Random Scale-free Networks Research Articles

Enhanced robustness of single-layer networks with redundant dependencies.

Dependency links in single-layer networks offer a convenient way of modeling nonlocal percolation effects in networked systems where certain pairs of nodes are only able to function together. We study the percolation properties of the weak variant of this model: Nodes with dependency neighbors may continue to function if at least one of their dependency neighbors is active. We show that this relaxation of the dependency rule allows for more robust structures and a rich variety of critical phenomena, as percolation is not determined strictly by finite dependency clusters. We study Erdős-Rényi and random scale-free networks with an underlying Erdős-Rényi network of dependency links. We identify a special "cusp" point above which the system is always stable, irrespective of the density of dependency links. We find continuous and discontinuous hybrid percolation transitions, separated by a tricritical point for Erdős-Rényi networks. For scale-free networks with a finite degree cutoff we observe the appearance of a critical point and corresponding double transitions in a certain range of the degree distribution exponent. We show that at a special point in the parameter space, where the critical point emerges, the giant viable cluster has the unusual critical singularity S-S_{c}∝(p-p_{c})^{1/4}. We study the robustness of networks where connectivity degrees and dependency degrees are correlated and find that scale-free networks are able to retain their high resilience for strong enough positive correlation, i.e., when hubs are protected by greater redundancy.

Effect of Segregation on Opinion Formation in Scale-Free Social Networks: An Agent-based Approach

We consider the effect of segregation on opinion formation in social networks with and without influential leaders in scale-free random networks, which is found in many social and natural phenomena. We have used agent-based modeling and simulation, focusing on the social impact model of opinion formation. Two simulation scenarios of this opinion formation model have been considered: (1) the original scenario which randomly assigns persuasion strengths to the agents, and (2) a centrality-based scenario, which assigns persuasion strengths proportional to the agents’ centralities. In the latter scenario, hubs are considered more influential leaders who are more connected to others and have higher persuasion strengths than others. The simulation results show a correlation between segregation and change of population opinion in the original model, but no correlation between both variables in the centrality-based scenario. The results lead us to conclude that with strong influential leaders in society, the effect of segregation in opinion formation is neglectable.

Modeling and analyzing cascading dynamics of the urban road traffic network

The cascade failure problem has gained an increasing interest in urban road traffic network. Considering the intrinsic characteristics of urban traffic network, the traffic load has its specific regularity and traffic driving has certain preferences, this paper introduces the average distance and the state change time of road junction. Using the weighted method to combine the two aspects, a trust value model is got to measure the road congestion. Then in order to avoid the congestion of the failed road junction again in a short time, the multiple overload of the failed road junction is redistributed according to the trust value model. Finally, the cascade failure model is obtained. The cascade failure model is verified in scale-free network and random network. The joint parameters of load threshold and load allocation factor that can control the cascade failure scale in a certain range are obtained, which improves the robustness of the networks.

Influence of opinion dynamics on rumor propagation in complex networks

In this paper, the influence of opinion dynamics on rumor propagation is studied by introducing the mechanism of stifler’s comments on rumors into the susceptible-infected-recovered (SIR) model. In this model, individuals can form a group together with their direct neighbors, through which the spreader can spread rumors, and the stiflers can express opinions and refute the rumors. The mechanism of rumor refuting can not only reduce the acceptance rate of the ignorant to rumor, but also increase the transition probability of the spreader to be a stifler. In this paper, we use the Erdös-Rényi (ER) random network, scale-free network and real social network as the underlying interaction structure to study the influence of stifler’s silence probability on the rumor spreading dynamics. First of all, we find that the process of rumor propagation can be roughly divided into two stages, i.e., the early stage of free propagation of rumors and the later stage of checks and balances between the stiflers and the spreaders, respectively. Secondly, it is found that the rumor will break out with the increase of stifler’s silence probability. Under a threshold of rumor outbreak, the increase of silence probability will not lead the number of spreaders to significantly increase, but will cause more ignorance to perceive the rumor and quickly turn into spreaders, and then change into stiflers under the guidance of other stiflers. When the silence probability reaches a threshold, the stiflers will not be able to control the spread of rumors, which will lead the number of stiflers to decrease and the rumors to break out. Finally, the early stage of rumor propagation in scale-free networks is shorter than that of random network, which makes rumor more difficult to break out. Our model comprehensively considers the influence of opinion dynamics on the spreading of rumors and more realistically simulates the rumor diffusion process, which provides a useful insight for the rumor control in real-world social networks.

