research article p2p lending risk contagion analysis based

Research ArticleP2P Lending Risk Contagion Analysis Based ona Complex Network Model

Qi Wei and Qiang Zhang

College of Finance and Statistics Hunan University Changsha Hunan 410082 China

Correspondence should be addressed to Qi Wei jr20071810529163com

Received 22 March 2016 Revised 2 June 2016 Accepted 19 June 2016

Academic Editor Guoqiang Hu

Copyright copy 2016 Q Wei and Q ZhangThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper analyzes two major channels of P2P lending risk contagion in Chinamdashdirect risk contagion between platforms andindirect risk contagion with other financial organizations as the contagion medium Based on this analysis the current studyconstructs a complex networkmodel of P2P lending risk contagion in China and performs dynamics analogue simulations in orderto analyze general characteristics of direct risk contagion among Chinarsquos online P2P lending platformsThe assumed conditions arethat other financial organizations act as the contagion medium with variations in the risk contagion characteristics set under thecondition of significant information asymmetry in Internet lending It is indicated that compared to direct risk contagion amongplatforms both financial organizations acting as the contagion medium and information asymmetry magnify the effect of riskcontagion It is also found that the superposition of media effects and information asymmetry is more likely to magnify the riskcontagion effect

1 Introduction

Based on Internet technology P2P lending realizes infor-mation exchange resource sharing and capital flow amongindividuals Many people who cannot obtain formal financ-ing from banks or other financial institutions seek to meettheir financing needs by network lending Since 2005 whenthe worldrsquos first P2P lending platform Zopa was founded inthe UK the new lending model has rapidly gained marketrecognition As of January 2016 there are 3917 P2P lendingplatforms in China with the cumulative turnover of 1495615billion yuan (statistical data came from Online LendingHouse) However together with this boom in the P2P lendingmarket certain risks increased as well most notably in theformof problematic platforms therewere 1351 such identifiedplatforms in January 2016

Apart from the fact that Chinarsquos P2P lending infras-tructure lacks necessary supervisory authorities is governedby imperfect laws and regulations contains outdated riskmanagement techniques and thus exposes P2P lendingplatforms to risks networking lending platformsrsquo risks areby themselves strong in transitive and spillover due to

cross-business and consolidated operations Given that manysmall-scale institutions with similar businesses may get introuble at the same time or take similar behavior to theimpact this may trigger a collective importance initiatingsystemic risks [1] For this reason the current study onlydiscusses circumstances of risk contagion amongP2P lendingplatforms and other financial institutions

Currently there exist few studies on network lendingrisk contagion effects Most scholars believe that due tothe lack of effective regulations and legal status definitionsof P2P lending platforms there exist widespread creditrisks moral risks operational risks and liquidity risksamong others [2 3] Some of these studies emphasize theimportance of credit systems for risk prevention advisingthe credit system closely to protect investorsrsquo interests andthus effectively control credit risks [4 5] For exampleKai argues that the development of P2P lending platformsengenders a significant challenge to the commercial banksrsquotraditional mode of operation and ultimately affects thewider financial environment and has a significant impact oncommercial banksrsquo existing financial platforms and channels[6]

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 5013954 8 pageshttpdxdoiorg10115520165013954

2 Discrete Dynamics in Nature and Society

Given continuous development of complex network tech-nologies they can arguably be applied to analyze the degreeof risk contagion in financial systems Allen and Gale foundthat the contagion of financial risks relies on the networkedstructure of a financial system that is the interrelatedrelationships between internal actors in the financial system[7] Cassar andDuffy discovered that risks spread faster whenbanks are globally rather than locally connected but alsowhen the problem of liquidity shortage is not outstanding[8] Iori and Jafarey found that when the initial scale of banksis homogeneous with the market concentration decreasingthe scale of bank failures tends to decrease [9] Aleksiejukand Hołyst argued that when banks are in a two-dimensionalor four-dimensional network the duration of their failuresand degree of risk contagion present exponential decay andpower-law distribution respectively [10] Based on a randomnetwork study Georg and Poschmann found that the generalimpact of the risk contagion effect is not minor but is aserious threat to system stability [11] Li found that whileconnectivity among banks reduces the risk contagion effect itcan also lead to liquidity problems [12] Jianxin et al analyzedfactors affecting the speed with which banksrsquo vulnerabilitycontagion spreads and found that under the same contagionrate and extinction rate a single point of contagion reachesa stabilized status slower than three points of contagion[13] Jianmin and Tingqiangrsquos study found that the morepopular the social network the higher the possibilities ofcredit risk contagion conversely as the heterogeneity of alending network increases the credit risk contagion ratedecreases Gang et al built a dynamical model for sup-ply chain network contagion risks finding that as supplychain network membersrsquo nodes contagion threshold valueincreases the average degree lowers and the reduction of thecontagion risk node density becomes more clear [14] Whileexisting literature focuses on the complex network theory andfinancial risk contagion studies regarding network financialrisk contagion especially with regard to P2P lending riskcontagion are relatively rare In addition there is currently norisk contagion networkmodel incorporating P2P lending risktransmission characteristics and descriptions of the dynamicprocess at play

In the P2P lending market the relationships betweenlending platforms and financing institutions may be seen as atype of social relation If risk contagion is investigated only bymeans of data collection and statistical surveys thismay yielda significantly erroneous impressionMoreover available datathat reflects the actual development and transactions inChinarsquos P2P lending market are quite limited in particulardata released from the platforms themselves is fairly scarceAcquisition of real and reliable data is a difficult job Inresponse to these challenges the current study aims to build acomplex network model of P2P lending risk contagion ana-lyze the process of risk contagion and reveal the mechanismof risk contagion These findings may hold significance forfurther improving Chinarsquos P2P lending supervision systemand perfecting its risk management system By combiningthe characteristics of P2P lending risk contagion this studyanalyzes general features of the risk contagion of P2P lendingplatforms and based on this it then considers changes that

may occur in the dynamics of risk contagion when otherfinancial institutions such as banks small loan companiessecurity agencies group company or securities institutionsact as vectors and when asymmetric information exists

2 Description of P2P Lending Risk Contagion

In terms of P2P lending risk contagion the present studymainly analyzes it from twoperspectives First there is a focuson risk contagion among P2P lending platforms Chinarsquos P2Plending industry is focused on providing services to the tailmarket which is not covered due to lack of either capacityor willingness by traditional financial institutions Investorsand borrowers serviced by the P2P lending industry are actorsin the long chain of financial service groups whose financialknowledge risk awareness and risk tolerance are relativelyinsufficient and thus they are prone to make individual andgroup irrational decisions Once one P2P lending platformexposes a risk the long chain tends to panic and quicklyspread the fear This then leads to the spread of risks to otherrelated or even unrelated P2P lending platforms

Second risk contagion mainly relies on financial institu-tions which have assets-based relationships with P2P lendingplatforms [15] Asset relationships are defined as relationshipsformed between P2P lending platforms and other financialinstitutions by means of associated debts credit guaranteescredit derivatives and so forth In China these financialinstitutions mainly include banks small loan companiessecurity agencies corporate groups and securities institu-tions [16] While one P2P lending platform might not bedirectly related to another both platforms may have anasset-based relationship with the same financial institutionTherefore risks exposed in the first P2P lending platformcan spread to the second one through the common financialinstitution This leads to the process of risk contagion

