Research Article A Fault Tolerance Optimization Model of ...

Research ArticleA Fault Tolerance Optimization Model of the China RailwayGeographic Network Topological Structure

Fenling Feng1 Ziwen Tang1 and Lei Wang12

1School of Traffic and Transportation Engineering Central South University Changsha 410075 China2Wuxi Railway Station Shanghai Railway Bureau Wuxi 214005 China

Correspondence should be addressed to Fenling Feng ffl0731163com

Received 17 July 2014 Revised 2 October 2014 Accepted 2 October 2014

Academic Editor Wuhong Wang

Copyright copy 2015 Fenling Feng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With degree distribution characteristics of the China Railway geographic network an optimization model from macro to microis established on the relative entropy Firstly Poisson distribution of the China Railway geographic network is verified and fittedSecondly the ldquoTwelfth Five-Year Planrdquo of the railway geographic network is chosen as an example on which a macro model isbuilt Finally the ldquoTwelfth Five-Year Planrdquo of Guangzhou Railway Grouprsquos geographic network is chosen as the other example onwhich a micro model is built and our optimization scheme is proposed Results reveal that for improving the railway network faulttolerance from the macroscopic aspect the ldquoTwelfth Five-Year Planrdquo railway network should strengthen the railway agglomerationdegree then the microscopic optimization model is able to improve the fault tolerance effectively

1 Introduction

Since the first railway was built in 1865 China Railwayhas kept developing After the implementation of reformand opening up policy in 1980s great progress was madeon the railway network On October 26 2012 a huge andcomplex railway network was established The Ministry ofRailway of PRC (the (former) Ministry of Railways of Chinawas dissolved in accordance with the decision by The 12thNational Peoplersquos Congress in March 2013 Of its dutiesdevelopment plan safety regulation and inspection weretaken up by State Railway Administration a new departmentadministrated by the Ministry of Transport of PRC Con-struction and management were taken up by China RailwayCorporation) website published ldquoThe 12th Five-year RailwayNetwork Planrdquo and the ldquoNational Rapid Rail Network Planrdquoboth of which described the development direction of ChinaRailway in the next five years A railway network shouldbe analyzed on its overall performance and its plan shouldbe evaluated from an external index The research on therailway network topology structure with statistical methodsregards the railway network as a complicated system which

will support planning constructing and implementing thenetwork with theoretical foundations The fault toleranceresearch on complex network topology structure is an impor-tant and essential aspect in the field of complex systemstudyTherefore the research on fault tolerance of the railwaytopology network is critical

Many scholars researched the fault tolerance of complexnetworks Albert et al [1] first studied the robustness ofrandom networks and scale-free networks under randomfailures and deliberate attacks Broder et al [2] discussedthe robustness of Internet networks under random failuresand deliberate attacks by researching on WWW networksCohen et al [3 4] discussed the robustness of Internetnetworks under random failures and malicious attacks basedon percolation theory Feng et al [5] studied connectionrobustness and recovery robustness based on the connectivityand recovery ability of the network Moreover many scholarscarried out researches on other networks such as Chinarsquosrailway interchange network [6] public transport networks[7] and complex military networks [8]

As for the optimization of fault tolerance in a complexnetwork Shargel et al [9] improved the resistance ability of

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 871074 8 pageshttpdxdoiorg1011552015871074

2 Discrete Dynamics in Nature and Society

Table 1 Degree distribution of the China Railway network

Degree 1 2 3 4 5 6 7 8 9 10Degree proportion of railwaynetwork in 2008 02077 02524 02204 01725 00895 00383 00096 00064 00032 mdash

Degree proportion of railwaynetwork in the ldquoTwelfthFive-Year Planrdquo

01024 01667 02571 02262 01429 00405 00238 00214 00119 00071

the BAmodel by adjusting its parameters which are identifiedfor networksrsquo better resistance ability Paul et al [10] madean analysis on the optimization of scale-free networks bycomparing the advancement in double power law networksand bimodal distribution networks and so forth in orderto improve the resistance ability against random failuresand deliberate attacks at the same time In terms of themeans of attacks Tanizawa et al [11] demonstrated the nodedegree distribution of networks with better resistance abilitywhen random failures and selective attacks arise regularlyWang and He [12 13] stated that entropy optimizationcan improve the network fault tolerance By changing thetopology structure the network fault tolerance capacity willbe improved They illustrated the feature that the strongerthe fault tolerance the less the hubs and the lower the nodedegree However with enhanced network fault toleranceand better transmission efficiency the network synchronousability seems to be weakened

Studies reviewed covering the complex network faulttolerance optimization mostly focus on adjusting one orfew microscopical parameters to improve the tolerance Thispaper proposes a new tolerance optimizationmethod using aset from macro to micro fault tolerance proposing to utilizethe optimal value of the overall network fault tolerance asthe degree distribution is macroscopic Then the methodwould be used from the proposed microscopic local networkfault tolerance optimization to the local specific networkoptimization