Biyolojik ve Biyolojik Olmayan Ağlar Üzerine

With a general classification, there are two types of networks in the world: Biological and non-biological networks. We are unable to change the structure of biological networks. However, for networks such as social networks, technological networks and transportation networks, the architectures of non-biological networks are designed and can be changed by people. Networks can be classified as random networks, small-world networks and scale-free networks. However, we have problems with small-world networks and scale free networks. As some authors ask, “how small is a small-world network and how does it compare to other models?” Even the issue of scale-free networks are whether abundant or rare is still debated. Our main goal in this study is to investigate whether biological and non-biological networks have basic defining features. Especially if we can determine the properties of biological networks in a detailed way, then we may have the chance to design more robust and efficient non-biological networks. However, this research results shows that discussions on the properties of biological networks are not yet complete.

A Simulation Approach of Information Propagation in Homogeneous Network

Modelling the behaviour of online social network users’ opinion propagation is of great significance. This paper proposes a method with individual’s own characteristics to model the opinion propagation in homogeneous network. We firstly described the detail of propagation model construction steps including opinion interaction rules, cumulative interaction influence value normalization and historical impact. Then we introduce the definition of individual tolerance to better model propagation. We conducted simulation experiments on two representative propagation networks: ER random network and BA scale-free network. In the end, we give the phenomena and explanation of each simulation result from three aspects.

How the intimate relationship affects the individuals' behaviors in a two-layer network

Game theories could help us to understand the origin of the emergence and persistence of cooperation among selfish individuals. Considering the intimate relationship among individuals, we study the prisoner's dilemma (PD) game in a two-layer network. Firstly, we calculate the closeness between any two players according to their connectivity. Further, we define and obtain one's comprehensive income index according to the different income values obtained by individuals in the two-layer network. Then, we formulate the mechanism of individual strategy change. Finally, we analyze the influence of different parameters on individual strategy change through numerical simulation results. Our results show that the heterogeneous network structure is more conducted to cooperation, but it takes a long time for the whole system to reach a stable state. With the increase of the measure coefficient θ, when the two-layer networks are, respectively, the BA scale-free network and ER random network, the final increase rate of cooperator's density in the whole system is the fastest.

Generating Massive Scale-free Networks

Recently, there has been substantial interest in the study of various random networks as mathematical models of complex systems. As real-life complex systems grow larger, the ability to generate progressively large random networks becomes all the more important. This motivates the need for efficient parallel algorithms for generating such networks. Naïve parallelization of sequential algorithms for generating random networks is inefficient due to inherent dependencies among the edges and the possibility of creating duplicate (parallel) edges. In this article, we present message passing interface-based distributed memory parallel algorithms for generating random scale-free networks using the preferential-attachment model. Our algorithms are experimentally verified to scale very well to a large number of processing elements (PEs), providing near-linear speedups. The algorithms have been exercised with regard to scale and speed to generate scale-free networks with one trillion edges in 6 minutes using 1,000 PEs.

