameters can flexibly update their states based on the latest and current sampled signals rather than transmitted signals, a novel state-dependent FPAETC protocol is devised to reduce the frequency of event-triggering and save more communication resources. Moreover, a more general time-dependent AETC protocol is proposed compared to the existing related results. Then, in combination with the graph theory, looped Lyapunov functional method, and fuzzy logic pinning technique, some relaxed criteria that establish the relationship between the synchronization and desired <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> performance for the considered system are derived. Simulation results validate the superiority of derived theoretical results.

A two-stage reconstruction method for complex networked system with hidden nodes.

Reconstructing the interacting topology from measurable data is fundamental to understanding, controlling, and predicting the collective dynamics of complex networked systems. Many methods have been proposed to address the basic inverse problem and have achieved satisfactory performance. However, a significant challenge arises when we attempt to decode the underlying structure in the presence of inaccessible nodes due to the partial loss of information. For the purpose of improving the accuracy of network reconstruction with hidden nodes, we developed a robust two-stage network reconstruction method for complex networks with hidden nodes from a small amount of observed time series data. Specifically, the proposed method takes full advantage of the natural sparsity of complex networks and the potential symmetry constraints in dynamic interactions. With robust reconstruction, we can not only locate the position of hidden nodes but also precisely recover the overall network structure on the basis of compensated nodal information. Extensive experiments are conducted to validate the effectiveness of the proposed method and superiority compared with ordinary methods. To some extent, this work sheds light on addressing the inverse problem, of which the system lacks complete exploration in the network science community.

Autonomous inference of complex network dynamics from incomplete and noisy data.

The availability of empirical data that capture the structure and behaviour of complex networked systems has been greatly increased in recent years; however, a versatile computational toolbox for unveiling a complex system's nodal and interaction dynamics from data remains elusive. Here we develop a two-phase approach for the autonomous inference of complex network dynamics, and its effectiveness is demonstrated by the tests of inferring neuronal, genetic, social and coupled oscillator dynamics on various synthetic and real networks. Importantly, the approach is robust to incompleteness and noises, including low resolution, observational and dynamical noises, missing and spurious links, and dynamical heterogeneity. We apply the two-phase approach to infer the early spreading dynamics of influenza A flu on the worldwide airline network, and the inferred dynamical equation can also capture the spread of severe acute respiratory syndrome and coronavirus disease 2019. These findings together offer an avenue to discover the hidden microscopic mechanisms of a broad array of real networked systems.

Complex Network Construction and Pattern Recognition of China’s Provincial Low-Carbon Economic Development with Long Time Series: Based on the Detailed Spatial Relationship

Low-carbon economic development has become the orientation of high-quality economic construction in the era of climate change. Because it involves multi-dimensional elements and is driven by the concept of regional integration and common development, the development pattern among China’s provinces should show the characteristics of complex networked spatial correlation. However, most of the existing research are for is based on attribute data and combined with traditional spatial econometric models, which cannot describe the complex networked spatial correlation. Concurrently, the network construction is mainly based on undirected-unweighted sparseness, which makes it difficult to truly restore the asymmetric detailed spatial relationship, making the related research about development pattern recognition and its evolution based on the detailed spatial relationship relatively lacking. Therefore, based on the perspective of relational data, this study constructs a multi-dimensional gravity model by using the comprehensive quality of China’s provincial low-carbon economic development, measured by the multi-dimensional evaluation index system and dynamic weight method, and the comprehensive distance based on geography, society, economy, and adjacency. And then the spatial correlation strength among China’s provincial low-carbon economic development is determined, and constructs a directed-weighted complex network. Then, it further explores the inter-provincial detailed spatial relationship of low-carbon economy development from the three dimensions of overall, individual and group, to recognize the development patterns and evolution rules of low-carbon economic development in different provinces, and clarify the development orientation. The results show that, during the study period, the spatial correlation of China’s provincial low-carbon economic development has become increasingly close, showing a complex networked correlation structure with multi-linearity, asymmetry, and geographical proximity as a whole. The division of general structure is consistent with China’s regional economic development strategy. The unbalanced development of China’s provincial low-carbon economy is the result of multi-center drive in the eastern and central regions, and the distance from the network center in the western regions. In general, China’s provincial low-carbon economic development patterns can be roughly divided into spillover, main bridge, auxiliary bridge, and benefit type. In the future, we should make full use of the power source role of spillover provinces, and the bridge support role of bridge provinces to drive the development of benefit provinces. Among them, Chongqing, Shaanxi, Hebei and Sichuan are expected to become new regional growth poles. The research on the inter-provincial spatial correlation effect and development pattern recognition of low-carbon economic development is expected to provide guidance for the sustainable development of low-carbon economy in China.

A system model of three-body interactions in complex networks: consensus and conservation

Networked complex systems in a wide range of physics, biology and social sciences involve synergy among multiple agents beyond pairwise interactions. Higher-order mathematical structures such as hypergraphs have been increasingly popular in modelling and analysis of complex dynamical behaviours. Here, we study a simple three-body consensus model, which favourably incorporates higher-order network interactions, higher-order dimensional states, the group reinforcement effect and the social homophily principle. The model features asymmetric roles of acting agents using modulating functions. We analytically establish sufficient conditions for nonlinear consensus and conservation of states for agents with both discrete-time and continuous-time dynamics. We show that higher-order interactions encoded in three-body edges give rise to consensus and conservation for systems with gravity-like and Heaviside-like modulating functions. Furthermore, we illustrate our theoretical results with numerical simulations and examine the system convergence time through a network depreciation process.

