Hyperdegree Distribution Research Articles

Robustness of interdependent directed higher-order networks against cascading failures

In the real world, directed networks are not just constructed as pairs of directed interactions, but also occur in groups of three or more nodes that form the higher-order structure of the network. From social networks to biological networks, there is growing evidence that real-world systems connect the functional relationships of multiple systems through interdependence. To understand the robustness of interdependent directed higher-order networks, we propose a new theoretical framework to model and analyze the robustness of such networks under random failures by percolation theory. We find that adding higher-order edges makes the network more vulnerable which quantifies and compares by two criteria: the size of the giant connected components and the percolation threshold. Increasing the hyperdegree distribution of heterogeneity or the hyperedge cardinality distribution of heterogeneity in interdependent directed higher-order networks will also make the network more vulnerable. Interestingly, the phase transition type changes from continuous to discontinuous with the increase of coupling strength, and partially interdependent directed higher-order networks exist hybrid phase transition. Moreover, by applying our theoretical analysis to real interdependent directed higher-order networks further validated our conclusion, it has implications for the design of flexible complex systems.

Higher-order interdependent percolation on hypergraphs

A fundamental concern on the robustness of hypergraphs lies in comprehending how the failure of individual nodes affects the hyperedges they are associated with. To address the issue, we propose a simple but novel percolation model that takes into account the dependency of hyperedges on their internal nodes, where the failure of a single node can lead to the dissolution of its associated hyperedge with a probability β. Based on a newly proposed analytical method of percolation theory on hypergraphs, our research reveals that the impact of mean cardinality on the system robustness varies with β. For a large value of β, a larger mean cardinality increases the fragility of hypergraphs, while for a small β, a larger mean cardinality enhances the robustness of hypergraphs. Additionally, our research uncovers divergent effects of hyperdegree distribution on system robustness between monolayer and double-layer hypergraphs. Specifically, monolayer hypergraphs with scale-free hyperdegree distribution exhibit higher robustness, while Poisson hyperdegree distributions lead to stronger robustness in double-layer hypergraphs. These findings provide valuable insights into the robustness of hypergraphs and its dependency on hyperdegree distributions and mean cardinality, contributing to a more comprehensive understanding of the complexities of robustness in complex systems. Furthermore, the development of the percolation model enriches our understanding of node-hyperedge interactions within complex systems.

Robustness of directed higher-order networks.

In complex systems, from human social networks to biological networks, pairwise interactions are insufficient to express the directed interactions in higher-order networks since the internal function is not only contained in directed pairwise interactions but rather in directed higher-order interactions. Therefore, researchers adopted directed higher-order networks to encode multinode interactions explicitly and revealed that higher-order interactions induced rich critical phenomena. However, the robustness of the directed higher-order networks has yet to receive much attention. Here, we propose a theoretical percolation model to analyze the robustness of directed higher-order networks. We study the size of the giant connected components and the percolation threshold of our proposed model by the theory and Monte-Carlo simulations on artificial networks and real-world networks. We find that the percolation threshold is affected by the inherent properties of higher-order networks, including the heterogeneity of the hyperdegree distribution and the hyperedge cardinality, which represents the number of nodes in the hyperedge. Increasing the hyperdegree distribution of heterogeneity or the hyperedge cardinality distribution of heterogeneity in higher-order networks will make the network more vulnerable, weakening the higher-order network's robustness. In other words, adding higher-order directed edges enhances the robustness of the systems. Our proposed theory can reasonably predict the simulations for percolation on artificial and real-world directed higher-order networks.

Study of Information Dissemination in Hypernetworks with Adjustable Clustering Coefficient

The structure of a model has an important impact on information dissemination. Many information models of hypernetworks have been proposed in recent years, in which nodes and hyperedges represent the individuals and the relationships between the individuals, respectively. However, these models select old nodes based on preference attachment and ignore the effect of aggregation. In real life, friends of friends are more likely to form friendships with each other, and a social network should be a hypernetwork with an aggregation phenomenon. Therefore, a social hypernetwork evolution model with adjustable clustering coefficients is proposed. Subsequently, we use the SIS (susceptible–infectious–susceptible) model to describe the information propagation process in the aggregation-phenomenon hypernetwork. In addition, we establish the relationship between the density of informed nodes and the structural parameters of the hypernetwork in a steady state using the mean field theory. Notably, modifications to the clustering coefficients do not impact the hyperdegree distribution; however, an increase in the clustering coefficients results in a reduced speed of information dissemination. It is further observed that the model can degenerate to a BA (Barabási–Albert) hypernetwork by setting the clustering coefficient to zero. Thus, the aggregation-phenomenon hypernetwork is an extension of the BA hypernetwork with stronger applicability.

