Scale-free Networks Research Articles

Emergent scale-free networks.

Many complex systems-from the Internet to social, biological, and communication networks-are thought to exhibit scale-free structure. However, prevailing explanations require that networks grow over time, an assumption that fails in some real-world settings. Here, we explain how scale-free structure can emerge without growth through network self-organization. Beginning with an arbitrary network, we allow connections to detach from random nodes and then reconnect under a mixture of preferential and random attachment. While the numbers of nodes and edges remain fixed, the degree distribution evolves toward a power-law with an exponent that depends only on the proportion p of preferential (rather than random) attachment. Applying our model to several real networks, we infer p directly from data and predict the relationship between network size and degree heterogeneity. Together, these results establish how scale-free structure can arise in networks of constant size and density, with broad implications for the structure and function of complex systems.

Study on the Stability of Complex Networks in the Stock Markets of Key Industries in China.

Investigating the significant "roles" within financial complex networks and their stability is of great importance for preventing financial risks. On one hand, this paper initially constructs a complex network model of the stock market based on mutual information theory and threshold methods, combined with the closing price returns of stocks. It then analyzes the basic topological characteristics of this network and examines its stability under random and targeted attacks by varying the threshold values. On the other hand, using systemic risk entropy as a metric to quantify the stability of the stock market, this paper validates the impact of the COVID-19 pandemic as a widespread, unexpected event on network stability. The research results indicate that this complex network exhibits small-world characteristics but cannot be strictly classified as a scale-free network. In this network, key roles are played by the industrial sector, media and information services, pharmaceuticals and healthcare, transportation, and utilities. Upon reducing the threshold, the network's resilience to random attacks is correspondingly strengthened. Dynamically, from 2000 to 2022, systemic risk in significant industrial share markets significantly increased. From a static perspective, the period around 2019, affected by the COVID-19 pandemic, experienced the most drastic fluctuations. Compared to the year 2000, systemic risk entropy in 2022 increased nearly sixtyfold, further indicating an increasing instability within this complex network.

Bifurcation and Chaos in a Fractional-Order Cournot Duopoly Game Model on Scale-Free Networks

In this study, a Cournot duopoly model describing Caputo fractional-order differential equations with piecewise constant arguments is discussed. We have obtained two-dimensional discrete dynamical system as a result of applying the discretization process to the model. By using the center manifold theory and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip bifurcation about the Nash equilibrium point. Phase portraits, bifurcation diagrams, and Lyapunov exponents show the existence of many complex dynamical behaviors in the model such as the stable equilibrium point, period-2 orbit, period-4 orbit, period-8 orbit, period-16 orbit, and chaos according to changing the speed of the adjustment parameter [Formula: see text]. The discrete Cournot duopoly game model is also considered on two scale-free networks with different numbers of nodes. It is observed that the complex dynamical networks exhibit similar dynamical behaviors such as the stable equilibrium point, flip bifurcation, and chaos depending on changing the coupling strength parameter [Formula: see text]. Moreover, flip bifurcation and transition chaos take place earlier in more heterogeneous networks. Calculating the largest Lyapunov exponents guarantees the transition from nonchaotic to chaotic states in complex dynamical networks.

Effect of interlayer dependence strength on continuous to discontinuous transitions in multi-layer networks

Studies on multilayer networks have aroused great interest in recent years because they allow a deeper description of the complexity present in many real systems. In this study, we introduce a dependency strength controller into multilayer interdependent networks, refining the traditional models that often inadequately represent cascading failures in real-world systems like urban transportation networks. This refined model enables a more realistic depiction of failure dynamics, where a node’s failure leads to its dependent node’s failure with a probability λ or survival with 1−λ. We derive exact equations that describe this process in a multilayer network and its structural evolution, and solve them numerically. We found the existence of a critical transition from continuous to discontinuous percolation as the strength of the dependence between layers increases. The critical percolation thresholds for these transitions are numerically derived for two-layer Erdős-Rényi networks and two-layer scale-free networks. This enhanced understanding of complex systems, incorporating initial disturbances and varying dependency strengths, contributes to designing more resilient multilayer complex systems.

