Average Degree Of Network Research Articles

Echo in complex networks.

Large populations of globally coupled or uncoupled oscillators have been recently shown to exhibit an intriguing echo behavior [Ott, Platig, Antonsen, and Girvan, Chaos: An Interdiscip. J. Nonlinear Sci. 18, 037115 (2008)CHAOEH1054-150010.1063/1.2973816; Chen, Tinsley, Ott, and Showalter, Phys. Rev. X 6, 041054 (2016)2160-330810.1103/PhysRevX.6.041054], wherein a system is perturbed by two successive pulses at times T and T+τ inducing a spontaneous increase in the order parameter at the given times. These two provoked increments in the order parameter are followed by an unprovoked spontaneous increment in the order parameter at time T+2τ termed as an echo. In this paper, the effects of network topology on the emergence of an echo are explored. Two principal network parameters, namely, average degree and network randomness, are varied for this purpose. The networks are rewired to increase randomness in the network connections using the Watts-Strogatz algorithm to generate small world networks [Watts and Strogatz, Nature (London) 393, 440 (1998)10.1038/30918]. Thus, the whole span of networks ranging from a regular ring to a completely random network is explored. The average degree of the underlying connectivity, starting from nearest neighbor connections, is also monotonically increased and its effects on the echo behavior are analyzed. We find that for rings with low average degrees and high coupling strengths a discernible echo is not observed. Remarkably, an echo reemerges in the presence of sufficient randomness in the connections for such networks. For a regular ring network, increasing the average degree after a critical value also yields a transition to echo behavior. However, for completely random networks echoes are present in networks of all average degrees. This suggests that randomizing connections can induce echoes in systems even when the average degree of connections is very low. Another subtle feature arises for intermediate randomness, where the system exhibits a nonmonotonic dependence of the echo size on average degree. The echo size was found to be minimum at an intermediate value of the average degree. Lastly we consider the influence of dynamically changing links on the echo size and demonstrate that time-varying connections destroy the echo in low average degree networks, while the echo persists under dynamic links in high average degree networks. So our results clearly demarcate the class of networks that are robust candidates for exhibiting echoes, as well as provide caveats for the observation of echoes in networks.

Interlayer link prediction in multiplex social networks: An iterative degree penalty algorithm

Online social network (OSN) applications provide different experiences; for example, posting a short text on Twitter and sharing photographs on Instagram. Multiple OSNs constitute a multiplex network. For privacy protection and usage purposes, accounts belonging to the same user in different OSNs may have different usernames, photographs, and introductions. Interlayer link prediction in multiplex network aims at identifying whether the accounts in different OSNs belong to the same person, which can aid in tasks including cybercriminal behavior modeling and customer interest analysis. Many real-world OSNs exhibit a scale-free degree distribution; thus, neighbors with different degrees may exert different influences on the node matching degrees across different OSNs. We developed an iterative degree penalty (IDP) algorithm for interlayer link prediction in the multiplex network. First, we proposed a degree penalty principle that assigns a greater weight to a common matched neighbor with fewer connections. Second, we applied node adjacency matrix multiplication for efficiently obtaining the matching degree of all unmatched node pairs. Thereafter, we used the approved maximum value method to obtain the interlayer link prediction results from the matching degree matrix. Finally, the prediction results were inserted into the priori interlayer node pair set and the above processes were performed iteratively until all unmatched nodes in one layer were matched or all matching degrees of the unmatched node pairs were equal to 0. Experiments demonstrated that our advanced IDP algorithm significantly outperforms current network structure-based methods when the multiplex network average degree and node overlapping rate are low.

Generalized Independence in the q-Voter Model: How Do Parameters Influence the Phase Transition?

We study the q-voter model with flexibility, which allows for describing a broad spectrum of independence from zealots, inflexibility, or stubbornness through noisy voters to self-anticonformity. Analyzing the model within the pair approximation allows us to derive the analytical formula for the critical point, below which an ordered (agreement) phase is stable. We determine the role of flexibility, which can be understood as an amount of variability associated with an independent behavior, as well as the role of the average network degree in shaping the character of the phase transition. We check the existence of the scaling relation, which previously was derived for the Sznajd model. We show that the scaling is universal, in a sense that it does not depend neither on the size of the group of influence nor on the average network degree. Analyzing the model in terms of the rescaled parameter, we determine the critical point, the jump of the order parameter, as well as the width of the hysteresis as a function of the average network degree and the size of the group of influence q.

