Research Article A Resilience Approach to Symbiosis ... · Parks Based on Cascading Failure Model YuZeng, 1,2 RenbinXiao, 1 andXiangmeiLi 3 Institute of Systems Engineering, Huazhong

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 372368 11 pageshttpdxdoiorg1011552013372368

Research ArticleA Resilience Approach to Symbiosis Networks of EcoindustrialParks Based on Cascading Failure Model

Yu Zeng12 Renbin Xiao1 and Xiangmei Li3

1 Institute of Systems Engineering Huazhong University of Science and Technology Wuhan 430074 China2 School of Science Hubei University of Technology Wuhan 430068 China3Wenhua College Huazhong University of Science and Technology Wuhan 430074 Hubei China

Correspondence should be addressed to Renbin Xiao rbxiao husteducn

Received 3 April 2013 Accepted 19 May 2013

Academic Editor Tingwen Huang

Copyright copy 2013 Yu Zeng et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Evaluation and improvement of resilience in ecoindustrial parks have been the pressing issues to be addressed in the study ofsafety In this paper eco-industrial systems are extracted as symbiosis networks by using social network analysis first We thenconstruct a novel cascading failure model and propose an evaluation method of node importance according to the features ofsymbiosis networks of eco-industrial parks Based on the cascading model an effective new method that is the critical thresholdis put forward to quantitatively assess the resilience of symbiosis networks of eco-industrial parks Some theoretical analysis isfurthermore provided to the critical threshold Finally we take Jinjie eco-industrial system in Shanxi Province of China as a case toinvestigate its resilience The key potential nodes are identified by using our model We also find the respective relation among theresilience of symbiosis networks and the parameters in our cascadingmodelTheoretical analysis results and numerical simulationsboth show the optimal value of the tunable load parameter with which the strongest resilience level against cascading failures canbe attained in symbiosis networks of eco-industrial parks

1 Introduction

Complex systems are everywhere in the nature [1ndash5] TheirSecurity could have a big impact on the economic devel-opment and social stability so it has attracted a great dealof attention [6ndash9] Complex systems can be described asall kinds of networks Complex networks have become anindispensable part of our life such as social ecologicalnetworks transportation networks and power grid networksParticularly cascading failure phenomenon of complex net-works has been the focus of many recent studies for thereason that cascading failures are common phenomenon inreal-life systems and can cause huge damages to our lifeand industry development [10ndash13] Symbiosis networks ofecoindustrial parks (abbreviated as EIPs hereafter) are alsotypical networks including cascading phenomenon In EIPsthe products produced by one enterprise could be the nutri-ents of other enterprises Through this kind of cooperationthey form the industrial symbiosis networks where there

exist fairly close industrial symbiosis relations between oramong products produced by different enterprises In thesymbiosis network of an EIP when an upstream enterpriseor link failure emerges the insufficiency of supplying goodsto the downstream enterprises will happen Even though thefailure emerges very locally in the ecoindustrial system it willalso quickly spread like a plague to large areas causing seriousdamage to the whole system and even resulting in the globalcollapse In this way the entire symbiosis network can belargely affected due to cascading failures Therefore it is verynecessary to study cascading failures in symbiosis networksof EIPs

Recently a plenty of EIPs are constructed in many placessuch as Denmark America Japan and Austria In ChinaGuangdong Nanhai Hunan Changsha Xinjiang ShiheziShanghai Wujing and others have also constructed someecoindustrial parks With the development of industrialecology the EIP is developing as a brand new research field[14ndash16] The research on the complexity of ecoindustrial

2 Mathematical Problems in Engineering

systems has become popular and gradually made a break-through Graph theory was adopted to introduce the flowof materials or energy in Choctaw ecoindustrial park first[17] Furthermore Sterr andOtt [18] introduced the flow pathof materials or energy in the industrial region network ofRhine-Neckar and made clear the relationship between theenterprise inside the network and that outside the networkGraedel et al [19] used statistical indexes of the foodweb fromnatural ecosystems for reference in his evaluations of EIPsThey assessed the business symbiosis in industrial systemsthrough species richness and connectedness Based on thework the potential contribution of industrial symbiosisnetworks to environmental innovation was explored [16] Inorder to measure the complexity characteristics of industrialsymbiosis networks network complexity of two cases theKalundborg system in Denmark and the Gongyi system inChina were studied The features of industrial symbiosisnetworks from the aspects of scale aggregation connected-ness complexity and node influences were analyzed [20]Different from these methods a methodology for translatingecological quantitative analysis techniques to an industrialcontext was proposed The methodology is demonstratedin the case of Burnside industrial park using the conceptsof connectance and diversity The demonstrated techniquescan potentially aid in gaining an understanding of industrialsymbiosis [21] In addition Posch [22] investigated whetherindustrial recycling networks or industrial symbiosis projectscan be used as a starting point for broader intercompanycooperation for sustainable development

Althoughmuch research on EIPs focuses on the exchangeanalysis of material or energy little work has been publishedon the topology structure of symbiosis networks of EIPsEspecially to date most researchers have investigated thecascading phenomenon in power grid networks and trafficnetworks [23 24] Little attention has been paid to the cas-cading phenomenon in symbiosis networks of EIPs uptil nowFacing the upsurge of ecoindustrial park construction a setof scientific appraise index systems needs to be established Achallenging problem in industrial ecology is whether indus-trial symbiosis systems bear complexity features if the systemsare described only from their mass or energy interlinkagesMoreover various members play various roles in symbiosisnetworks of EIPs If a critical component fail the failurewill spread very quickly and it could trigger a more seriousdamage to symbiosis networks of EIPs due to cascadingfailures

Taking the above existing problems into account we usecomplex network theory to go into the interior of symbiosisnetworks and analyze the resilience against cascading failuresin symbiosis networks of EIPs in this paper We abstract EIPsas symbiosis networks and construct a novel cascadingmodelfor symbiosis network according to the EIPrsquos characteristicsWe use the ecological degree in EIPs for reference and raisea measure to quantitatively evaluate the power and statusof nodes Based on the cascading model an effective newconcept that is the critical threshold is put forward toquantitatively assess the resilience of symbiosis networks ofEIPs We take Jinjie ecoindustrial system in Shanxi Provinceof China as a case to investigate its resilience Through our

proposed model some potential critical nodes which aresensitive to the ecological degree and the global connectivityof the symbiosis network but not so important intuitivelyare found The respective relation among the resilience ofsymbiosis networks and the parameters is also found in ourcascading model Moreover according to theoretical analysisresults and numerical simulations we obtain an interestingresult that is when the value of the tunable load parameter is1 the symbiosis network displays the strongest resilience levelagainst cascading failures These results in this paper may bevery helpful for symbiosis networks of EIPs to enhance theresilience against cascading failures Our work also may havepractical implications for promoting the construction of EIPs

The remainder of this paper is organised as followsSymbiosis networks of ecoindustrial parks are establishedwith social network analysis in Section 2 According to theconstruction method an illustrative example is also given inthis section Section 3 proposes the cascading failure modeland an evaluation method of node importance accordingto the features of ecoindustrial parks Section 4 gives sometheoretical analysis and numerical simulations to our modelFinally Section 5 concludes with some discussions

2 Symbiosis Network Construction ofEcoindustrial Parks

As one direct practical form of industrial ecology ecoin-dustrial parks are becoming important research contents ofindustrial ecology The ecoindustrial park (EIP) is defined asan industrial park in which enterprises cooperate with eachother by using each otherrsquos by-products and wastes throughthis kind of cooperation they form a symbiosis networkNetwork analysis is a methodology to study objects as apart of a larger system so it is a brand new feeling to studyEIPs through this method Here the symbiosis network ofan EIP is a network that consists of nodes (components)and the connections between them In order to investigatethe status of all components in EIPs and the resilience ofsymbiosis networks of EIPs we will extract ecoindustrialparks as symbiosis networkswith social network analysisThedetailed method of building symbiosis network models is asfollows

(1) Nodes and edges of symbiosis networks of EIPsWe confirmed the members of the parks first Thearea products by-products wastes and materialsare the foundation nodes Using the administrativeboundary as the network boundary we add somemembers outside the parks who exchange resourceswith members within the parks as the supplementnodes Then we investigate the flow between mem-bers such as exchanges of products by-productswastes and materials When there is a flow of matteror energy between any two members in the parks wesay that there is an edge between them

(2) Adjacency matrices of symbiosis networks of EIPson the basis of confirmed nodes and edges amongmembers we establish a relational data collectionWeuse the directive dichotomous assessment system to

Mathematical Problems in Engineering 3

judge whether the relation exists or not and call thesystem the adjacency matrix The adjacency matrixis a square matrix in which if there is one relationbetween longitudinal and transverse members thecrossover box should be filled with 1 that is 119886

119894119895= 1

otherwise 0 that is 119886119894119895= 0 Obviously if the matrix

represents an undirected graph then it is symmetricas 119886119894119895= 119886119895119894

(3) Symbiosis network models of EIPs when one nodefails it not only could lead to the insufficient supplyof matter to the downstream enterprises but alsodecrease the matter outflow of the upstream enter-prises at the inverse direction Members in symbiosisnetworks cooperate with each other So we regardsymbiosis networks of EIPs as undirected networksFurthermore considering that the exchange of dataof matter between nodes is difficult to obtain andthe measurement standard could also not be unifiedwe therefore extract ecoindustrial parks as undirectedand unweighted symbiosis networks which consist ofnodes and edges