The present paper focuses on the current situation inChinarsquos P2P lending market According to data on P2P lend-ing platforms issued by theOnline LendingHouse therewerea total of 3917 identified P2P lending platforms in China as ofJanuary 2016 of which 1351 platforms were recorded as thoseoperating improperly This study gathered and organizedinformation on existing P2P lending platforms includingtheir location background subject types interest rate levelsmain business types and guarantee- and trusteeship-relatedfinancing institutions Risk contagion was then primarilyinvestigated via two channels and the financial institutionsthat have business relationships with P2P lending platformswere adopted as node sets In the first channel if platformA and platform B are in the same region and have identicalsubject types almost the same interest rate levels and mainbusiness types it can be said that investors in that networklending market may make poor evaluations or have lowexpectations of platform B once a risk appears in platformA leading to an increase in risks of platform B that is riskcontagion does occur In this case there exists an edge con-necting platformA to platform B In the second channel dataprocessing revealed existence of 275 financing institutionsthat have business relationships with existing P2P lendingplatforms that is platform A shares an edge with financing

Discrete Dynamics in Nature and Society 3

(a) Risk contagion topology of network lending

middot middot middot






(b) Network hierarchy diagram of riskcontagion of network lending

Figure 1 Risk contagion topology of network lending

institution C which in turn has a business relationship withplatform A Accordingly a complex network consisting of4192 nodes and 39982 edges was constructed as shown inFigure 1

The topology consists of two different types of nodesThefirst type is a P2P loan platform indicated by I (markedin yellow in Figure 1(a)) which we define as individualsinitially exposed to a risk The second type refers to theother financial institutions represented in the figure by

(marked in green in Figure 1(a)) which we define as anintermediate vector As we can see in the figure there is adirect risk of contagion among the network loan platformsAt the same time platforms can also spread risks to theiraffiliated financial institutions in which case these financialinstitutions may then spread the risk to other network loanplatforms Here for the purpose of simplification we assumethat there is no significant risk contagion among financialinstitutions From calculations the average structural degreeof a P2P lending market is 953 the focusing coefficient is0142 and the scaling parameter is 28433 Figure 2 showsthe structural degree distribution of the network lendingmarket in which we can observe that the structural degreedistribution of this market in China follows the power-lawdistribution Due to substantial development of Chinarsquos P2Plending market the number of P2P lending platforms in theP2P lendingmarket indeed increases continuously andnewlycreated P2P lending platforms also choose existing larger P2P

P(k

)

k

101

100

10minus1

10minus2

10minus3

10minus4

100 101 102 103

Figure 2Distribution of relation degrees in the P2P lendingmarket

lending platforms and financial institutions to build businessrelations to improve their credibility level and further attractmore investorsThe above two characteristics also correspondto two important properties (new platforms and preferentialselection) of the scale-free network Accordingly this paperassumes that the network lending market structure can beapproximately regarded as a scale-free network

At the same time the focusing coefficient of the networktopology of the P2P lending market is smaller than thatof the traditional bank network topology which showsthat the traditional bank network topology possesses betternodes connectivity than the P2P lending market It is alsoappropriate to the actual situation of Chinarsquos P2P lendingmarket at the early development stage Since Chinarsquos P2Plending market started recently and is currently still at anearly stage disclosed information and market databases arerelatively insufficient According to the database used in thispaper it can be found that the structural degree distributionof Chinarsquos P2P lending market corresponds to the power-lawdistribution which shows that the connectivity ofmost nodesin the market is relatively low However there exist a fewnodes with very high node degrees that are very crucial tothe network connectivity and that are also key nodes in therisk contagion process of network lending such as powerfulnetwork lending platforms and large financial institutions

From these findings it can be argued that there existtwo key differences between the contagion process of P2Plending platforms and traditional financial institutions Firstrisk contagion occurs not only amongnetwork loan platformsbut also through other financial institutions Essentiallymoreintermediate vectors are added to the contagion process Sec-ond in contrast to traditional financial institutions the super-vision of network lending companies is relatively weak theirinternal control and information disclosure mechanisms arerelatively imperfect and there is a problem with informationasymmetry Therefore there is a time lag between internalrisk perception and external risk exposure This lag thencreates further difficulties for financial institutions and other


P2P lending platforms when dealing with risks In this paperpriority attention is paid to these two features the riskintermediate vector and time lag in risk exposure whendiscussing risk contagion among P2P lending platforms inChina

3 Complex Network-Based Internet LendingRisk Contagion Model

31 Model Setting In this paper we denote an Internetlending platform in a financial system as 119882 and we denoteInternet lending platforms number as 119873 We denote theldquootherrdquo category of financial institutions that have businesscontacts with Internet lending platforms in the financialsystem as 119865 regarded as the default propagation mediumin the whole risk contagion process and we denote theirnumber as 119872 Regarding the traditional virus contagionmodel both Internet lending platform 119882 and the ldquootherrdquocategory of financial institutions 119865 in the financial systemcan only be in one of the following two states tangiblestate 119878 representing that while there is no risk exposure onan Internet lending platform or financial institution thereis possibility of being contaminated and contagion state 119868representing that there has been risk exposure on an Internetlending platform or financial institution and that there is alsopossibility of contagion to other Internet lending platformsor financial institutions It is assumed that initially there is norisk exposure for financial institutions 119865

We assume that there is a direct risk of contagion amongInternet lending platforms owing to investorsrsquo psychologicalanticipations at time 119905 if there exists a relatively close investorrelationship between a non-risk exposure Internet lendingplatform 119882 and another Internet lending platform withrisk exposure in the system the non-risk exposure Internetlending platform 119882 changes from the tangible state to thecontagion state with the infection rate of 120574 at time 119905 + 1

[17] What should be noted here is the following (1) theInternet lending platform 119882 contaminated at time 119905 willalways maintain the state of risk exposure until 119905 + 1 + 119879 witheffective internal and external emergency troubleshootingthe Internet lending platform 119882 will recover to the non-risk exposure state with probability 120575 where the effectivespreading rate is defined as 120582 = 120574120575 (2) If there exists abusiness relationship between the financial institution 119865 andInternet lending platform 119882 which is in the state of riskexposure at time 119905 the financial institution 119865 which is in thenon-risk exposure state will be contaminated at time 119905 + 1

with probability 1205741 falling into the risk exposure state

To simplify analysis it is assumed that the numberof nodes in the whole network remains unchanged afterinitial growth that is the current number of nodes remainsunchanged meaning that the networkrsquos topological structurealso remains unchanged Subsequently the dynamic spread-ing process of risk contagion in the P2P lending market isinvestigated based on the mean-field theory

32 Spreading Behavior Dynamics of Internet Lending RiskThedegrees of nodes of a scale-free network are different that

is the numbers of other Internet lending platforms and finan-cial institutions having risk or asset correlation relationshipswith a given Internet lending platform in the system differwhich is in accordance with the current situation in ChinarsquosInternet lending market We denote the relative density ofthe Internet lending platform 119882 with risk exposure havingdegree 119896 as 120588