Multiple probabilistic networks with scale-free property

It is well-known that the random scale-free networks are ubiquitous in the world and are applied in many areas of scientific research. Most previous networks are obtained from a single probability parameter, while our networks are produced by multiple probability parameters. This paper aims at generating a family of random scale-free networks by graphic operations based on probabilistic behaviors. These random scale-free networks can span a network space S(p, q, r, t) with three probabilistic parameters p, q, and r holding on p + q + r = 1 with 0 ≤ p, q, r ≤ 1 at each time step t. Each network N(p, q, r, t) of S(p, q, r, t) is a dynamic network that will be produced by N(p, q, r, t − 1) based on three types of operations, called the type-A operation, the type-B operation, and the type-C operation. We will show the topological structures of each network N(p, q, r, t) by its average degree, degree distribution, diameter, and clustering coefficient, and, furthermore, compute the degree exponent γ=1+ln(4−r)ln⁡2, as well as power-law distribution, in order to reveal the scale-free behavior of N(p, q, r, t), which induces the whole space S(p, q, r, t) to be scale-free. Our findings are able to enrich the fundamental structure properties of complex networks, in particular scale-free networks.

Constructions and properties of a class of random scale-free networks.

Complex networks have abundant and extensive applications in real life. Recently, researchers have proposed a large variety of complex networks, in which some are deterministic and others are random. The goal of this paper is to generate a class of random scale-free networks. To achieve this, we introduce three types of operations, i.e., rectangle operation, diamond operation, and triangle operation, and provide the concrete process for generating random scale-free networks N(p,q,r,t), where probability parameters p,q,r hold on p+q+r=1 with 0≤p,q,r≤1. We then discuss their topological properties, such as average degree, degree distribution, diameter, and clustering coefficient. First, we calculate the average degree of each member and discover that each member is a sparse graph. Second, by computing the degree distribution of our network N(p,q,r,t), we find that degree distribution obeys the power-law distribution, which implies that each member is scale-free. Next, according to our analysis of the diameter of our network N(p,q,r,t), we reveal the fact that the diameter may abruptly transform from small to large. Afterward, we give the calculation process of the clustering coefficient and discover that its value is mainly determined by r.

Edge-based SEIR dynamics with recovery rate in latent period on complex networks

Disease and information spreading on social and information networks have often been described by ordinary differential equations. A recent research by the authors [Y. Wang et al., Commun. Nonlinear Sci. Numer. Simulat. 45, 35 (2017).] presented an analysis of susceptible-exposed-infected-recovered (SEIR) model with and without infectious force in latent period. We present a full analysis in the more general scenario where the exposed nodes can get vaccinated or recovered. The basic reproduction number and the final epidemic size are theoretically derived. Compared to the standard SEIR model without recovery rate in latent period, our results reveal that both the recovery rate in latent period and the length of latent period can increase the epidemic threshold and inhibit the epidemic outbreak. In addition, the model predictions agree well with the continuous-time stochastic simulations in Erdős–Rényi random graphs and scale-free configuration networks.

The influence of social network structure on stock price disclosure

We propose a new heterogeneous agent model of a dynamic continuous double auction stock market embedding multiple social network to investigate the influence of different network structures on stock price disclosure. According to the four network formation algorithms (preferential connection, random connection, weighted random connection and random reconnection), we construct scale-free random network, weighted random network and small-world network respectively. We find that the small-world characteristic of social network speed up the diffusion of private information in market, and then improve market efficiency and promote price discovery. Furthermore, to some extent, simulation results indicate that the network structure can explain the occurrence of anomalies like underreaction and overreaction in the financial market.

Optimal Microbiome Networks: Macroecology and Criticality.