How the structure of egocentric Facebook networks is associated with exposure to risky content for maltreated versus comparison youth

The current study examined the size and connectedness of egocentric Facebook networks as predictors of exposure to risky content among a sample of maltreated and comparison youth (n = 118). Social network measures (i.e., size, density, average degree, percent of isolates) were computed from the mutual friend list. A content analysis of posts by friends captured references to alcohol use, marijuana use, partying, and sexual content. Multiple-group path models showed that the larger size of the Facebook network and higher average degree predicted references to marijuana use only for comparison youth, whereas for maltreated youth a higher percent of isolates predicted more references to sexual content by Facebook friends. Structural measures of online networks may have potential utility for identifying those at risk.

Assortativity provides a narrow margin for enhanced cooperation on multilayer networks

Research at the interface of statistical physics, evolutionary game theory, and network science has in the past two decades significantly improved our understanding of cooperation in structured populations. We know that networks with broad-scale degree distributions favor the emergence of robust cooperative clusters, and that temporal networks might preclude defectors to exploit cooperators, provided the later can sever their bad ties soon enough. In recent years, however, research has shifted from single and isolated networks to multilayer and interdependent networks. This has revealed new paths to cooperation, but also opened up new questions that remain to be answered. We here study how assortativity in connections between two different network layers affects public cooperation. The connections between the two layers determine to what extent payoffs in one network influence the payoffs in the other network. We show that assortative linking between the layers—connecting hubs of one network with the hubs in the other—does enhance cooperation under adverse conditions, but does so with a relatively modest margin in comparison to random matching or disassortative matching between the two layers. We also confirm previous results, showing that the bias in the payoffs in terms of contributions from different layers can help public cooperation to prevail, and in fact more so than the assortativity between layers. These results are robust to variations in the network structure and average degree, and they can be explained well by the distribution of strategies across the networks and by the suppression of individual success levels that is due to the payoff interdependence.

Structural Robustness to Noise in Consensus Networks: Impact of Degrees and Distances, Fundamental Limits, and Extremal Graphs

We investigate how the graph topology influences the robustness to noise in undirected linear consensus networks. We measure the structural robustness by using the smallest possible value of steady-state population variance of states under the noisy consensus dynamics with edge weights from the unit interval. We derive tight upper and lower bounds on the structural robustness of networks based on the average distance between nodes and the average node degree. Using the proposed bounds, we characterize the networks with different types of robustness scaling under increasing size. Furthermore, we present a fundamental tradeoff between the structural robustness and the average degree of networks. While this tradeoff implies that a desired level of structural robustness can only be achieved by graphs with a sufficiently large average degree, we also show that there exist dense graphs with poor structural robustness. We then show that random <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -regular graphs (the degree of each node is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> ) with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> nodes typically have near-optimal structural robustness among all the graphs with size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> and average degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> for sufficiently large <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> . We also show that when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> increases properly with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> , random <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -regular graphs maintain a structural robustness within a constant factor of the complete graph's while also having the minimum average degree required for such robustness.

Two competitive products diffusion in heterogeneous consumer social networks with repeat purchase

This paper studies the dynamics of two competitive products diffusion in heterogeneous consumer social networks with repeat purchase. We demonstrate a threshold for the diffusion rate above which a single product can persistently diffuse in heterogeneous consumer social networks without considering advertising strategy if the other product fails to diffuse, and there exists a unique positive equilibrium state where two competitive products coexist and persistently diffuse in heterogeneous consumer social networks without considering advertising strategy. We also prove that there exists at least one positive equilibrium state where two products coexist and persistently diffuse in heterogeneous consumer social networks if considering advertising strategy. The numerical simulations show that the higher the average degree of heterogeneous consumer social networks, the faster the two competitive products diffuse, and the shorter the time required to reach the stable state.