Mechanistic and Statistical Analysis of Possible Defense Mechanisms for Probabilistic Failures in Robustness, Stability and Sustainability of Static and Dynamic Complex Networks

Complex networks are ubiquitous even in the form of defense networks in many fields such as ecological immune systems, biological immune systems, brain networks, air defense systems, national security, cyber security, peace and conflict studies, and so on. In the real world, complex networks exist in both static and dynamic forms. By understanding probabilistic failures and their possible defense mechanisms in static and dynamic complex networks, scientists and researchers can make decisions in developing and implementing methodologies of many fields as we can say, from the fields of biology to fields of technology. In this research, static and dynamic scale-free and small world networks have been generated as complex networks with network generation algorithms. Besides that, probabilistic failures and their possible defense mechanisms have also been generated and designed for previously generated complex networks. On these generated probabilistic failures and their possible defense mechanism, several mechanistic and statistical analysis methods have been applied. To generate these static and dynamic complex networks and their probabilistic failures and possible defense mechanism, two modified network generation algorithms along with several network measurements algorithms have been implemented where parameters have been chosen to produce possible types of static and dynamic scale-free and small world networks, and their corresponding probabilistic failures and possible defense mechanisms. Besides analyzing probabilistic failures and possible defense mechanisms for generated complex networks; the robustness, stability and sustainability of those networks have been studied and analyzed for probabilistic failures and their possible defense mechanisms.

Sampling unknown large networks restricted by low sampling rates

Graph sampling plays an important role in data mining for large networks. Specifically, larger networks often correspond to lower sampling rates. Under the situation, traditional traversal-based samplings for large networks usually have an excessive preference for densely-connected network core nodes. Aim at this issue, this paper proposes a sampling method for unknown networks at low sampling rates, called SLSR, which first adopts a random node sampling to evaluate a degree threshold, utilized to distinguish the core from periphery, and the average degree in unknown networks, and then runs a double-layer sampling strategy on the core and periphery. SLSR is simple that results in a high time efficiency, but experiments verify that the proposed method can accurately preserve many critical structures of unknown large scale-free networks with low sampling rates and low variances.

Assessing resiliency in scale-free supply chain networks: a stress testing approach based on entropy measurements and value-at-risk analysis

Supply chain (SC) resiliency and risk management have garnered increasing attention recently. While several studies have explored the use of scale-free network models to design and optimise SC networks, there remains a lack of a generalised stress-testing method that can be applied to various types and sizes of SCs. To address this, we propose a novel approach to assess the resiliency of scale-free SC networks based on entropy measurements. By considering various sources of disruptions, including demand fluctuations, capacity constraints, and transportation disruptions, we create a stress test that measures the network's resiliency. Our method categorises the SC into plants, ports, and customers and analyses the impact of disruptions on individual nodes within similar categories. We utilise real-world SC logistics data and map it into a scale-free network representation. By calculating the global vulnerability of the entire SC network and the localised impact of node vulnerability, we gain insights into how disruptions propagate and affect interconnected nodes. We use value-at-risk (VaR) and conditional-VaR (CVaR) to identify high-risk nodes, highlighting nodes at risk of extreme performance fluctuations or failures. The analytical results have led to developing six propositions, which serve as references for practitioners and help generalise the findings.

Quickcent: a fast and frugal heuristic for harmonic centrality estimation on scale-free networks

Quickcent: a fast and frugal heuristic for harmonic centrality estimation on scale-free networks

Modelling attack and defense games in infrastructure networks with interval-valued intuitionistic fuzzy set payoffs

