Scale-free Networks Research Articles

A possible quadruple point in networks of directed networks under targeted attacks

Many real systems are known to interact with one another, forming networks of networks (NONs). Plenty of attention has been poured into the research on the robustness in NONs in the past decade. Previous studies focus on undirected networks, or directed networks under random attacks. While many real networks are directed and how networks of directed networks (NODNs) respond to targeted attacks remains unknown. We thus develop a general analytical tool for analyzing the robustness of NODNs under two kinds of targeted attacks: degree-based attacks and in-degree (out-degree)-based attacks. The analytical tool can perfectly predict the sizes of the final giant strongly connected components and the phase transitions on the NODNs in response to targeted attacks. By applying the tool to synthesis networks, we find that a quadruple point intersected by four different phase regions could appear in the random regular NODNs. To the best of our knowledge, it is the first time that a quadruple point is found in the studies of complex networks. In addition, we find triple points intersected by three phases in networks of directed scale-free networks, and critical points that connect two phases in networks of directed Erdös–Rényi networks. The discovery of these tipping points could help understand network robustness and enable better design of networked systems.

Research on Financial Risk Monitoring and Early Warning in China Based on Big Data Analysis

With the continuous development of China's economy, the financial industry plays a more and more important role in the national economy. At the same time, the financial risk(FR) are increasing and becoming increasingly complex. While sharing the huge benefits brought by financial globalization and financial deepening, China is also faced with greater FR from home and abroad, which greatly increase the possibility of financial crisis in China. In order to enhance the initiative of risk management and ensure the smooth operation of asset quality, we must rely on a powerful system tool, make use of the international advanced risk early warning management concept, and use Internet thinking and data mining technology to dig deep into hidden risk factors, discover hidden risks in advance and deal with them in time, so as to ensure the automatic monitoring ability of credit business. Based on the theoretical research on the generation, accumulation and diffusion of systematic FR, this paper explores the introduction of big data methods into the research on systematic FR in China, comprehensively uses traditional and emerging indicators and models in big data network analysis methods, analyzes the contagion effects of regional systematic FR from multiple perspectives and levels, improves research methods, and enriches research perspectives. The introduction of SIRS model on scale-free network provides a new powerful tool for the study of regional systemic FR.

Spatiotemporal Pattern Formation of Metal Electrodissolution Reactions Using Multielectrode Arrays in Lab-on-Chip Cells

Electrochemical reactions can generate complex spatiotemporal patterns due to the inherent nonlinearities in the reaction kinetics, which often couple with various mass transfer and electrical coupling effects. We explore spatiotemporal pattern formation using multielectrode arrays embedded in flow channels. The kinetic equations are expressed as a superposition of local kinetics with a network of electrical coupling due to ohmic potential drops in the electrolyte. The networks are revealed by analysis of the synchronization patterns with the use of an oscillatory chemical reaction (nickel electrodissolution) and are further confirmed by direct decoding using phase model analysis. In dual electrode configuration, a variety coupling schemes, (uni- or bidirectional positive or negative) were identified depending on the relative placement of the reference and counter electrodes (e.g., placed at the same or the opposite ends of the ow channel). The synchronization patterns and the network configuration are compared for the linear and the branched channel configurations. With straight channel and multiple electrodes, a scale-free network was formed. With branched channels, the global to local coupling can be changed by the position of electrodes in the channel, and introduction of different concentration of electrolyte can mediate coupling between only the downstream electrodes. In fluidic network with cross channels, nonlinearity can be induced by obstacles in the channel which open ways to investigate the Braess’s paradox. The networked, electrode potential (current) spike generating electrochemical reactions hold potential for construction of an in-situ information processing unit to be used in electrochemical devices in sensors and batteries.

Epidemic spreading dynamics on two-layer complex networks

Recent studies have shown that many real-world systems can be described by multi-layer complex networks. In this paper, the concept of layers is introduced to construct a traffic-driven SIR epidemic spreading model on “logical-physical” layered network. Based on the peak density of infected nodes and the ultimate density of recovered nodes, we investigate the features of epidemic spreading on layered network. Through numerical simulations, it is shown that traffic flow greatly influences the intensity and scope of epidemic spreading. By comparing the effects of four kinds of two-layer networks generated by ER random network model and BA scale-free network model on epidemic spreading, we found that the homogeneity of logical or physical network structure can promote the spread of epidemic more than heterogeneous networks. This work may be of service to design traffic-driven epidemic prevention and control strategies.

