Degree Distribution Of Network Research Articles

Quantifying and predicting evolutionary networks

The process of network evolution is typically characterized by the emergence of topological structures and the intricate interplay between determinism and stochasticity. Our work introduces a novel structural analysis framework to observe fluctuations in the network evolution process, profoundly influenced by the underlying network generation mechanisms. Based on theoretical reasoning and empirical examination, utilizing synthetic, static, and temporal networks, we arrive at two principal conclusions. Firstly, we reveal that the degree and distance distributions of networks exhibit two dynamic phenomena, convergence and divergence on high- and low-frequency curves, depend on the reflections of the network generation mechanism at both topological extreme values and the overall distribution. Secondly, we develop a novel link prediction method based on the convergence of topological fluctuations, which significantly outperforms benchmark algorithms in both real static and temporal networks. These findings are of considerable significance to the study of real network evolution mechanisms, contributing to a more comprehensive understanding of network behavior over time.

Feature analysis of 5G traffic data based on visibility graph

IntroductionAs 5G networks become widespread and their application scenarios expand, massive amounts of traffic data are continuously generated. Properly analyzing this data is crucial for enhancing 5G services.MethodsThis paper uses the visibility graph method to convert 5G traffic data into a visibility graph network, conducting a feature analysis of the 5G traffic data. Using the AfreecaTV dataset as the research object, this paper constructs visibility networks at different scales and observes the evolution of degree distribution with varying data volumes. The paper employs the Hurst index to evaluate the 5G traffic network and uses community detection to study the networks converted from 5G traffic data of different applications.ResultsExperimental results reveal significant differences in node degree distribution and topological structures of 5G traffic data across different application scenarios, such as star structures and multiple subnetwork structures. It is found that the node degree distribution of 5G traffic networks exhibits heterogeneity, reflecting the uneven growth of node degrees during network expansion. The Hurst index analysis discovers that the 5G traffic network retains the long-term dependence and trends of the original data. Through community detection, it is observed that networks converted from 5G traffic data of different applications exhibit diverse community structures, such as high centrality nodes, star-like community structures, modularity, and multilayer characteristics.DiscussionThese findings indicate that 5G traffic networks in different application scenarios exhibit complex and diverse characteristics. The heterogeneity of node degree distribution and differences in topological structures reflect the imbalance in node connection methods during network expansion. The results of the Hurst index show that the 5G traffic network inherits the long-term dependence of the original data, providing a basis for analyzing the dynamic characteristics of the network. The diverse community structures reveal the inherent modularity and hierarchy of the network, which helps to understand the performance and optimization directions of 5G networks in different applications.

Nonuniversal critical dynamics on planar random lattices with heterogeneous degree distributions

The weighted planar stochastic (WPS) lattice introduces a topological disorder that emerges from a multifractal structure. Its dual network has a power-law degree distribution and is embedded in a two-dimensional space, forming a planar network. We modify the original recipe to construct WPS networks with degree distributions interpolating smoothly between the original power-law tail, P(q)∼q−α with exponent α≈5.6, and a square lattice. We analyze the role of the disorder in the modified WPS model, considering the critical behavior of the contact process (CP). We report a critical scaling depending on the network degree distribution. The scaling exponents differ from the standard mean-field behavior reported for CP on infinite-dimensional (random) graphs with power-law degree distribution. Furthermore, the disorder present in the WPS lattice model is in agreement with the Luck-Harris criterion for the relevance of disorder in critical dynamics. However, despite the same wandering exponent ω=1/2, the disorder effects observed for the WPS lattice are weaker than those found for uncorrelated disorder.

Three-dimensional shape and connectivity of physical networks

Data describing the three-dimensional structure of physical networks is increasingly available, leading to a surge of interest in network science to explore the relationship between the shape and connectivity of physical networks. We contribute to this effort by standardizing and analyzing 15 data sets from different domains. Each network is made of tube-like objects bound together at junction points, which we treat as nodes, with the connections between them considered as links. We divide these networks into three categories: lattice-like networks, trees, and linked trees. The degree distribution of these physical networks is bounded, with most nodes having degrees one or three. Characterizing the physical properties of links, we show that links have an elongated shape and tend to follow a nearly straight trajectory, while a small fraction of links follow a winding path. These typical node and link properties must be reflected by physical network models. We also measure how confined a link is in space by comparing its trajectory to a randomized null model, showing that links that are central in the abstract network tend to be physically confined by their neighbors. The fact that the shape and connectivity of the physical networks are intertwined highlights that their three-dimensional layout must be taken into account to understand the evolution and function of physical networks.