119896(119905) with the steady state expressed as 120588

119896 The

contagion of risks between the Internet lending platform 119882

and the ldquootherrdquo category of financial institutions 119865 can beregarded as uniform that is determined by the contagionrates 120574 and 120574

1 We then obtain the dynamics mean-field

reaction equation of the above system as follows [18]

120597119905120588119896

(119905) = minus120588119896119879

(119905) + 120582119896 [1 minus 120588119896

(119905)] 120579 (120588 (119905))

+ 1205741

[1 minus 120588119896

(119905)] 120599 (119905)

120597119905120599119896

(119905) = minus120599 (119905) + 120574 [1 minus 120599 (119905)] 120579 (120588 (119905))

(1)

where 120588119896119879

(119905) is the relative density of risk contagion of theInternet lending platform119882with degree 119896 at 119905minus119879 and 120579(120588(119905))

is the probability of connection between an arbitrary givenedge of the whole network and the Internet lending platformwith risk exposure with the steady state value of 120579 Whencalculating 120579(120588(119905)) it is necessary to take nonuniformityof the scale-free network into consideration as well as theprobability of arbitrarily selecting an edge pointing to thenode 119882 where the edge direction degree 119904 is proportional to119904119875(119904) that is the connection probability between a randomedge and a node having a larger degree is higher In additionthe degree of the Internet lending platformwith risk exposureis higher and thus the probability of risk exposure to otherInternet lending platforms and financial institutions causedby it is higher The expression for 120579(120588(119905)) is

120579 (120588 (119905)) = sum

119896

119896119875 (119896) 120588119896

sum 119904119875 (119904) (2)

where 119875(119896) is the degree distribution function of the Internetlending platform119882 in a scale-free networkThemean densityof the contaminated Internet lending platform 119882 in thenetwork structure can be expressed as follows

120588 (119905) = sum

119896

119875 (119896) 120588119896 (3)

Using 120597119905120588119896(119905) = 0 and 120597

119905120599(119905) = 0 and according to (1) we

obtain

minus120588119896119879

+ 120582119896 (1 minus 120588119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

minus120599 + 120574 (1 minus 120599) 120579 = 0

(4)

According to (4) when the network structure (1) is in asteady state the relative steady state density 120588

119896of the Internet

lending platform 119882 with risk exposure due to contaminationis a function of 120582 120574 120574

1 and 119879 and 120579 is an implicit function

of 120582 120574 1205741 and 119879


According to the structure characteristics of the scale-freenetwork substituting 120588

119896119879= 120588119896(119879 + 1) into (4) obtains

minus120588119896

119879 + 1+ 120582119896 (1 minus 120588

119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

120599 =120574120579

1 + 120574120579

(5)

According to (5) the relative steady state density 120588119896of

the Internet lending platform 119882 with risk exposure havingdegree 119896 can be derived as

120588119896

=(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(6)

Using ⟨119896⟩ = sum119904

119904119875(119904) and (2) it can be seen that

120579 = sum

119896

119896119875 (119896) 120588119896

sum119904

119904119875 (119904)

=1

⟨119896⟩sum

119896

119896119875 (119896)(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(7)

In order to make risks infectious across the whole network(7) needs to satisfy 120579 = 0 that is the following formula mustbe satisfied

119889

119889120579(

1

⟨119896⟩sum

119896

119896119875 (119896)

sdot(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

)

1003816100381610038161003816100381610038161003816100381610038161003816120579=0

ge 1

(8)

By further simplifying this the spreading critical value 120582119888of

the scale-free network can be obtained as

120582119888

=(1 minus 120574120574

1) ⟨119896⟩

(119879 + 1) ⟨1198962⟩ (9)

In (9) ⟨1198962⟩ = sum

1198961198962119875(119896) When 119879 = 120574 = 120574

1= 0 that is

under the condition of no spreading delay and a spreadingmedium it can be seen that 120582

119888= ⟨119896⟩⟨119896

2⟩ Therefore

under the condition of a spreading delay and in the presenceof a spreading medium the spreading clinical value of thescale-free network is significantly reduced That is to saywhen there is a lack of information transparency provided byan Internet lending platform and business-related financialinstitution the risk contagion probability of the Internetlending platform may increase

In order to make the network model more in accordancewith practice a scale-free network of a limited scale wastaken into consideration [19] In the scale-free network witha limited quantity of Internet lending platforms the degreeof the Internet lending platform 119882 is distributed as 119875(119896) =

21198982119896minus3 where 119898 is the minimum value of other Internet

lending platforms or financial institutions connected to eachInternet lending platform 119882 in the network From this it canbe derived that ⟨119896⟩ = int

infin

119898119896119875(119896)119889119896 = 2119898 By introducing

the maximum connection degree 119870119888of the limited scale-

free network when 119870119888

rarr infin ⟨1198962⟩ asymp int

119870119888

1198981198962119875(119896)119889119896 =

21198982 ln (119870

119888119898) According to this the risk contagion clinical

value in the limited scale-free network can be calculated as

120582119888

asymp(1 minus 120574120574

1)

(119879 + 1) 119898 ln (119870119888119898)

(10)

According to existing research results on scale-free networksit is apparent that 119870

119888is determined by the total quantity

of Internet lending platforms 119882 in the limited scale-freenetwork that is 119870

119888asymp 119898119873

12 By substituting 119870119888

asymp 11989811987312

into (10) the following expression can be acquired

120582119888

asymp2 (1 minus 120574120574

1)

(119879 + 1) 119898 ln119873 (11)

According to (11) the risk contagion critical value 120582119888of the

whole limited scale-free network is related to 119879 120574 and 1205741

In the limited scale-free network when only the spreadingdelay state is taken into consideration the contagion criticalvalue is 120582

119888= 2((119879 + 1)119898 ln119873) and if only the existence

of a contagion medium is taken into consideration the riskcontagion critical value is 120582

119888= 2(1 minus 120574120574

1)(119898 ln119873) It can

further be seen thatwhenboth contagiondelay and contagionmedium are taken into consideration the risk contagionprobability of the limited scale-free network increases

Further when 120582 gt 120582119888 substituting 119875(119896) and ⟨119896⟩ into (7)

obtains

1

119898

= int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198962 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

= 119860 ln1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579)+

119861

119898

(12)

In (12)

119860 =(119879 + 1) 120582 (1 + 120574120579)

2

(1 + 120574120579 + 1205741205741120579)2

119861 =1205741205741

1 + 120574120579 + 1205741205741120579

119862 = minus(119879 + 1)

21205822120579 (1 + 120574120579)

3

(1 + 120574120579 + 1205741205741120579)2

(13)

Therefore a formula can be derived to express 120579

ln(1 +1 + 120574120579 + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579))

=(1 + 120574120579 + 120574120574

1120579) (1 + 120574120579 + 120574120574

1120579 minus 120574120574

1)

119898 (119879 + 1) 120582 (1 + 120574120579)2

(14)


It is clear that (14) has a unique solution Based on this wesubstitute 119875(119896) into (3) to obtain

120588 = sum

119870

119875 (119896) 120588119896

= sum

119896

21198982119896minus3

(119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

= 21198982120579 int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198963 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