SHI1I2R competitive information spreading model in online and offline two-layer networks in emergencies

Previous research on information propagation has primarily focused on either online social networks or offline social networks. Additionally, there remains a gap in understanding the propagation of competitive information in emergency scenarios. This paper introduces an SHI1I2R information spreading model to investigate the dynamics of competitive information dissemination and interactions in online and offline two-layer networks during emergencies. To enhance the accuracy of information propagation modeling, we depart from the conventional assumption of constant spreading and decay rates and instead propose nonlinear spreading and interest decay rates within the SHI1I2R model. Employing a microscopic Markov chain approach, we conduct theoretical analysis and derive the spreading threshold of information in two-layer networks. Subsequently, we perform numerical simulations to validate the effectiveness of the SHI1I2R model and explore the factors influencing information propagation. Our findings indicate that information spreads more rapidly and extensively in two-layer networks compared to one-layer networks. Furthermore, the final scale of information dissemination and the speed of propagation follow the order: random network > small-world network > scale-free network > regular network.

Global stability and optimal control of epidemics in heterogeneously structured populations exhibiting adaptive behavior

We present a novel SIRS model on scale-free networks that takes into account behavioral memory and time delay to depict an adaptive behavioral feedback mechanism, which can better characterize the actual spread of epidemics. We conduct rigorous analysis on the dynamics of the model, including the basic reproduction number R0, uniform persistence and the global asymptotic stability of equilibria. The model has the sharp threshold property, namely, if R0 is less than 1 then the disease-free equilibrium is globally asymptotically stable while if R0 is larger than 1 then the endemic equilibrium is globally asymptotically stable. We further perform an optimal control study for the model to seek effective vaccination and treatment strategies. The existence and uniqueness of these optimal control strategies are demonstrated. Finally, we perform some stochastic network simulations that yield quantitative agreement with the deterministic mean-field approach. Our findings indicate that time delay does not affect R0, but behavioral memory does.

Election of MPR Nodes and Detection of Malicious Nodes Based on a Byzantine Fault in the OLSR Protocol Case of a Scale-Free Network

V2X (Vehicle-to-Everything) communications play a crucial role in enabling the efficient and reliable exchange of information among vehicles, infrastructure, and other entities in smart transportation systems. However, the inherent vulnerabilities and dynamic nature of V2X networks present significant challenges for ensuring secure and trustworthy communication. By enhancing the security of the OLSR (Optimized Link State Routing) protocol through secure MultiPoint Relays (MPRs) Selection, this research aims to provide a robust approach that enhances the overall security posture of V2X networks, enabling safe and secure interactions between vehicles and their environment. The proposed method is based on the Byzantine general’s problem, which is the principle used in blockchain. Compared to the classical flooding mechanism, this technique greatly reduces network traffic overhead and improves the efficiency of bandwidth utilization. The results demonstrated that the proposed algorithm performed better than the well-used UM-OLSR implementation. The outcome proved that our MPR election algorithm guarantees a better packet delivery ratio, and it also performs very well in the detection and isolation of malicious nodes, leading to increased security of the OLSR protocol control plane.

Inter-layer synchronization on a two-layer network of unified chaotic systems: The role of network nodal dynamics

The interplay between synchronization across layers and structures within layers still remains far unclear in multi-layer networks. Via our newly-constructed two-layer network consisting of the unified chaotic systems where the link within layer is linearly coupled and the link across layers is formed via sharing a common signal, this paper investigates the inter-layer synchronization on two-layer networks from the perspective of network nodal dynamics, as well as explore the influence of intra-layer network structures on inter-layer synchronization. Here we derive an analytical sufficient condition for inter-layer synchronization, which is only dependent on network nodal dynamics, showing a handful of Lorenz-type oscillators including the classic one are able to successfully realize inter-layer synchronization of the network for any connected intra-layer topological structures and diagonal coupling matrices. Further, four regular networks, small-world, scale-free and random networks are respectively used to run numerical simulations. Numerical results verify the presented theoretical results, and also show that Lü-type as well as Chen-type oscillators networks fail to realize inter-layer synchronization no matter which kinds of topologies and coupling matrices within layers. Specifically, for the case of identical topologies within layers, the chain-type and ring-type structures can stimulate larger range of Lorenz-type oscillators into synchronization across layers under the assumption of diagonal coupling matrices, but the fully-connected and star-type structures fail, and the non-diagonal intra-layer coupling matrix on the whole weaken the inter-layer synchronizability. For the case of nonidentical topologies within layers, the connection structure within layers can hardly enlarge the range of oscillators for synchronization across layers, even narrow the range, but some structures may enlarge the range of coupling strengthes for inter-layer synchronization.