Network characteristics for neighborhood field algorithms

Evolutionary algorithms (EAs) have been successfully applied to solve numerous optimization problems. Neighborhood field optimization algorithm (NFO) has been proposed to integrate the neighborhood field in EAs, which utilizes local cooperation behaviors to explore new solutions. In this paper, certain new NFO variants are proposed based on the cooperation of descendent neighbors. The competitive and cooperative behaviors of NFO variants provide a remarkable ability to accelerate information exchanges and achieve global search. Experimental results show that NFO variants perform better than basic and other state-of-the-art EAs under different benchmark functions. For NFO and other EAs, it is difficult to quantify benefits of local cooperation in the optimization process. For this purpose, the cooperation behaviors are analyzed in a new network approach in this paper. In the proposed NFO variants, population graph shows a scale-free network with power-law distribution. Network characteristics, i.e., degree distribution, cluster coefficient and average degree, are used to quantify the cooperation behaviors. Experimental results show that network characteristics can effectively indicate the optimization performance of NFO variants in terms of convergence rate and population diversity. NFO variants with large cluster coefficients and significant heterogeneous characteristics can achieve a significant performance improvement on numerous problems.

Early recovery of interhemispheric functional connectivity after corpus callosotomy.

To investigate whether interhemispheric functional connectivity (FC) recovers in the first year after total callosotomy. Eight epilepsy patients undergoing total callosotomy were recruited. Resting-state functional magnetic resonance imaging was acquired before and after surgery. The precallosotomy and postcallosotomy interhemispheric and intrahemispheric FC was analyzed by using graph theory and voxel-mirrored homotopic connectivity (VMHC). The seizure outcome was scored using the Engel surgical outcome scale. After callosotomy (mean postoperative interval = 4months), the network density, average node degree, characteristic path length, and global efficiency of the whole interhemispheric networks were significantly decreased, compared to those in the precallosotomy networks. However, postcallosotomy interhemispheric FC and homotopic VMHC were not significantly reduced in bilateral frontal and temporal lobes. The network density and average node degree of the intrahemispheric networks were significantly increased. The characteristic path length and global efficiency of intrahemispheric networks were unchanged. The interhemispheric FC may be preserved or recover early within the first postoperative year after total callosotomy, particularly in the frontal and anterior temporal lobes.

Dynamics Modeling and Analysis of SIS Epidemic Spreading in Cluster Networks

In this paper, we propose and study an SIS epidemic model with clustering characteristics based on networks. Using the method of the existence of positive equilibrium point, we obtain the formula of the basic reproduction number R0. Furthermore, by constructing Lyapunov function, we also prove that the disease-free equilibrium of the model is globally asymptotically stable when R0&lt;1. When R0&gt;1, there is only one positive equilibrium point which is globally asymptotically stable. It is also shown that the infection proportion and the basic reproduction number R0 increases as the clustering coefficient increases when the average degree of networks is fixed.

Heterogeneity of Networks Promotes Cooperation in the Prisoner’s Dilemma and the Snowdrift Game

Complex networks are ubiquitous in real-world systems. In the last decade, the investigation of complex networks has received much attention. Most of this research suggests that the topological properties of networks have an important influence on the evolution of cooperation. In this paper, we used the prisoner’s dilemma game and snowdrift game as models to study the problem of cooperation evolution on heterogeneous random networks. We found that the heterogeneity of the networks plays a more important role in promoting cooperation than the average clustering coefficient and average path length of the networks do. The higher the heterogeneity of the networks is, the more conducive the emergence and spread of cooperation is. In addition, we found that increasing the average degree of networks would be detrimental for cooperation to thrive.

An Analysis of Social Networks Based on Tera-Scale Telecommunication Datasets

With the popularization of mobile phone usage, telecommunication networks have turned into a socially binding medium. Considering the traces of human communication held inside these networks, telecommunication networks are now able to provide a proxy for human social networks. To study degree characteristics and structural properties in large-scale social networks, we gathered a tera-scale dataset of call detail records that contains $\approx 5 \times 10^7$≈5×107 nodes and $\approx 3.6 \times 10^{10}$≈3.6×1010 links for three GSM (mobile) networks, as well as $\approx 1.4 \times 10^7$≈1.4×107 nodes and $\approx 1.9 \times 10^9$≈1.9×109 links for one PSTN (fixed-line) network. In this paper, we first empirically evaluate some statistical models against the degree distribution of the country's call graph and determine that a Pareto log-normal distribution provides the best fit, despite claims in the literature that power-law distribution is the best model. We then question how network operator, size, density, and location affect degree distribution to understand the parameters governing it in social networks. Our empirical analysis indicates that changes in density, operator and location do not show a particular correlation with degree distribution; however, the average degree of social networks is proportional to the logarithm of network size. We also report on the structural properties of the communication network. These novel results are useful for managing and planning communication networks.

