EXTENDED CLUSTERING COEFFICIENTS：GENERALIZATION OF CLUSTERING COEFFICIENTS IN SMALL-WORLD NETWORKS

The clustering coefficient C of a network, which is a measure of direct connectivity between neighbors of the various nodes, ranges from 0 （for no connectivity） to 1 （for full connectivity）. We define extended clustering coefficients C（h） of a small-world network based on nodes that are at distance h from a source node, thus generalizing distance-1 neighborhoods employed in computing the ordinary clustering coefficient C = C（1）. Based on known results about the distance distribution Pδ（h） in a network, that is, the probability that a randomly chosen pair of vertices have distance h, we derive and experimentally validate the law Pδ（h）C（h） ≤ c log N / N, where c is a small constant that seldom exceeds 1. This result is significant because it shows that the product Pδ（h）C（h） is upper-bounded by a value that is considerably smaller than the product of maximum values for Pδ（h） and C（h）. Extended clustering coefficients and laws that govern them offer new insights into the structure of small-world networks and open up avenues for further exploration of their properties.

Assessing the Eutrophication of Shengzhong Reservoir Based on Grey Clustering Method

Reservoir water environment is a grey system. The grey clustering method is applied to assessing the reservoir water enviromnent to establish a relatively complete model suitable for the reservoir eutrophication evaluation and appropriately evaluate the quality of reservoir water, providing evidence for reservoir management. According to China＇s lakes and reservoir eutrophication criteria and the characteristics of China＇s eutrophication, as well as certain evaluation indices, the degree of eutrophication is classified into six categories with the utilization of grey classified whitening weight function to represent the boundaries of classification, to determine the clustering weight and clustering coefficient of each index in grey classifications, and the classification of each clustering object. The comprehensive evaluation of reservoir eutrophication is established on such a foundation, with Sichuan Shengzhong Reservoir as the survey object and the analysis of the data attained by several typical monitoring points there in 2006. It is found that eutrophication of Tiebian Power Generation Station, Guoyuanchang and Dashiqiao Bridge is the heaviest, Tielusi and Qinggangya the second, and Lijiaba the least. The eutrophication of this reservoir is closely relevant to the irrational exploitation in its surrounding areas, especially to the aggravation of the non-point source pollution and the increase of net-culture fishing. Therefore, it is feasible to use grey clustering in environment quality evaluation, and the point lies in the correct division of grey whitening function

Exact Solution for Clustering Coefficient of Random Apollonian Networks

By means of the series method, we obtain the exact analytical solution of clustering coefficient in random Apollonian networks [Phys. Rev. E 71 （2005）046141]. Our exact analytical result is identical with the simulation, whereas in the original work, there is a deviation of about 4% between their approximate analytical result and the simulation.

Small-worldness of brain networks after brachial plexus injury: a resting-state functional magnetic resonance imaging study

Research on brain function after brachial plexus injury focuses on local cortical functional reorganization,and few studies have focused on brain networks after brachial plexus injury.Changes in brain networks may help understanding of brain plasticity at the global level.We hypothesized that topology of the global cerebral resting-state functional network changes after unilateral brachial plexus injury.Thus,in this cross-sectional study,we recruited eight male patients with unilateral brachial plexus injury（right handedness,mean age of 27.9±5.4years old）and eight male healthy controls（right handedness,mean age of 28.6±3.2）.After acquiring and preprocessing resting-state magnetic resonance imaging data,the cerebrum was divided into 90 regions and Pearson’s correlation coefficient calculated between regions.These correlation matrices were then converted into a binary matrix with affixed sparsity values of 0.1–0.46.Under sparsity conditions,both groups satisfied this small-world property.The clustering coefficient was markedly lower,while average shortest path remarkably higher in patients compared with healthy controls.These findings confirm that cerebral functional networks in patients still show smallworld characteristics,which are highly effective in information transmission in the brain,as well as normal controls.Alternatively,varied small-worldness suggests that capacity of information transmission and integration in different brain regions in brachial plexus injury patients is damaged.

Small World Properties Generated by a New Algorithm Under Same Degree of All Nodes

Based on the model of the same degree of all nodes we proposed before, a new algorithm, the so-called “spread all over vertices” （SAV） algorithm, is proposed for generating small-world properties from a regular ring lattices. During randomly rewiring connections the SAV is used to keep the unchanged number of links. Comparing the SAV algorithm with the Watts-Strogatz model and the “spread all over boundaries” algorithm, three methods can have the same topological properties of the small world networks. These results offer diverse formation of small world networks. It is helpful to the research of some applications for dynamics of mutual oscillator inside nodes and interacting automata associated with networks.