Here we take the Jinjie ecoindustrial park in ShanxiProvince of China as a case to build a symbiosis net-work model The EIP consists of four industrial systemsthe industrial system of coal power the industrial systemof coal chemical the industrial system of salt chemicaland the industrial system of glass building materials Thisecoindustrial system is designed to transform resources insitu develop comprehensive utilization of materials andproduce minimal emissions A lot of exchanges of matteror energy exist in Jinjie ecoindustrial system and it is arelatively complex industrial symbiosis network We chooseproducts by-products wastes and materials from a certainscale of enterprises whose annual output values are morethan 2 million yuan as 75 nodes of the symbiosis networkThese nodes come from four different industrial systemsrespectively including outsourcing coal semicoke coal tarcoal gangue coal mine methyl alcohol solid sulfur con-densed water salt polyvinyl chloride caustic soda methaneoxides quartz sand industrial silicon glass glass brickswaste heat waste water waste residues and waste gas Bycollecting data and investigating the interrelation betweenmembers of Jinjie ecoindustrial park we obtain the adjacencymatrix According to the adjacency matrix of the symbiosisnetwork we could obtain the symbiosis network model ofJinjie ecoindustrial park under UCINET 60 as is shown inFigure 1

3 Model of Cascading Failure in SymbiosisNetworks of Ecoindustrial Parks

In EIPs the products produced by one enterprise could be thenutrients of other enterprises Components in EIPs cooperatewith each other If an upstream enterprise or link failureemerges it not only could lead to the insufficient supply ofmatter to the downstream enterprises but also decrease thematter outflow of the upstream enterprises at the inversedirection The fairly close industrial symbiosis relations exist

in symbiosis networks In order to show the cascadingphenomenon in symbiosis networks of ecoindustrial parkswe will focus on cascades triggered by the removal of acomponent For the symbiosis network of an EIP initiallythe network is in a stationary state in which the load ofeach node is smaller than its capacityThe production systemwith the EIP maintains the normal operationsm but theremoval of a node will change the load balance and lead tothe load redistribution over other nodes Once the capacityof these nodes is insufficient to handle the extra load thismust be induce the node further breaking triggering acascade of overload failures and eventually a large drop inthe performance of the network Therefore the removal ofone node can cause serious consequences In the followingwe will construct a cascading failure model according to thecharacteristics of EIPs

31 A New Cascading Failure Load Model We define anetwork 119866 = (119881 119864) by a nonempty set of nodes 119881 =

1 2 119873 and a nonempty set of edges 119864 = 119894119895 | 119894 119895 isin 119881In most of the previous cascade failure models the load ona node (or an edge) was generally estimated by its degreeor its betweenness [25 26] The degree method is inferiorowing to its consideration of only a single node degreeThusit will losemuch information inmany actual applicationsThebetweenness method is not beneficial to its consideration oftopological information for the whole network In symbiosisnetworks of EIPs considering that the load of a componentis associated with its degree and the degrees of its neighborcomponents due to the close relation between supply anddemand we assume that the initial load 119871

119894of node 119894

being dependent on the degree of node 119894 and the degreesof its neighbor nodes the expression of which is definedas follows

119871119894= 119896120573

119894+ ( sum

119898isin119867119894

119896119898)

120573

(1)

where 119896119894and119867

119894are the degree of the node 119894 and the set of its

neighbor nodes respectively and 120573 is the tunable parameterwhich controls the strength of the initial load of a node

We know that each node of symbiosis networks has acapacity which is the largest load that the node can handleIn symbiosis networks of EIPs the load capacity of each nodeis severely limited by cost Therefore we assume that theload capacity 119862

119894of node 119894 is proportional to its initial load

that is

119862119894= (1 + 120572) 119871 119894 (

0) 119894 = 1 2 119873 (2)

where 120572 gt 0 is the tolerance parameter which characterizesthe resistance to the attacks For the symbiosis network of anEIP the tolerance parameter 120572 also reflects the constructioncost of the EIP

After node 119894 is removed from the network its load willbe redistributed to its neighbor node 119895 We assume that


6

5

4

202223

65

666764 63

62

69

68

61

7533 31

32

4748

49

30

36

28

38

3

2

34

35

37747242

39

60

40

43

4645

55

58

5354

51

56

5273

59

50

5744

29

27

26

25

2421

19

15

101312

18

8

9

1

7

11

16

1714

41

70

71

Figure 1 The symbiosis network representation of Jinjie ecoindustrial park

P(L)

1

0Ci 120574Ci L

Figure 2 The removal probability 119875(119871) of node 119894 at which load 119871lasts for time 119879

the additional load Δ119871119894119895received by the neighbor node 119895 is

proportional to its initial load that is

Δ119871119894119895=

119871119895

sum119891isin119867119894

119871119891

times 119871119894 (3)

=

119896119895

120573+ (sum119891isin119867119895

119896119891)

120573

sum119891isin119867119894

[119896119891

120573+ (sum119897isin119867119891

119896119897)

120573

]

times[

[

119896119894

120573+ ( sum

119891isin119867119894

119896119891)

120573

]

]

(4)

The method of immediate removal of an instantaneouslyoverloaded node is widely adopted in most of the previouscascade failure models [25 26] This method does notconsider the overloaded node removal mechanism in reallife According to the actual situation of EIPs a certainquantity of instantaneous node overload is permissible and

the failure of a node is mainly caused by the accumulativeeffect of overload especially for EIPs In symbiosis networksof EIPs the insufficient supply of goods induced by a smalloccasional incident usually has no distinct effect on thenormal operation of the whole symbiosis networks It isbecause that there exists a certain monitoring and controlsystem in such networks Once the node overload is detectedsome effective measures are taken within the response timeto decrease the capacity constraint violation and even canmake the node load less than its capacity In other wordsthe overloaded node is not necessarily removed Thereforewe introduce a new method of the overloaded node removalto reflect ecoindustrial systems

Suppose that 119875(119871) is the removal probability of a node atwhich load 119871 lasts for a period of time119879 as shown in Figure 2The probability density of 119875(119871) obeys uniform distribution ofthe interval [0 119879] The value of 119879 has some relation to theresponse time of industrial symbiosis systems The detailedexpression of 119875(119871) is as follows

119875 (119871) =

0 119871 le 119862119894

119871 minus 119862119894

(120574 minus 1) 119862119894

119862119894lt 119871 lt 120574119862

119894

1 119871 ge 120574119862119894

(5)

where the parameter 120574 (120574 gt 1) may be different for varioussymbiosis networks and the value of 120574 determines to a certainextent the protection of those overload nodes Obviously thebigger the value of 120574 the higher the cost of avoiding cascadingfailures In (5) the three cases correspond to three states ofnodes normal state overload state and failure state


In the simulations the following iterative rule is adoptedat each time 119905

ℎ119894 (119905) =

ℎ119894 (119905 minus 1) +

119875 (119871119894 (119905))

119879

119871119894 (119905) gt 119862119894

0 119871119894 (119905) le 119862119894

119894 = 1 2 119873

(6)

where ℎ119894(119905) is the removal probability of node 119894 at time 119905 and

119871119894(119905) is the load of node 119894 at time 119905We obtain 119875(119871

119894(119905)) according to Figure 2 and (5) and

derive the malfunction probability ℎ119894(119905) from (6) According

to the above process the loads of all nodes are calculatedat each time after an initial removal of a node induced bya malfunction We then determine the nodes which will beremoved from the network by comparing the malfunctionprobability and a random 120575 isin (0 1) The cascading failurestops until the loads of all nodes are not larger than theircorresponding capacity

32 Node Importance Evaluation Based on Cascading ModelIn the symbiosis network of an EIP various nodes havedifferent effects on the operation and management of ecoin-dustrial system especially for the ecological degree which isemployed to measure the relation of exchanging by-productsand wastes The failures of critical nodes are more likely tocause the collapse of the whole ecoindustrial system due tocascading failures If we can provide priority protection forthose critical nodes in advance the resilience and securityof symbiosis networks of EIPs can be improved To betterunderstand the cascading failure characteristics in this sec-tion we will discuss and propose a new evaluationmethod ofnode importance considering cascading failure for symbiosisnetworks of EIPs

After the cascading process is induced by removing anode the total number of residual nodes is called the networkarea In order to find a suitable evaluation method of nodeimportance we define the following indicator by adopting thenetwork area first

119877119894= 1 minus

1198731015840

119894

119873

(7)

where119873 is the number of all nodes before cascading failuresand 1198731015840

119894is the number of the nodes which can maintain

the normal operations after cascading failures induced byremoving node 119894 in the whole symbiosis network Obviouslythe value of indicator 119877

119894reflects the damage degree of node 119894

to the global connectivity of symbiosis networksThen the set 119881 of all nodes in the symbiosis network

is divided into two categories 119881 = 1198811cup 1198812 where the set

1198812= 1198991 1198992 119899

119873119890 consists of waste nodes and by-product

nodes which are all called ecological nodes and the set 1198811=

1198981 1198982 119898

119873119901 consists of all other nodes which are called

product nodes The number of all nodes is 119873 = 119873119890+ 119873119901

We propose a parameter which is called the rate of ecologicalconnectance 119877

119890 It is defined as follows

119877119890=

sum119886isin1198812

119896119886

119873(119873 minus 1) 2

(8)

where 119896119886is the degree of node 119886 and the summation is over

the degrees of all ecological nodes From (8) the higher thevalue of 119877

119890is the more the flow of the reciprocal utilization

of by-products and wastes within EIP isWe know that the development aim of EIPs is maximum

utilization of matter and energy To reflect the feature ofsymbiosis networks of EIPs which is different from othernetworks nowwe define another indicator which can be usedto characterize the nodersquos impact on the ecological degree ofsymbiosis networks as follows