How connectivity and search for producers impact production in Industry 4.0 networks

Technological evolutions lead to changes in production processes; the Fourth Industrial Revolution has been called Industry 4.0, as it integrates Cyber-Physical Systems and the Internet of Things into supply chains. Large complex networks are the core structure of Industry 4.0: any node in a network can demand a task, which can be answered by one node or a set of them, collaboratively, when they are connected. In this paper, the aim is to verify how (i) network's connectivity (average degree) and (ii) the number of levels covered in nodes search impacts the total of production tasks completely performed in the network. To achieve the goal of this paper, two hypotheses were formulated and tested in a computer simulation environment developed based on a modeling and simulation study. Results showed that the higher the network's average degree is (their nodes are more connected), the greater are the number of tasks performed; in addition, generally, the greater are the levels defined in the search for nodes, the more tasks are completely executed. This paper's main limitations are related to the simulation process, which led to a simplification of production process. The results found can be applied in several Industry 4.0 networks, such as additive manufacturing and collaborative networks, and this paper is original due to the use of simulation to test this kind of hypotheses in an Industry 4.0 production network.

Information spreading on multirelational networks

The spreading dynamics of information has attracted much attention in recent years, yet the effects of multirelational networks have not been systematically studied. In this paper, we propose an information spreading model on multirelational networks, in which assumes that the information transmitted through friend relationships induces the susceptible nodes to become infected, while the information transmitted through hostile relationships proves vaccination to susceptible nodes. Through extensive numerical simulations, we find that the information spreading is suppressed on multirelational networks especially when the average degree of hostile network is large. However the information outbreak threshold is only dependent on the topology of friend network. The above phenomena are quantificationally predicted by a generalized edge-based compartmental theory.

Dynamic analysis and simulation study of knowledge flow under the perspective of industrial transfer

AbstractBased on the actual situation of the knowledge flow during industrial transfer, a dynamic model of knowledge flow in a complex network is built. The mechanism of the knowledge flow in the network is analysed, and an analogue simulation is conducted using the Netlogo platform to identify the effects of the network's average degree, the number of enterprises involved in the transfer, the network size, and the effective transmission rate of the knowledge on the efficiency of the knowledge flow in the network. According to the final steady state of the knowledge, the technology spillover effect of the industrial transfer is then discussed. The research results show that the average degree of the network, the number of enterprises involved in the transfer, and the effective transmission rate of the knowledge have significant influences on the efficiency of the knowledge flow in the network, and increasing the network size is beneficial to increasing the number of final knowledge owners. Thus, the final steady state of the knowledge flow plays a decisive role in the technology spillover effect of industrial transfer.

Network Neuroscience Reveals Distinct Neuromarkers of Flow During Media Use

Flow is characterized by a high level of intrinsic reward that results from a balance between task difficulty and individual ability. The Synchronization Theory of Flow offers an explanation for the neural basis of this process. It predicts an energetically-optimized, brain-network organization between cognitive control and reward regions when task difficulty and individual ability are balanced. While initial results provide support for structural predictions, the many-to-many connectivity and energetic optimality hypotheses remain untested. Our study addresses this gap. Subjects played a video game while undergoing functional magnetic resonance imaging. We experimentally manipulated task difficulty and individual ability. Using graph theoretical analyses, we show that the balanced-difficulty condition (compared to low- or high-difficulty) was associated with the highest average network degree in the fronto-parietal control network (implicated in cognitive control) and had the lowest global efficiency value, indicating low metabolic cost, and thereby testing Synchronization Theory’s core predictions.