Infrastructure networks are critical components of contemporary society, and numerous approaches have been suggested for the selection of strategies to protect these networks. However, for uncertain environments, research on attack and defense game models for infrastructure networks is limited. Therefore, after reviewing the existing approaches, a method based on interval-valued intuitionistic fuzzy set (IVIFS) theory is proposed for attack and defense games in critical infrastructure networks. First, we present the process of constructing the game model proposed in this paper, which mainly includes the formulation of the cost model, the strategies, and the method of generating IVIFS payoffs. Next, the Nash equilibria of the game are identified by a pair of nonlinear programming models based on IVIFS theory. Finally, experiments are conducted on a target scale-free network, and an investigation into the variation patterns of the Nash equilibria under different circumstances is also conducted. We provide explanations for these variation patterns by considering payoffs from the perspective of mathematical programming models. Furthermore, we find that compared to the existing attack and defense game model with crisp payoffs, the model proposed in this paper leads to superior Nash equilibria. Our work is a preliminary attempt to analyse attack and defense games for infrastructure networks based on IVIFS theory, providing a method for assessing payoffs in uncertain environments for the attacker and defender. This topic deserves further study.

Effects of network heterogeneity on phases of the quenched contact process in directed complex networks

We investigate the quenched contact process within directed complex networks, including directed random networks and directed scale-free networks. Additionally, we explore the contact process within directed general small-world networks and directed hierarchical modular networks. We observed the presence of a Griffiths phase characterized by continuously varying critical exponents of the order parameter. Notably, the directionality of the networks plays a significant role in influencing the Griffiths phase within the quenched contact process. We suggest that the presence of strongly directed connecting clusters is responsible for the changes in the continuously varying critical exponents of the order parameter within directed networks.

Dynamic vaccination strategies in dual-strain epidemics: A multi-agent-based game-theoretic approach on scale-free hybrid networks

This paper proposes a multi-agent-based game-theoretic framework to explore the dynamics of decision-making among agents in a situation involving a dual-strain epidemic, where the emergence of the second strain is directly related to the first strain. It considers the impact of imperfect vaccination on dual strains within the SIRV framework, in a hybrid network that combines scale-free and grid-based structures. This research presents three models based on the level of decision-making of agents: Individual Decision Making (IDM), Collective Decision Making (CDM), and Stochastic Decision Making (SDM) model. In the IDM model, agents can continuously update their strategies based on the infection and vaccination statuses of their immediate neighbors. The CDM model offers a macro-level approach where agents’ willingness to vaccinate varies uniformly, while the SDM model maintains a constant level of willingness for vaccination across agents. This study demonstrates that agents using the IDM model tend to initiate vaccination earlier in scenarios of high infection rates, effectively mitigating the spread of the disease in subsequent epidemic seasons. However, when both infection and vaccination rates heavily influence decision-making, there is a noticeable decrease in vaccination uptake, thereby aggravating the spread of infection. Additionally, the authors found that in scenarios of high infection, an over reliance on vaccination without adequate initial uptake can jeopardize disease control efforts. Another important finding of this study is that despite the CDM and SDM models showing a quicker emergence of the second strain and reaching equilibrium points faster than the IDM model, the overall infection rate remains lower in the IDM model. This research underscores the complexity of managing dual-strain epidemics through vaccination strategies and highlights the significance of micro-level decision-making in epidemic control.

The topology of spatial networks affects stability in experimental metacommunities.

Understanding the drivers of community stability has been a central goal in ecology. Traditionally, emphasis has been placed on studying the effects of biotic interactions on community variability, and less is understood about how the spatial configuration of habitats promotes or hinders metacommunity stability. To test the effects of contrasting spatial configurations on metacommunity stability, I designed metacommunities with patches connected as random or scale-free networks. In these microcosms, two prey and one protist predator dispersed, and I evaluated community persistence, tracked biomass variations, and measured synchrony between local communities and the whole metacommunity. After 30 generations, scale-free metacommunities had lower global biomass variability and higher persistence, suggesting higher stability. Synchrony between patches was lower in scale-free metacommunities. Patches in scale-free metacommunities showed a positive relationship between variability and patch connectivity, indicating higher stability in isolated communities. No clear relationship between variability and patch connectivity was observed in random networks. These results suggest the increased heterogeneity in connectivity of scale-free networks favours the prevalence of isolated patches of the metacommunity, which likely act as refugia against competition-the dominant interaction in this system-resulting in higher global stability. These results highlight the importance of accounting for network topology in the study of spatial dynamics.