119868119894= 1 minus

1198771015840

119890

119877119890

= 1 minus

sum119887isin1198812cap119881

1015840

119894

119896119887

sum119886isin1198812

119896119886

(9)

where119877119890and1198771015840

119890are the rates of ecological connectance before

and after cascading failures in symbiosis networks of EIPsrespectively 1198811015840

119894is the set of residual nodes under the normal

operations in the whole network after cascading breakdownsinduced by removing node 119894 That is the numerator of thefraction is the summation over the degree of all residualecological nodes after cascading breakdowns caused by theremoval of node 119894 and the denominator is the summationover the degree of all ecological nodes before cascadingbreakdownsThe indicator 119868

119894characterizes the damage degree

of node 119894 to the rate of ecological connectanceThe higher thevalue of indicator 119868

119894 the greater the impact of node 119894 on the

ecological degree of EIPsIn order to compare indicators119877

119894and 119868119894 we apply the two

methods to the symbiosis network of Jinjie ecoindustrial parkin Section 2 and analyze its cascading failure characteristicsIn the symbiosis network of Jinjie ecoindustrial park theset 1198812has 14 nodes such as waste water waste residue and

waste gas and the set 1198811consists of other 61 nodes such as

outsourcing coal salt and quartz sand In the simulationswe choose 120572 = 02 120573 = 02 and 120574 = 03 Every node inthe network is removed one by one and the correspondingresults are calculated for example removing the node 119894 andcalculating119877

119894and 119868119894by (7) and (9) after the cascading process

is overFigure 3 shows the values of two indicators for each

node We see that the change in 119868119894is more obvious than 119877

119894

For example the values of indicators 1198776 1198777are the same

It indicates that the removals of node 6 and node 7 leadto the same damage degree to the global connectivity ofthe symbiosis network However the difference in 119868

6and 1198687

illustrates that the two nodesrsquo removal has a different impacton the ecological degree of the symbiosis network Similarlyalthough the values of 119877

74and 119877

75are same the removals of

node 74 and node 75 cause different damage on ecologicaldegree according to the values of 119868

74and 11986875 The same cases

also happen in other nodes Furthermore we also find thatthe removals of nodes 1 2 6 7 9ndash28 33ndash39 42ndash46 48ndash5153 56ndash59 61ndash68 74 and 75 do not cause cascading failures Itimplies that their importance is relatively weak Despite thisthe removals of these nodes result in different impacts on theecological degree according to indicator 119868

119894 So we can declare

that indicator 119868119894is more helpful in finding out the potential

key nodes than indicator 119877119894 In other words indicator 119868

119894is a

more suitable measure to evaluate the node importance and


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800

01

02

03

04

05

06

07

08

09

1

Node i

IiRi

I iR

i

Figure 3 Comparison of two indicators for each node

reflect the status and role of nodes for symbiosis networks ofEIPs It is because that this method can help us to find somepotential critical nodes which are sensitive to the ecologicaldegree of symbiosis networks and the efficiency but not soimportant intuitively

With the above analysis we choose indicator 119868119894as the

measurement of node importance All nodes are sortedaccording to the descending order of the values of indicator119868119894 Table 1 represents the detailed cascading failure processes

induced by the nodes that take the top 5 places in rankingThe result shows that node 29 is the most important nodein the symbiosis network of Jinjie ecoindustrial park becausethe failure of node 29 (outsourcing coal) leads to the collapseof the whole symbiosis network We also find that node30 (waste water) node 4 (salt) node 47 (waste residue)node 69 (quartz sand) and node 31 (waste heat) rank 2nd3rd 4th and 5th respectively The removals of these nodesresult in great changes in the ecological degree They alsooccupy a very important status The simulation results arereasonably consistent with the actual situation Thereforein the operation and management of Jinjie ecoindustrialpark we should pay more attention to the enterprises inadvance that product or deal with these critical nodes suchas waster water salt waste residue waste heat quartz sandand especially outsourcing coal

In addition we also find that the highest degree orload node is not necessarily one of the most importantnodes and the low degree or load node may also occupy animportant status For instance the load and the degree ofnode 30 are both low Its removal leads to neighbor nodes27 75 malfunction first This triggers a cascade of overloadfailures and eventually results in the whole network almostcollapsing As a result there is a large drop in the ecologicaldegree of the whole network after the cascading process isover It is shown that node 30 is actually a potential keynode Therefore the high or low ecological degree does

not absolutely depend on the node degree or node loadEspecially for symbiosis networks of EIPs the exchange andrecycling of by-products and wastes is the most essential anddirect factor

By applying our model to the symbiosis network of Jinjieecoindustrial park one can see that our new evaluationmethod of node importance characterizes the status of nodewellThis measure can help us to identify some potential crit-ical nodes in symbiosis networks of EIPsThrough protectingthese nodes in the operation and management in advanceandmaking an effort to strengthen their construction we canachieve the comprehensive utilization of matter and energybetween components and even achieve the optimal resourceallocation The ecological construction will be promotedsystematically with the step-by-step breakthrough methodThe cost of pollution treatment not only will be decreasedbut also the considerable economic and social benefits willbe obtained Thus the resilience and security of symbiosisnetworks of EIPs are improved especially at the same timethe development of EIPs will be promoted

4 Resilience Analysis for Symbiosis Networksof EIPs on Model

It is known that the safety of EIPs severely affects theeconomic development and the social stability Thereforeit is very necessary to evaluate and improve resilience ofsymbiosis networks of EIPs In this section we will go intothe interior of symbiosis networks and propose the conceptof critical threshold to characterize the resilience of symbiosisnetworks against cascading failures by using our cascadingmodel

41 A Quantitative Measurement of Resilience for SymbiosisNetworks of EIPs Based on Cascading Model 119862119865

119894denotes the

avalanche size that is the number of broken nodes inducedby removing node 119894 Obviously 0 le 119862119865

119894le 119873 minus 1 We

remove every node in a network one by one and calculate thecorresponding results for example removing the node 119894 andcalculating 119862119865

119894after the cascading process is over Then the

normalized avalanche size is defined as follows

119862119865lowast=

sum119894isin119881119862119865119894

119873(119873 minus 1)

(10)

where 119881 represents the set of all nodes in the symbiosisnetwork of an EIP

From the proposed model the load capacity 119862119894of the

node 119894 is proportional to its initial load and precisely 119862119894=

(1 + 120572)119871119894(0) 119894 = 1 2 119873 Given a value of the tunable

parameter 120573 when the value of the tolerance parameter 120572is sufficiently small we can imagine that it is easy for thewhole network to fully collapse in the case of an arbitrarynode failure because the capacity of each node is limited Onthe other hand for sufficiently large120572 since all nodes have thelarger extra capacities to tolerate the load no cascading failureoccurs and the network maintains its normal and efficientfunctioning but at the same time it will need greater networkcost because the load capacity of each node is limited by the


Table 1 The cascading failure process and node weight ranking of the top 5 placesNode number Cascading failure process 119877 119868 Ranking Weight

29

27 75 rarr 28 30ndash32 rarr 35 43 rarr 48ndash50 59 62ndash65 rarr 1 5 6 33 34 36 3739ndash41 46 51 58 rarr 7ndash9 42 44 45 47 61 66ndash69 rarr 2 4 15ndash17 56 57

74 rarr 3806800 1 1 00843

30

27 75 rarr 28 29 31 32 rarr 35 43 48ndash50 59 62ndash65 rarr 1 5 6 33 34 36 3739 40 41 46 51 58 rarr 7ndash9 42 44 45 47 61 66ndash69 rarr 2 4 15ndash17 56 57

74 rarr 3806800 09976 2 00841

4

2 45 rarr 38 44 rarr 32ndash34 46 48 75 rarr 35 36 43 47 49 50 5962ndash65 rarr 1 5 6 31 37 39 40 41 51 58 74 rarr 7 8 9 27 42 61 rarr 15ndash17

28ndash30 5706133 09551 3 00805

4746 74 rarr 31 75 rarr 36 43 rarr 1 32ndash34 44 45 59 rarr 2 4 27 35 39 48ndash50

62ndash65 rarr 5 6 28ndash30 37 38 40 41 51 58 rarr 42 61 rarr 57 05333 09396 4 00786

69 33 56 75 rarr 66 67 68 rarr 56 01067 06573 5 00554

cost of the node Thus with the increase of the parameter120572 there should be some crossover behavior of the systemfrom large scale breakdown to no breakdown going throughsmall scale ones Therefore we use the crossover behavior toquantify the network resilience that is the critical threshold120572119888 at which a phrase transition occurs from normal state

to collapse That is to say when 120572 gt 120572119888 the ecoindustrial

system maintains its normal and efficient functioning whilewhen 120572 lt 120572

119888 119862119865lowast suddenly increases from 0 and cascading

failure emerges because the capacity of each node is limitedpropagating the whole or part network to stop workingTherefore 120572

119888is the least value of protection strength to

avoid cascading failure which reflects the minimum anti-interference ability of the network Apparently the lowerthe value of 120572