A novel evolving scale-free model with tunable attractiveness

In this paper, a new evolving model with tunable attractiveness is presented. Based on the Barabasi-Albert （BA） model, we introduce the attractiveness of node which can change with node degree. Using the mean-field theory, we obtain the analytical expression of power-law degree distribution with the exponent γ∈ （3, ∞）. The new model is more homogeneous and has a lower clustering coefficient and bigger average path length than the BA model.

Evolutionary prisoner's dilemma game on highly clustered community networks

This paper studies the evolutionary prisoner＇s dilemma game on a highly clustered community network in which the clustering coefficient and the community size can be tuned. It finds that the clustering coefficient in such a degree-homogeneous network inhibits the emergence of cooperation for the entire range of the payoff parameter. Moreover, it finds that the community size can also have a marked influence on the evolution of cooperation, with a larger community size leading to not only a lower cooperation level but also a smaller threshold of the payoff parameter above which cooperators become extinct.

Evolution of two properties for scale-free network

Fitness of node can denote its competing power and clustering denotes the transitivity of network. Because the fitness of node is uncertain or fuzzy in some social networks, an explicit form of the degree distribution on fuzzy fitness is derived within a mean field approach. It is a weighted sum of different fuzzy fitness. It can be found that the fuzzy fitness of nodes may lead to multiscaling. Moreover, the clustering coefficient of node decays as power law and clustering coefficient of network behavior not-decrease-but-increase’ phenomenon after some time. Some computer simulation results of these models illustrate these analytical results.

Two Novel Methods to Enhance Network Synchronizability

In this paper, we propose two methods to enhance the synchronizability of a class of complex networks which do not hold the positive correlation between betweenness centrality （BC） and degree of a node, and observe other topology characteristics of the network affected by the methods. Numerical simulations show that both methods can effectively enhance the synchronizability of this kind of networks. Furthermore, we show that the maximal BC of all edges is an important factor to affect the network synchronizability, although it is not the unique factor.

An Ideal Assortative Network and Synchronization

This paper proposes a novel complex network with assortative property based on multi-center networks. The average path length and clustering coefficient of the network are calculated, and the impact on the network topology is investigated. A simple dynamic system established on the proposed network is used to analyze how the assortative property of the network affects synchronization.

A Mixed Mechanism of Weighted-Driven and Inner Selection in Networks

For most networks, the weight of connection is changing with their attachment and inner affinity. By introducing a mixed mechanism of weighted-driven and inner selection, the model exhibits wide range power-law distributions of node strength and edge weight, and the exponent can be adjusted by not only the parameter δ but also the probability q. Furthermore, we investigate the weighted average shortest distance, clustering coefficient, and the correlation of our network. In addition, the weighted assortativity coefficient which characterizes important information of weighted topological networks has been discussed, but the variation of coefficients is much smaller than the former researches.

Three vertex degree correlations of fixed act-size collaboration networks

A rate equation approach was presented for the exact computation of the three vertex degree correlations of the fixed act-size collaboration networks.Measurements of the three vertex degree correlations were based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree k,with an appropriate boundary condition.The rate equation proposed can be generalized in more sophisticated growing network models,and also extended to deal with related correlation measurements.Finally,in order to check the theoretical prediction,a numerical example was solved to demonstrate the performance of the degree correlation function.

Quasispecies distribution of Eigen model

We have studied sharp peak landscapes of the Eigen model from a new perspective about how the quasispecies are distributed in the sequence space. To analyse the distribution more carefully, we bring in two tools. One tool is the variance of Hamming distance of the sequences at a given generation. It not only offers us a different avenue for accurately locating the error threshold and illustrates how the configuration of the distribution varies with copying fidelity q in the sequence space, but also divides the copying fidelity into three distinct regimes. The other tool is the similarity network of a certain Hamming distance do, by which we can gain a visual and in-depth result about how the sequences axe distributed. We find that there are several local similarity optima around the centre （global similarity optimum） in the distribution of the sequences reproduced near the threshold. Furthermore, it is interesting that the distribution of clustering coefficient C（k） follows lognormal distribution and the curve of clustering coefficient C of the network versus do appears to be linear near the threshold.

Identifying Functional Modules in Complex Networks

In this paper, we propose a new method that enables us to detect and describe the functional modules in complex networks. Using the proposed method, we can classify the nodes of networks into different modules according to their pattern of intra- and extra-module links. We use our method to analyze the modular structures of the ER random networks. We find that different modules of networks have different structure properties, such as the clustering coefficient. Moreover, at the same time, many nodes of networks participate different modules. Remarkably, we find that in the ER random networks, when the probability p is small, different modules or different roles of nodes can be Mentified by different regions in the c-p parameter space.

Synchronization in Triangled Complex Networks

Using a tunable clustering coeffcient model withoutchanging the degree distribution, we investigate the effect of clustering coefficient on synchronization of networks with both unweighted and weighted couplings. For several typical categories of complex networks, the more triangles are in the networks, the worse the synchronizability of the networks is.