Tracing the network of scientific collaboration

Scholar hyperdegree distribution can be modeled as a power law distribution.

Dynamical evolution behavior of scientific collaboration hypernetwork

Scientific collaboration has a complex hypernetwork structure. How to construct scientific collaboration in a complex system is an open issue. In this paper, a non-uniform dynamic collaborative evolution model is proposed. In the proposed method, each scholar is viewed as a node, and each cooperation relationship is regarded as a hyperedge. This model includes three processes: adding hyperedges, entering nodes, and forming hyperedges by new nodes. It is theoretically proved that the hyperdegree distribution of nodes follows the power law distribution. Furthermore, the effects of different parameters on the proposed model are numerically simulated in this paper. The experimental results are consistent with the theoretical ones. In addition, experiments show that the influence of nodes and hyperedges will affect the selection of old nodes when new nodes enter the network. This paper not only considers the construction of hyperedges with old nodes but also considers the possibility that new nodes construct new hyperedges among themselves. This model provides a reference for the research of the evolution process of scientific collaboration hypernetworks.

Risk propagation and evolution analysis of multi-level handlings at automated terminals based on double-layer dynamic network model

The propagation and evolution of risks in the system adversely affects the safety of automated terminal handlings. Therefore, it is necessary to explore the formation and development process of handling risks, predict the path and pattern of risk propagation and summarize the evolution features to improve the efficiency of multi-level handlings at automated terminals (ATMH). This paper proposes a double-layer dynamics risk propagation and evolution model. First, a complex network of risk relationships is established, and the Gillespie algorithm is employed to simulate the risk propagation dynamics based on the metric of state transition value and the dynamic mean-field theory. Next, an evolution hypernetwork is constructed and the average smooth hyperdegree distribution is derived using Poisson process and continuity methods. The simulation results show that the risk propagation link group obtained by the empirical testing is credible. The theoretical predictions of risk evolution agree well with the numerical simulations. The greater the probability of risk propagation, the more active the evolution. The model provides a certain theoretical reference for the risk management of automated terminals.

Dynamic evolution of shipping network based on hypergraph

This study aims to use evolutionary hypergraphs to explore the dynamic evolution mechanism of the shipping network. Firstly, an evolution model of the shipping network is constructed, in which nodes represent ports while hyperedges represent routes between ports. The evolution process includes the addition of new nodes, linking of old nodes, and rewiring of links. Then, based on Poisson process theory and continuum method, we obtained that the stationary average hyperdegree distribution obeys a shifted power law. Finally, the effects of different parameters on the model are tested. The results show that ports with a developed hinterland economy and great influence have obvious advantages in the evolution process, gathering a large amount of resource allocation. This model can provide a theoretical reference for the reasonable allocation of routes and the improvement of network efficiency.

Emergence of scaling in evolving hypernetworks

Based on hypernetwork theory, we construct a new evolving hypernetwork, which incorporating new nodes and new hyperedges growing, old nodes and old hyperedges disappearing, or rewiring of hyperedges from some nodes to anothers. Besides the growth of hyperedges by adding new nodes, it is also possible that a new hyperedge can be constructed between old nodes in hypernetworks. This evolving hypernetwork model with both increasing and decreasing is more realistic than the evolving model only with increasing. By employing Poisson process theory and continuous method, we obtain stationary average hyperdegree distribution and degree distribution of the hypernetwork. Analytical result shows that the evolving hypernetwork following a generalized power-law distribution, has a phenomenon of “the rich get richer” and a wide range of universality. The theoretical prediction of the stationary average hyperdegree distribution and degree distribution are in good agreement with the real numerical simulation results.

Exploring the dynamic growth mechanism of social networks using evolutionary hypergraph

An evolutionary hypernetwork model is proposed to describe the non-uniform evolution of social networks, in which nodes represent individuals while hyperedges represent relationships among individuals. The number of nodes in a hyperedge is a random integer. And the evolving process includes the addition of new nodes, linking of old nodes, and rewiring of links. By using Poisson process theory and the continuous method, we proved that the stationary average hyperdegree distribution follows the shifted power law (SPL). The theoretical analysis agree with the numerical simulations. Our model is universal, the fitness model in complex networks and scale-free model in hypernetworks can all be regarded as degradation cases of the model.

Information dissemination in dynamic hypernetwork

Recently, the information diffusion models based on static hypernetworks have been proposed, in which the nodes and hyperedges represent the individuals and the social groups consisting of several individuals, respectively. However, the social networks represented by hypernetwork should be a dynamic network because of social activities. Hence, we establish an SIS model to present the dynamics of information spreading in a dynamic social hypernetworks with three dynamic processes. Then the theoretical analysis and simulation of the model are carried out, and theoretical results are accordant with the simulation results. In addition, the effects of different parameters on the information dissemination dynamics are analyzed in RP and CP strategy respectively. Most noteworthy, reorganization of social relations leads to a decrease in the proportion of informed individuals in stable state. By analyzing the effect of hyperedge reorganization on hyperdegree distribution, we discover the reorganization result in the increasing number of isolated nodes. These isolated nodes is hard to be joined in new social relationships, and cannot get information from the social network.