119888 the stronger the resilience of the network

against cascading failureThe critical threshold 120572119888controls all

the properties of the symbiosis network of an EIP and alsoquantifies the symbiosis network resilience by a transitionfrom the normal state to collapse

42 Theoretical Analysis To explore the resilience againstcascading failures for symbiosis networks of EIPs sometheoretical analysis is given below to discuss the criticalthreshold 120572

119888 Taking two cases of 120574 = 1 and 120574 gt 1 into

account we analyze the observed cascading phenomenon onsymbiosis networks The failure probability of an overloadnode is denoted by ℎ Based on our removal rule of anoverload node in order to avoid the occurrence of cascadingfailures for a neighbor node 119895 of the removed node 119894 thefollowing condition should be satisfied

119871119895+ Δ119871119894119895lt 119862119895 120574 = 1

119871119895+ Δ119871119894119895minus 119862119895

120574119862119895minus 119862119895

lt ℎ 120574 gt 1

(11)

According to (1) and (3) in our cascading model theabove condition (11) can be expressed as

[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

120574 = 1

([

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

)

times((120574 minus 1) (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

)

minus1

lt ℎ

120574 gt 1

(12)

From (12) we get

1 +

[119896119894

120573+ (sum119899isin119867119894

119896119899)

120573

]

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt 1 + 120572 120574 = 1

1+[119896119894

120573+(sum119899isin119867119894

119896119899)

120573

] sum119899isin119867119894

[119896119899

120573+(sum119891isin119867119899

119896119891)

120573

]

ℎ120574 minus ℎ + 1

lt1 + 120572

120574 gt 1

(13)

sum

119899isin119867119894

119896119899

120573=

119870

sum

=119898

119896119894119875 (119896 | 119896119894)119896120573 (14)

where 119898 and 119870 are the minimum degree and the maximumdegree respectively 119875(119896 | 119896

119894) is the conditional probability

that the node of degree 119896119894has a neighbor of degree 119896 It


is satisfied with the normalized condition and the balancecondition which is as follows

sum

119875 (119896 | 119896119894) = 1

119896119894119875 (119896 | 119896119894) 119875 (119896

119894) =

119896119875 (119896119894|119896) 119875 (

119896)

(15)

Based on the condition (15) equation (14) can be expressedas follows

sum

119899isin119867119894

119896119899

120573= 119896119894

119870

sum

=119898

119896119875 (

119896)119896120573

⟨119896⟩

= 119896119894

⟨119896120573+1⟩

⟨119896⟩

(16)

In the case of 120573 = 1 we get

sum

119899isin119867119894

119896119899= 119896119894

⟨1198962⟩

⟨119896⟩

(17)

Hence based on (16) and (17) the condition (13) can berewritten as follows

1 +

119896119894

120573minus1⟨119896⟩

119896120573+1

lt 1 + 120572 120574 = 1

1 + 119896119894

120573minus1⟨119896⟩ 119896

120573+1

ℎ120574 minus ℎ + 1

lt 1 + 120572 120574 gt 1

(18)

By the inequality (18) the critical threshold 120572119888can be gotten

by considering the ranges of 120573 lt 1 120573 = 1 and 120573 gt 1respectively as follows

when 120574 = 1 we have

120572119888=

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 lt 1

⟨119896⟩

⟨1198962⟩

120573 = 1

119870120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 gt 1

(19)

and when 120574 gt 1 we have

120572119888=

1 + 119898120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 lt 1

1 + ⟨119896⟩ ⟨1198962⟩

ℎ120574 minus ℎ + 1

minus 1 120573 = 1

1 + 119870120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 gt 1

(20)

For (18) we consider in the case of 120573 lt 1 that

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

=

⟨119896⟩

(1119873)sum119873

119894=11198962

119894(119896119894119898)120573minus1

gt

⟨119896⟩

(1119873)sum119873

119894=11198962

119894

=

⟨119896⟩

⟨1198962⟩

(21)

From the inequality (21) we find that 120572119888(120573 lt 1) gt 120572

119888(120573 =

1) when 120574 = 1 and 120574 gt 1 Similarly 120572119888(120573 gt 1) gt 120572

119888(120573 = 1)

when 120574 = 1 and 120574 gt 1 Compared with other values of theparameter 120573 the lowest value of 120572

119888at 120573 = 1 indicates that

symbiosis systems reach the strongest resilience level againstcascading failures We know that the tunable parameter 120573controls the strength of the initial load of a node Thereforethe strongest resilience at120573 = 1 implies an optimal initial loaddistribution rule which may be useful for the EIP to avoidcascading failure triggered by production problem

Moreover according to (20) we find that120572119888has a negative

relationship with the value 120574 when the value 120573 is given Thatis to say a bigger value 120574 leads to a stronger resilience againstcascading failures The value of 120574 determines to a certainextent the protection of those overload nodes The biggerthe value of the parameter 120574 the higher the cost of avoidingcascading failures This result will be very helpful for thechoice on an optimal value of 120574

43 Numerical Simulation The network structure plays animportant role in the resilience of networks For the symbiosisnetwork of an EIP if the tolerance parameter is able to beeffectively selected not only the construction cost of theecoindustrial park will be saved but also the overall resilienceof the symbiosis network will be enhanced Thus manyexisting problems in the current planning and constructionfor many ecoindustrial parks will be solved Therefore wefurthermore explore the relation between the critical thresh-old 120572119888and the other parameters in our cascading model

Figures 4ndash6 report the dynamical results of cascading failuresin the symbiosis network of Jinjie ecoindustrial park withdifferent values of the parameterswhere cascading failures aretriggered by removing the nodes one by one in the set 119881

We focus on the effect of the tolerance parameter 120572 for thenormalized avalanche size 119862119865lowast for different values of 120574 firstFigure 4 illustrates119862119865lowast after cascading failures of all removednodes as a function of the tolerance parameter 120572 under sixdifferent values of 120574 from Figures 4(a) 4(b) 4(c) 4(d) 4(e)and 4(f) From Figure 4 one can see that the value of 119862119865lowastincreases rapidly as the decrease of the tolerance parameterfrom 1 to 0 which indicates that the cascading failures ofsymbiosis networks triggered by removing nodes can lead toserious results

Figure 5 represents the relation between the criticalthreshold 120572

119888and the tunable parameter 120573 for different values

of 120574 One can see that120572119888attains theminimal valuewhen120573 = 1

on all the curves for different 120574 in Figure 5 From Figures 4-5we also can see an interesting phenomenon in our cascadingmodelThe symbiosis network displays its strongest resilienceagainst cascading failures decided by the critical threshold120572119888at 120573 = 1 under six different values of 120574 Therefore the

theoretical analysis result is further verified by the simulationresults These numerical simulation results agree well withthe finding of theoretical analysis result In addition we seethat the parameter 120574 plays an important role in the symbiosisnetwork resilience against cascading failures in Figures 4-5

The effect of the parameter 120574 on the critical threshold120572119888is further reported in Figure 6 It is easy to find that the

obtained 120572119888originating from the same value 120573 has a negative


0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)

Figure 4 119862119865lowast as a function of the tolerance parameter 120572 for different values of 120573 (a) (b) (c) (d) (e) and (f) show six results when 120574 = 1011 13 15 17 and 19 respectively


02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19

Figure 5 Relation between the critical threshold 120572119888and the tunable

parameter 120573 for different values of 120574

relationship with the parameter 120574 From Figure 6 we cansee that the curves of simulation results are in good agree-ment with the above theoretical analysis results and alsojust consistent with the actual situation This relationshipcan thus supply information for monitoring and control toavoid cascading failures in the symbiosis network of Jinjieecoindustrial park Additionally as is shown in Figure 6 allvalues on the curve 120573 = 1 are smaller than the ones on othercurves in the case of the same 120574 which also verify the resultthat the symbiosis network reach their strongest resiliencelevel at 120573 = 1 Therefore our model is reasonable and usefulto improve the resilience of symbiosis networks of EIPs in thecontrol and defense of the cascading failures

5 Conclusions

In this paper we construct a cascading model with tunableparameters for symbiosis networks of EIPs and explorecascading failure phenomenon in EIPs Considering thatindustrial developments are usually planned to increase therecycling of wastes and by-products we introduce the rate ofecological connectance of EIPs and propose a new evaluationmethod of node importance considering cascading failureaccording to the features of symbiosis networks of EIPsThis evaluation method can help us to find some potentialcritical nodes which are sensitive to the ecological degree andthe functioning efficiency of networks but not so importantintuitively Furthermore based on the proposed cascadingmodel we propose the critical threshold to quantify andfurther investigate the resilience of symbiosis networks ofecoindustrial parks To illustrate the effectiveness and fea-sibility of the proposed model we extract Jinjie ecoindus-trial system in Shanxi Province of China as an undirectedand unweighted symbiosis network The cascading reactionbehaviors on the symbiosis network are investigated underthe cascading model

09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c

Figure 6 Relation between the critical threshold 120572119888and the

parameter 120574

Our results show those critical nodes in the symbiosisnetwork of Jinjie EIP In addition some interesting andinstructive results are obtained in our proposed cascadingmodel The theoretical analysis results and numerical sim-ulations both show that the symbiosis network displays thestrongest resilience level against cascading failure at 120573 = 1

under the proposed cascading model This result implies anoptimal initial load distribution rule whichmay be useful forthe EIP to avoid cascading failure triggered by productionproblem It is also seen that the critical threshold 120572