Perfect synchronization in complex networks with higher-order interactions.

Achieving perfect synchronization in a complex network, specially in the presence of higher-order interactions (HOIs) at a targeted point in the parameter space, is an interesting, yet challenging task. Here we present a theoretical framework to achieve the same under the paradigm of the Sakaguchi-Kuramoto (SK) model. We analytically derive a frequency set to achieve perfect synchrony at some desired point in a complex network of SK oscillators with higher-order interactions. Considering the SK model with HOIs on top of the scale-free, random, and small world networks, we perform extensive numerical simulations to verify the proposed theory. Numerical simulations show that the analytically derived frequency set not only provides stable perfect synchronization in the network at a desired point but also proves to be very effective in achieving a high level of synchronization around it compared to the other choices of frequency sets. The stability and the robustness of the perfect synchronization state of the system are determined using the low-dimensional reduction of the network and by introducing a Gaussian noise around the derived frequency set, respectively.

Bifractality of fractal scale-free networks.

The presence of large-scale real-world networks with various architectures has motivated active research towards a unified understanding of diverse topologies of networks. Such studies have revealed that many networks with scale-free and fractal properties exhibit the structural multifractality, some of which are actually bifractal. Bifractality is a particular case of the multifractal property, where only two local fractal dimensions d_{f}^{min} and d_{f}^{max}(>d_{f}^{min}) suffice to explain the structural inhomogeneity of a network. In this work we investigate analytically and numerically the multifractal property of a wide range of fractal scale-free networks (FSFNs) including deterministic hierarchical, stochastic hierarchical, nonhierarchical, and real-world FSFNs. Then we demonstrate how commonly FSFNs exhibit the bifractal property. The results show that all these networks possess the bifractal nature. We conjecture from our findings that any FSFN is bifractal. Furthermore, we find that in the thermodynamic limit the lower local fractal dimension d_{f}^{min} describes substructures around infinitely high-degree hub nodes and finite-degree nodes at finite distances from these hub nodes, whereas d_{f}^{max} characterizes local fractality around finite-degree nodes infinitely far from the infinite-degree hub nodes. Since the bifractal nature of FSFNs may strongly influence time-dependent phenomena on FSFNs, our results will be useful for understanding dynamics such as information diffusion and synchronization on FSFNs from a unified perspective.

Characterizing force-chain network in aggregate blend using discrete element method and complex network theory

The force chain network in aggregate blend indicates load transfer path and is the main carrier to resist external loads. The objective of this study is to analyze the force chain network characteristics of aggregate blend on the basis of complex network theory. The virtual aggregate blend models with three stone matrix asphalt (SMA) gradations and three dense asphalt concrete (AC) gradations were constructed using discrete element method (DEM). The force chain network topology structure of each aggregate blend was established according to the aggregate contact parameters extracted from DEM software and the introduced complex network theory. The characteristics of each force chain network were analyzed by the various network evaluation indicators. The results showed that there are two or more contacts between most particle pairs. The average coordination number and the average contact number of aggregate blend are not equal. The degree distribution of each gradation shows a good power-law distribution. The force chain network of aggregate blend is a scale-free network. Big particles have greater average strength and smaller average weight distribution difference compared with small particles. The reference lower limit has better clustering performance for SMA-16, and the reference upper limit has better clustering performance for AC-25. The average path length of SMA-16 is longer than that of AC-25, indicating that SMA-16 has better load transfer ability and skeleton properties.