2 Poisson Distribution of ChinarsquosRailway Network

According to researches by He Cheng [13] Zhao et al [14]and other scholars the degree distribution of Chinarsquos Railwaygeographic network is similar to a tree structure aligningwith Poissondistribution Toprove that the network structureis Poisson distributed we select the railway network in theyear of 2008 and in the ldquoTwelfth Five-Year Planrdquo and thesecond and the third nodesrsquo degree weighsmore in the degreedistribution of the China which agrees with characteristics ofthe Poisson distribution see Table 1 and Figure 1

Fitting with Poisson distribution function the functionof degree distribution of the 2008 China Railway geographynetwork is 119875(119909 = 119896) = (22882119896119896)119890minus22882

And the function of the degree distribution of theldquoTwelfth Five-Year Planrdquo is 119875(119909 = 119896) = (35853119896119896)119890minus35853

For the comparison of China Railway geography networkdegree distribution see Figure 2

3 Fault Tolerance MacroscopicOptimization Model

Cohen et al (2000) stated that the stronger the networkheterogeneity the stronger the fault tolerance Traditionallyfrom the perspective of information theory informationentropy is commonly used as a measure of the networkheterogeneity

Relative entropy measures the distance between tworandom distributions In statistics it corresponds to the log-arithm expectation of likelihood ratios The relative entropyor Kullback-Leibler distance of two probability density func-tions 119901(119909) and 119902(119909) is defined as follows

119863(119901

119902) = sum

119909isin119860

119901 (119909) log119901 (119909)

119902 (119909) (1)

Obviously uniform network has the worst heterogeneityTherefore this study uses the relative information entropybased on the real network and takes the regular networkdegree distribution as the measurement which is alsoadopted to research on fault tolerance The relative entropymeasurement is more representative revealing more compa-rability than traditional information entropy ones

All the nodes in the homogeneous network are evenlydistributed The proportion of every node degree is 1119899which is for any node degree 119909119894 isin 119860 119894 = 1 2 119899 119901(119909119894) =1119899 so the information entropy of the uniform network is

119867max = minus119899

sum

119894=1

1

119899log2

1

119899= minus1198991

119899log2

1

119899= log2119899 (2)

We set the node degree distribution density function ofthe railway network as 119901(119909) so according to the relativeentropy formula fault tolerance measurement 119862 is

119862 = 119863 (119901119902)

= sum

119909isin119860

119901 (119909) log119901 (119909)

119902 (119909)

= sum

119909isin119860

119901 (119909) log (119899119901 (119909))

(3)

Discrete Dynamics in Nature and Society 3

1 2 3 4 5 6 7 8 90

005

01

015

02

025

03

0352008

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre

es

(a)

1 2 3 4 5 6 7 8 9 100

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre

es The 12th Five-Year Plan

01

02

03

(b)

Figure 1 Degree distribution of the China Railway geography network

1 2 3 4 5 6 7 8 90

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre

es

2008

01

02

03

The actual distributionThe fitting distribution

(a)

1 2 3 4 5 6 7 8 9 100

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre

esThe 12th Five-Year Plan

01

02

03


(b)

Figure 2 Comparison between actual degree distribution and fitting degree distribution of China Railway geography network

Based on the above discussion the fault tolerance macro-scopic optimization model is established the object of whichis the ldquoTwelfth Five-Year Planrdquo of the China Railway geogra-phy network

31 The Basic Model Due to the fact that the optimum of therelative entropy is considered as the optimal fault tolerancethe study uses the relative entropy as the objective functionto find the fault tolerance optimization by adjusting the scale

in the network under the condition that keeps the costefficiency and other actual factors and constraints constantBased on all the assumptions the basic model is as follows

max 119867 (Relative entropy)

st Cost constraints

Efficiency constraints

Practical constraints

(4)


32 Model Implementations

(1) The Objective Function Because the degree distributionof China railway geographic network fits in the Poissondistribution we can determine the relative entropy functionwhich is

119867 = sum

119896isin119860

119901 (119896) log (119899119901 (119896))

119867 = sum

119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582)

(5)

The objective function for the maximum relative entropyis

max119867 = maxsum119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582) (6)

(2) Cost Constraints Limited by the budgets railway con-struction expansion scale is constrained which reflects on themodel as the following formula indicates whereas the totalnumber of nodes is 119899 which is a constant and the plannedscale is 1198990 Therefore cost constraints are

119899 lt 1198861198990 (119886 gt 1) (7)

(3) Constraints in Efficiency The efficiency of the railwaynetwork is mainly determined by the average distanceBecause the degree distribution of China Railway geographynetwork is Poisson distributed we can conclude that thenetwork is random A large number of researches state thatthe average shortest distance and the logarithm of scale arepositive correlated which means 119899 lt 1198861198990 (119886 gt 1) When theaverage distance is constant cost constraints equal efficiencyconstraints