Predicting epidemic threshold in complex networks by graph neural network.

To achieve precision in predicting an epidemic threshold in complex networks, we have developed a novel threshold graph neural network (TGNN) that takes into account both the network topology and the spreading dynamical process, which together contribute to the epidemic threshold. The proposed TGNN could effectively and accurately predict the epidemic threshold in homogeneous networks, characterized by a small variance in the degree distribution, such as Erdős-Rényi random networks. Usability has also been validated when the range of the effective spreading rate is altered. Furthermore, extensive experiments in ER networks and scale-free networks validate the adaptability of the TGNN to different network topologies without the necessity for retaining. The adaptability of the TGNN is further validated in real-world networks.

Enhancing the robustness of interdependent networks by positively correlating a portion of nodes

Cascading failures caused by interdependencies make modern coupled systems extremely fragile to failures. In existing network robustness enhancing methods, maximizing interlayer degree-degree correlations has been proven to be an effective way to improve the robustness of interdependent networks under random failures. Here, we propose a portion of nodes positively correlated strategy (PNC) to improve network robustness by positively correlating a portion of nodes, in which the nodes that are positively correlated are selected in descending order of degree, starting at a cutoff value. Based on percolation theory, we verify the effectiveness of PNC on different networks. And find that, when the nodes with the highest degree are preferentially correlated, i.e. the cutoff value takes the maximum degree, this strategy achieves a state-of-the-art optimization effect. In particular, for interdependent scale-free networks with power-law exponent γ that satisfies 2<γ<3 , we theoretically demonstrate that the highest degree preferentially correlated mode can maximize network robustness by changing the coupling state of the near 0 proportion of nodes. For γ = 3, such a mode can make the network turn into a second-order phase transition at the collapse point. Finally, we discuss the relationship between the robustness of optimized networks and common links in real-world networks.

Stochastic modeling of malware propagation on reduced scale-free topology-based wireless sensor networks: dynamics, resilience, and countermeasures

Stochastic modeling of malware propagation on reduced scale-free topology-based wireless sensor networks: dynamics, resilience, and countermeasures

Barely Supercritical Percolation on Poissonian Scale-free Networks

We study the giant component problem slightly above the critical regime for percolation on Poissonian random graphs in the scale-free regime, where the vertex weights and degrees have a diverging second moment. Critical percolation on scale-free random graphs has been observed to have incredibly subtle features that are markedly different compared to those in random graphs with a converging second moment. In particular, the critical window for percolation depends sensitively on whether we consider single- or multi-edge versions of the Poissonian random graph. In this paper, and together with our companion paper [3], we build a bridge between these two cases. Our results characterize the part of the barely supercritical regime where the size of the giant components are approximately same for the single- and multi-edge settings. The methods for establishing concentration of giant for the single- and multi-edge versions are quite different. While the analysis in the multi-edge case is based on scaling limits of exploration processes, the single-edge setting requires identi fication of a core structure inside certain high-degree vertices that forms the giant component.

Structural properties of extended pseudo-fractal scale-free network with higher network efficiency

Abstract Complex networks, as an important model for studying complex systems, have been gradually studied in various extension models. Firstly, the network is proved to be sparse by calculating the density of the extending pseudo-fractal network. Secondly, it is proved that the network nodes conform the power-law distribution by enumerate the degree sequence of nodes with power law index $ 2 \lt \gamma \leq 2.585 $ shows that the network has scale-free feature. Thirdly, it is also demonstrated that the average clustering coefficient $ \overline{C_{g}} $ can reach a theoretical lower limit value 0.8182 and the increase of extension coefficient m makes the $ \overline{C_{g}} $ higher than traditional fractal networks. Fourthly, we derive the analytical expression and limit expression of the average path length, and conclude that it has small-world effect. Meanwhile, it is shown that the average path length $ \overline{D_{g}} $ is logarithmically related to Ng growth relationship in Fg. Finally, it is concluded that this extended pseudo-fractal network becomes more effective under the effect of m.