The agglomeration phenomenon influence on the scaling law of the scientific collaboration system

This paper presents a scientific collaboration hypernetwork evolution model with adjustable clustering coefficient in which the authors are regarded as nodes and the hyperedges are the cooperative articles. Firstly, we build the scientific cooperation hypernetwork through the real paper data which comes from the database of arxive. The empirical results show that the node's hyperdegree follows the power law distribution, but the hyperedge's node-degree has the exponent distribution. In addition, we establish the scientific collaboration clustering hypernetwork evolution model by employing the Poisson process theory and the continuous method for studying the agglomeration phenomenon of scientific cooperation system. The theoretical analysis shows that the node's hyperdegree distribution has scale-free characteristics. The power index of our model is independent of the clustering coefficient, and the theoretical analyses agree with the conducted numerical simulations. Moreover, our model not only describes the scale-free property, but also depicts the phenomenon of agglomeration. Both the properties are usually coexist in the scientific cooperation, our model is more actual.

The evolutionary mechanism of high-speed railway system based on hypernetwork theory

Railway system as an important transportation pattern can improve regional economic growth and accelerate the development of transportation structure. In this paper, we propose an evolving model to describe the evolutionary mechanism of high-speed railway system by using hypernetwork theory. The stations are represented as nodes while the lines are represented as hyperedges. The evolving process includes two ingredients: growing and linking which is driven by both random and preferential attachment. We analytically deduce the node hyperdegree distribution which is shown to follow a shifted power-law distribution. Furthermore, we test the impact of parameters on the model. Then the empirical investigations of China Railway High-speed (CRH) are demonstrated which can be explained by the proposed model. The model can reveal the macro characteristics of railway system and provide reference for the further development of high-speed railway.

Research on mobile social non-uniform hypernetwork evolving model

The modelling studies for the mobile social network will help to understand the characteristics of network structure and evolution mechanisms. In order to improve the accuracy of the description of mobile social hypernetwork model, this paper takes the WeChat as a sample and puts forward a non-uniform hypernetwork (hypergraph-Oriented) model with the attractiveness and the ages of nodes. In the model, nodes arrive at the system in accordance with Poisson process and are gradually ageing as the growth of time. We analyse the relevant characteristics of hypernetwork evolution and obtain the stationary average hyperdegree distribution of the hypernetwork by the characteristic equation. Numerical simulations of the models agree with the analytical results well. The whole research aims to provide support for the dynamics rule of mobile social hypernetwork and to provide a new perspective for the study of mobile social hypernetwork.

Research on Mobile Social Non-uniform Hypernetwork Evolving model

The modelling studies for the mobile social network will help to understand the characteristics of network structure and evolution mechanisms. In order to improve the accuracy of the description of mobile social hypernetwork model, this paper takes the WeChat as a sample and puts forward a non-uniform hypernetwork (hypergraph-Oriented) model with the attractiveness and the ages of nodes. In the model, nodes arrive at the system in accordance with Poisson process and are gradually ageing as the growth of time. We analyse the relevant characteristics of hypernetwork evolution and obtain the stationary average hyperdegree distribution of the hypernetwork by the characteristic equation. Numerical simulations of the models agree with the analytical results well. The whole research aims to provide support for the dynamics rule of mobile social hypernetwork and to provide a new perspective for the study of mobile social hypernetwork.

Exploring the evolutionary mechanism of complex supply chain systems using evolving hypergraphs

A new evolutionary model is proposed to describe the characteristics and evolution pattern of supply chain systems using evolving hypergraphs, in which nodes represent enterprise entities while hyperedges represent the relationships among diverse trades. The nodes arrive at the system in accordance with a Poisson process, with the evolving process incorporating the addition of new nodes, linking of old nodes, and rewiring of links. Grounded in the Poisson process theory and continuum theory, the stationary average hyperdegree distribution is shown to follow a shifted power law (SPL), and the theoretical predictions are consistent with the results of numerical simulations. Testing the impact of parameters on the model yields a positive correlation between hyperdegree and degree. The model also uncovers macro characteristics of the relationships among enterprises due to the microscopic interactions among individuals.