(4) The Average Node Degree Distribution ConstraintsAccording to the current circumstances the main nodedegree in the geographical network is 2 or 3 This paperemphasizes mainly hubs and ignores the secondary railwaystations then the node degree for most of the selected sites is3 or 4 As for those whose node degree is larger than 4 thenode degree size is negatively correlated with the proportionTherefore the average node degree falls into the scope of (34) which also determines the overall structure of the railwayIn order to keep the railway structure constant we set theaverage node degree as

3 lt 120582 lt 4 (8)

(5) Planning Strategy Constraints The ldquoTwelfth Five-YearPlanrdquo focuses on developing Western China and the key forthe construction plan is to build new railway sites as well asextension of these sites We will ensure that the proportionof these sites does not decrease Therefore we set 119904 as theproportion of the sites whose node degree is less than or equal

to 3 and the strategic constraints are shown in the followingequation

sum

119896isin123

119901 (119896) gt 119904

Namely sum119896isin123

120582119896

119896119890minus120582gt 119904

(9)

In conclusion assuming the maximum node degreeremains the same the macro optimization model for faulttolerance is as follows

max 119867(120582 119899) = sum119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582)

st 119899 lt 1198861198990

3 lt 120582 lt 4

sum

119896isin123

120582119896

119896119890minus120582gt 119904

(10)

33 The Model Results Under conditions that cost efficiencykeeps fixed railway structure remains the same strategicconstrains stay unchanged and fault tolerance in the ldquoTwelfthFive-Year Planrdquo degree gets maximized the average nodedegree is calculated to be 37464 which is improved com-pared with the original value of 35853 The improvementrequires that new railway lines if possible connect withthe existing stations and lines between new sites should beincreased

4 Fault Tolerance MicroscopicOptimization Model

This paper establishes a microscopic optimization modelof fault tolerance in which the Guangzhou Railway Grouprailway network is the object It targets finding the maximumrelative entropy by keeping cost constrains and efficiencyconstrains constant while maximizing node degree achievingthe macroscopic optimization And the basic model is asfollows

max 119867(Relative entropy)

st Cost constraints

Degree constraints


(11)

41 Algorithm Ideas According to the basic model deter-mine the algorithm

Step 1 (initialization) According to the real network dia-gram geographically adjacent map and distance map areconstructed The ldquoTwelfth Five-Year Planrdquo is regarded as theinitial policy sets The number of iterations of the networkbased on the scale of the networks is determined


A

DC

B A

DC

B

A

C

B A

C

B

(1)

(2)

Figure 3 Exchanging lines (1) and changing lines (2)

sm

zjj

dz syhk

hazj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy

Existing linesPlanning lines

Figure 4 The ldquoTwelfth Five-Year Planrdquo of railway geographicnetwork diagram of Guangzhou Railway Bureau

Step 2 (construction of strategy candidate groups) Select onepair of lines from the planned routes exchange or changethem as Figure 3 suggests and iterate all the situations Thestrategy candidate groups are established

Step 3 (finding the optimal efficiency) The average distancereflects the distance attribute in the network which repre-sents the efficiency of the network The optimized networkrequires better efficiency than the original

Step 4 (seeking the optimal costs) In order to reduce the costthe actual distance increase after the optimization should bewithin a certain limit

Step 5 (macroscopic optimal screening) In addition to con-sidering the fault tolerance of the regional railway bureau weshould also improve the national rail network fault toleranceTherefore it is necessary to make the optimized node degreeclose to the average node degree in themacroscopic optimiza-tion

Step 6 (choosing the optimal fault tolerance) After all thework above select the optimal strategy with the maximumrelative entropy from the remaining cluster

Step 7 The optimal strategy is exchanged to replace theinitial adjacency matrix and to generate a new policy setIf the number of iterations is less than the measured valuethen restart from Step 2 If it is greater than the number ofiterations the final new strategy sets should be evaluated bythe level of efficiency and cost evaluation taking the optimalstrategy as the final policy set

42 Guangzhou Railway Bureau Railway Network in theldquoTwelfth Five-Year Planrdquo An Example Guangzhou RailwayBureau railway network in the ldquoTwelfth Five-Year Planrdquocontains 28 nodes 39 edges which contain 33 existing linesand six planned lines (Figure 4)

Assume the planned network average distance is 1198970 theactual distance is 1198710 the average degree is 119897 the averagedistance of optimized network is 119897 the actual distance is 119871and the average degree is 119896 then take constraints magnitudes1198861 1198862 1198863 as the rise of the optimized network relative to theoriginal planning network There are constraints

(119897 minus 1198970)

1198970

lt 1198861

(119871 minus 1198710)

1198710

lt 1198862

100381610038161003816100381610038161003816100381610038161003816

(119896 minus 1198960)

1198960

100381610038161003816100381610038161003816100381610038161003816

lt 1198863

(12)

It means that both the average distances and the actualdistance have upper limit after optimization and the averagedegree is controlled in a certain range Based on the actualdistance adjacency matrix the average distance is 7514calculated from local railway in geographic network For thedistance of those pairs of sites is greater than 400 km it isestimated to be 400 kilometers And the actual distance is1270771 km with an average degree of 37464 Constraintsare assumed a 5 margin