Dynamic survivability in oscillator systems

Whether the dynamic systems can display normal performance or complete key mission is the hot spot in recent years. In accordance with this, dynamic survivability is defined for the first time and mission-completion-probability is given as an index in this study. Taking the behavior of diffusively coupled oscillator systems with Erdös-Rényi random graph and Barabási-Albert scale-free network structures as examples, we show the network structures, systems’ parameters and attack strategies exhibit a profound influence on the dynamic survivability. Erdös-Rényi random graph and Barabási-Albert scale-free systems display better or worse dynamic survivability under different attack strategies respectively. Our outcomes fill the gap in the survivability study and are helpful for enhancing the dynamic survivability of real systems.

Strategically positioning non-competitive individuals can rescue cooperation in scale-free networks

Cooperation rescue in populations has long been a challenging problem, and previous studies have suggested that strategically placing cooperators in important positions in structured populations at the initial moment can obtain the opportunity of cooperation expansion. How to effectively lay out the location and place personalized nodes occupies a very important position in solving social dilemmas. However, most previous studies have focused on how the distribution of initial cooperators or defectors in homogeneous groups affects evolutionary outcomes, and relatively few studies have examined the placement of individuals with invariant heterogeneous characteristics. To enrich this research, we extend the cooperation rescue model to investigate the importance of the positional layout of two types of individuals with heterogeneous social value orientation characteristics in BA scale-free networks. Our study specifically investigates the impact of the distribution of non-competitive individuals on cooperation. We introduce non-competitive individuals into a BA scale-free network with only competitive individuals using different ranking rules. Our findings demonstrate that the presence of non-competitive individuals can promote cooperation. Furthermore, we determine the minimum number of nodes required to sustain cooperation under different ranking rules. In summary, our study emphasizes the importance of non-competitive individuals being placed in positions of importance for cooperation to flourish.

Predicting aging transition using Echo state network.

It is generally known that in a mixture of coupled active and inactive nonlinear oscillators, the entire system may stop oscillating and become inactive if the fraction of active oscillators is reduced to a critical value. This emerging phenomenon, called the "aging transition," can be analytically predicted from the view point of cluster synchronization. One can question whether a model-free, data-driven framework based on neural networks could be used to foretell when such a system will cease oscillation. Here, we demonstrate how a straightforward ESN with trained output weights can accurately forecast both the temporal evaluation and the onset of collapse in coupled paradigmatic limit-cycle oscillators. In particular, we have demonstrated that an ESN can identify the critical fraction of inactive oscillators in a large all-to-all, small-world, and scale-free network when it is trained only with two nodes (one active and the other inactive) selected from three different pre-collapse regimes. We further demonstrate that ESN can anticipate aging transition of the network when trained with the mean-field dynamics of active and inactive oscillators.

Information dissemination evolution under group feedback

In this paper, based on the feedback mechanism from the perspective of network groups, the evolutionary characteristics and laws of group networks under information dissemination are studied. First, the network is divided into groups of different sizes, and each group is given a dynamically changing group activity and a positive degree of response to different categories of information. Second, a feedback-based model of information dissemination in group networks is developed, which takes into account the differences between same-group and cross-group dissemination of information. Next, the model is applied to a scale-free network and a small-world network for simulation experiments. The experimental results show that, under the feedback mechanism, the main factor affecting the final evolutionary results of each group size in the small-world network is group positivity, which has little relationship with the initial size; similarly, the main factor affecting the average degree of each group in the scale-free network is also group positivity, which has nothing to do with the initial average degree. Finally, the method is applied to a real network to verify the rationality and effectiveness of the proposed model.

Evolution of residents' cooperative behavior in neighborhood renewal: An agent-based computational approach

The cooperative behavior of residents is complex and influenced by their complicated social relationships. This complexity is especially noticeable in neighborhood renewal, so the government does not know how to promote residents' cooperative behavior. Therefore, this study proposes an agent-based model (ABM) to investigate the development of residents' cooperative behavior in neighborhood renewal. Based on a questionnaire survey among residents of old neighborhoods in China, the parameters of ABM were determined in this study. Then, controlled experiments were conducted to investigate the effects of general trust among residents and government control of neighborhood renewal on cooperation patterns in renewal projects. In addition, this study examines the effects of different types of social network structures (small-world, scale-free, and random networks) on the evolution of residents' cooperative behaviors. The simulation results show that when residents' initial willingness to agree to renewal projects is high, their close social relationships need to be managed by the government to achieve better outcomes. Conversely, if initial willingness is low, residents' close relationships may pose a challenge to the government. In addition, government-led renewal projects should be encouraged to a greater extent. This study confirms that the different social network structures have an influence on the development of residents' cooperative behavior. The results of this study provide concrete evidence for understanding the factors that contribute to the emergence of residents' cooperative behavior and for studying the effects of government intervention on neighborhood renewal projects. In addition, the results of this study provide theoretical support for future studies of residents' social network structures.