An SEIR network epidemic model with manual and digital contact tracing allowing delays

We consider an SEIR epidemic model on a network also allowing random contacts, where recovered individuals could either recover naturally or be diagnosed. Upon diagnosis, manual contact tracing is triggered such that each infected network contact is reported, tested and isolated with some probability and after a random delay. Additionally, digital tracing (based on a tracing app) is triggered if the diagnosed individual is an app-user, and then all of its app-using infectees are immediately notified and isolated. The early phase of the epidemic with manual and/or digital tracing is approximated by different multi-type branching processes, and three respective reproduction numbers are derived. The effectiveness of both contact tracing mechanisms is numerically quantified through the reduction of the reproduction number. This shows that app-using fraction plays an essential role in the overall effectiveness of contact tracing. The relative effectiveness of manual tracing compared to digital tracing increases if: more of the transmission occurs on the network, when the tracing delay is shortened, and when the network degree distribution is heavy-tailed. For realistic values, the combined tracing case can reduce R0 by 20%–30%, so other preventive measures are needed to reduce the reproduction number down to 1.2–1.4 for contact tracing to make it successful in avoiding big outbreaks.

Regulation of phonon localization on thermal transport in complex networks.

The regulation of thermal transport is a challenging topic in complex networks. At present, the hidden physical mechanism behind thermal transport is poorly understood. This paper addresses this issue by proposing a complex network model that focuses on the thermal transport regulation through the manipulation of the network's degree distribution and clustering coefficient. Our findings indicate that increasing the degree distribution regulation parameter σ leads to reduced phonon localization and improved thermal transport efficiency. Conversely, increasing the clustering coefficient c results in enhanced phonon localization and reduced thermal transport efficiency. Meanwhile, by calculating the pseudodispersion relation of the network, we find that the maximum (or the second smallest) eigenfrequency decreases with increasing σ (or c). Finally, we elucidate that phonon localization plays a pivotal role in the thermal transport of the network, as demonstrated through density of states and the participation ratio.

Evolution of cooperation on reinforcement-learning driven-adaptive networks.

Complex networks are widespread in real-world environments across diverse domains. Real-world networks tend to form spontaneously through interactions between individual agents. Inspired by this, we design an evolutionary game model in which agents participate in a prisoner's dilemma game (PDG) with their neighboring agents. Agents can autonomously modify their connections with neighbors using reinforcement learning to avoid unfavorable environments. Interestingly, our findings reveal some remarkable results. Exploiting reinforcement learning-based adaptive networks improves cooperation when juxtaposed with existing PDGs performed on homogeneous networks. At the same time, the network's topology evolves from homogeneous to heterogeneous states. This change occurs as players gain experience from past games and become more astute in deciding whether to join PDGs with their current neighbors or disconnect from the least profitable neighbors. Instead, they seek out more favorable environments by establishing connections with second-order neighbors with higher rewards. By calculating the degree distribution and modularity of the adaptive network in a steady state, we confirm that the adaptive network follows a power law and has a clear community structure, indicating that the adaptive network is similar to networks in the real world. Our study reports a new phenomenon in evolutionary game theory on networks. It proposes a new perspective to generate scale-free networks, which is generating scale-free networks by the evolution of homogeneous networks rather than typical ways of network growth and preferential connection. Our results provide new aspects to understanding the network structure, the emergence of cooperation, and the behavior of actors in nature and society.