The human microbiome is an extremely complex ecosystem considering the number of bacterial species, their interactions, and its variability over space and time. Here, we untangle the complexity of the human microbiome for the Irritable Bowel Syndrome (IBS) that is the most prevalent functional gastrointestinal disorder in human populations. Based on a novel information theoretic network inference model, we detected potential species interaction networks that are functionally and structurally different for healthy and unhealthy individuals. Healthy networks are characterized by a neutral symmetrical pattern of species interactions and scale-free topology versus random unhealthy networks. We detected an inverse scaling relationship between species total outgoing information flow, meaningful of node interactivity, and relative species abundance (RSA). The top ten interacting species are also the least relatively abundant for the healthy microbiome and the most detrimental. These findings support the idea about the diminishing role of network hubs and how these should be defined considering the total outgoing information flow rather than the node degree. Macroecologically, the healthy microbiome is characterized by the highest Pareto total species diversity growth rate, the lowest species turnover, and the smallest variability of RSA for all species. This result challenges current views that posit a universal association between healthy states and the highest absolute species diversity in ecosystems. Additionally, we show how the transitory microbiome is unstable and microbiome criticality is not necessarily at the phase transition between healthy and unhealthy states. We stress the importance of considering portfolios of interacting pairs versus single node dynamics when characterizing the microbiome and of ranking these pairs in terms of their interactions (i.e., species collective behavior) that shape transition from healthy to unhealthy states. The macroecological characterization of the microbiome is useful for public health and disease diagnosis and etiognosis, while species-specific analyses can detect beneficial species leading to personalized design of pre- and probiotic treatments and microbiome engineering.

Epidemic spreading dynamics on complex networks with adaptive social-support

Using conditional quenched mean-field (cQMF) method, we develop a continuous-time susceptible–infected–susceptible epidemic model on quenched complex networks with combined self-recovery and social support from healthy network nodes. We determined the lower bound of the epidemic threshold which was found to depend directly on social support α. Our analytical results of the generalized cQMF model were in good agreement with numerical simulations on Erdő s–Rényi (ER) random graphs and different scale-free (SF) network configurations. We observed an interaction effect between network type (ER vs. SF) and social support level (α), finding a significant difference between the final epidemic sizes observed on ER versus SF networks when α was large, but not when α level was small or zero. Our findings suggest that social support from non-infected individuals contributes substantially towards the inhibition of an epidemic outbreak, and that the type of network structure plays a role in determining the final epidemic size but only when the social support level is sufficiently high.

Novel Parallel Algorithms for Fast Multi-GPU-Based Generation of Massive Scale-Free Networks

A novel parallel algorithm is presented for generating random scale-free networks using the preferential attachment model. The algorithm, named cuPPA, is custom-designed for “single instruction multiple data (SIMD)” style of parallel processing supported by modern processors such as graphical processing units (GPUs). To the best of our knowledge, our algorithm is the first to exploit GPUs, and also the fastest implementation available today, to generate scale-free networks using the preferential attachment model. A detailed performance study is presented to understand the scalability and runtime characteristics of the cuPPA algorithm. Also another version of the algorithm called cuPPA-Hash tailored for multiple GPUs is presented. On a single GPU, the original cuPPA algorithm delivers the best performance, but is challenging to port to multi-GPU implementation. For multi-GPU implementation, cuPPA-Hash has been used as the parallel algorithm to achieve a perfect linear speedup up to 4 GPUs. In one of the best cases, when executed on an NVidia GeForce 1080 GPU, the original cuPPA generates a scale-free network of two billion edges in less than 3 s. On multi-GPU platforms, cuPPA-Hash generates a scale-free network of 16 billion edges in less than 7 s using a machine consisting of 4 NVidia Tesla P100 GPUs.

Role of hubs in the synergistic spread of behavior.

The spread of behavior in a society has two major features: the synergy of multiple spreaders and the dominance of hubs. While strong synergy is known to induce mixed-order transitions (MOTs) at percolation, the effects of hubs on the phenomena are yet to be clarified. By analytically solving the generalized epidemic process on random scale-free networks with the power-law degree distribution , we clarify how the dominance of hubs in social networks affects the conditions for MOTs. Our results show that, for , an abundance of hubs drive MOTs, even if a synergistic spreading event requires an arbitrarily large number of adjacent spreaders. In particular, for , we find that a global cascade is possible even when only synergistic spreading events are allowed. These transition properties are substantially different from those of cooperative contagions, which are another class of synergistic cascading processes exhibiting MOTs.