Reviving a failed network through microscopic interventions

From mass extinction to cell death, complex networked systems often exhibit abrupt dynamic transitions between desirable and undesirable states. These transitions are often caused by topological perturbations (such as node or link removal, or decreasing link strengths). The problem is that reversing the topological damage, namely, retrieving lost nodes or links or reinforcing weakened interactions, does not guarantee spontaneous recovery to the desired functional state. Indeed, many of the relevant systems exhibit a hysteresis phenomenon, remaining in the dysfunctional state, despite reconstructing their damaged topology. To address this challenge, we develop a two-step recovery scheme: first, topological reconstruction to the point where the system can be revived and then dynamic interventions to reignite the system’s lost functionality. By applying this method to a range of nonlinear network dynamics, we identify the recoverable phase of a complex system, a state in which the system can be reignited by microscopic interventions, for instance, controlling just a single node. Mapping the boundaries of this dynamical phase, we obtain guidelines for our two-step recovery. Perturbations and disturbances can bring complex networks into undesirable states in which global functionality is suppressed. Now, a recovery scheme explains how to revive a damaged network by controlling only a small number of nodes.

Inferring Network Structure and Estimating Dynamical Process From Binary-State Data via Logistic Regression

Inferring the structures and the dynamics of the complex networked systems based on time series data is a challenging problem. The existing reconstruction methods often rely on the knowledge of the dynamics on networks. In many cases, a prior knowledge of the dynamics is unknown, so it is natural to ask: is it possible to reconstruct network and estimate the dynamical processes on complex networks only rely on the observed data? In this article, we develop a framework to reconstruct the structures of networks with binary-state dynamics, in which the knowledge of the original dynamical processes is unknown. Within the reconstruction framework, the transition probabilities of binary dynamical processes are described by the Sigmoid function in logistic regression, we then apply the mean-field approximation to enable maximum likelihood estimation (MLE), which gives rise to that the network structure can be inferred by solving the linear system of equations. Meanwhile, the original dynamical processes can be simulated by estimating the parameters in the Sigmoid function. Our framework has been validated by a variety of binary dynamical processes on synthetic and empirical networks, indicating that our method can not only reveal the network structures but also estimate the dynamical processes. Moreover, the high accuracy of our method is highlighted by comparing it with the existing methods.

Adaptive Event-Triggered Nonfragile State Estimation for Fractional-Order Complex Networked Systems With Cyber Attacks

This article addresses the adaptive event-triggered nonfragile state estimation for the fractional-order complex networked systems subject to randomly occurring nonlinearities and adversarial network attacks, in which the order of fractional derivative operator satisfies <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0 &lt; p &lt; 1$ </tex-math></inline-formula> . To reduce unnecessary transmission burden as much as possible in an allowable range, an adaptive event-triggered scheme (AETS) is introduced to determine whether the data released by the sensor should be transmitted to the nonfragile state estimator. First, based on the considerations of the designed AETS and stochastic cyber-attacks, one constructs a newly fractional-order estimation error system model. Then, by employing the Lyapunov functional approach and the properties of Mittag–Leffler (M–L) functions, a sufficient condition is obtained to ensure the augmented error system stochastic mean-square stability; moreover, by making use of matrix’s singular value decomposition (SVD), the desired nonfragile state estimator is designed, and the estimator gains can be obtained by finding the feasible solutions of linear matrix inequality (LMI). Finally, a numerical example and Chua’s circuit model example are given to illustrate the feasibility of the designed nonfragile estimator.

Consensus formation in networks with neighbor-dependent synergy and observer effect

• Opinion formation model with social observer effect proposed. • Model featuring neighbor-dependent synergy through Petri net method. • Strongly connectivity of commensurate graph of a finite state machine. • Convergence analysis under non-empty intersection of comfortable ranges. • Observer effect, hypergraph structure, confidence bounds revealed by simulations. In this paper, we study the consensus formation over a directed hypergraph, which is an important generalization of standard graph structure by allowing possible neighbor-dependent synergy. The proposed model is situated in the social dynamics providing key features including social observer effect and bounded confidence. Under the minimal siphon condition of a directed hypergraph (Petri net), we show that global consensus can be reached with the final consensus value residing in the common comfortable range if it is non-empty. To achieved this, we establish an equivalent condition for the commensurate graph of a finite state machine to be strongly connected. Convergence analysis is performed based on the proposed nonlinear dynamic system model and Petri net method. The consensus result holds for any non-negative confidence bound, which distinguishes from traditional bounded confidence opinion models as we measure the difference among neighbors rather than the gap between neighbors and the ego. Numerical studies are conducted to unravel some insights in relation to the influence of observers, hypergraph architecture, and confidence bounds on opinion evolution. The results and methodologies presented here facilitate research of social consensus and also offer a way to make sense of synergy in networked complex systems.

Sequential Recovery of Complex Networks Suffering From Cascading Failure Blackouts

Over the past decades, prevention and mitigation against cascading failure blackouts in complex networked systems have been extensively studied. In reality, the functional and structural restoration of complex networks suffering from large-scale cascading failures may involve a sequence of repairing actions. In this paper, we propose a novel sequential recovery model that takes into consideration both the operating mechanism of complex power transmission networks and potential cascading failures triggered during the recovery process. From a complex network perspective, we introduce a new tool called the sequential recovery graph (SRG) to identify the critical nodes and the order of their restorations which lead to better performance in the recovery process. We further develop a practical sequential recovery strategy based on SRG. We perform simulations on three benchmark power grids and five baseline methods for performance comparison. Simulation results and complexity analysis demonstrate the effectiveness and low computational complexity of the proposed SRG-based strategy.