119888has a

negative relationship with the value 120574 We know that thevalue of the tolerance parameter reflects the constructioncost of an EIP If the tolerance parameter is able to beeffectively selected in construction and management of EIPswe not only can save the construction cost of EIPs but alsoenhance the resilience for symbiosis networks of EIPs Itis very helpful for the construction of symbiosis networksof EIPs to select the optimal value of the parameters toattain the optimal resilienceWith these resultsmany existingproduction problems in the planning and constructionwill besolved in EIPs

Our work may have practical implications for protectingthe key nodes effectively and avoiding cascading-failure-induced disasters in symbiosis networks Moreover the pro-posed cascadingmodel has great generality for characterizingcascading-failure-induced disasters in the real world andprovides a valuable method to many real-life networks Itis expected that the presented model may help us betterunderstand the cascading phenomena in nature and enhancethe resilience of the networks

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (no 60974076)


References

[1] H Li and R B Xiao ldquoA multi-agent virtual enterprise modeland its simulationwith Swarmrdquo International Journal of Produc-tion Research vol 44 no 9 pp 1719ndash1737 2006

[2] SMWagner andNNeshat ldquoA comparison of supply chain vul-nerability indices for different categories of firmsrdquo InternationalJournal of Production Research vol 50 no 11 pp 2877ndash28912012

[3] S P Wen Z G Zeng and T W Huang ldquo119867infin

Filtering forneutral systems with mixed delays and multiplicative noisesrdquoIEEE Transactions on Circuits and Systems Part II vol 59 no 11pp 820ndash824 2012

[4] S P Wen Z G Zeng and T W Huang ldquoRobust probabilisticsampling 119867

infinoutput tracking control for a class of nonlinear

networked systems with multiplicative noisesrdquo Journal of theFranklin Institute vol 350 pp 1093ndash1111 2013

[5] S P Wen Z G Zeng and T W Huang ldquoObserver-based119867infin

control of discrete time-delay systems with random com-munication packet losses and multiplicative noisesrdquo AppliedMathematics and Computation vol 219 no 12 pp 6484ndash64932013

[6] 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

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

[8] A X C N Valente A Sarkar and H A Stone ldquoTwo-peak andthree-peak optimal complex networksrdquo Physical Review Lettersvol 92 no 11 4 pages 2004

[9] K Burnard and R Bhamra ldquoOrganisational resilience develop-ment of a conceptual framework for organisational responsesrdquoInternational Journal of Production Research vol 49 no 18 pp5581ndash5599 2011

[10] K-I Goh B Kahng andDKim ldquoFluctuation-driven dynamicsof the Internet topologyrdquo Physical Review Letters vol 88 no 10pp 1087011ndash1087014 2002

[11] R Albert I Albert and G L Nakarado ldquoStructural vulnerabil-ity of the North American power gridrdquo Physical Review E vol69 no 2 Article ID 025103 4 pages 2004

[12] H Pearson ldquoProtecting critical assets the interdiction medianproblemwith fortificationrdquo Physical Review E vol 37 no 19 pp352ndash367 2004

[13] D-O Leonardo J I Craig B J Goodno and A BostromldquoInterdependent response of networked systemsrdquo Journal ofInfrastructure Systems vol 13 no 3 pp 185ndash194 2007

[14] E A Lowe and L K Evans ldquoIndustrial ecology and industrialecosystemsrdquo Journal of Cleaner Production vol 3 no 1-2 pp47ndash53 1995

[15] J Korhonen ldquoIndustrial ecology in the strategic sustainabledevelopment model strategic applications of industrial ecol-ogyrdquo Journal of Cleaner Production vol 12 no 8ndash10 pp 809ndash823 2004

[16] M Mirata and T Emtairah ldquoIndustrial symbiosis networksand the contribution to environmental innovation the case ofthe Landskrona industrial symbiosis programmerdquo Journal ofCleaner Production vol 13 no 10-11 pp 993ndash1002 2005

[17] A J P Carr ldquoChoctaw Eco-Industrial Park an ecologicalapproach to industrial land-use planning and designrdquo Land-scape and Urban Planning vol 42 no 2ndash4 pp 239ndash257 1998

[18] T Sterr and T Ott ldquoThe industrial region as a promising unitfor eco-industrial developmentmdashreflections practical experi-ence and establishment of innovative instruments to supportindustrial ecologyrdquo Journal of Cleaner Production vol 12 no8ndash10 pp 947ndash965 2004

[19] T Graedel B R Allenby and H Shi Industrial EcologyTsinghua University Press Beijing China 2004 (in Chinese)

[20] Y M Song and L Shi ldquoComplexity measurements for Kalund-borg and Gongyi industrial symbiosis networksrdquo Journal ofTsinghua University vol 48 no 9 pp 61ndash64 2008 (Chinese)

[21] R A Wright R P Cote J Duffy and J Brazner ldquoDiversityand connectance in an industrial context the case of burnsideindustrial parkrdquo Journal of Industrial Ecology vol 13 no 4 pp551ndash564 2009

[22] A Posch ldquoIndustrial recycling networks as starting pointsfor broader sustainability-oriented cooperationrdquo Journal ofIndustrial Ecology vol 14 no 2 pp 242ndash257 2010

[23] I Dobson B A Carreras V E Lynch and D E NewmanldquoComplex systems analysis of series of blackouts cascadingfailure critical points and self-organizationrdquo Chaos vol 17 no2 Article ID 026103 13 pages 2007

[24] J F Zheng Z Y Gao and X M Zhao ldquoModeling cascadingfailures in congested complex networksrdquo Physica A vol 385 no2 pp 700ndash706 2007

[25] A E Motter and Y Lai ldquoCascade-based attacks on complexnetworksrdquo Physical Review E vol 66 no 6 Article ID 0651024 pages 2002

[26] D-O Leonardo and S M Vemuru ldquoCascading failures incomplex infrastructure systemsrdquo Structural Safety vol 31 no2 pp 157ndash167 2009

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


systems has become popular and gradually made a break-through Graph theory was adopted to introduce the flowof materials or energy in Choctaw ecoindustrial park first[17] Furthermore Sterr andOtt [18] introduced the flow pathof materials or energy in the industrial region network ofRhine-Neckar and made clear the relationship between theenterprise inside the network and that outside the networkGraedel et al [19] used statistical indexes of the foodweb fromnatural ecosystems for reference in his evaluations of EIPsThey assessed the business symbiosis in industrial systemsthrough species richness and connectedness Based on thework the potential contribution of industrial symbiosisnetworks to environmental innovation was explored [16] Inorder to measure the complexity characteristics of industrialsymbiosis networks network complexity of two cases theKalundborg system in Denmark and the Gongyi system inChina were studied The features of industrial symbiosisnetworks from the aspects of scale aggregation connected-ness complexity and node influences were analyzed [20]Different from these methods a methodology for translatingecological quantitative analysis techniques to an industrialcontext was proposed The methodology is demonstratedin the case of Burnside industrial park using the conceptsof connectance and diversity The demonstrated techniquescan potentially aid in gaining an understanding of industrialsymbiosis [21] In addition Posch [22] investigated whetherindustrial recycling networks or industrial symbiosis projectscan be used as a starting point for broader intercompanycooperation for sustainable development

Althoughmuch research on EIPs focuses on the exchangeanalysis of material or energy little work has been publishedon the topology structure of symbiosis networks of EIPsEspecially to date most researchers have investigated thecascading phenomenon in power grid networks and trafficnetworks [23 24] Little attention has been paid to the cas-cading phenomenon in symbiosis networks of EIPs uptil nowFacing the upsurge of ecoindustrial park construction a setof scientific appraise index systems needs to be established Achallenging problem in industrial ecology is whether indus-trial symbiosis systems bear complexity features if the systemsare described only from their mass or energy interlinkagesMoreover various members play various roles in symbiosisnetworks of EIPs If a critical component fail the failurewill spread very quickly and it could trigger a more seriousdamage to symbiosis networks of EIPs due to cascadingfailures

Taking the above existing problems into account we usecomplex network theory to go into the interior of symbiosisnetworks and analyze the resilience against cascading failuresin symbiosis networks of EIPs in this paper We abstract EIPsas symbiosis networks and construct a novel cascadingmodelfor symbiosis network according to the EIPrsquos characteristicsWe use the ecological degree in EIPs for reference and raisea measure to quantitatively evaluate the power and statusof nodes Based on the cascading model an effective newconcept that is the critical threshold is put forward toquantitatively assess the resilience of symbiosis networks ofEIPs We take Jinjie ecoindustrial system in Shanxi Provinceof China as a case to investigate its resilience Through our

proposed model some potential critical nodes which aresensitive to the ecological degree and the global connectivityof the symbiosis network but not so important intuitivelyare found The respective relation among the resilience ofsymbiosis networks and the parameters is also found in ourcascading model Moreover according to theoretical analysisresults and numerical simulations we obtain an interestingresult that is when the value of the tunable load parameter is1 the symbiosis network displays the strongest resilience levelagainst cascading failures These results in this paper may bevery helpful for symbiosis networks of EIPs to enhance theresilience against cascading failures Our work also may havepractical implications for promoting the construction of EIPs

The remainder of this paper is organised as followsSymbiosis networks of ecoindustrial parks are establishedwith social network analysis in Section 2 According to theconstruction method an illustrative example is also given inthis section Section 3 proposes the cascading failure modeland an evaluation method of node importance accordingto the features of ecoindustrial parks Section 4 gives sometheoretical analysis and numerical simulations to our modelFinally Section 5 concludes with some discussions