(15)

By combining (12) and (15) we can obtain the density of theInternet lending platform 119882 when it is contaminated duringthe steady state and with risk exposure

120588 = 21198982120579 [

(119879 + 1) 120582 (1 + 120574120579) minus (119879 + 1) 120582119896 (1 + 120574120579)

119898 (1 + 120574120579 + 1205741205741120579)

+1205741205741

21198982 (1 + 120574120579 + 1205741205741120579)

] = 120579

sdot2119898 (119879 + 1) 120582 (1 + 120574120579) (1 minus 120579) + 120574120574

1

1 + 120574120579 + 1205741205741120579

(16)

According to (14) and (16) it is clear that in the presence ofa spreading delay and contagion medium in a limited scale-free network there are significant changes in the networksrsquorisk contagion dynamic behavior

4 Simulation Experiment

In order to further depict general characteristics of Internetlending risk contagion in China and take changes in therisk contagion effect under conditions of asymmetry of thecontagion medium and information into consideration weconducted visualization of the network structure in lightof the aforementioned analysis and results We set 119898 = 2in the limited scale-free network [20] Chinarsquos first networklending platform PPDAIcom appeared in 2007 followingwhich other P2P lending platforms appeared one by oneConsidering the current developmental conditions of Chinarsquosnetwork lending market this study found that when thenetwork model is constructed the initial number of nodesis four In addition the nodes of financing institutions werefound to be connected to the first network lending platformand therefore from a relatively conservative perspectiveit can be assumed that when adding a new node threecorresponding new edges are added

Figure 3 depicts under different risk spreading delayeffects and contagion probabilities changes in the contagiondensity 120588 alongside changes in the effective spreading rate 120582

in the limited scale-free networkIt can be seen from Figure 3 that under the condition of

taking different values of 119879 and 120574 (1205741) the system achieves

a relatively steady state This is basically in accordance withthe spreading situation of the basic risk contagion modelin a limited scale-free network however by comparing thiswith the contagion rate and contagion density under different119879 and 120574 (120574

1) values it becomes evident that if the risk

0 002 004 006 008 01 0120

001

002

003

004

005

006

007

008

009

01

T = 1 r1 = 02 r2 = 04T = 0 r1 = 02 r2 = 04

T = 1 r1 = 015 r2 = 02T = 0 r1 = 015 r2 = 02

120588

120582

Figure 3 Changes in the contagion density 120588 as a function ofchanges in the effective spreading rate 120582 in the limited scale-freenetwork

contagion delay is ignored that is 119879 = 0 and if thereexists a risk contagion medium in the network structure thatis there is the other category of financial institutions withbusiness relationships with the Internet lending platform therisk contagion probability in the whole network system isincreased which is in accordance with practical situationsWhen taking the risk contagion delay into consideration atthe same contagion medium state the risk contagion delayeffect increases the risk contagion probability in the wholenetworkThe practical outcome is that if one Internet lendingplatform 119882 is contaminated by risk given the asymmetry ofinformation the risk exposure condition cannot be acknowl-edged by investors and financial institutions in time Thisin turn leads to failure of countermeasures for risk isolationby related investors and financial institutions thus increasinginvestorsrsquo and the ldquootherrdquo category financial institutionsrsquodifficulties with handling risks and reducing their ability toeliminate risks Ultimately this increases the probability ofrisk contagion across the whole system

Figure 4 depicts changes in the risk contagion density 120588

alongside changes in the risk contagion probability 120574 (1205741) in

the limited scale-free network under different values of therisk contagion delay and effective contagion rate

Figure 4 shows that when there exists a risk spreadingdelay effect and a contagion medium at the same time andwhen the risk contagion rate is 120582 = 001 then there exists apower-law relationship between the risk contagion density 120588

and 120574 (1205741) of the Internet lending platform 119882 in the steady

state Across the whole network structure the existence ofa risk contagion medium increases the probability of riskcontagion of the network systemwhich also increaseswith anincrease in the probability of risk contagion 120574 among differentInternet lending platforms as well as 120574

1among Internet

lending platforms and financial institutions In the state of theother same conditions a delay in risk contagion significantly


0

0002

0004

0006

0008

001

0012

0014

0016

120588

0 01 02 03 04 05 06 07 08 09

120574 (1205741)

120582 = 001 T = 3

120582 = 001 T = 2120582 = 001 T = 1

120582 = 001 T = 0

Figure 4 Changes in the contagion density 120588 with 120574 (1205741) in the

limited scale-free network

increases the probability of risk contagion in the systemComparing the risk contagion delay effect and contagionmedium the latter was found to exert more significantinfluence on the risk contagion probability of the system asa whole

Comparing the case when only either the risk contagiondelay or the contagion medium is taken into considerationwith that when both are taken into consideration the latterwas found to exert more significant influence on the proba-bility of risk contagion of the whole network system On onehand when both of these conditions exist at the same timethis increases the number of Internet lending platforms orfinancial institutions with risk exposure caused by one-timecontagion in the network On the other hand it also reducesthe coping capacity of Internet lending platforms or financialinstitutions when facing risks thus significantly increasingthe probability of risk contagion across the whole Internetlending system

5 Conclusions

This paper analyzes the network structure and characteristicsof risk infection in Chinarsquos online network lending systemsthrough establishment of a dynamic model of risk contagionAt the same time based on the characteristics of Chinarsquosnetwork lending market the paper analyzes the impact thatthe risk intermediate vector and contagion lag have onnetwork lending risk contagion as a whole The followingconclusions are reached

(1) With no risk contagion delay taken into accountrisk contagion media exist in the network That isthe existence of other financing institutions that havebusiness relationshipswith P2P lending platforms canincrease the danger of risk contagion in the wholenetwork In light of this it is recommended that

appropriate regulatory authorities establish risk con-tagion firewalls between financing institutions andP2P lending platforms rather than cut off businessrelationships between themowing to the existing pos-sibility of aggravating risk contagion In combinationwith the risk contagion monitoring system whenthe contagion probability and density exceed presetthreshold values leading financing institutions andP2P lending platforms should be isolated in a timelymanner to achieve the goals of early warning andisolation

(2) With the risk contagion delay taken into accountthe danger of risk contagion in the whole networkincreases under the same contagion media condi-tions Although the communication of informationseems to be adequate in the network lending processboth the truthfulness and actual transparency ofinformation provided are relatively low in Chinarsquoscurrent network lending process due to the countryrsquosincomplete credit system and banksrsquo undisclosedcredit information system Moreover informationasymmetry is also very severe and the lack of infor-mation transparency can directly lead to time lags inrisk exposure Since both P2P lending platforms andborrowers can hide the truth deliberately or uninten-tionally for other P2P lending platforms investorsand financing institutions involved it becomes moredifficult to respond when risk information is dis-closed Therefore regulatory authorities should con-struct and disseminate more detailed informationdisclosure guidelines to both borrowers andplatformsin the network lending process to abide by and thusenhance information transparency In other wordsthey should take appropriate measures not only toreduce the probability of risk in advance but alsoto allow other market participants to acquire riskinformation as early as possible and thus be able toundertake prevention measures and respond to risksin a timely manner