Construction Research of Maintenance Support System

In order to guide the construction of maintenance support system,the theory of complex network to maintenance support area is introduced,and a construction idea of maintenance support system based on complex network is put forward.Firstly,from the perspective of the similarity,maintenance support system is abstracted into complex network to form maintenance support network.Secondly,the basic concepts and parameters of maintenance support network are also introduced.Thirdly,the maintenance support system in certain period is abstracted into a maintenance support network,and the network makes some changes.Finally,the correlative parameters of the network are calculated.The results show that the changed network is more conducive to the maintenance support.This provides a new thought and method to construct maintenance support system.

Topological Properties of Complex Networks with Different Types of Nodes

Nodes play different roles or have different functions in many natural and social networks.In this paper,a simple model with different types of nodes and deterministic selective linking rule is proposed.The structural properties by theoretical predictions are investigated that the given model exhibits a power-law distribution.

The Clustering Coefficient and the Diameter of Small-world Networks

The small-world network, proposed by Watts and Strogatz, has been extensively studied for the past over ten years. In this paper, a generalized smMl-world network is proposed, which extends severM small-world network models. Furthermore, some properties of a special type of generalized small-world network with given expectation of edge numbers have been investigated, such as the degree distribution and the isoperimetric number. These results are used to present a lower and an upper bounds for the clustering coefficient and the diameter of the given edge number expectation generalized small-world network, respectively. In other words, we prove mathematically that the given edge number expectation generalized small-world network possesses large clustering coefficient and small diameter.

Nonpharmaceutical interventions contribute to the control of COVID-19 in China based on a pairwise model

Nonpharmaceutical interventions(NPIs),particularly contact tracing isolation and household quarantine,play a vital role in effectively bringing the Coronavirus Disease 2019(COVID-19)under control in China.The pairwise model,has an inherent advantage in characterizing those two NPIs than the classical well-mixed models.Therefore,in this paper,we devised a pairwise epidemic model with NPIs to analyze COVID-19 outbreak in China by using confirmed cases during February 3rde22nd,2020.By explicitly incorporating contact tracing isolation and family clusters caused by household quarantine,our model provided a good fit to the trajectory of COVID-19 infections.We calculated the reproduction number R=1.345(95%CI:1.230-1.460)for Hubei province and R=1.217(95%CI:1.207-1.227)for China(except Hubei).We also estimated the peak time of infections,the epidemic duration and the final size,which are basically consistent with real observation.We indicated by simulation that the traced high-risk contacts from incubated to susceptible decrease under NPIs,regardless of infected cases.The sensitivity analysis showed that reducing the exposure of the susceptible and increasing the clustering coefficient bolster COVID-19 control.With the enforcement of household quarantine,the reproduction number R and the epidemic prevalence declined effectively.Furthermore,we obtained the resumption time of work and production in China(except Hubei)on 10th March and in Hubei at the end of April 2020,respectively,which is broadly in line with the actual time.Our results may provide some potential lessons from China on the control of COVID-19 for other parts of the world.

A quantification model of Traditional Chinese Medicine syndromes in children with idiopathic precocious puberty and early puberty

OBJECTIVE:To establish a quantification model of Traditional Chinese Medicine(TCM)syndromes by sampling patients undergoing idiopathic precocious puberty(IPP)and early puberty.METHODS:A questionnaire for classifying and quantifying TCM syndromes was designed and administered.All the results were analyzed;the relationship between 3 types of syndrome and 47symptoms were summated.Meanwhile,the frequency distribution of each symptom or sign was aggregated.Fuzzy mathematics was used to develop a quantification model ofTCM syndromes.RESULTS:We found that precocious puberty had 3types of syndrome,including hyperactivity of fire due to Yin deficiency(Syndrome I),depressed liver Qi transforming into fire(Syndrome II),and end retention of damp heat(Syndrome III).In the IPP group,Syndrome I was the most common principal syndrome(100%).Forty-six patients(43.81%)werediagnosed with Syndrome I accompanied by Syndrome II and 11(10.48%)were diagnosed with Syndrome I accompanied by Syndrome III.In the early puberty group,Syndrome I was also the main syndrome(98.39%).The degrees of most symptoms were mild to moderate.Reddened tongue was the most common tongue manifestation(62.86%prevalence)in the IPP group.The most common pulse manifestations were slippery pulse,thread pulse,and taut pulse.The Asymptotic Normalization Coefficient(ANC)method was used to quantify the TCM syndromes in 167 cases.Diagnostic accuracy rate reached 91%,comparable to expert diagnosis.CONCLUSION:We find that there are 3 types of syndrome in the IPP group and in the early puberty group.Syndrome I(hyperactivity of fire due to Yin deficiency)is the main syndrome in the two groups.ANC may be an appropriate for quantification model ofTCM syndromes.