2 Symbiosis Network Construction ofEcoindustrial Parks

As one direct practical form of industrial ecology ecoin-dustrial parks are becoming important research contents ofindustrial ecology The ecoindustrial park (EIP) is defined asan industrial park in which enterprises cooperate with eachother by using each otherrsquos by-products and wastes throughthis kind of cooperation they form a symbiosis networkNetwork analysis is a methodology to study objects as apart of a larger system so it is a brand new feeling to studyEIPs through this method Here the symbiosis network ofan EIP is a network that consists of nodes (components)and the connections between them In order to investigatethe status of all components in EIPs and the resilience ofsymbiosis networks of EIPs we will extract ecoindustrialparks as symbiosis networkswith social network analysisThedetailed method of building symbiosis network models is asfollows

(1) Nodes and edges of symbiosis networks of EIPsWe confirmed the members of the parks first Thearea products by-products wastes and materialsare the foundation nodes Using the administrativeboundary as the network boundary we add somemembers outside the parks who exchange resourceswith members within the parks as the supplementnodes Then we investigate the flow between mem-bers such as exchanges of products by-productswastes and materials When there is a flow of matteror energy between any two members in the parks wesay that there is an edge between them

(2) Adjacency matrices of symbiosis networks of EIPson the basis of confirmed nodes and edges amongmembers we establish a relational data collectionWeuse the directive dichotomous assessment system to



119894119895= 1










119894of node 119894


119871119894= 119896120573

119894+ ( sum

119898isin119867119894

119896119898)

120573

(1)

where 119896119894and119867





that is

119862119894= (1 + 120572) 119871 119894 (

0) 119894 = 1 2 119873 (2)




6

5

4

202223

65

666764 63

62

69

68

61

7533 31

32

4748

49

30

36

28

38

3

2

34

35

37747242

39

60

40

43

4645

55

58

5354

51

56

5273

59

50

5744

29

27

26

25

2421

19

15

101312

18

8

9

1

7

11

16

1714

41

70

71


P(L)

1

0Ci 120574Ci L




Δ119871119894119895=

119871119895

sum119891isin119867119894

119871119891

times 119871119894 (3)

=

119896119895

120573+ (sum119891isin119867119895

119896119891)

120573

sum119891isin119867119894

[119896119891

120573+ (sum119897isin119867119891

119896119897)

120573

]

times[

[

119896119894

120573+ ( sum

119891isin119867119894

119896119891)

120573

]

]

(4)




119875 (119871) =

0 119871 le 119862119894

119871 minus 119862119894

(120574 minus 1) 119862119894

119862119894lt 119871 lt 120574119862

119894

1 119871 ge 120574119862119894

(5)




ℎ119894 (119905) =

ℎ119894 (119905 minus 1) +

119875 (119871119894 (119905))

119879

119871119894 (119905) gt 119862119894

0 119871119894 (119905) le 119862119894

119894 = 1 2 119873

(6)








119877119894= 1 minus

1198731015840

119894

119873

(7)







1198812= 1198991 1198992 119899



1198981 1198982 119898





119877119890=

sum119886isin1198812

119896119886

119873(119873 minus 1) 2

(8)






119868119894= 1 minus

1198771015840

119890

119877119890

= 1 minus

sum119887isin1198812cap119881

1015840

119894

119896119887

sum119886isin1198812

119896119886

(9)

where119877119890and1198771015840
















119894



6and 1198687


74and 119877








119894is a



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800

01

02

03

04

05

06

07

08

09

1

Node i

IiRi

I iR

i












119894denotes the


119894le 119873 minus 1 We




119862119865lowast=

sum119894isin119881119862119865119894

119873(119873 minus 1)

(10)








29


74 rarr 3806800 1 1 00843

30


74 rarr 3806800 09976 2 00841

4


28ndash30 5706133 09551 3 00805



69 33 56 75 rarr 66 67 68 rarr 56 01067 06573 5 00554














119871119895+ Δ119871119894119895lt 119862119895 120574 = 1

119871119895+ Δ119871119894119895minus 119862119895

120574119862119895minus 119862119895

lt ℎ 120574 gt 1

(11)


[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

120574 = 1

([

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

)

times((120574 minus 1) (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

)

minus1

lt ℎ

120574 gt 1

(12)

From (12) we get

1 +

[119896119894

120573+ (sum119899isin119867119894

119896119899)

120573

]

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt 1 + 120572 120574 = 1

1+[119896119894

120573+(sum119899isin119867119894

119896119899)

120573

] sum119899isin119867119894

[119896119899

120573+(sum119891isin119867119899

119896119891)

120573

]

ℎ120574 minus ℎ + 1

lt1 + 120572

120574 gt 1

(13)

sum

119899isin119867119894

119896119899

120573=

119870

sum

=119898

119896119894119875 (119896 | 119896119894)119896120573 (14)






sum

119875 (119896 | 119896119894) = 1

119896119894119875 (119896 | 119896119894) 119875 (119896

119894) =

119896119875 (119896119894|119896) 119875 (

119896)

(15)


sum

119899isin119867119894

119896119899

120573= 119896119894

119870

sum

=119898

119896119875 (

119896)119896120573

⟨119896⟩

= 119896119894

⟨119896120573+1⟩

⟨119896⟩

(16)


sum

119899isin119867119894

119896119899= 119896119894

⟨1198962⟩

⟨119896⟩

(17)


1 +

119896119894

120573minus1⟨119896⟩

119896120573+1

lt 1 + 120572 120574 = 1

1 + 119896119894

120573minus1⟨119896⟩ 119896

120573+1

ℎ120574 minus ℎ + 1

lt 1 + 120572 120574 gt 1

(18)




120572119888=

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 lt 1

⟨119896⟩

⟨1198962⟩

120573 = 1

119870120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 gt 1

(19)


120572119888=

1 + 119898120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 lt 1

1 + ⟨119896⟩ ⟨1198962⟩

ℎ120574 minus ℎ + 1

minus 1 120573 = 1

1 + 119870120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 gt 1

(20)


119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

=

⟨119896⟩

(1119873)sum119873

119894=11198962

119894(119896119894119898)120573minus1

gt

⟨119896⟩

(1119873)sum119873

119894=11198962

119894

=

⟨119896⟩

⟨1198962⟩

(21)


119888(120573 =


119888(120573 = 1)

















0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)



02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894119895= 1










119894of node 119894


119871119894= 119896120573

119894+ ( sum

119898isin119867119894

119896119898)

120573

(1)

where 119896119894and119867





that is

119862119894= (1 + 120572) 119871 119894 (

0) 119894 = 1 2 119873 (2)




6

5

4

202223

65

666764 63

62

69

68

61

7533 31

32

4748

49

30

36

28

38

3

2

34

35

37747242

39

60

40

43

4645

55

58

5354

51

56

5273

59

50

5744

29

27

26

25

2421

19

15

101312

18

8

9

1

7

11

16

1714

41

70

71


P(L)

1

0Ci 120574Ci L




Δ119871119894119895=

119871119895

sum119891isin119867119894

119871119891

times 119871119894 (3)

=

119896119895

120573+ (sum119891isin119867119895

119896119891)

120573

sum119891isin119867119894

[119896119891

120573+ (sum119897isin119867119891

119896119897)

120573

]

times[

[

119896119894

120573+ ( sum

119891isin119867119894

119896119891)

120573

]

]

(4)




119875 (119871) =

0 119871 le 119862119894

119871 minus 119862119894

(120574 minus 1) 119862119894

119862119894lt 119871 lt 120574119862

119894

1 119871 ge 120574119862119894

(5)




ℎ119894 (119905) =

ℎ119894 (119905 minus 1) +

119875 (119871119894 (119905))

119879

119871119894 (119905) gt 119862119894

0 119871119894 (119905) le 119862119894

119894 = 1 2 119873

(6)








119877119894= 1 minus

1198731015840

119894

119873

(7)







1198812= 1198991 1198992 119899



1198981 1198982 119898





119877119890=

sum119886isin1198812

119896119886

119873(119873 minus 1) 2

(8)






119868119894= 1 minus

1198771015840

119890

119877119890

= 1 minus

sum119887isin1198812cap119881

1015840

119894

119896119887

sum119886isin1198812

119896119886

(9)

where119877119890and1198771015840
















119894



6and 1198687


74and 119877








119894is a



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800

01

02

03

04

05

06

07

08

09

1

Node i

IiRi

I iR

i












119894denotes the


119894le 119873 minus 1 We




119862119865lowast=

sum119894isin119881119862119865119894

119873(119873 minus 1)

(10)








29


74 rarr 3806800 1 1 00843

30


74 rarr 3806800 09976 2 00841

4


28ndash30 5706133 09551 3 00805



69 33 56 75 rarr 66 67 68 rarr 56 01067 06573 5 00554














119871119895+ Δ119871119894119895lt 119862119895 120574 = 1

119871119895+ Δ119871119894119895minus 119862119895

120574119862119895minus 119862119895

lt ℎ 120574 gt 1

(11)


[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

120574 = 1

([

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

)

times((120574 minus 1) (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

)

minus1

lt ℎ

120574 gt 1

(12)

From (12) we get

1 +

[119896119894

120573+ (sum119899isin119867119894

119896119899)

120573

]

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt 1 + 120572 120574 = 1

1+[119896119894

120573+(sum119899isin119867119894

119896119899)

120573

] sum119899isin119867119894

[119896119899

120573+(sum119891isin119867119899

119896119891)

