Research Article Optimal Design on Robustness of Scale-Free Networks Based on Degree Distribution

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1 Scientific Prograing Volue 16, Article ID , 7 pages Research Article Optial Design on Robustness of Scale-Free etworks Based on Degree Distribution Jianhua Zhang, 1 Shuliang Wang, 1 and Yixing Wang,3 1 School of Electrical Engineering and Autoation, Jiangsu oral University, Xuzhou 1116, China School of Governent, Beijing oral University, Beijing 1875, China 3 People s Governent of Wuduan Town, Peixian, Xuzhou 1638, China Correspondence should be addressed to Jianhua Zhang; zhangjianhua198@16.co Received 3 April 16; Revised 7 June 16; Accepted 5 July 16 Acadeic Editor: eng Guo Copyright 16 Jianhua Zhang et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. This paper uses -nor degree and coefficient of variation on degree to analyze the basic characteristics and to discuss the robustness of scale-free networks. And we design two optial nonlinear ixed integer prograing schees to investigate the optial robustness and analyze the characteristic paraeters of different schees. In this paper, we can obtain the optial values of the corresponding paraeters of optial designs, and we find that coefficient of variation is a better easure than -nor degree and two-step degree to study the robustness of scale-free networks. eanwhile, we discover that there is a tradeoff aong the robustness, the degree, and the cost of scale-free networks, and we find that when average degree equals 6, this point is a tradeoff point between the robustness and cost of scale-free networks. 1. Introduction Coplexnetworkshavebecoeoreandoreiportant to our daily life [1 5], such as biology syste, electrical power grid, transportation network, and pipeline networks. The security and reliability of these networks have been the concern of ore and ore scientists. Since the discovery of sall-world networks, the sall-word property [] and scale-free property [3] have attracted continuous attention fro all over the world. It has been recognized that any networks have scale-free property which eans that the degree distribution follows a power-law distribution [4, 5]. Recently, there are any novel concepts and approaches in any subjects, such as inforation science, control science, statistical and nonlinear physics, and atheatics and social science which are used to investigate the characteristics in any fields, especially coplex network [6, 7]. The critical point is that the inforation flow between topological nodes and other physical quantities is iportant to network security, so we ust keep the inforation exchange unipeded. Because of the ubiquity of scale-free networks in natural and anade systes, the security and reliability of these networks have attracted great interest [8, 9]. The work by Albert deonstrated that scale-free networks possess the robust yet fragile property and he found that it is robust against rando failures of nodes but fragile to intentional attacks [3 5]. Cascading failure can occur in any coplex systes; an intuitive thinking suggests that the possibility of breakdown of networks triggered by attacks or failures cannot be ignored in scale-free networks. Avalanche of breakdown is a serious threat to the network when nodes and links are sensitive to overloading. The reoval of nodes which resulted fro rando breakdown or intentional attack can change the balance of flows and lead to redistributing loads all over the network; soeties the redistribution of loads cannot be tolerated and ight trigger a cascade of overload failure [1];finallythenetworkwouldbecollapsing.Butitcanalso propagate and cut down inforation transission of the whole network in soe cases [11]. Therefore the robustness is an iportant aspect to investigate the characteristics of scalefree network. Based on the statistical property of coputer network, it is known that coputer network is a scale-free network.