Evolutionary public good games based on the long-term payoff mechanism in heterogeneous networks

Over the past two decades, it has been of great concern to solve the puzzle of social dilemmas in spatial public goods games. It has been demonstrated that the introduction of memory effects can effectively promote cooperative emergence to some extent. In this paper, we focus on the impact of scale-free and high aggregation properties on the public goods game of memory heterogeneous groups. Here, we propose an optimal strategy selection mechanism based on long-term historical payoffs. In our model, a memory loss function ω is introduced to characterize the memory length of players, and accordingly, pairs determine the optimal strategy by comparing historical gains for the next game cycle. Experiments conducted on scale-free networks and small world networks show that the strategy selection mechanism based on long-term payoff can keep the proportion of cooperators at a high level. In addition, this study demonstrates that the heterogeneity of memory length in the network promotes the enhancement of network reciprocity and robustness, which is beneficial to further reveal the mechanism of cooperative behaviour emergence in complex networks.

Droplet finite-size scaling of the majority-vote model on scale-free networks.

We discuss the majority vote model coupled with scale-free networks and investigate its critical behavior. Previous studies point to a nonuniversal behavior of the majority vote model, where the critical exponents depend on the connectivity. At the same time, the effective dimension D_{eff} is unity for a degree distribution exponent 5/2<γ<7/2. We introduce a finite-size theory of the majority vote model for uncorrelated networks and present generalized scaling relations with good agreement with Monte Carlo simulation results. Our finite-size approach has two sources of size dependence: an external field representing the influence of the mass media on consensus formation and the scale-free network cutoff. The critical exponents are nonuniversal, dependent on the degree distribution exponent, precisely when 5/2<γ<7/2. For γ≥7/2, the model is in the same universality class as the majority vote model on Erdős-Rényi random graphs. However, for γ=7/2, the critical behavior includes additional logarithmic corrections.

Complexity of networks: A structural heterogeneity approach

The complexity measurement of networks is one of the hot topics in complex networks. How to accurately describe the complexity difference between networks is helpful to the study of network structure. This paper proposes a method for measuring the network complexity considering the local and global structural heterogeneity. This method introduces the k-order neighbors to examine the local structural heterogeneity. Besides, the standard deviation is used to evaluate the global structural heterogeneity. Based on this, the complexity measurement model is established. In order to testify the efficiency of the method, the ER random networks, BA scale-free networks and real networks (including four social networks and two infrastructure networks) are used for experiments. The results show that the proposed method is sensitive and can effectively describe the subtle difference of structure complexity between networks.

Study on the Evolution of Risk Contagion in Urban River Ecological Management Projects Based on SEIRS

The risk transmission mechanisms of urban river ecological management engineering projects are examined in this study. Using the Susceptible Exposed Infectious Recovered Susceptible (SEIRS) model for risk transmission, a model of risk propagation delay for urban river ecological management engineering projects on scale-free networks is developed, which takes into account the effects of risk propagation and delay. We conducted a steady-state analysis of the model and obtained the basic reproduction number R. When R &gt; 1, the equilibrium point of risk outbreak is stable, and when R &lt; 1, the equilibrium point of risk disappearance is stable. Numerical simulations of the model were conducted using the MATLAB2022b to reveal the dynamic propagation patterns of risk in urban river ecological management engineering projects. The research results show that the steady-state density of the infected nodes in the network increases with the increase in the effective propagation rate and the propagation delay time; the propagation delay reduces the risk propagation threshold in the network and accelerates the occurrence of the equilibrium state of risk outbreak. There is a correlation between the transmission rate of latent nodes and the transmission rate of infected nodes, and the effective transmission rate of latent nodes has a greater influence on risk propagation. The spread of risk in the network can be effectively controlled and mitigated with targeted immunity for susceptible nodes. This article, based on the theory of complex networks and the mean-field theory, takes into account the propagation delay and spreading of latent nodes. Building a D-SEIRS model for risk propagation broadens the research perspective on urban river ecological management risk propagation.