120573

]

ℎ120574 minus ℎ + 1

lt1 + 120572

120574 gt 1

(13)

sum

119899isin119867119894

119896119899

120573=

119870

sum

=119898

119896119894119875 (119896 | 119896119894)119896120573 (14)






sum

119875 (119896 | 119896119894) = 1

119896119894119875 (119896 | 119896119894) 119875 (119896

119894) =

119896119875 (119896119894|119896) 119875 (

119896)

(15)


sum

119899isin119867119894

119896119899

120573= 119896119894

119870

sum

=119898

119896119875 (

119896)119896120573

⟨119896⟩

= 119896119894

⟨119896120573+1⟩

⟨119896⟩

(16)


sum

119899isin119867119894

119896119899= 119896119894

⟨1198962⟩

⟨119896⟩

(17)


1 +

119896119894

120573minus1⟨119896⟩

119896120573+1

lt 1 + 120572 120574 = 1

1 + 119896119894

120573minus1⟨119896⟩ 119896

120573+1

ℎ120574 minus ℎ + 1

lt 1 + 120572 120574 gt 1

(18)




120572119888=

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 lt 1

⟨119896⟩

⟨1198962⟩

120573 = 1

119870120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 gt 1

(19)


120572119888=

1 + 119898120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 lt 1

1 + ⟨119896⟩ ⟨1198962⟩

ℎ120574 minus ℎ + 1

minus 1 120573 = 1

1 + 119870120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 gt 1

(20)


119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

=

⟨119896⟩

(1119873)sum119873

119894=11198962

119894(119896119894119898)120573minus1

gt

⟨119896⟩

(1119873)sum119873

119894=11198962

119894

=

⟨119896⟩

⟨1198962⟩

(21)


119888(120573 =


119888(120573 = 1)

















0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)



02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6

5

4

202223

65

666764 63

62

69

68

61

7533 31

32

4748

49

30

36

28

38

3

2

34

35

37747242

39

60

40

43

4645

55

58

5354

51

56

5273

59

50

5744

29

27

26

25

2421

19

15

101312

18

8

9

1

7

11

16

1714

41

70

71


P(L)

1

0Ci 120574Ci L




Δ119871119894119895=

119871119895

sum119891isin119867119894

119871119891

times 119871119894 (3)

=

119896119895

120573+ (sum119891isin119867119895

119896119891)

120573

sum119891isin119867119894

[119896119891

120573+ (sum119897isin119867119891

119896119897)

120573

]

times[

[

119896119894

120573+ ( sum

119891isin119867119894

119896119891)

120573

]

]

(4)




119875 (119871) =

0 119871 le 119862119894

119871 minus 119862119894

(120574 minus 1) 119862119894

119862119894lt 119871 lt 120574119862

119894

1 119871 ge 120574119862119894

(5)




ℎ119894 (119905) =

ℎ119894 (119905 minus 1) +

119875 (119871119894 (119905))

119879

119871119894 (119905) gt 119862119894

0 119871119894 (119905) le 119862119894

119894 = 1 2 119873

(6)








119877119894= 1 minus

1198731015840

119894

119873

(7)







1198812= 1198991 1198992 119899



1198981 1198982 119898





119877119890=

sum119886isin1198812

119896119886

119873(119873 minus 1) 2

(8)






119868119894= 1 minus

1198771015840

119890

119877119890

= 1 minus

sum119887isin1198812cap119881

1015840

119894

119896119887

sum119886isin1198812

119896119886

(9)

where119877119890and1198771015840
















119894



6and 1198687


74and 119877








119894is a



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800

01

02

03

04

05

06

07

08

09

1

Node i

IiRi

I iR

i












119894denotes the


119894le 119873 minus 1 We




119862119865lowast=

sum119894isin119881119862119865119894

119873(119873 minus 1)

(10)








29


74 rarr 3806800 1 1 00843

30


74 rarr 3806800 09976 2 00841

4


28ndash30 5706133 09551 3 00805



69 33 56 75 rarr 66 67 68 rarr 56 01067 06573 5 00554














119871119895+ Δ119871119894119895lt 119862119895 120574 = 1

119871119895+ Δ119871119894119895minus 119862119895

120574119862119895minus 119862119895

lt ℎ 120574 gt 1

(11)


[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

120574 = 1

([

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

)

times((120574 minus 1) (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

)

minus1

lt ℎ

120574 gt 1

(12)

From (12) we get

1 +

[119896119894

120573+ (sum119899isin119867119894

119896119899)

120573

]

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt 1 + 120572 120574 = 1

1+[119896119894

120573+(sum119899isin119867119894

119896119899)

120573

] sum119899isin119867119894

[119896119899

120573+(sum119891isin119867119899

119896119891)

120573

]

ℎ120574 minus ℎ + 1

lt1 + 120572

120574 gt 1

(13)

sum

119899isin119867119894

119896119899

120573=

119870

sum

=119898

119896119894119875 (119896 | 119896119894)119896120573 (14)






sum

119875 (119896 | 119896119894) = 1

119896119894119875 (119896 | 119896119894) 119875 (119896

119894) =

119896119875 (119896119894|119896) 119875 (

119896)

(15)


sum

119899isin119867119894

119896119899

120573= 119896119894

119870

sum

=119898

119896119875 (

119896)119896120573

⟨119896⟩

= 119896119894

⟨119896120573+1⟩

⟨119896⟩

(16)


sum

119899isin119867119894

119896119899= 119896119894

⟨1198962⟩

⟨119896⟩

(17)


1 +

119896119894

120573minus1⟨119896⟩

119896120573+1

lt 1 + 120572 120574 = 1

1 + 119896119894

120573minus1⟨119896⟩ 119896

120573+1

ℎ120574 minus ℎ + 1

lt 1 + 120572 120574 gt 1

(18)




120572119888=

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 lt 1

⟨119896⟩

⟨1198962⟩

120573 = 1

119870120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 gt 1

(19)


120572119888=

1 + 119898120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 lt 1

1 + ⟨119896⟩ ⟨1198962⟩

ℎ120574 minus ℎ + 1

minus 1 120573 = 1

1 + 119870120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 gt 1

(20)


119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

=

⟨119896⟩

(1119873)sum119873

119894=11198962

119894(119896119894119898)120573minus1

gt

⟨119896⟩

(1119873)sum119873

119894=11198962

119894

=

⟨119896⟩

⟨1198962⟩

(21)


119888(120573 =


119888(120573 = 1)

















0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)



02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ119894 (119905) =

ℎ119894 (119905 minus 1) +

119875 (119871119894 (119905))

119879

119871119894 (119905) gt 119862119894

0 119871119894 (119905) le 119862119894

119894 = 1 2 119873

(6)








119877119894= 1 minus

1198731015840

119894

119873

(7)







1198812= 1198991 1198992 119899



1198981 1198982 119898





119877119890=

sum119886isin1198812

119896119886

119873(119873 minus 1) 2

(8)






119868119894= 1 minus

1198771015840

119890

119877119890

= 1 minus

sum119887isin1198812cap119881

1015840

119894

119896119887

sum119886isin1198812

119896119886

(9)

where119877119890and1198771015840
















119894



6and 1198687


74and 119877








119894is a



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800

01

02

03

04

05

06

07

08

09

1

Node i

IiRi

I iR

i












119894denotes the


119894le 119873 minus 1 We




119862119865lowast=

sum119894isin119881119862119865119894

119873(119873 minus 1)

(10)








29


74 rarr 3806800 1 1 00843

30


74 rarr 3806800 09976 2 00841

4


28ndash30 5706133 09551 3 00805



69 33 56 75 rarr 66 67 68 rarr 56 01067 06573 5 00554














119871119895+ Δ119871119894119895lt 119862119895 120574 = 1

119871119895+ Δ119871119894119895minus 119862119895

120574119862119895minus 119862119895

lt ℎ 120574 gt 1

(11)


[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

120574 = 1

([

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

)

times((120574 minus 1) (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

)

minus1

lt ℎ

120574 gt 1

(12)

From (12) we get

1 +

[119896119894

120573+ (sum119899isin119867119894

119896119899)

120573

]

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt 1 + 120572 120574 = 1

1+[119896119894

120573+(sum119899isin119867119894

119896119899)

120573

] sum119899isin119867119894

[119896119899

120573+(sum119891isin119867119899

119896119891)

120573

]

ℎ120574 minus ℎ + 1

lt1 + 120572

120574 gt 1

(13)

sum

119899isin119867119894

119896119899

120573=

119870

sum

=119898

119896119894119875 (119896 | 119896119894)119896120573 (14)






sum

119875 (119896 | 119896119894) = 1

119896119894119875 (119896 | 119896119894) 119875 (119896

119894) =

119896119875 (119896119894|119896) 119875 (

119896)

(15)


sum

119899isin119867119894

119896119899

120573= 119896119894

119870

sum

=119898

119896119875 (

119896)119896120573

⟨119896⟩

= 119896119894

⟨119896120573+1⟩

⟨119896⟩

(16)


sum

119899isin119867119894

119896119899= 119896119894

⟨1198962⟩

⟨119896⟩

(17)


1 +

119896119894

120573minus1⟨119896⟩

119896120573+1

lt 1 + 120572 120574 = 1

1 + 119896119894

120573minus1⟨119896⟩ 119896

120573+1

ℎ120574 minus ℎ + 1

lt 1 + 120572 120574 gt 1

(18)