Wave-Function Network Description and Kolmogorov Complexity of Quantum Many-Body Systems

Programmable quantum devices are now able to probe wave functions at unprecedented levels. This is based on the ability to project the many-body state of atom and qubit arrays onto a measurement basis which produces snapshots of the system wave function. Extracting and processing information from such observations remains, however, an open quest. One often resorts to analyzing low-order correlation functions—that is, discarding most of the available information content. Here, we introduce wave-function networks—a mathematical framework to describe wave-function snapshots based on network theory. For many-body systems, these networks can become scale-free—a mathematical structure that has found tremendous success and applications in a broad set of fields, ranging from biology to epidemics to Internet science. We demonstrate the potential of applying these techniques to quantum science by introducing protocols to extract the Kolmogorov complexity corresponding to the output of a quantum simulator and implementing tools for fully scalable cross-platform certification based on similarity tests between networks. We demonstrate the emergence of scale-free networks analyzing experimental data obtained with a Rydberg quantum simulator manipulating up to 100 atoms. Our approach illustrates how, upon crossing a phase transition, the simulator complexity decreases while correlation length increases—a direct signature of buildup of universal behavior in data space. Comparing experiments with numerical simulations, we achieve cross-certification at the wave-function level up to timescales of 4 μs with a confidence level of 90% and determine experimental calibration intervals with unprecedented accuracy. Our framework is generically applicable to the output of quantum computers and simulators with access to the system wave function and requires probing accuracy and repetition rates accessible to most currently available platforms. Published by the American Physical Society 2024

The bass diffusion model: agent-based implementation on arbitrary networks

ABSTRACT The goal of this study is to model Bass diffusion and its extensions on complex networks, including scale-free networks with arbitrary power-law exponent and assortative degree correlations. For this purpose we employ a combination of the free software packages networkX and NetLogo. Some new results obtained in the agent-based simulations (and differing from those in mean-field approximation) are the following. The introduction of assortative correlations in scale-free networks has the effect of delaying the adoption peak in the Bass model, compared to the uncorrelated case. The peak time depends strongly also on the maximum degree effectively present in the network. For diffusion models with threshold on signed network, a high level of clustering tends to cause adoption blockades. By analysing statistical ensembles of assortative networks generated via Newman rewiring one observes a remarkable strong correlation between the average degree of first neighbours k ˉ nn ( k ) and the average clustering coefficient C ˉ ( k ) .

The effectiveness of intervention measures on MERS-CoV transmission by using the contact networks reconstructed from link prediction data.

Mitigating the spread of infectious diseases is of paramount concern for societal safety, necessitating the development of effective intervention measures. Epidemic simulation is widely used to evaluate the efficacy of such measures, but realistic simulation environments are crucial for meaningful insights. Despite the common use of contact-tracing data to construct realistic networks, they have inherent limitations. This study explores reconstructing simulation networks using link prediction methods as an alternative approach. The primary objective of this study is to assess the effectiveness of intervention measures on the reconstructed network, focusing on the 2015 MERS-CoV outbreak in South Korea. Contact-tracing data were acquired, and simulation networks were reconstructed using the graph autoencoder (GAE)-based link prediction method. A scale-free (SF) network was employed for comparison purposes. Epidemic simulations were conducted to evaluate three intervention strategies: Mass Quarantine (MQ), Isolation, and Isolation combined with Acquaintance Quarantine (AQ + Isolation). Simulation results showed that AQ + Isolation was the most effective intervention on the GAE network, resulting in consistent epidemic curves due to high clustering coefficients. Conversely, MQ and AQ + Isolation were highly effective on the SF network, attributed to its low clustering coefficient and intervention sensitivity. Isolation alone exhibited reduced effectiveness. These findings emphasize the significant impact of network structure on intervention outcomes and suggest a potential overestimation of effectiveness in SF networks. Additionally, they highlight the complementary use of link prediction methods. This innovative methodology provides inspiration for enhancing simulation environments in future endeavors. It also offers valuable insights for informing public health decision-making processes, emphasizing the importance of realistic simulation environments and the potential of link prediction methods.