(3) Compared to the time lag effects of risk contagion theeffects of the vectorswere found to bemore significantin terms of the possibility of systematic risk contagion

(4) The combined impacts that the contagion lag andintermediate vectors have on the possibility of sys-tematic risk contagion are clearer than the impactcaused by either of them separately When bothfactors exist the number of platforms and otherfinancial institutions vulnerable to a single wave ofrisk contagion increases Simultaneously the copingability of those institutions is lowered and this largelyincreases the possibility of risk contagion across thewhole networked lending system

For the purposes of concision this paper has not dealtwith risk contagion among traditional financial institutionsalthough it is acknowledged that the possibility of riskcontagion in the network lending infrastructure is likely to befurther increased if the spread of risk among these traditional


institutions is also taken into consideration The details ofthis potential impact will be further analyzed in the followingstudies

Competing Interests

The authors declare that they have no competing interests

References

[1] Financial Research Institute of the Peoplersquos Bank of China NewFinancial Times Authoritative Interpretation of Internet Bank-ing CITIC Publishing Group Beijing China 2015 (Chinese)

[2] X Bo and P Zhuangchen ldquoResearch on the development statusand risk disclosure of P2P network lending model in ChinardquoFuture and Development no 6 pp 86ndash74 2014 (Chinese)

[3] L Lili ldquoDiscussion on the risk and supervision of P2P networklending in Chinardquo Credit Reference no 11 pp 29ndash32 2013(Chinese)

[4] C Xi and J Xingchen ldquoOn credit and credit risk control ofP2P net loan platformrdquo Financial Times no 6 pp 63ndash64 2014(Chinese)

[5] M Qingjun and H Minyu ldquoRisk and system analysis inthe development of P2P net loan platformrdquo Wuhan FinanceMonthly no 6 pp 45ndash48 2014 (Chinese)

[6] J Kai ldquoStudy on the influence of Internet banking on the tradi-tional financial modelrdquo Journal of Shandong Youth University ofPolitical Science no 4 pp 118ndash121 2014 (Chinese)

[7] F Allen and D Gale ldquoFinancial contagionrdquo Journal of PoliticalEconomy vol 108 no 1 pp 1ndash33 2000

[8] A Cassar and N Duffy ldquoContagion of financial crises underlocal and global networksrdquo in Agent-Based Methods in Eco-nomics and Finance Simulations F Luna and A Perrone EdsKluwer Academic Norwell Mass USA 2002

[9] G Iori and S Jafarey ldquoCriticality in a model of banking crisesrdquoPhysicaA StatisticalMechanics and Its Applications vol 299 no1-2 pp 205ndash212 2001

[10] A Aleksiejuk and J A Hołyst ldquoA simple model of bankbankruptciesrdquo Physica A Statistical Mechanics and Its Applica-tions vol 299 no 1-2 pp 198ndash204 2001

[11] C P Georg and J Poschmann ldquoSystemic risk in a networkmodel of interbank markets with central bank activityrdquo JenaEconomic Research Papers 2010-033 Jena Germany 2010

[12] S Li ldquoContagion risk in an evolving network model of bankingsystemsrdquo Advances in Complex Systems vol 14 no 5 pp 673ndash690 2011

[13] C Jianxin L Weiqi and P Sulin ldquoA set of population model ofbank risk contagion based on the dynamic model proposed byautomatonrdquo System Engineering Theory and Practice no 3 pp543ndash548 2013 (Chinese)

[14] Z Gang Y Yingbao and B Xu ldquoDynamic model for the riskspreading in supply chain network and its applicationrdquo SystemsEngineeringmdashTheory amp Practice vol 35 no 8 pp 2014ndash20242015 (Chinese)

[15] Z Qiang and W Qi ldquoAn empirical study on the Risk SpilloverEffect of Chinarsquos net lending platform to commercial banksrdquoChinese Review of Financial Studies no 3 2015 (Chinese)

[16] F Shulian L Zhiping and Z Peng ldquoResearch on the strategy oftraditional bank Internet Bankingrdquo Financial in North Chinano 10 pp 25ndash29 2014 (Chinese)

[17] J Guoping and W Yaqi ldquoSpreading of epidemics in complexnetworks with infective medium and spreading delayrdquo Acta-PhysicaSinica vol 59 no 10 pp 6725ndash6733 2010

[18] Z Yang and T Zhou ldquoEpidemic spreading in weighted net-works an edge-based mean-field solutionrdquo Physical Review Evol 85 Article ID 056106 2012

[19] X Chengyi M Junhai and C Zengqiang ldquoSIR epidemicmodel with infectionmedium on complex networksrdquo Journal ofSystems Engineering vol 25 no 6 pp 818ndash823 2010 (Chinese)

[20] W Jing W Yaqi and Y Haibin ldquoAn evolution model ofmicroblog user relationship networks based on complex net-work theoryrdquo ActaPhysicaSinica vol 63 no 20 pp 1ndash7 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

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


Given continuous development of complex network tech-nologies they can arguably be applied to analyze the degreeof risk contagion in financial systems Allen and Gale foundthat the contagion of financial risks relies on the networkedstructure of a financial system that is the interrelatedrelationships between internal actors in the financial system[7] Cassar andDuffy discovered that risks spread faster whenbanks are globally rather than locally connected but alsowhen the problem of liquidity shortage is not outstanding[8] Iori and Jafarey found that when the initial scale of banksis homogeneous with the market concentration decreasingthe scale of bank failures tends to decrease [9] Aleksiejukand Hołyst argued that when banks are in a two-dimensionalor four-dimensional network the duration of their failuresand degree of risk contagion present exponential decay andpower-law distribution respectively [10] Based on a randomnetwork study Georg and Poschmann found that the generalimpact of the risk contagion effect is not minor but is aserious threat to system stability [11] Li found that whileconnectivity among banks reduces the risk contagion effect itcan also lead to liquidity problems [12] Jianxin et al analyzedfactors affecting the speed with which banksrsquo vulnerabilitycontagion spreads and found that under the same contagionrate and extinction rate a single point of contagion reachesa stabilized status slower than three points of contagion[13] Jianmin and Tingqiangrsquos study found that the morepopular the social network the higher the possibilities ofcredit risk contagion conversely as the heterogeneity of alending network increases the credit risk contagion ratedecreases Gang et al built a dynamical model for sup-ply chain network contagion risks finding that as supplychain network membersrsquo nodes contagion threshold valueincreases the average degree lowers and the reduction of thecontagion risk node density becomes more clear [14] Whileexisting literature focuses on the complex network theory andfinancial risk contagion studies regarding network financialrisk contagion especially with regard to P2P lending riskcontagion are relatively rare In addition there is currently norisk contagion networkmodel incorporating P2P lending risktransmission characteristics and descriptions of the dynamicprocess at play