(119897 minus 7514)

7514lt 5

(119871 minus 1270771)

1270771lt 5

10038161003816100381610038161003816100381610038161003816

(119896 minus 37464)

37464

10038161003816100381610038161003816100381610038161003816

lt 5

(13)

After calculation the initial relative entropy for localrailway network is 05375 Obtained by the local optimizationmodel the optimal relative entropy converges to 05392indicating that the local optimization model can improvenetwork fault toleranceThis optimal fault tolerance indicatesthat a candidate cluster network due to the impact of theactual terrain may be difficult to achieve optimization of thenew line Therefore it is required to be selected based on costand efficiency indicators



sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy

(a)

sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy


(b)

sm

zjj

dz syhkha

zj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mzsg

yz

hh ld zz cl

yiy

cs

yuy

hy


(c)

Figure 5 Local railway network optimization geographic figure

We calculate the 18 candidate clusters Then we findout the cost and the efficiency are increased compared withthe original planning of the network The average distanceincreases compared with the actual distance It may reducedue to the distance decrease Normalization method isadopted for processing the maximum and minimum

Let the relative increase in the amount the actual dis-tance or the average distance be 119909 Let the amount of theincrease normalized be 119909119904 see Table 2 Consider

119909119904 =119909 minus 119909max119909max minus 119909min

(14)

Table 2 shows the relative increase of normalized actualdistance and average distance

Consolidated costs and optimized network efficiency areevaluated relative to the comprehensive increment in theamount of the actual distance of the average distance If therelative increase of the actual distance is 119886119904 and the relativeincrease of the average distance is 119886119901 then the comprehensiveincrement is 119886119911 = 119886119904 + 119886119901 Assuming the cost and efficiencyare of equal importance to the network comprehensiveincrement of the actual distance is weighted equal to that ofthe average distance

The candidate program is sorted from the superior to theinferior by the integrated incremental order in accordancewith the policy set (Table 3)This paper selects the best of thethree programs namely program 17 program 8 and program14

43 Optimization Programs The selected three optimiza-tions are original plans for the Guangzhou Railway Bureauwith the following adjustments made Option One the cd-csand hh-hy line are replaced by hh-yz and ld-hy line OptionTwo the hh-hy and cl-hy line are replaced by hh-yz and yz-sgline Option Three the cl-hy and hh-hy line are replaced byhh-yz and yz-gz line see Figure 5

Table 2 The actual distance and the relative increase in the averagedistance scales

Programnumber

Relative increase ofactual distance

Relative increase ofaverage distance

1 0459839 03999952 0793302 03999953 1 07999924 0378309 05 0668881 03999956 006378 03999957 0903425 07999928 0331941 09 0223749 019999610 053596 059999311 0689747 012 03729 013 076239 059999314 0157675 019999615 049307 039999516 0570349 019999617 0 018 0724523 1

Using the network diagram we can determine the adja-cencymatrix and the linear distancematrix to calculate initialrelative entropy as 05375 The microscopic optimizationmodel can be calculated by the fault tolerance of convergenceto the optimal 05392 and the visible microscopic optimiza-tion model can improve the network fault tolerance To fulfillthe optimal fault tolerance of the network a large cluster issupposed to be selected according to the actual situation

In an optimized network fault tolerance inspection theaverage clustering coefficient and average distance are used


0 1 2 3 4 5 644

44545

45546

46547

475

The number of failed node

The a

vera

ge d

istan

ce

Before optimizationAfter optimization

(a)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(b)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(c)

Figure 6 Micro fault tolerance before and after optimization

Table 3 The order of integrated incremental

Programnumber



Integratedincremental

17 0 0 08 0331941 0 033194114 0157675 0199996 035767112 03729 0 037294 0378309 0 03783099 0223749 0199996 04237456 006378 0399995 046377511 0689747 0 068974716 0570349 0199996 07703451 0459839 0399995 085983415 049307 0399995 08930655 0668881 0399995 106887610 053596 0599993 11359532 0793302 0399995 119329713 076239 0599993 13623837 0903425 0799992 170341718 0724523 1 17245233 1 0799992 1799992

as indicators of fault tolerance As Figure 5 illustrates themicronetwork optimization model is an improved version

compared with the original network Under the same degreeof random failures on average the clustering coefficientdecreases more slowly and the average distance increasesmore slowly as well and then collapses after a long durationFigure 6 shows the specific performance as follows

Average clustering coefficient

(1) compared with the expansion of the scope of theattack the rate of improved networkrsquos overall declineis slower than that of the original network

(2) the improved network has a longer duration beforecollapse

(3) with a small number of attack modes the improvednetwork has no significant change compared with theoriginal network

Average distance

(1) as the scope of the attack expanded the improvednetwork one increases much slowly

(2) with a small number of attack modes the improvednetwork has no significant change compared with theoriginal one which increases drastically

Therefore it is concluded that the improved network hasa better fault tolerance than the original one