2 Scientific Prograing Supercoputer is a coputer with a high-level coputational capacity copared to a general-purpose coputer. With the developents of supercoputers, supercoputer networks will also be fored in the future; therefore supercoputer networks also possess scale-free property, and the reliability and robustness of supercoputer network becoe ore and ore iportant. The optial design of supercoputer networks can iprove the reliability and robustness of supercoputer networks and also enhance speed of the inforation transission and exchange of supercoputer networks. Hence the optial design can present the theoretical and practical significances for the design and construction of supercoputer networks. Thispaperisorganizedasfollows.Sectiondiscussesthe basic characteristics by analyzing the characteristic paraeters of scale-free networks. Two-nor degree and its optial prograing of degree distribution are studied in Section 3. Section 4 investigates the coefficient of variation and its optial design of degree distribution. Finally, a conclusion is presented in Section 5.. Basic Characteristics of Scale-Free etwork Recently, any researchers have constructed several schees to easure the robustness of scale-free networks, such as the average degree, average two-step degree (ATSD) [1], and entropy [13, 14]. In this paper, we construct two odels defined as the average -nor degree (ATD) and coefficient of variation (CV) to discuss the robustness of scale-free network. Fro otter and Lai [11], we know that the node V i of coplex network has a load capacity as follows: C(V i )=(1+λ) L(V i ), i=1,,...,, (1) where λ>is the tolerance paraeter, L(V i ) is the initial load of node V i,and is the total nuber of vertices. The network is designed as G = V,E, V = {V 1, V,...,V } is the set of vertices, and E V V is the set of edges which can exchange inforation fro vertices to others. According to [1, 13, 15], one knows that the robustness can be iproved with the increase of the heterogeneity, and the larger the average degree is, the better the robustness is. any coplex networks are scale-free networks, and the degree distribution of nodes is power-law distribution [3 5], the density function being as follows: f (k) ck, (, 3). () Based on the discrete property of degree and continuous approxiation principle, we can declare 1= k= ck =c k= =c 1 1, 1 k c k dk (3) P(k >K) K=5 K=1 K= Figure 1: The characteristics of cuulative probability with =1 and = 1. where and aretheinialandtheaxiudegree in finite network. Fro the literature [16], we can get the axiu degree of scale-free networks: Substituting (4) into (3), we can obtain c 1/( 1). (4) 1 1 ( 1 1). (5) According to () and (5), we can obtain the density function of degree distribution as follows: f (k) (1 ) k 1 ( 1 1). (6) Forula (6) introduces the characteristics of the density function on degree of scale-free networks. We know that the scale-free network is heterogeneous network, and we observe the cuulative probability on degree and investigate the robustness of scale-free networks; forula (7) presents the cuulative probability of scale-free network: K P {k >K} =1 P{k K} =1 f (k) dk K =1 c k dk=1 K1 1 1 ( 1 1). Figure 1 introduces the properties of the cuulative probability on degree distribution of the scale-free network, and we can declare that the cuulative probability decreases with the increase of the paraeter ; eanwhile, with the increase of K, the cuulative probability decreases.and (7)

3 Scientific Prograing ATD = 1 = = 5 Figure : The property of average degree versus with different. = 1 = = 5 Figure 3: The characteristics of ATD versus with different. we find that the cuulative probability is very sall when K = ;thatistosay,therearefewverticeswhichare degree surpassing K = ; this phenoenon reflects the heterogeneity of scale-free networks. According to (4), (6), and continuous approxiation principle, we can obtain the average degree as follows: E (k) = 1 i= d i = c( ) kf (k) dk=c k 1 dk = ( 1) (1 ( )/( 1) ). ( ) (1 1 ) Figure describes the property of the average degree of scale-free networks, and we discover that the average degree decreases with the increase of the scale exponent and the average degree decreases with the decrease of the nuber of nodes. Fro the real case, we know that the larger the average degree is, the better the robustness is, so the saller the paraeter is, the better the robustness is. 3. Optial Robustness Design Based on ATD A coplex network odel G= V,E, V={V 1, V,...,V } is the set of nodes, and E V V is the set of the edges. eanwhile the adjacent atrix is defined as follows: H=(h ij ), h ij = { 1 (V i, V j ) E { (V { i, V j ) E. (8) (9) Hence, the degree on adjacent nodes of node V i can be obtained by D i = V j G i d j, (1) where G i ={V ω V V i V (V ω, V i ) E}, i,ω=1,,...,, and d i is the degree of node V i. We calculate the degree of adjacent nodes of networks and we can get the following forula: D=Hd= i=1 D i = d j = i=1v j G i i=1 d i = d, (11) where d=(d 1,d,...,d ) T and D=(D 1,D,...,D ) T are the degree vector and the adjacent degree vector. Based on the above analysis, we define D asthe-nordegreeof the network; therefore we define the average -nor degree (ATD) as follows: ATD = 1 D = 1 d = 1 1 di. (1) According to continuous approxiation, we can get the quantitative value of ATD as follows: ATD 1 k f (k) dk i=1 = 1 3 (3 )/( 1) (13) Fro Figure 3, we know that ATD decreases with the increase of the paraeter, and the saller the paraeter