In the P2P lending market the relationships betweenlending platforms and financing institutions may be seen as atype of social relation If risk contagion is investigated only bymeans of data collection and statistical surveys thismay yielda significantly erroneous impressionMoreover available datathat reflects the actual development and transactions inChinarsquos P2P lending market are quite limited in particulardata released from the platforms themselves is fairly scarceAcquisition of real and reliable data is a difficult job Inresponse to these challenges the current study aims to build acomplex network model of P2P lending risk contagion ana-lyze the process of risk contagion and reveal the mechanismof risk contagion These findings may hold significance forfurther improving Chinarsquos P2P lending supervision systemand perfecting its risk management system By combiningthe characteristics of P2P lending risk contagion this studyanalyzes general features of the risk contagion of P2P lendingplatforms and based on this it then considers changes that

may occur in the dynamics of risk contagion when otherfinancial institutions such as banks small loan companiessecurity agencies group company or securities institutionsact as vectors and when asymmetric information exists

2 Description of P2P Lending Risk Contagion

In terms of P2P lending risk contagion the present studymainly analyzes it from twoperspectives First there is a focuson risk contagion among P2P lending platforms Chinarsquos P2Plending industry is focused on providing services to the tailmarket which is not covered due to lack of either capacityor willingness by traditional financial institutions Investorsand borrowers serviced by the P2P lending industry are actorsin the long chain of financial service groups whose financialknowledge risk awareness and risk tolerance are relativelyinsufficient and thus they are prone to make individual andgroup irrational decisions Once one P2P lending platformexposes a risk the long chain tends to panic and quicklyspread the fear This then leads to the spread of risks to otherrelated or even unrelated P2P lending platforms

Second risk contagion mainly relies on financial institu-tions which have assets-based relationships with P2P lendingplatforms [15] Asset relationships are defined as relationshipsformed between P2P lending platforms and other financialinstitutions by means of associated debts credit guaranteescredit derivatives and so forth In China these financialinstitutions mainly include banks small loan companiessecurity agencies corporate groups and securities institu-tions [16] While one P2P lending platform might not bedirectly related to another both platforms may have anasset-based relationship with the same financial institutionTherefore risks exposed in the first P2P lending platformcan spread to the second one through the common financialinstitution This leads to the process of risk contagion

The present paper focuses on the current situation inChinarsquos P2P lending market According to data on P2P lend-ing platforms issued by theOnline LendingHouse therewerea total of 3917 identified P2P lending platforms in China as ofJanuary 2016 of which 1351 platforms were recorded as thoseoperating improperly This study gathered and organizedinformation on existing P2P lending platforms includingtheir location background subject types interest rate levelsmain business types and guarantee- and trusteeship-relatedfinancing institutions Risk contagion was then primarilyinvestigated via two channels and the financial institutionsthat have business relationships with P2P lending platformswere adopted as node sets In the first channel if platformA and platform B are in the same region and have identicalsubject types almost the same interest rate levels and mainbusiness types it can be said that investors in that networklending market may make poor evaluations or have lowexpectations of platform B once a risk appears in platformA leading to an increase in risks of platform B that is riskcontagion does occur In this case there exists an edge con-necting platformA to platform B In the second channel dataprocessing revealed existence of 275 financing institutionsthat have business relationships with existing P2P lendingplatforms that is platform A shares an edge with financing














P(k

)

k

101

100

10minus1

10minus2

10minus3

10minus4

100 101 102 103
















119896 The





120597119905120588119896

(119905) = minus120588119896119879

(119905) + 120582119896 [1 minus 120588119896

(119905)] 120579 (120588 (119905))

+ 1205741

[1 minus 120588119896

(119905)] 120599 (119905)

120597119905120599119896

(119905) = minus120599 (119905) + 120574 [1 minus 120599 (119905)] 120579 (120588 (119905))

(1)

where 120588119896119879



120579 (120588 (119905)) = sum

119896

119896119875 (119896) 120588119896

sum 119904119875 (119904) (2)


120588 (119905) = sum

119896

119875 (119896) 120588119896 (3)

Using 120597119905120588119896(119905) = 0 and 120597


obtain

minus120588119896119879

+ 120582119896 (1 minus 120588119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

minus120599 + 120574 (1 minus 120599) 120579 = 0

(4)





of 120582 120574 1205741 and 119879



119896119879= 120588119896(119879 + 1) into (4) obtains

minus120588119896

119879 + 1+ 120582119896 (1 minus 120588

119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

120599 =120574120579

1 + 120574120579

(5)



120588119896

=(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(6)

Using ⟨119896⟩ = sum119904


120579 = sum

119896

119896119875 (119896) 120588119896

sum119904

119904119875 (119904)

=1

⟨119896⟩sum

119896

119896119875 (119896)(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(7)


119889

119889120579(

1

⟨119896⟩sum

119896

119896119875 (119896)

sdot(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

)

1003816100381610038161003816100381610038161003816100381610038161003816120579=0

ge 1

(8)



120582119888

=(1 minus 120574120574

1) ⟨119896⟩

(119879 + 1) ⟨1198962⟩ (9)

In (9) ⟨1198962⟩ = sum

1198961198962119875(119896) When 119879 = 120574 = 120574

1= 0 that is


119888= ⟨119896⟩⟨119896

2⟩ Therefore





infin

119898119896119875(119896)119889119896 = 2119898 By introducing




119870119888

1198981198962119875(119896)119889119896 =

21198982 ln (119870



120582119888

asymp(1 minus 120574120574

1)

(119879 + 1) 119898 ln (119870119888119898)

(10)




119888asymp 119898119873


asymp 11989811987312


120582119888

asymp2 (1 minus 120574120574

1)

(119879 + 1) 119898 ln119873 (11)






119888= 2(1 minus 120574120574

1)(119898 ln119873) It can



obtains

1

119898

= int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198962 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

= 119860 ln1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579)+

119861

119898

(12)

In (12)

119860 =(119879 + 1) 120582 (1 + 120574120579)

2

(1 + 120574120579 + 1205741205741120579)2

119861 =1205741205741

1 + 120574120579 + 1205741205741120579

119862 = minus(119879 + 1)

21205822120579 (1 + 120574120579)

3

(1 + 120574120579 + 1205741205741120579)2

(13)


ln(1 +1 + 120574120579 + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579))

=(1 + 120574120579 + 120574120574

1120579) (1 + 120574120579 + 120574120574

1120579 minus 120574120574

1)

119898 (119879 + 1) 120582 (1 + 120574120579)2

(14)



120588 = sum

119870

119875 (119896) 120588119896

= sum

119896

21198982119896minus3

(119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

= 21198982120579 int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198963 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

(15)


120588 = 21198982120579 [

(119879 + 1) 120582 (1 + 120574120579) minus (119879 + 1) 120582119896 (1 + 120574120579)

119898 (1 + 120574120579 + 1205741205741120579)

+1205741205741

21198982 (1 + 120574120579 + 1205741205741120579)

] = 120579

sdot2119898 (119879 + 1) 120582 (1 + 120574120579) (1 minus 120579) + 120574120574

1

1 + 120574120579 + 1205741205741120579

(16)









0 002 004 006 008 01 0120

001

002

003

004

005

006

007

008

009

01

T = 1 r1 = 02 r2 = 04T = 0 r1 = 02 r2 = 04

T = 1 r1 = 015 r2 = 02T = 0 r1 = 015 r2 = 02

120588

120582









1among Internet



0

0002

0004

0006

0008

001

0012

0014

0016

120588

0 01 02 03 04 05 06 07 08 09

120574 (1205741)