5 Conclusions

In general the primary aimof a railway network plan is to cre-ate a smooth traffic flow With the consideration of randomnetwork breakdowns and failures this paper maximizes thefault tolerance of railway networks and proposes themethodsin enhancing the network from both macro and microaspects The study reveals that (1) the average node degreeof ldquoTwelfth Five-Year Planrdquo has greatly increased comparedwith Chinarsquos 2008 network (2) in order to maximize thefault tolerance the average node degree in ldquoTwelfth Five-Year Planrdquo should be raised (3) Guangzhou Railway Bureaursquosrailway network fault tolerance has been increased after thenetworkrsquos micro optimization

This paper focuses on analyzing geographic railway net-work Aligningwith the power-law distribution some studiesargued that the traffic networks are scale-free networksrequiring different optimization schemesWith relevant datafuture researches and analysis about the fault tolerance willbe conducted from the aspects of the traffic networks andtransfer networks

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Project is supported by the Fundamental Research Fundsfor the Central Universities (Grant no 2010QZZD021) andthe Ministry of Railway Science and Technology ResearchDevelopment Program (Grant no 2012X012-A) China

References

[1] R Albert H Jeong and A-L Barabasi ldquoError and attacktolerance of complex networksrdquo Nature vol 406 no 6794 pp378ndash382 2000

[2] A Broder R Kumar F Maghoul et al ldquoGraph structure in thewebrdquo Computer Networks vol 33 no 1ndash6 pp 309ndash320 2000

[3] R Cohen K Erez D Ben-Avraham and S Havlin ldquoResilienceof the internet to random breakdownsrdquo Physical Review Lettersvol 85 no 21 pp 4626ndash4628 2000

[4] R Cohen K Erez D Ben-Avraham and S Havlin ldquoBreakdownof the internet under intentional attackrdquoPhysical Review Lettersvol 86 no 16 pp 3682ndash3685 2001

[5] T Feng H Li Z Yuan and J Ma ldquoAnalysis method of robust-ness for topology of Bernoulli Node modelrdquo Acta ElectronicaSinica vol 20 no 7 pp 1673ndash1678 2011 (Chinese)

[6] WWang J Liu X Jiang and YWang ldquoTopology properties onChinese railway networkrdquo Journal of Beijing Jiaotong Universityvol 34 no 3 pp 148ndash152 2010 (Chinese)

[7] H Duan Z Li and Y Zhang ldquoRobustness analysis model ofurban public transport networkrdquo Natural Science vol 38 no 3pp 70ndash75 2010 (Chinese)

[8] Y Chen J Zhao and H Qi ldquoRobustness analysis for complexmilitary networkrdquo Fire Control ampCommand Control vol 35 no5 pp 23ndash25 2010 (Chinese)

[9] B Shargel H Sayama I R Epstein and Y Bar-Yam ldquoOpti-mization of robustness and connectivity in complex networksrdquoPhysical Review Letters vol 90 no 6 Article ID 068701 2003

[10] G Paul T Tanizawa S Havlin andH E Stanley ldquoOptimizationof robustness of complex networksrdquo European Physical JournalB vol 38 no 2 pp 187ndash191 2004

[11] T Tanizawa G Paul R Cohen S Havlin and H E StanleyldquoOptimization of network robustness to waves of targeted andrandom attacksrdquo Physical Review E vol 71 no 4 Article ID047101 2005

[12] BWang Evolutionmechanism and some dynamical processes oncomplex network [PhD thesis] Dalian University of Technol-ogy 2006 (Chinese)

[13] C He Complex Network Properties of the Chinese RailwayNetwork Sun Yat-Sen University Guangzhou China 2007(Chinese)

[14] WZhaoH SHe Z C Lin andKQ Yang ldquoStudy of propertiesof Chinese railway passenger transport networkrdquo Acta PhysicaSinica vol 55 no 8 pp 3906ndash3911 2006 (Chinese)

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Table 1 Degree distribution of the China Railway network

Degree 1 2 3 4 5 6 7 8 9 10Degree proportion of railwaynetwork in 2008 02077 02524 02204 01725 00895 00383 00096 00064 00032 mdash

Degree proportion of railwaynetwork in the ldquoTwelfthFive-Year Planrdquo

01024 01667 02571 02262 01429 00405 00238 00214 00119 00071

the BAmodel by adjusting its parameters which are identifiedfor networksrsquo better resistance ability Paul et al [10] madean analysis on the optimization of scale-free networks bycomparing the advancement in double power law networksand bimodal distribution networks and so forth in orderto improve the resistance ability against random failuresand deliberate attacks at the same time In terms of themeans of attacks Tanizawa et al [11] demonstrated the nodedegree distribution of networks with better resistance abilitywhen random failures and selective attacks arise regularlyWang and He [12 13] stated that entropy optimizationcan improve the network fault tolerance By changing thetopology structure the network fault tolerance capacity willbe improved They illustrated the feature that the strongerthe fault tolerance the less the hubs and the lower the nodedegree However with enhanced network fault toleranceand better transmission efficiency the network synchronousability seems to be weakened