An empirical assessment of whether urban green ecological networks have the capacity to store higher levels of carbon

Carbon–neutral growth is a crucial long-term climatic aim in the context of global warming. This paper introduces complex network theory and explores its potential application to achieve this goal. Specifically, we investigate the spatial and temporal distribution of nodes and sources in the ecological network, and examine whether a relationship exists between the topological index of network nodes and the landscape pattern index of ecological source areas. We also determine the contribution of nodes to the carbon stock of the entire network by exploring the correlation between the carbon stock of nodes and sources to develop an optimization strategy based on the synergistic effect of node-source carbon enhancement. Finally, we test the effect of network optimization through robustness. Our results show that: (1) The correlation topological feature index analysis reveals that the degree distribution of the node network's topological characteristics becomes dispersed and modular, exhibiting the characteristics of small-world networks according to a large clustering coefficient. The heterogeneity and extent of ecological source landscapes have increased by modularity index but remain distributed and locally fragmented; (2) According to correlation analysis, by enhancing the eccentricity of the node topology, the patch cohesion index (COHESION) of the ecological source site can maximize the contribution of the node to the enhancement of the carbon stock benefits of the source site; (3) According to the tests on the robustness of nodes and edges and the robustness of network links, network stability is improved and carbon sink capacity is enhanced. Simultaneously, the restoration and rejuvenation of ecological space through national ecological construction projects can effectively improve the carbon sink within the organized region, contributing to the carbon neutrality aim. This research gives scientific and quantifiable references for potential ecological construction projects for sustainable cities and the optimization of urban ecological space structure.

The spark of synchronization in heterogeneous networks of chaotic maps.

We investigate the emergence of synchronization in heterogeneous networks of chaotic maps. Our findings reveal that a small cluster of highly connected maps is responsible for triggering the spark of synchronization. After the spark, the synchronized cluster grows in size and progressively moves to less connected maps, eventually reaching a cluster that may remain synchronized over time. We explore how the shape of the network degree distribution affects the onset of synchronization and derive an expression based on the network construction that determines the expected time for a network to synchronize. Understanding how the network design affects the spark of synchronization is particularly important for the control and design of more robust systems that require some level of coherence between a subset of units for better functioning. Numerical simulations in finite-sized networks are consistent with this analysis.

Assessing mutualistic metacommunity capacity by integrating spatial and interaction networks

We develop a spatially realistic model of mutualistic metacommunities that exploits the joint structure of spatial and interaction networks. Assuming that all species have the same colonisation and extinction parameters, this model exhibits a sharp transition between stable non-null equilibrium states and a global extinction state. This behaviour allows defining a threshold on colonisation/extinction parameters for the long-term metacommunity persistence. This threshold, the ‘metacommunity capacity’, extends the metapopulation capacity concept and can be calculated from the spatial and interaction networks without needing to simulate the whole dynamics. In several applications we illustrate how the joint structure of the spatial and the interaction networks affects metacommunity capacity. It results that a weakly modular spatial network and a power-law degree distribution of the interaction network provide the most favourable configuration for the long-term persistence of a mutualistic metacommunity. Our model that encodes several explicit ecological assumptions should pave the way for a larger exploration of spatially realistic metacommunity models involving multiple interaction types.

The nature and nurture of network evolution

Although the origin of the fat-tail characteristic of the degree distribution in complex networks has been extensively researched, the underlying cause of the degree distribution characteristic across the complete range of degrees remains obscure. Here, we propose an evolution model that incorporates only two factors: the node’s weight, reflecting its innate attractiveness (nature), and the node’s degree, reflecting the external influences (nurture). The proposed model provides a good fit for degree distributions and degree ratio distributions of numerous real-world networks and reproduces their evolution processes. Our results indicate that the nurture factor plays a dominant role in the evolution of social networks. In contrast, the nature factor plays a dominant role in the evolution of non-social networks, suggesting that whether nodes are people determines the dominant factor influencing the evolution of real-world networks.

Contrasts in Sustainability between Hub-Based and Point-to-Point Airline Networks

Airline hubs are often defined as nodes with a high degree of connectivity. Connectivity is measured by the “degree” of the node. The degree distribution of hub networks tends to have a convex shape (curved towards the origin), while point-to-point networks have a higher number of high-degree nodes and a concave shape. This study aims to classify airline networks based on their hub orientation, expanding our understanding of network differences. The analysis in this paper involves fitting a power-law distribution, determining the range of degree distribution, and calculating the distribution of betweenness. These analyses provide insight into the classification of each airline. Each measurement helps to clarify the ambiguity in other scores. The goal is to establish a small set of rules that can clearly distinguish between the main types of networks. The classification includes four types of networks: One-hub, P2P(point-to-point), Multi-hub, and Complex networks. There is a well-recognized empirical distinction between hub networks, which have a few places with large betweenness, and point-to-point cases, which have a larger number of places with moderate betweenness. The significance of these results in terms of geographic importance is demonstrated by sorting 284 different airline networks based on these dimensions. These findings are expected to provide valuable information about the resilience and recovery of a network, as networks with many long-range connections are particularly vulnerable to a decrease in traffic. Additionally, these results have implications for the ability of networks to recover from a downturn.