4 4 Scientific Prograing ATD Figure 4: The optional value ATD, the optional solutions, and versus. is, the better the robustness is. eanwhile, with the increase of the nuber of nodes, we obtain that ATD decreases which is different fro the result given in Figure, so we can declare that, with the increase of the nuber of vertices, the robustness decreases to soe extent. So we can declare that the robustness of the scale-free network can be easured by any easureents and different easureents have different results. ext, we consider the optial average -nor degree; we construct the following nonlinear ixed integer prograing schee: ax s.t. ATD = const; <<3, 1 1, Z. (14) Figure 4 portrays the characteristics of optial design of scale-free network based on ATD and describes the optial valueatdandtheoptionalsolutionsof and with different average degree and = 1. FroFigure4, we find that, with the increase of, the inial degree increases and the scaling exponent has the behavior of oscillation. eanwhile, with the increase of the average degree, the optial average -nor degree has the trend to becoe larger, so the robustness becoes better. oreover there is an oscillation about ATD proposed in Figure 4, which is because of the fact that the scale-free networks are heterogeneous networks. Furtherore, when = 7, these three paraeters have the obvious oscillation, which indicates that this point is a tradeoff point between the robustness and cost. 4. Optial Robustness Design Based on CV In this section, we discuss the robustness of scale-free networksbasedoncv.frosection,weknowthatthe average degree is a robustness easureent of network, but the average degree is a one-sided easureent and it is an overall easure. We know that the larger the cuulative degree is, the larger the average degree is, so the better the robustness is, but the average degree does not portray the relationships between degree and the average degree. Siilar to star networks, if the load of one edge exceeds its axial load, the edge would be collapsing, so there would be an isolated node in this star network. In order to investigate the reliability of networks, we introduce variance degree [17, 18] of scale-free networks as follows: V (k) =E[k ], (15) where V( ) represents the variance, E( ) represents the atheatical expectation, and k is the degree of the node. According to the characteristics of variance, we know that V( ) is the easureent which describes the average distance fro the degree k to the average degree of vertices; that is to say, the variance degree describes the relationship between individual easure and overall easure. So we can better coprehend degree distribution according to variance degree: the larger thevariancedegreeis,theworsethedegreedistribution is; on the contrary, the saller the variance degree is, the ore unifor the degree distribution is, so the better the robustness is. In ters of forulas (4) and (6) and continuous approxiation principle, we can estiate the variance degree as follows: V (k) =E[k ] =E[k ] [E (k)] = 1 k=k [ 1 k] k= k f (k) dk [ kf (k) dk] = ( 1) ( (3 )/( 1) 1) (3 ) (1 1 ) [ ( 1) (1 ( )/( 1) ) ]. ( ) (1 1 ) (16) Forula (16) introduces the characteristics of the variance degree, and we obtain the relationships between variance degree and the paraeters of scale-free network. Because the variance degree is the distance fro the degree to the average degree, we know that the variance degree is an individual easureent of scale-free network. According to the property of variance, we find that the saller the variance degree is, the better the robustness is with fixed average degree. Fro Section, we know that the average degree is the overall easureent which neglects the individual

5 Scientific Prograing 5 CV Figure 5: Behaviors of the variant coefficient versus scale exponent. behaviors and the relationships between overall characteristics and individual properties of nodes, so the variance degree is investigated to consider the relationship between the.accordingtothesetwoaspects,anycoplexartificial networks would be constructed to suit the needs of society. Fro the average degree and the variance degree, we know that the average degree depicts the overall connectivity and the variance degree portrays the individual connectivity, so we ust cobine these two aspects to construct anotherbettereasurewhichiscalledcvtoinvestigatethe robustness of scale-free network. CV is proportional to the variance degree and inversely proportional to the average degree; it considers not only the overall behaviors of the network but also the individual behaviors of network, so it is a better easure than the average degree and the variance degree. Fro the definition and the characteristics of the average degree and variance degree, we know that the saller the CV is, the better the robustness is of the scale-free network. Figure 5 portrays