Study of the variable growth hypernetworks influence on the scaling law

This paper presents the influence of two kinds of variable growth hypernetworks on the scaling law of hypernetworks. Our model can be degenerated to the original evolving model. On one hand, we establish the variable growth of hyperedges evolving hypernetworks model and obtain the stationary average hyperdegree distribution of the model by employing the Poisson process theory and the continuous method. The theoretical analyses in this paper agree with the conducted numerical simulations. The results show that the transient average hyperdegree distribution of the hypernetwork follows the scale-free law. However, the stationary average hyperdegree distribution does not, which indicates that the number of hyperedges at each time step affects the scaling law of hypernetworks. On the other hand, we establish the new nodes variable growth hypernetworks model and obtain analytical solutions of the transient average hyperdegree distribution. We simulate to compare the different node growth rate effect on the hyperdegree distribution characteristics. The results show that the growth rate does not affect the scaling law properties of the hypernetworks. Only when λ=1 can we obtain analytical solutions of stationary average hyperdegree distribution of the hypernetwork which is not scale-free law.

Non-uniform Evolving Hypergraphs and Weighted Evolving Hypergraphs.

Firstly, this paper proposes a non-uniform evolving hypergraph model with nonlinear preferential attachment and an attractiveness. This model allows nodes to arrive in batches according to a Poisson process and to form hyperedges with existing batches of nodes. Both the number of arriving nodes and that of chosen existing nodes are random variables so that the size of each hyperedge is non-uniform. This paper establishes the characteristic equation of hyperdegrees, calculates changes in the hyperdegree of each node, and obtains the stationary average hyperdegree distribution of the model by employing the Poisson process theory and the characteristic equation. Secondly, this paper constructs a model for weighted evolving hypergraphs that couples the establishment of new hyperedges, nodes and the dynamical evolution of the weights. Furthermore, what is obtained are respectively the stationary average hyperdegree and hyperstrength distributions by using the hyperdegree distribution of the established unweighted model above so that the weighted evolving hypergraph exhibits a scale-free behavior for both hyperdegree and hyperstrength distributions.

Cascading failure analysis in hyper-network based on the hypergraph

In this paper, we analyze the diffusion patterns of cascading failure, which happen in the express hypernetwork and electronic hypernetwork respectively. The cascading failure of the express hypernetwork is diffused by the node, and the cascading failure of the electronic hypernetwork is diffused by the hyper-edge. According to hyper-graph theory, we propose two methods to characterize these cascading failures, which are 2-section graph analytical method and line-graph analytical method. We analyze the characteristics of the cascading failures based on node by using the 2-section graph analytical method and based on hyper-edge by using line-graph analytical method, respectively. We construct a k uniform scale-free hypernetwork and analyze the cascading failure process of this hypernetwork based on the couple map lattice according to our methods. The simulation results show that the scale-free hypernetworks are both robust and vulnerable for attack. It is found that the cascading failure based on the node of k uniform scale-free hypernetwork is associated with the hyper-degree distribution of nodes, and the scale-free hypernetwork is robust for random attack and vulnerable for deliberate attack. The more nodes a hyper-edge has, the better robustness the hypernetwork has.The cascading failure based on the hyper-edge is different from the cascading failure based on the node. The cascading failure based on the hyper-edge is associated with the hyper-edge degree distribution. The hyper-edge degree distribution of the scale-free hypernetwork is not entirely the power-low distribution. When the cascading failure is diffused by the hyper-edge, the hypernetwork is vulnerable for random attack and robustness for deliberate attack if there are 3 or 5 nodes in a hyper-edge. Moreover, the hypernetwork becomes robust for the random attack if there are 7 nodes in a hyper-edge. Furthermore, the k uniform scale-free hypernetwork is more robust than the same size Barabasi-Albert scale-free network for the same attack. The cascading failure process based on the hyper-edge is slower than based on the node. We find that the edge number is another influential factor of robustness. The network is more robust if it has more edges for fixed node number. The line-graph has more edges than the 2-section graph in the same size scale-free hypernetwork, so the cascading failure of node is slower than that of hyper-edge.

Brand effect versus competitiveness in hypernetworks.

A few of evolving models in hypernetworks have been proposed based on uniform growth. In order to better depict the growth mechanism and competitive aspect of real hypernetworks, we propose a model in term of the non-uniform growth. Besides hyperdegrees, the other two important factors are introduced to underlie preferential attachment. One dimension is the brand effect and the other is the competitiveness. Our model can accurately describe the evolution of real hypernetworks. The paper analyzes the model and calculates the stationary average hyperdegree distribution of the hypernetwork by using Poisson process theory and a continuous technique. We also address the limit in which this model has a condensation. The theoretical analyses agree with numerical simulations. Our model is universal, in that the standard preferential attachment, the fitness model in complex networks and scale-free model in hypernetworks can all be seen as degenerate cases of the model.