Efficient network exploration by means of resetting self-avoiding random walkers

The self-avoiding random walk (SARW) is a stochastic process whose state variable avoids returning to previously visited states. This non-Markovian feature has turned SARWs a powerful tool for modeling a plethora of relevant aspects in network science, such as network navigability, robustness and resilience. We analytically characterize self-avoiding random walkers that evolve on complex networks and whose memory suffers stochastic resetting, that is, at each step, with a certain probability, they forget their previous trajectory and start free diffusion anew. Several out-of-equilibrium properties are addressed, such as the time-dependent position of the walker, the time-dependent degree distribution of the non-visited network and the first-passage time distribution, and its moments, to target nodes. We examine these metrics for different resetting parameters and network topologies, both synthetic and empirical, and find a good agreement with simulations in all cases. We also explore the role of resetting on network exploration and report a non-monotonic behavior of the cover time: frequent memory resets induce a global minimum in the cover time, significantly outperforming the well-known case of the pure random walk, while reset events that are too spaced apart become detrimental for the network discovery. Our results provide new insights into the profound interplay between topology and dynamics in complex networks, and shed light on the fundamental properties of SARWs in nontrivial environments.

Topology switching during window thresholding fMRI-based functional networks of patients with major depressive disorder: Consensus network approach.

We present a novel method for analyzing brain functional networks using functional magnetic resonance imaging data, which involves utilizing consensus networks. In this study, we compare our approach to a standard group-based method for patients diagnosed with major depressive disorder (MDD) and a healthy control group, taking into account different levels of connectivity. Our findings demonstrate that the consensus network approach uncovers distinct characteristics in network measures and degree distributions when considering connection strengths. In the healthy control group, as connection strengths increase, we observe a transition in the network topology from a combination of scale-free and random topologies to a small-world topology. Conversely, the MDD group exhibits uncertainty in weak connections, while strong connections display small-world properties. In contrast, the group-based approach does not exhibit significant differences in behavior between the two groups. However, it does indicate a transition in topology from a scale-free-like structure to a combination of small-world and scale-free topologies. The use of the consensus network approach also holds immense potential for the classification of MDD patients, as it unveils substantial distinctions between the two groups.

Defining the Scale to Build Complex Networks with a 40-Year Norwegian Intraplate Seismicity Dataset.

We present a new complex network-based study focused on intraplate earthquakes recorded in southern Norway during the period 1980-2020. One of the most recognized limitations of spatial complex network procedures and analyses concerns the definition of adequate cell size, which is the focus of this approach. In the present study, we analyze the influence of observational errors of hypocentral and epicentral locations of seismic events in the construction of a complex network, looking for the best cell size to build it and to develop a basis for interpreting the results in terms of the structure of the complex network in this seismic region. We focus the analysis on the degree distribution of the complex networks. We observed a strong result of the cell size for the slope of the degree distribution of the nodes, called the critical exponent γ. Based on the Abe-Suzuki method, the slope (γ) showed a negligible variation between the construction of 3- and 2-dimensional complex networks. The results were also very similar for a complex network built with subsets of seismic events. These results suggest a weak influence of observational errors measured for the coordinates latitude, longitude, and depth in the outcomes obtained with this particular methodology and for this high-quality dataset. These results imply stable behavior of the complex network, which shows a structure of hubs for small values of the cell size and a more homogeneous degree distribution when the cell size increases. In all the analyses, the γ parameter showed smaller values of the error bars for greater values of the cell size. To keep the structure of hubs and small error bars, a better range of the side sizes was determined to be between 8 to 16 km. From now on, these values can be used as the most stable cell sizes to perform any kind of study concerning complex network studies in southern Norway.