the characteristics of CV and we know that it can reflect the relationships between the average degree and the variance degree versus scaling exponent. CV, which avoids soe shortcoings of the average degree and the variance degree,is a better easure to investigate the robustness of networks. If we want to iprove the robustness of scale-free network, we ust iprove the average degree and reduce the variance degree. Fro Figure 5, we find that CV decreases with the increase of the scaling exponent, and we can tell that, with the decrease of CV, the robustness increases and the resilience to rando failure or intentional attacks becoes stronger. According to the properties of real networks and considering the cost of networks, we design the optial odel to investigate the robustness of scale-free network. Fro the relationship between the robustness and the heterogeneity and studying the eanings of literatures [1, 13], we design another nonlinear ixed integer prograing schee about CV Figure 6: The optional value CV, the optional solutions, and versus with = 1. CV to obtain the optional values of all the paraeters as follows: in CV s.t. = const; <<3, 1 1, Z. (17) We portray the characteristics of optial design (17) in Figure 6, which describes the optial value CV and the optional solutions of and with different. Fro Figure 6, we discover that, with the increase of, the inial degree increases and the scaling exponent has thebehaviorofoscillation.eanwhile,withtheincreaseof, CV has an obvious oscillation and has the trend to increase. Fro Figure 6, we can obtain the optial values of different paraeters, and we also can design the real network according to these values to iprove the robustness of the real network. Furtherore we find that when =,4,6,8, CV has sall values and the scale-free networks have the hoogenous degree distribution; therefore the networks have the better robustness. In particular when = 4,6,8, the siulation results are ore to eet the actual situations, and, fro Figure 6, we find that when = 4, the CV has the sallest value; therefore the scale-free networks have the best robustness aong these actual situations. 5. Discussion of Optial Robustness In the paper, we investigate two optial odels on degree of scale-free network. Fro our optial design, we find that different odels have different behaviors in easuring the

6 6 Scientific Prograing CV -ATSD -ATD Figure 7: The axiu degree of different optial robustness odels versus average degree. robustness of network, and the robustness can be investigated by the axiu and iniu degree of scale-free network. With the fixed average degree, we find that the saller the CV is, the better the robustness is; eanwhile, the saller the axiu degree is, the ore hoogeneous the degree distribution is and therefore the better the robustness is; oreover the larger the iniu degree is, the ore hoogeneous the degree distribution is and the better the robustness is. Hence the axiu and the iniu degree are the other two easures on robustness of scale-free network. In this section, we discuss the robustness of optial odels according to axiu and iniu degree of scale-free network. With fixed average degree, we know that the robustness becoes better with the decrease of the axiu degree. Figure 7 introduces the characteristics of the axiu degrees of the optial odels versus the average degree. In this section, we use -CV, -ATSD, and -ATD to represent the axiu degrees of odels aboutoptialcv,atsd,andatd,respectively.figure7 illustrates that -ATSD is larger than the other two cases with froto7andsallerthantheothertwocases with fro 8 to 9. eanwhile Figure 7 tells us that -CV has sall values all the tie, and it has a slight increase. So we can declare that -CV aong the is the best selection to assess the robustness of scale-free network; that is to say, CV is a better easure than ATSD and ATD. Figure 8 gives us another evidence which indicates the fact that CV is a better easure than ATSD and ATD. We use -CV, -ATSD, and -ATD to represent the iniu degrees of optial odels about CV, ATSD, and ATD, respectively. Figure 8 indicates that, with fixed average degree, the iniu degree of optial ATSD is the sallest degree in these three optial odels and the optial CV have the largest sallest degree in these three cases. Based on Figures 7 and 8, copared with the largest CV -ATSD -ATD Figure 8: The iniu degree of different optial odels versus average degree CV -ATSD -ATD Figure 9: The scaling exponent values of different optial odels versus average degree. degree of iniu degree of optial ATSD and ATD, with fixed average degree, optial CV schee has saller axiu degree and larger iniu degree; that is to say, the degree distribution of CV is ore hoogeneous than that of the other two cases, so we can declare that CV is a better robustness easure than the other two odels. Figure 9 depicts characteristics of the optial scaling exponent of three different optial odels and Figure 9 shows that the scaling exponent value of optial CV odel is larger than that of the other two optial odels. And in few cases, the optial scaling exponent values of different odels are equal, and, in soe cases, they have little differences.