Mitigation Strategy for the Cascading Failure of Complex Networks Based on Node Capacity Control Function

A single node failure capacity control function taking into account attack strength, attack times, control node load intensity and the degree of attacked node is proposed in this paper to mitigate the cascading failure of complex networks under random attack. An optimal probability allocation mechanism of redundant resources is established by targeting the load of each neighbor node. Then, the node failure capacity control function and allocation mechanism are used to define the phase transition critical factor and robustness indicator that used the attack strength, control node load intensity and the degree of attacked node as the parameters. Based on the above analysis, the phase transition critical factor model of degree distribution of scale-free network and random network is derived, and the dynamic change law between the parameters and phase transition critical state as well as robust performance of classical network and real network is analyzed. The theoretical and experimental results show that in the controllable region, the smaller the degree of attacked node, the greater the control node load intensity and the more difficult the phase transition critical state to be achieved, and the better the effect of mitigating cascading failure. Besides, the robustness of the network with cascading failure is mutually affected by the control node load capacity, the degree of attacked node and phase transition critical factor within a certain range, which thus embarks on a new perspective to mitigate the failure.

The Opinion Dynamics on the Evolving Complex Network by Achlioptas Process

The interactions of opinions on the complex networks are significantly impacted by the structure of the networks. Previous studies of this kind mainly investigated the opinion dynamics on the fixed networks as a kind of synchronization. In this study, we focus on how the opinions evolving on the growing networks. We provide isolated nodes with different initial opinions at the beginning. The Achlioptas Process is introduced to link the nodes eventually. The opinions of two nodes influence each other linearly if there is a link between the two nodes. We establish both random graph and scale-free network in this paper. The finite-size scaling is discussed. We discover explosive transition of the speed for the opinions to achieve a consensus on some networks. Meanwhile, the stability of the networks to suppress the random damage is highly enhanced by the Achlioptas Process which is used to link all the nodes as a network. The encouraging results are obtained on different structures of networks.

Dynamics of Cooperation in Minority Games in Alliance Networks

Alliance networks are the underlying structures of social systems in business, management, and society. The sustainability and dynamics of a social system rely on the structural evolutions of the topologies. Understanding the evolution sheds light on the dynamics and sustainability of a social system. Minority game models have been successfully applied across social science, economy, management, and engineering. They provide simple yet applicable modeling to articulate the evolutionary cooperation dynamics of competitive players in binary decision situations. By extending the minority games played in alliance networks, the cooperation in structured systems of different network topologies is analyzed. In this model, local and global score strategies are considered with and without cooperation rewiring options. The cooperation level, the score, and the topological properties are investigated. The research uses a numerical simulation approach on random networks, scale-free networks, and small-world networks. The results suggest that the network rewiring strategy leads to higher systemic performance with a higher score and a higher level of stability in decision-making. Competitive decision-making can lead to a higher level of cooperation from a poor initial start. However, stubbornness in decision-making can lead to a poor situation when cooperation is discouraged. Players with local or global information adopt local and global score strategies. The results show that local strategies might lead to imbalance, while a global strategy might achieve a relatively stable outcome. This work contributes to bridge minority games in structured networks to study the cooperation between formation and evolution, and calls for future minority game modeling on social networks.

Effects of the planarity and heterogeneity of networks on evolutionary two-player games**Project supported by the National Natural Science Foundation of China (Grant Nos. 11575072 and 11475074) and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2017-172).

We study the effects of the planarity and heterogeneity of networks on evolutionary two-player symmetric games by considering four different kinds of networks, including two types of heterogeneous networks: the weighted planar stochastic lattice (a planar scale-free network) and the random uncorrelated scale-free network with the same degree distribution as the weighted planar stochastic lattice; and two types of homogeneous networks: the hexagonal lattice and the random regular network with the same degree k0 = 6 as the hexagonal lattice. Using extensive computer simulations, we found that both the planarity and heterogeneity of the network have a significant influence on the evolution of cooperation, either promotion or inhibition, depending not only on the specific kind of game (the Harmony, Snowdrift, Stag Hunt or Prisoner’s Dilemma games), but also on the update rule (the Fermi, replicator or unconditional imitation rules).