120572119888=

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 lt 1

⟨119896⟩

⟨1198962⟩

120573 = 1

119870120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 gt 1

(19)


120572119888=

1 + 119898120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 lt 1

1 + ⟨119896⟩ ⟨1198962⟩

ℎ120574 minus ℎ + 1

minus 1 120573 = 1

1 + 119870120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 gt 1

(20)


119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

=

⟨119896⟩

(1119873)sum119873

119894=11198962

119894(119896119894119898)120573minus1

gt

⟨119896⟩

(1119873)sum119873

119894=11198962

119894

=

⟨119896⟩

⟨1198962⟩

(21)


119888(120573 =


119888(120573 = 1)

















0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)



02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800

01

02

03

04

05

06

07

08

09

1

Node i

IiRi

I iR

i












119894denotes the


119894le 119873 minus 1 We




119862119865lowast=

sum119894isin119881119862119865119894

119873(119873 minus 1)

(10)








29


74 rarr 3806800 1 1 00843

30


74 rarr 3806800 09976 2 00841

4


28ndash30 5706133 09551 3 00805



69 33 56 75 rarr 66 67 68 rarr 56 01067 06573 5 00554














119871119895+ Δ119871119894119895lt 119862119895 120574 = 1

119871119895+ Δ119871119894119895minus 119862119895

120574119862119895minus 119862119895

lt ℎ 120574 gt 1

(11)


[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

120574 = 1

([

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

)

times((120574 minus 1) (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

)

minus1

lt ℎ

120574 gt 1

(12)

From (12) we get

1 +

[119896119894

120573+ (sum119899isin119867119894

119896119899)

120573

]

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt 1 + 120572 120574 = 1

1+[119896119894

120573+(sum119899isin119867119894

119896119899)

120573

] sum119899isin119867119894

[119896119899

120573+(sum119891isin119867119899

119896119891)

120573

]

ℎ120574 minus ℎ + 1

lt1 + 120572

120574 gt 1

(13)

sum

119899isin119867119894

119896119899

120573=

119870

sum

=119898

119896119894119875 (119896 | 119896119894)119896120573 (14)






sum

119875 (119896 | 119896119894) = 1

119896119894119875 (119896 | 119896119894) 119875 (119896

119894) =

119896119875 (119896119894|119896) 119875 (

119896)

(15)


sum

119899isin119867119894

119896119899

120573= 119896119894

119870

sum

=119898

119896119875 (

119896)119896120573

⟨119896⟩

= 119896119894

⟨119896120573+1⟩

⟨119896⟩

(16)


sum

119899isin119867119894

119896119899= 119896119894

⟨1198962⟩

⟨119896⟩

(17)


1 +

119896119894

120573minus1⟨119896⟩

119896120573+1

lt 1 + 120572 120574 = 1

1 + 119896119894

120573minus1⟨119896⟩ 119896

120573+1

ℎ120574 minus ℎ + 1

lt 1 + 120572 120574 gt 1

(18)




120572119888=

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 lt 1

⟨119896⟩

⟨1198962⟩

120573 = 1

119870120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 gt 1

(19)


120572119888=

1 + 119898120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 lt 1

1 + ⟨119896⟩ ⟨1198962⟩

ℎ120574 minus ℎ + 1

minus 1 120573 = 1

1 + 119870120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 gt 1

(20)


119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

=

⟨119896⟩

(1119873)sum119873

119894=11198962

119894(119896119894119898)120573minus1

gt

⟨119896⟩

(1119873)sum119873

119894=11198962

119894

=

⟨119896⟩

⟨1198962⟩

(21)


119888(120573 =


119888(120573 = 1)

















0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)



02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











29


74 rarr 3806800 1 1 00843

30


74 rarr 3806800 09976 2 00841

4


28ndash30 5706133 09551 3 00805



69 33 56 75 rarr 66 67 68 rarr 56 01067 06573 5 00554














119871119895+ Δ119871119894119895lt 119862119895 120574 = 1

119871119895+ Δ119871119894119895minus 119862119895

120574119862119895minus 119862119895

lt ℎ 120574 gt 1

(11)


[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

120574 = 1

([

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

+[

[

119896119894

120573+ (sum

119899isin119867119894

119896119899)

120573

]

]

times

119896119895

120573+ (sum119898isin119867119895

119896119898)

120573

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

)

times((120574 minus 1) (1 + 120572)[

[

119896119895

120573+ ( sum

119898isin119867119895

119896119898)

120573

]

]

)

minus1

lt ℎ

120574 gt 1

(12)

From (12) we get

1 +

[119896119894

120573+ (sum119899isin119867119894

119896119899)

120573

]

sum119899isin119867119894

[119896119899

120573+ (sum119891isin119867119899

119896119891)

120573

]

lt 1 + 120572 120574 = 1

1+[119896119894

120573+(sum119899isin119867119894

119896119899)

120573

] sum119899isin119867119894

[119896119899

120573+(sum119891isin119867119899

119896119891)

120573

]

ℎ120574 minus ℎ + 1

lt1 + 120572

120574 gt 1

(13)

sum

119899isin119867119894

119896119899

120573=

119870

sum

=119898

119896119894119875 (119896 | 119896119894)119896120573 (14)






sum

119875 (119896 | 119896119894) = 1

119896119894119875 (119896 | 119896119894) 119875 (119896

119894) =

119896119875 (119896119894|119896) 119875 (

119896)

(15)


sum

119899isin119867119894

119896119899

120573= 119896119894

119870

sum

=119898

119896119875 (

119896)119896120573

⟨119896⟩

= 119896119894

⟨119896120573+1⟩

⟨119896⟩

(16)


sum

119899isin119867119894

119896119899= 119896119894

⟨1198962⟩

⟨119896⟩

(17)


1 +

119896119894

120573minus1⟨119896⟩

119896120573+1

lt 1 + 120572 120574 = 1

1 + 119896119894

120573minus1⟨119896⟩ 119896

120573+1

ℎ120574 minus ℎ + 1

lt 1 + 120572 120574 gt 1

(18)




120572119888=

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 lt 1

⟨119896⟩

⟨1198962⟩

120573 = 1

119870120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 gt 1

(19)


120572119888=

1 + 119898120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 lt 1

1 + ⟨119896⟩ ⟨1198962⟩

ℎ120574 minus ℎ + 1

minus 1 120573 = 1

1 + 119870120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 gt 1

(20)


119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

=

⟨119896⟩

(1119873)sum119873

119894=11198962

119894(119896119894119898)120573minus1

gt

⟨119896⟩

(1119873)sum119873

119894=11198962

119894

=

⟨119896⟩

⟨1198962⟩

(21)


119888(120573 =


119888(120573 = 1)

















0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)



02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sum

119875 (119896 | 119896119894) = 1

119896119894119875 (119896 | 119896119894) 119875 (119896

119894) =

119896119875 (119896119894|119896) 119875 (

119896)

(15)


sum

119899isin119867119894

119896119899

120573= 119896119894

119870

sum

=119898

119896119875 (

119896)119896120573

⟨119896⟩

= 119896119894

⟨119896120573+1⟩

⟨119896⟩

(16)


sum

119899isin119867119894

119896119899= 119896119894

⟨1198962⟩

⟨119896⟩

(17)


1 +

119896119894

120573minus1⟨119896⟩

119896120573+1

lt 1 + 120572 120574 = 1

1 + 119896119894

120573minus1⟨119896⟩ 119896

120573+1

ℎ120574 minus ℎ + 1

lt 1 + 120572 120574 gt 1

(18)




120572119888=

119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 lt 1

⟨119896⟩

⟨1198962⟩

120573 = 1

119870120573minus1

⟨119896⟩

⟨119896120573+1⟩

120573 gt 1

(19)


120572119888=

1 + 119898120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 lt 1

1 + ⟨119896⟩ ⟨1198962⟩

ℎ120574 minus ℎ + 1

minus 1 120573 = 1

1 + 119870120573minus1

⟨119896⟩ ⟨119896120573+1⟩

ℎ120574 minus ℎ + 1

minus 1 120573 gt 1

(20)


119898120573minus1

⟨119896⟩

⟨119896120573+1⟩

=

⟨119896⟩

(1119873)sum119873

119894=11198962

119894(119896119894119898)120573minus1

gt

⟨119896⟩

(1119873)sum119873

119894=11198962

119894

=

⟨119896⟩

⟨1198962⟩

(21)


119888(120573 =


119888(120573 = 1)

















0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)



02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 01

120573 = 02

120573 = 03

120573 = 04

120573 = 05

120573 = 06

120573 = 07

120573 = 08

120573 = 09

120573 = 10

120574 = 10

CFlowast

120572

(a)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 11

CFlowast

120572

(b)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 13

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(c)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 15CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(d)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574 = 17

CFlowast

120572

(e)

0 01 02 03 04 05 06 07 08 09 10

0102030405060708091

120574 = 19

CFlowast

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120572

(f)



02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 04 06 08 1 12 14 16 18 20

005

01

015

02

025

03

035

04

045

05

120573

120572c

120574 = 10

120574 = 11

120574 = 13

120574 = 15

120574 = 17

120574 = 19




5 Conclusions


09 1 11 12 13 14 15 16 17 18 19 20

005010150202503035040450505506

120573 = 02

120573 = 04

120573 = 06

120573 = 08

120573 = 10

120573 = 12

120573 = 14

120573 = 16

120573 = 18

120573 = 20

120574

120572c


parameter 120574



119888has a



Acknowledgment



References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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