Complete and partial synchronization in empirical brain networks

The chimera state, characterized by the coexistence of synchronization and asynchronization in a network, associates with special brain functions and disorders. This paper studies the chimera state in empirical networks reconstructed from the human, macaque, marmoset, and mouse brain data and also a constructed random network. The structural properties, such as clustering coefficient, efficiency, and modularity index of the networks, are computed and compared. The degree distributions of networks are also represented, yielding a more different degree distributon for the mouse network with the others. To determine the collective behavior of the network, the global and local order parameters and the master stability function are applied. Under increasing coupling strength, all the networks generically display the sequence of transitions from asynchronous state, over chimeras to the synchronous regime. There are also multistable regions during the transitions. The results represent that based on the coupling strength required to reach a fully synchronized state, human, marmoset, and macaque networks have relatively similar dynamical regions of synchronization. Mouse and random networks, however, require considerably stronger and weaker couplings, respectively. Changing the dynamics of the individual neurons shows that the asynchronous and chimera regions extend as the dynamics of the neurons become more complex.

Critical transition of thermal rectification on complex networks

Thermal rectification is a mechanism that controls the direction of heat conduction, allowing it to flow freely in one direction and hindering it in the opposite direction. In this study, we propose a heat conduction model on a complex network where the node masses are non-uniformly distributed according to mi∼kiα. Our findings show that the existence of a critical point, α=1, determines the working mode of thermal rectification. For α>1, the working mode of thermal rectification is positive, whereas for α<1, the working mode is negative. Additionally, we discovered that this critical transition is a general phenomenon and does not vary with changes in network size, average degree, or degree distribution. By conducting theoretical analyses based on phonon spectra, we also identified the physical mechanism of the critical transition. These results provide a new approach to implement and enrich thermal diodes, opening up new possibilities for more efficient thermal management.

Dependency cluster analysis of urban road network based on percolation

It is known that failure dependencies in complex systems may cause cascading failures among components and dramatically reduce system reliability. Urban road network is a typical example where failure dependencies often exist between roads, leading to failure (congestion) propagations. Thus, identifying strong failure dependencies and their clusters in road networks can help city managers to relieve congestions. However, previous road network reliability studies seldom considered to uncover existing dependency clusters. Here, we propose a causal inference approach to detect dependency relationships among road sections based on information theory using real-time velocity data in Beijing. On the obtained traffic causal network, percolation is used to identify smaller dependency clusters, which will merge into a giant dependency cluster at the percolation criticality. Critical dependency links contributing to the formation of the giant dependency cluster have been also detected. Moreover, the power-law distribution of finite-size dependency cluster sizes at criticality has been investigated, where the estimated exponent can be related to the log-normal degree distribution of the causal network. Finally, we illustrate how the identified dependency clusters impact the road network reliability. Our findings can help to uncover potential cascading failure patterns in road networks, and promote reliability management of transportation systems.

A matrix completion bootstrap method for estimating scale-free network degree distribution

Scale-free networks are very common in real practice. The key to analyze scale-free networks is the statistical inference of degree distribution. However, one observed network only allows us to calculate network statistics such as nodal degree, but does not provide enough information for further inference such as constructing confidence intervals. Borrowing the spirit of bootstrap, by generating network samples as bootstrap samples, we are then able to quantify statistical accuracy of various network statistics. In this paper, we propose a novel network bootstrap method named 1-BNB where bootstrap samples are generated via 1-bit matrix completion. We focus on constructing confidence interval for network degree distribution. Extensive simulation studies are conducted to demonstrate the finite sample performance of our newly propose method. A collaboration network among statisticians, a social network and an electrical grid network are studied for illustration purpose.

A robust method for fitting degree distributions of complex networks

Abstract This work introduces a method for fitting to the degree distributions of complex network datasets, such that the most appropriate distribution from a set of candidate distributions is chosen while maximizing the portion of the distribution to which the model is fit. Current methods for fitting to degree distributions in the literature are inconsistent and often assume a priori what distribution the data are drawn from. Much focus is given to fitting to the tail of the distribution, while a large portion of the distribution below the tail is ignored. It is important to account for these low degree nodes, as they play crucial roles in processes such as percolation. Here, we address these issues, using maximum likelihood estimators to fit to the entire dataset or close to it. This methodology is applicable to any network dataset (or discrete empirical dataset), and we test it on over 25 network datasets from a wide range of sources, achieving good fits in all but a few cases. We also demonstrate that numerical maximization of the likelihood performs better than commonly used analytical approximations. In addition, we have made available a Python package which can be used to apply this methodology.