120582 = 001 T = 3

120582 = 001 T = 2120582 = 001 T = 1

120582 = 001 T = 0





5 Conclusions










Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















P(k

)

k

101

100

10minus1

10minus2

10minus3

10minus4

100 101 102 103
















119896 The





120597119905120588119896

(119905) = minus120588119896119879

(119905) + 120582119896 [1 minus 120588119896

(119905)] 120579 (120588 (119905))

+ 1205741

[1 minus 120588119896

(119905)] 120599 (119905)

120597119905120599119896

(119905) = minus120599 (119905) + 120574 [1 minus 120599 (119905)] 120579 (120588 (119905))

(1)

where 120588119896119879



120579 (120588 (119905)) = sum

119896

119896119875 (119896) 120588119896

sum 119904119875 (119904) (2)


120588 (119905) = sum

119896

119875 (119896) 120588119896 (3)

Using 120597119905120588119896(119905) = 0 and 120597


obtain

minus120588119896119879

+ 120582119896 (1 minus 120588119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

minus120599 + 120574 (1 minus 120599) 120579 = 0

(4)





of 120582 120574 1205741 and 119879



119896119879= 120588119896(119879 + 1) into (4) obtains

minus120588119896

119879 + 1+ 120582119896 (1 minus 120588

119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

120599 =120574120579

1 + 120574120579

(5)



120588119896

=(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(6)

Using ⟨119896⟩ = sum119904


120579 = sum

119896

119896119875 (119896) 120588119896

sum119904

119904119875 (119904)

=1

⟨119896⟩sum

119896

119896119875 (119896)(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(7)


119889

119889120579(

1

⟨119896⟩sum

119896

119896119875 (119896)

sdot(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

)

1003816100381610038161003816100381610038161003816100381610038161003816120579=0

ge 1

(8)



120582119888

=(1 minus 120574120574

1) ⟨119896⟩

(119879 + 1) ⟨1198962⟩ (9)

In (9) ⟨1198962⟩ = sum

1198961198962119875(119896) When 119879 = 120574 = 120574

1= 0 that is


119888= ⟨119896⟩⟨119896

2⟩ Therefore





infin

119898119896119875(119896)119889119896 = 2119898 By introducing




119870119888

1198981198962119875(119896)119889119896 =

21198982 ln (119870



120582119888

asymp(1 minus 120574120574

1)

(119879 + 1) 119898 ln (119870119888119898)

(10)




119888asymp 119898119873


asymp 11989811987312


120582119888

asymp2 (1 minus 120574120574

1)

(119879 + 1) 119898 ln119873 (11)






119888= 2(1 minus 120574120574

1)(119898 ln119873) It can



obtains

1

119898

= int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198962 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

= 119860 ln1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579)+

119861

119898

(12)

In (12)

119860 =(119879 + 1) 120582 (1 + 120574120579)

2

(1 + 120574120579 + 1205741205741120579)2

119861 =1205741205741

1 + 120574120579 + 1205741205741120579

119862 = minus(119879 + 1)

21205822120579 (1 + 120574120579)

3

(1 + 120574120579 + 1205741205741120579)2

(13)


ln(1 +1 + 120574120579 + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579))

=(1 + 120574120579 + 120574120574

1120579) (1 + 120574120579 + 120574120574

1120579 minus 120574120574

1)

119898 (119879 + 1) 120582 (1 + 120574120579)2

(14)



120588 = sum

119870

119875 (119896) 120588119896

= sum

119896

21198982119896minus3

(119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

= 21198982120579 int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198963 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

(15)


120588 = 21198982120579 [

(119879 + 1) 120582 (1 + 120574120579) minus (119879 + 1) 120582119896 (1 + 120574120579)

119898 (1 + 120574120579 + 1205741205741120579)

+1205741205741

21198982 (1 + 120574120579 + 1205741205741120579)

] = 120579

sdot2119898 (119879 + 1) 120582 (1 + 120574120579) (1 minus 120579) + 120574120574

1

1 + 120574120579 + 1205741205741120579

(16)









0 002 004 006 008 01 0120

001

002

003

004

005

006

007

008

009

01

T = 1 r1 = 02 r2 = 04T = 0 r1 = 02 r2 = 04

T = 1 r1 = 015 r2 = 02T = 0 r1 = 015 r2 = 02

120588

120582









1among Internet



0

0002

0004

0006

0008

001

0012

0014

0016

120588

0 01 02 03 04 05 06 07 08 09

120574 (1205741)

120582 = 001 T = 3

120582 = 001 T = 2120582 = 001 T = 1

120582 = 001 T = 0





5 Conclusions










Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















119896 The





120597119905120588119896

(119905) = minus120588119896119879

(119905) + 120582119896 [1 minus 120588119896

(119905)] 120579 (120588 (119905))

+ 1205741

[1 minus 120588119896

(119905)] 120599 (119905)

120597119905120599119896

(119905) = minus120599 (119905) + 120574 [1 minus 120599 (119905)] 120579 (120588 (119905))

(1)

where 120588119896119879



120579 (120588 (119905)) = sum

119896

119896119875 (119896) 120588119896

sum 119904119875 (119904) (2)


120588 (119905) = sum

119896

119875 (119896) 120588119896 (3)

Using 120597119905120588119896(119905) = 0 and 120597


obtain

minus120588119896119879

+ 120582119896 (1 minus 120588119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

minus120599 + 120574 (1 minus 120599) 120579 = 0

(4)





of 120582 120574 1205741 and 119879



119896119879= 120588119896(119879 + 1) into (4) obtains

minus120588119896

119879 + 1+ 120582119896 (1 minus 120588

119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

120599 =120574120579

1 + 120574120579

(5)



120588119896

=(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(6)

Using ⟨119896⟩ = sum119904


120579 = sum

119896

119896119875 (119896) 120588119896

sum119904

119904119875 (119904)

=1

⟨119896⟩sum

119896

119896119875 (119896)(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(7)


119889

119889120579(

1

⟨119896⟩sum

119896

119896119875 (119896)

sdot(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

)

1003816100381610038161003816100381610038161003816100381610038161003816120579=0

ge 1

(8)



120582119888

=(1 minus 120574120574

1) ⟨119896⟩

(119879 + 1) ⟨1198962⟩ (9)

In (9) ⟨1198962⟩ = sum

1198961198962119875(119896) When 119879 = 120574 = 120574

1= 0 that is


119888= ⟨119896⟩⟨119896

2⟩ Therefore





infin

119898119896119875(119896)119889119896 = 2119898 By introducing




119870119888

1198981198962119875(119896)119889119896 =

21198982 ln (119870



120582119888

asymp(1 minus 120574120574

1)

(119879 + 1) 119898 ln (119870119888119898)

(10)




119888asymp 119898119873


asymp 11989811987312


120582119888

asymp2 (1 minus 120574120574

1)

(119879 + 1) 119898 ln119873 (11)






119888= 2(1 minus 120574120574

1)(119898 ln119873) It can



obtains

1

119898

= int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198962 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

= 119860 ln1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579)+

119861

119898

(12)

In (12)