Studies reviewed covering the complex network faulttolerance optimization mostly focus on adjusting one orfew microscopical parameters to improve the tolerance Thispaper proposes a new tolerance optimizationmethod using aset from macro to micro fault tolerance proposing to utilizethe optimal value of the overall network fault tolerance asthe degree distribution is macroscopic Then the methodwould be used from the proposed microscopic local networkfault tolerance optimization to the local specific networkoptimization

2 Poisson Distribution of ChinarsquosRailway Network

According to researches by He Cheng [13] Zhao et al [14]and other scholars the degree distribution of Chinarsquos Railwaygeographic network is similar to a tree structure aligningwith Poissondistribution Toprove that the network structureis Poisson distributed we select the railway network in theyear of 2008 and in the ldquoTwelfth Five-Year Planrdquo and thesecond and the third nodesrsquo degree weighsmore in the degreedistribution of the China which agrees with characteristics ofthe Poisson distribution see Table 1 and Figure 1

Fitting with Poisson distribution function the functionof degree distribution of the 2008 China Railway geographynetwork is 119875(119909 = 119896) = (22882119896119896)119890minus22882

And the function of the degree distribution of theldquoTwelfth Five-Year Planrdquo is 119875(119909 = 119896) = (35853119896119896)119890minus35853

For the comparison of China Railway geography networkdegree distribution see Figure 2

3 Fault Tolerance MacroscopicOptimization Model

Cohen et al (2000) stated that the stronger the networkheterogeneity the stronger the fault tolerance Traditionallyfrom the perspective of information theory informationentropy is commonly used as a measure of the networkheterogeneity

Relative entropy measures the distance between tworandom distributions In statistics it corresponds to the log-arithm expectation of likelihood ratios The relative entropyor Kullback-Leibler distance of two probability density func-tions 119901(119909) and 119902(119909) is defined as follows

119863(119901

119902) = sum

119909isin119860

119901 (119909) log119901 (119909)

119902 (119909) (1)

Obviously uniform network has the worst heterogeneityTherefore this study uses the relative information entropybased on the real network and takes the regular networkdegree distribution as the measurement which is alsoadopted to research on fault tolerance The relative entropymeasurement is more representative revealing more compa-rability than traditional information entropy ones

All the nodes in the homogeneous network are evenlydistributed The proportion of every node degree is 1119899which is for any node degree 119909119894 isin 119860 119894 = 1 2 119899 119901(119909119894) =1119899 so the information entropy of the uniform network is

119867max = minus119899

sum

119894=1

1

119899log2

1

119899= minus1198991

119899log2

1

119899= log2119899 (2)

We set the node degree distribution density function ofthe railway network as 119901(119909) so according to the relativeentropy formula fault tolerance measurement 119862 is

119862 = 119863 (119901119902)

= sum

119909isin119860

119901 (119909) log119901 (119909)

119902 (119909)

= sum

119909isin119860

119901 (119909) log (119899119901 (119909))

(3)


1 2 3 4 5 6 7 8 90

005

01

015

02

025

03

0352008

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre

es

(a)

1 2 3 4 5 6 7 8 9 100

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre


01

02

03

(b)


1 2 3 4 5 6 7 8 90

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre

es

2008

01

02

03


(a)

1 2 3 4 5 6 7 8 9 100

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre


01

02

03


(b)






st Cost constraints



(4)




119867 = sum

119896isin119860

119901 (119896) log (119899119901 (119896))

119867 = sum

119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582)

(5)


max119867 = maxsum119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582) (6)


119899 lt 1198861198990 (119886 gt 1) (7)



3 lt 120582 lt 4 (8)



sum

119896isin123

119901 (119896) gt 119904


120582119896

119896119890minus120582gt 119904

(9)


max 119867(120582 119899) = sum119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582)

st 119899 lt 1198861198990

3 lt 120582 lt 4

sum

119896isin123

120582119896

119896119890minus120582gt 119904

(10)





st Cost constraints

Degree constraints


(11)




A

DC

B A

DC

B

A

C

B A

C

B

(1)

(2)


sm

zjj

dz syhk

hazj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy











(119897 minus 1198970)

1198970

lt 1198861

(119871 minus 1198710)

1198710

lt 1198862

100381610038161003816100381610038161003816100381610038161003816

(119896 minus 1198960)

1198960

100381610038161003816100381610038161003816100381610038161003816

lt 1198863

(12)


(119897 minus 7514)

7514lt 5

(119871 minus 1270771)

1270771lt 5

10038161003816100381610038161003816100381610038161003816

(119896 minus 37464)

37464

10038161003816100381610038161003816100381610038161003816

lt 5

(13)




sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy

(a)

sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy


(b)

sm

zjj

dz syhkha

zj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mzsg

yz

hh ld zz cl

yiy

cs

yuy

hy


(c)





(14)