7 Scientific Prograing 7 eanwhile Figure 9 indicates that the average degree and the two-step degree are very lopsided in investigating the robustness on degree of scale-free network; hence CV is a better odel in studying the robustness of scale-free network. 6. Conclusion Coplex networks have been paid increasing attention in the past two decades, and any valuable results have been obtained fro several aspects [19, ]. In this paper, we reestiate the paraeter of the density function on degree anddiscussthebasicbehaviorsofscale-freenetwork.based on the density function, we discuss the characteristics of cuulative probability of degree and introduce the properties oftheaveragedegree.eanwhilewegivetwobasicrobustness easures of scale-free networks, which are called - nor degree and CV, and we also construct two nonlinear ixed integer prograing schees according to the average -nor degree and CV. oreover the optial values of threeoptialodelscanbeobtainedinthispaperandwe copare the optial values of the corresponding paraeters of different odels. Furtherore we give several evidences of optial values of optial designs to illustrate the fact that CV is a better easure than ATSD and ATD to investigate the robustness of scale-free networks. In the future, we will continue to investigate the robustness and optial design of scale-free networks; eanwhile we will also investigate the robustness and optial design of the actual networks. Copeting Interests The authors declare that they have no copeting interests. Acknowledgents This work was jointly supported by the ational atural Science Foundation of China ( , , , and ) and the atural Science Foundation for Youths of Jiangsu Province of China (BK1441). [8] L.Zhao,K.Park,andY.-C.Lai, Attackvulnerabilityofscalefree networks due to cascading breakdown, Physical Review E, vol.7,no.3,articleid3511,4. [9]J.W.Wang,L.L.Rong,L.Zhang,andZ.Z.Zhang, Attack vulnerability of scale-free networks due to cascading failures, Physica A,vol.387,no.6,pp ,8. [1] S. Boccaletti, V. Latora, Y. oreno,. Chavez, and D.-U. Hwang, Coplex networks: structure and dynaics, Physics Reports,vol.44,no.4-5,pp ,6. [11] A. E. otter and Y.-C. Lai, Cascade-based attacks on coplex networks, Physical Review E, vol. 66, no. 6, ArticleID651,. [1] Z.-H. Wu and H.-J. Fang, Cascading failures of coplex networksbasedontwo-stepdegree, Chinese Physics Letters,vol. 5, no. 1, pp , 8. [13]B.Wang,H.W.Tang,C.H.Guo,andZ.L.Xiu, Entropy optiization of scale-free networks robustness to rando failures, Physica A,vol.363,no.,pp ,6. [14] J. Wu, Y.-J. Tan, H.-Z. Deng, and D.-Z. Zhu, oralized entropy of rank distribution: a novel easure of heterogeneity of coplex networks, Chinese Physics, vol. 16, no. 6, pp , 7. [15]. E. J. ewan, S. H. Strogatz, and D. J. Watts, Rando graphs with arbitrary degree distributions and their applications, Physical Review E,vol.64,no.,ArticleID6118,1. [16] R. Cohen, K. Erez, D. Ben-Avraha, and S. Havlin, Resilience of the Internet to rando breakdowns, Physical Review Letters, vol. 85, no. 1, pp ,. [17] E. Tsukaoto and S. Shirayaa, Influence of the variance of degree distributions on the evolution of cooperation in coplex networks, Physica A, vol. 389, no.3, pp , 1. [18] T. A. Snijders, The degree variance: an index of graph heterogeneity, Social etworks,vol.3,no.3,pp ,1981. [19] J. Wang, itigation of cascading failures on coplex networks, onlinear Dynaics, vol. 7, no. 3, pp , 1. [] J. Wang, itigation strategies on scale-free networks against cascading failures, Physica A: Statistical echanics and its Applications,vol.39,no.9,pp.57 64,13. References [1] S. H. Strogatz, Exploring coplex networks, ature,vol.41, no. 685, pp , 1. [] D. J. Watts and S. H. Strogatz, Collective dynaics of sallworld networks, ature, vol. 393, no. 6684, pp , [3] A.-L. Barabasi and R. Albert, Eergence of scaling in rando networks, Science,vol.86,no. 5439,pp , [4] R. Albert and A.-L. Barabasi, Statistical echanics of coplex networks, Reviews of odern Physics, vol. 74, no. 1, pp ,. [5]R.Albert,H.Jeong,andA.L.Barabasi, Attackanderror tolerance of coplex networks, ature, vol. 46, pp ,. [6] S..DorogovtsevandJ.F.F.endes, Evolutionofnetworks, Advance in Physics,vol.51,no.4,pp ,. [7]. E. ewan, The structure and function of coplex networks, SIA Review, vol. 45, no., pp , 3.

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