119860 =(119879 + 1) 120582 (1 + 120574120579)

2

(1 + 120574120579 + 1205741205741120579)2

119861 =1205741205741

1 + 120574120579 + 1205741205741120579

119862 = minus(119879 + 1)

21205822120579 (1 + 120574120579)

3

(1 + 120574120579 + 1205741205741120579)2

(13)


ln(1 +1 + 120574120579 + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579))

=(1 + 120574120579 + 120574120574

1120579) (1 + 120574120579 + 120574120574

1120579 minus 120574120574

1)

119898 (119879 + 1) 120582 (1 + 120574120579)2

(14)



120588 = sum

119870

119875 (119896) 120588119896

= sum

119896

21198982119896minus3

(119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

= 21198982120579 int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198963 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

(15)


120588 = 21198982120579 [

(119879 + 1) 120582 (1 + 120574120579) minus (119879 + 1) 120582119896 (1 + 120574120579)

119898 (1 + 120574120579 + 1205741205741120579)

+1205741205741

21198982 (1 + 120574120579 + 1205741205741120579)

] = 120579

sdot2119898 (119879 + 1) 120582 (1 + 120574120579) (1 minus 120579) + 120574120574

1

1 + 120574120579 + 1205741205741120579

(16)









0 002 004 006 008 01 0120

001

002

003

004

005

006

007

008

009

01

T = 1 r1 = 02 r2 = 04T = 0 r1 = 02 r2 = 04

T = 1 r1 = 015 r2 = 02T = 0 r1 = 015 r2 = 02

120588

120582









1among Internet



0

0002

0004

0006

0008

001

0012

0014

0016

120588

0 01 02 03 04 05 06 07 08 09

120574 (1205741)

120582 = 001 T = 3

120582 = 001 T = 2120582 = 001 T = 1

120582 = 001 T = 0





5 Conclusions










Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896119879= 120588119896(119879 + 1) into (4) obtains

minus120588119896

119879 + 1+ 120582119896 (1 minus 120588

119896) 120579 + 120574

1(1 minus 120588

119896) 120599 = 0

120599 =120574120579

1 + 120574120579

(5)



120588119896

=(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(6)

Using ⟨119896⟩ = sum119904


120579 = sum

119896

119896119875 (119896) 120588119896

sum119904

119904119875 (119904)

=1

⟨119896⟩sum

119896

119896119875 (119896)(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

(7)


119889

119889120579(

1

⟨119896⟩sum

119896

119896119875 (119896)

sdot(119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

)

1003816100381610038161003816100381610038161003816100381610038161003816120579=0

ge 1

(8)



120582119888

=(1 minus 120574120574

1) ⟨119896⟩

(119879 + 1) ⟨1198962⟩ (9)

In (9) ⟨1198962⟩ = sum

1198961198962119875(119896) When 119879 = 120574 = 120574

1= 0 that is


119888= ⟨119896⟩⟨119896

2⟩ Therefore





infin

119898119896119875(119896)119889119896 = 2119898 By introducing




119870119888

1198981198962119875(119896)119889119896 =

21198982 ln (119870



120582119888

asymp(1 minus 120574120574

1)

(119879 + 1) 119898 ln (119870119888119898)

(10)




119888asymp 119898119873


asymp 11989811987312


120582119888

asymp2 (1 minus 120574120574

1)

(119879 + 1) 119898 ln119873 (11)






119888= 2(1 minus 120574120574

1)(119898 ln119873) It can



obtains

1

119898

= int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198962 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

= 119860 ln1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579)+

119861

119898

(12)

In (12)

119860 =(119879 + 1) 120582 (1 + 120574120579)

2

(1 + 120574120579 + 1205741205741120579)2

119861 =1205741205741

1 + 120574120579 + 1205741205741120579

119862 = minus(119879 + 1)

21205822120579 (1 + 120574120579)

3

(1 + 120574120579 + 1205741205741120579)2

(13)


ln(1 +1 + 120574120579 + 120574120574

1120579

119898 (119879 + 1) 120582120579 (1 + 120574120579))

=(1 + 120574120579 + 120574120574

1120579) (1 + 120574120579 + 120574120574

1120579 minus 120574120574

1)

119898 (119879 + 1) 120582 (1 + 120574120579)2

(14)



120588 = sum

119870

119875 (119896) 120588119896

= sum

119896

21198982119896minus3

(119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

= 21198982120579 int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198963 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

(15)


120588 = 21198982120579 [

(119879 + 1) 120582 (1 + 120574120579) minus (119879 + 1) 120582119896 (1 + 120574120579)

119898 (1 + 120574120579 + 1205741205741120579)

+1205741205741

21198982 (1 + 120574120579 + 1205741205741120579)

] = 120579

sdot2119898 (119879 + 1) 120582 (1 + 120574120579) (1 minus 120579) + 120574120574

1

1 + 120574120579 + 1205741205741120579

(16)









0 002 004 006 008 01 0120

001

002

003

004

005

006

007

008

009

01

T = 1 r1 = 02 r2 = 04T = 0 r1 = 02 r2 = 04

T = 1 r1 = 015 r2 = 02T = 0 r1 = 015 r2 = 02

120588

120582









1among Internet



0

0002

0004

0006

0008

001

0012

0014

0016

120588

0 01 02 03 04 05 06 07 08 09

120574 (1205741)

120582 = 001 T = 3

120582 = 001 T = 2120582 = 001 T = 1

120582 = 001 T = 0





5 Conclusions










Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588 = sum

119870

119875 (119896) 120588119896

= sum

119896

21198982119896minus3

(119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579

= 21198982120579 int

+infin

119898

(119879 + 1) 120582119896 (1 + 120574120579) + 1205741205741

1198963 [1 + 120574120579 + (119879 + 1) 120582119896120579 (1 + 120574120579) + 1205741205741120579]

119889119896

(15)


120588 = 21198982120579 [

(119879 + 1) 120582 (1 + 120574120579) minus (119879 + 1) 120582119896 (1 + 120574120579)

119898 (1 + 120574120579 + 1205741205741120579)

+1205741205741

21198982 (1 + 120574120579 + 1205741205741120579)

] = 120579

sdot2119898 (119879 + 1) 120582 (1 + 120574120579) (1 minus 120579) + 120574120574

1

1 + 120574120579 + 1205741205741120579

(16)









0 002 004 006 008 01 0120

001

002

003

004

005

006

007

008

009

01

T = 1 r1 = 02 r2 = 04T = 0 r1 = 02 r2 = 04

T = 1 r1 = 015 r2 = 02T = 0 r1 = 015 r2 = 02

120588

120582









1among Internet



0

0002

0004

0006

0008

001

0012

0014

0016

120588

0 01 02 03 04 05 06 07 08 09

120574 (1205741)

120582 = 001 T = 3

120582 = 001 T = 2120582 = 001 T = 1

120582 = 001 T = 0





5 Conclusions










Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0002

0004

0006

0008

001

0012

0014

0016

120588

0 01 02 03 04 05 06 07 08 09

120574 (1205741)

120582 = 001 T = 3

120582 = 001 T = 2120582 = 001 T = 1

120582 = 001 T = 0





5 Conclusions










Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Competing Interests


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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