Programnumber



1 0459839 03999952 0793302 03999953 1 07999924 0378309 05 0668881 03999956 006378 03999957 0903425 07999928 0331941 09 0223749 019999610 053596 059999311 0689747 012 03729 013 076239 059999314 0157675 019999615 049307 039999516 0570349 019999617 0 018 0724523 1




0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(a)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(b)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(c)



Programnumber




17 0 0 08 0331941 0 033194114 0157675 0199996 035767112 03729 0 037294 0378309 0 03783099 0223749 0199996 04237456 006378 0399995 046377511 0689747 0 068974716 0570349 0199996 07703451 0459839 0399995 085983415 049307 0399995 08930655 0668881 0399995 106887610 053596 0599993 11359532 0793302 0399995 119329713 076239 0599993 13623837 0903425 0799992 170341718 0724523 1 17245233 1 0799992 1799992







Average distance





5 Conclusions





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8 90

005

01

015

02

025

03

0352008

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre

es

(a)

1 2 3 4 5 6 7 8 9 100

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre


01

02

03

(b)


1 2 3 4 5 6 7 8 90

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre

es

2008

01

02

03


(a)

1 2 3 4 5 6 7 8 9 100

005

015

025

035

The node degrees

The p

ropo

rtio

n of

the n

ode d

egre


01

02

03


(b)






st Cost constraints



(4)




119867 = sum

119896isin119860

119901 (119896) log (119899119901 (119896))

119867 = sum

119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582)

(5)


max119867 = maxsum119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582) (6)


119899 lt 1198861198990 (119886 gt 1) (7)



3 lt 120582 lt 4 (8)



sum

119896isin123

119901 (119896) gt 119904


120582119896

119896119890minus120582gt 119904

(9)


max 119867(120582 119899) = sum119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582)

st 119899 lt 1198861198990

3 lt 120582 lt 4

sum

119896isin123

120582119896

119896119890minus120582gt 119904

(10)





st Cost constraints

Degree constraints


(11)




A

DC

B A

DC

B

A

C

B A

C

B

(1)

(2)


sm

zjj

dz syhk

hazj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy











(119897 minus 1198970)

1198970

lt 1198861

(119871 minus 1198710)

1198710

lt 1198862

100381610038161003816100381610038161003816100381610038161003816

(119896 minus 1198960)

1198960

100381610038161003816100381610038161003816100381610038161003816

lt 1198863

(12)


(119897 minus 7514)

7514lt 5

(119871 minus 1270771)

1270771lt 5

10038161003816100381610038161003816100381610038161003816

(119896 minus 37464)

37464

10038161003816100381610038161003816100381610038161003816

lt 5

(13)




sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy

(a)

sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy


(b)

sm

zjj

dz syhkha

zj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mzsg

yz

hh ld zz cl

yiy

cs

yuy

hy


(c)





(14)






Programnumber



1 0459839 03999952 0793302 03999953 1 07999924 0378309 05 0668881 03999956 006378 03999957 0903425 07999928 0331941 09 0223749 019999610 053596 059999311 0689747 012 03729 013 076239 059999314 0157675 019999615 049307 039999516 0570349 019999617 0 018 0724523 1




0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(a)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(b)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(c)



Programnumber




17 0 0 08 0331941 0 033194114 0157675 0199996 035767112 03729 0 037294 0378309 0 03783099 0223749 0199996 04237456 006378 0399995 046377511 0689747 0 068974716 0570349 0199996 07703451 0459839 0399995 085983415 049307 0399995 08930655 0668881 0399995 106887610 053596 0599993 11359532 0793302 0399995 119329713 076239 0599993 13623837 0903425 0799992 170341718 0724523 1 17245233 1 0799992 1799992







Average distance





5 Conclusions





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119867 = sum

119896isin119860

119901 (119896) log (119899119901 (119896))

119867 = sum

119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582)

(5)


max119867 = maxsum119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582) (6)


119899 lt 1198861198990 (119886 gt 1) (7)



3 lt 120582 lt 4 (8)



sum

119896isin123

119901 (119896) gt 119904


120582119896

119896119890minus120582gt 119904

(9)


max 119867(120582 119899) = sum119896isin119860

120582119896

119896119890minus120582 log(119899120582

119896

119896119890minus120582)

st 119899 lt 1198861198990

3 lt 120582 lt 4

sum

119896isin123

120582119896

119896119890minus120582gt 119904

(10)





st Cost constraints

Degree constraints


(11)




A

DC

B A

DC

B

A

C

B A

C

B

(1)

(2)


sm

zjj

dz syhk

hazj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy











(119897 minus 1198970)

1198970

lt 1198861

(119871 minus 1198710)

1198710

lt 1198862

100381610038161003816100381610038161003816100381610038161003816

(119896 minus 1198960)

1198960

100381610038161003816100381610038161003816100381610038161003816

lt 1198863

(12)


(119897 minus 7514)

7514lt 5

(119871 minus 1270771)

1270771lt 5

10038161003816100381610038161003816100381610038161003816

(119896 minus 37464)