Analysis of social interaction network properties and growth on Twitter

The complex topology of real networks allows its actors to change their functional behavior. Network models provide better understanding of the evolutionary mechanisms being accountable for the growth of such networks by capturing the dynamics in the ways network agents interact and change their behavior. Considerable amount of research efforts is required for developing novel network modeling techniques to understand the structural properties of such networks, reproducing similar properties based on empirical evidence, and designing such networks efficiently. In this study, we first demonstrate how to construct social interaction networks using social media data and then present the key findings obtained from the network analytics. We analyze the characteristics and growth of online social interaction networks, examine the network properties and derive important insights based on the theories of network science literature. We observed that the degree distributions of such networks follow power-law that is indicative of the existence of fewer nodes in the network with higher levels of interactions, and many other nodes with less interaction. While the network elements and average user degree grow linearly each day, densities of such networks tend to become zero. Largest connected components exhibit higher connectivity (density) when compared with the whole graph. Network radius and diameter become stable over time evidencing the small-world property. We also observe increased transitivity and higher stability of the power-law exponents as the networks grow. Since the data is specific to the Purdue University community, we also observe two very big events, namely Purdue Day of Giving and Senator Bernie Sanders’ visit to Purdue University as part of Indiana Primary Election 2016. The social interaction network properties that are revealed in this study can be useful in disseminating targeted information during planned special events.

Cascading failure in multilayer networks with dynamic dependency groups**Project supported by the National Natural Science Foundation of China (Grant No. 61601053).

The cascading failure often occurs in real networks. It is significant to analyze the cascading failure in the complex network research. The dependency relation can change over time. Therefore, in this study, we investigate the cascading failure in multilayer networks with dynamic dependency groups. We construct a model considering the recovery mechanism. In our model, two effects between layers are defined. Under Effect 1, the dependent nodes in other layers will be disabled as long as one node does not belong to the largest connected component in one layer. Under Effect 2, the dependent nodes in other layers will recover when one node belongs to the largest connected component. The theoretical solution of the largest component is deduced and the simulation results verify our theoretical solution. In the simulation, we analyze the influence factors of the network robustness, including the fraction of dependent nodes and the group size, in our model. It shows that increasing the fraction of dependent nodes and the group size will enhance the network robustness under Effect 1. On the contrary, these will reduce the network robustness under Effect 2. Meanwhile, we find that the tightness of the network connection will affect the robustness of networks. Furthermore, setting the average degree of network as 8 is enough to keep the network robust.

Ecological Innovation Decision Behavior of Enterprises in the Strategic Emerging Industrial Clusters Based on Cognitive Neuroscience

For the decision makers of eco-innovation in the strategic emerging industrial clusters, the behavioral preference function is formed during their previous cognitive practice and learning process. The processes of Learning and memory can shape the structure and function of the brain. Meanwhile, the specific structure and function of the brain also affects and optimizes the learning behavior and cognitive process. Therefore, it is of great significance to grasp and explain the decision-making behavior of the enterprise's eco-innovation in the strategic emerging industrial clusters from the perspective of neuroscience. Firstly, we analyzed the influence of the mental model of the entrepreneurs in the strategic emerging industrial clusters on the choice of decision-making mode for company's eco-innovation. Secondly, we analyzed the process of decision of company's eco-innovation in the strategic emerging industrial clusters based on the mental model, and established an benefit model of eco-technology R&D, and analyzed the eco-innovation decision-making process of companies in the strategic emerging industrial clusters by simulation. The result shows that companies in the strategic emerging industrial clusters will decide whether to adopt eco-innovation behaviors according to their own mental model. At the same time, the knowledge accumulation obtained by the enterprises in the strategic emerging industrial clusters is inversely related to the average path length of the cluster eco-innovation network and is positively related to knowledge spillover benefits under the influence of the entrepreneurs’ mental model. The R&D revenue obtained by enterprises is inversely related to the average path length of the cluster eco-innovation network and is positively related to the average degree and aggregation coefficient of the cluster eco-innovation network.

Channel Noise-Enhanced Synchronization Transitions Induced by Time Delay in Adaptive Neuronal Networks with Spike-Timing-Dependent Plasticity

In this paper, we numerically study the effect of channel noise on synchronization transitions induced by time delay in adaptive scale-free Hodgkin–Huxley neuronal networks with spike-timing-dependent plasticity (STDP). It is found that synchronization transitions by time delay vary as channel noise intensity is changed and become most pronounced when channel noise intensity is optimal. This phenomenon depends on STDP and network average degree, and it can be either enhanced or suppressed as network average degree increases depending on channel noise intensity. These results show that there are optimal channel noise and network average degree that can enhance the synchronization transitions by time delay in the adaptive neuronal networks. These findings could be helpful for better understanding of the regulation effect of channel noise on synchronization of neuronal networks. They could find potential implications for information transmission in neural systems.