37464

10038161003816100381610038161003816100381610038161003816

lt 5

(13)




sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy

(a)

sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy


(b)

sm

zjj

dz syhkha

zj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mzsg

yz

hh ld zz cl

yiy

cs

yuy

hy


(c)





(14)






Programnumber



1 0459839 03999952 0793302 03999953 1 07999924 0378309 05 0668881 03999956 006378 03999957 0903425 07999928 0331941 09 0223749 019999610 053596 059999311 0689747 012 03729 013 076239 059999314 0157675 019999615 049307 039999516 0570349 019999617 0 018 0724523 1




0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(a)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(b)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(c)



Programnumber




17 0 0 08 0331941 0 033194114 0157675 0199996 035767112 03729 0 037294 0378309 0 03783099 0223749 0199996 04237456 006378 0399995 046377511 0689747 0 068974716 0570349 0199996 07703451 0459839 0399995 085983415 049307 0399995 08930655 0668881 0399995 106887610 053596 0599993 11359532 0793302 0399995 119329713 076239 0599993 13623837 0903425 0799992 170341718 0724523 1 17245233 1 0799992 1799992







Average distance





5 Conclusions





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A

DC

B A

DC

B

A

C

B A

C

B

(1)

(2)


sm

zjj

dz syhk

hazj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy











(119897 minus 1198970)

1198970

lt 1198861

(119871 minus 1198710)

1198710

lt 1198862

100381610038161003816100381610038161003816100381610038161003816

(119896 minus 1198960)

1198960

100381610038161003816100381610038161003816100381610038161003816

lt 1198863

(12)


(119897 minus 7514)

7514lt 5

(119871 minus 1270771)

1270771lt 5

10038161003816100381610038161003816100381610038161003816

(119896 minus 37464)

37464

10038161003816100381610038161003816100381610038161003816

lt 5

(13)




sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy

(a)

sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy


(b)

sm

zjj

dz syhkha

zj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mzsg

yz

hh ld zz cl

yiy

cs

yuy

hy


(c)





(14)






Programnumber



1 0459839 03999952 0793302 03999953 1 07999924 0378309 05 0668881 03999956 006378 03999957 0903425 07999928 0331941 09 0223749 019999610 053596 059999311 0689747 012 03729 013 076239 059999314 0157675 019999615 049307 039999516 0570349 019999617 0 018 0724523 1




0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(a)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(b)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(c)



Programnumber




17 0 0 08 0331941 0 033194114 0157675 0199996 035767112 03729 0 037294 0378309 0 03783099 0223749 0199996 04237456 006378 0399995 046377511 0689747 0 068974716 0570349 0199996 07703451 0459839 0399995 085983415 049307 0399995 08930655 0668881 0399995 106887610 053596 0599993 11359532 0793302 0399995 119329713 076239 0599993 13623837 0903425 0799992 170341718 0724523 1 17245233 1 0799992 1799992







Average distance





5 Conclusions





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy

(a)

sm

zjj

dz syhk

hazj

cd

mm yjzq

zh

gz

sz

dg lc

st

mz

sgyz

hh ld zz cl

yiy

cs

yuy

hy


(b)

sm

zjj

dz syhkha

zj

cd

mm yj

zq

zh

gz

sz

dg lc

st

mzsg

yz

hh ld zz cl

yiy

cs

yuy

hy


(c)





(14)






Programnumber



1 0459839 03999952 0793302 03999953 1 07999924 0378309 05 0668881 03999956 006378 03999957 0903425 07999928 0331941 09 0223749 019999610 053596 059999311 0689747 012 03729 013 076239 059999314 0157675 019999615 049307 039999516 0570349 019999617 0 018 0724523 1




0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(a)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(b)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(c)



Programnumber




17 0 0 08 0331941 0 033194114 0157675 0199996 035767112 03729 0 037294 0378309 0 03783099 0223749 0199996 04237456 006378 0399995 046377511 0689747 0 068974716 0570349 0199996 07703451 0459839 0399995 085983415 049307 0399995 08930655 0668881 0399995 106887610 053596 0599993 11359532 0793302 0399995 119329713 076239 0599993 13623837 0903425 0799992 170341718 0724523 1 17245233 1 0799992 1799992







Average distance





5 Conclusions





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(a)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(b)

0 1 2 3 4 5 644

44545

45546

46547

475


The a

vera

ge d

istan

ce


(c)



Programnumber




17 0 0 08 0331941 0 033194114 0157675 0199996 035767112 03729 0 037294 0378309 0 03783099 0223749 0199996 04237456 006378 0399995 046377511 0689747 0 068974716 0570349 0199996 07703451 0459839 0399995 085983415 049307 0399995 08930655 0668881 0399995 106887610 053596 0599993 11359532 0793302 0399995 119329713 076239 0599993 13623837 0903425 0799992 170341718 0724523 1 17245233 1 0799992 1799992







Average distance





5 Conclusions





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusions





Acknowledgment


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Fault Tolerance Optimization Model of ...

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Transcript of Research Article A Fault Tolerance Optimization Model of ...