Weight-driven growing networks

Transcription

1 PHYSICAL REVIEW E 71, Weight-driven groing netorks T. Antal* and P. L. Krapivsky Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA Received 12 August 2004; published 8 February 2005 We study groing netorks in hich each link carries a certain eight randomly assigned at birth and fixed thereafter. The eight of a node is defined as the sum of the eights of the links attached to the node, and the netork gros via the simplest eight-driven rule: A nely added node is connected to an already existing node ith the probability hich is proportional to the eight of that node. We sho that the node eight distribution n has a universal tail, that is, it is independent of the link eight distribution: n 3 as. Results are particularly neat for the exponential link eight distribution hen n is algebraic over the entire eight range. DOI: /PhysRevE PACS numbers: C, a, q, Sn I. INTRODUCTION The netork structure lies beneath many physical, biological, social, economic, and other real-orld systems. Examples include resistor netorks, metabolic netorks, communication netorks like the Internet, information netorks like the World Wide Web, transportation netorks, food ebs, etc. Some netorks are engineered, hile others like the World Wide Web are created in a chaotic manner, viz., by the uncoordinated actions of many individuals, and yet they sho a great deal of self-organization 1. This surprising order of seemingly disordered netorks as noticed long ago in the context of random graphs 2,3, and it is also apparent in the models of groing random netorks that have been thoroughly studied in the past fe years see revies 4 6 and references therein. Netorks or graphs are defined as sets of nodes joined by links. A link in a graph merely indicates that a given pair of nodes is connected. If, hoever, connections differ in strength, one may formalize this property by assigning a eight to each link. Many real-orld netorks are intrinsically eighted. In a collaboration netork, the eight of a link beteen co-authors measures the strength of the collaboration such as the number of jointly authored publications 7,8. In the airline transportation netork, the eight of a link beteen to airports gives the passenger capacity on this route 9,10. Thus mathematically the eighted netork is a graph in hich each link carries a certain number hich is called the eight. Weighted netorks appear in the literature under different names. For instance, the multigraph that is, the graph in hich to nodes can be joined by multiple links can be replaced by the graph in hich a link beteen to nodes carries the integer eight equal to the number of links in the original multigraph joining those to nodes. Numerous engineering and mathematical studies of the flos in netorks also treat eighted netorks the eight usually termed capacity represents *On leave from Institute for Theoretical Physics HAS, Eötvös University, Budapest, Hungary. Electronic address: paulk@bu.edu the maximum alloed flo 11,12. Resistor netorks see, e.g., form another very important class of eighted netorks. The models of eighted netorks are usually close to the uneighted ones as far as the underlying graph structure is concerned. In other ords, the link eights are passive variables. This allos us to study eighted netorks using the knoledge of uneighted netorks as the starting point 16,17. In reality, the link eights can of course affect the graph structure. The range of possible models in hich the link eights are active variables is extremely broad see, e.g., hile the mechanisms driving the evolution of the real-orld netorks are still hardly knon. In such a situation, one ants to study a minimal rather than detailed model. Here e introduce and investigate a minimal model of the eight-driven groth. In Sec. II, e introduce the minimal model precisely, and determine the node eight distribution and the joint node eight-degree distribution. We then compute the incomponent eight distribution. In Sec. III, e discuss some generalizations of the minimal model. We conclude in Sec. IV. II. THE MODEL The model is defined as follos. Each link carries a positive eight hich is dran from a certain distribution. We shall assume that the eights are positive 23. The eight is assigned to the link hen the link is created, and it remains fixed thereafter. The eight of a node is the sum of the eights of the links attached to the node. When a ne node is added, it is linked to a single target node ith probability proportional to the eight of the target node. We assume that only one link emanates from each nely introduced node, so the resulting netork is a tree; the general case hen a fe links emanate from each node is discussed in Sec. III. The eight of a node also termed the node strength by some authors increases hen a ne link is attached to it, yet, as the size of the netork gros, the node eight distribution approaches a stationary distribution. Our first goal is to determine this distribution /2005/712/ /$ The American Physical Society

2 T. ANTAL AND P. L. KRAPIVSKY PHYSICAL REVIEW E 71, A. Weight distribution Let N be the total number of nodes in the netork and N Nd the number of nodes hose eight lies in the range,+d. When N is large, it can be treated as a continuous variable, and N N satisfies dn dn = 1 dx xxn x N 0 +, 1 here N= 0 dn N is the total eight of all nodes 25. The term in Eq. 1 hich is proportional to xn x / accounts for nodes ith eight x hich gain a link of eight x thereby creating nodes of eight. The term N / is the corresponding loss term. The last term accounts for the nely introduced node hich has the same eight as its link. In the large N limit, N N=Nn, i.e., the eight distribution approaches a stationary N independent distribution n. Hence, Eq. 1 simplifies to + n dx xxnx +, 2 =0 here = 0 dn is the average node eight. The total eight of all nodes is tice the total eight of all links since each link connects to nodes and the same obviously holds for the average eights. Therefore, =2 d. =20 We no specialize Eq. 2 to the uniform link eight distribution =1, 1 3 0, 1. Using relation =2=1 and the shorthand notation F =1+ 0 dxxnx, e recast Eq. 2 into =F, 1 1+n 4 F F 1, 1. One can solve Eq. 4 recursively. Using F=n, e find that for 1, the cumulative distribution satisfies 1 +1F=F here F=dF/d. Solving this equation subject to the boundary condition F0=1 yields F =1+ 1 e. Therefore, the eight distribution is n = 1+ 2 e for 1. On the next interval 12, e ought to solve 1 +1 df d = F 1 e 1. The solution should match the previous one; this gives the boundary condition F1=e/2. The resulting cumulative distribution is F=e 1 e+1 /1+, from hich e obtain the node eight distribution FIG. 1. The node eight distribution n emerging for uniform and exponential link eight distributions. n = 1+ e 1 e for 1 2. Interestingly, the eight distribution loses continuity at the cutoff value =1 of the uniform link eight distribution: n1 0=e/4 hile n1+0=e 2/4. Proceeding, one finds the node eight distribution on the interval 23, etc. The distribution is analytic everyhere apart from the integer values; at the integer value =k2, the node eight distribution is continuously differentiable k 2 times see Fig The analytic expressions for the node eight distribution are very cumbersome for large. Fortunately, the asymptotic behavior is very simple. To extract the asymptotic, e expand the right-hand side of Eq. 4 using the Taylor series, F F 1 = F 1 2 F + = n 1 n +. 2 Plugging this expansion into Eq. 4, e obtain hich is solved to give dn +3n =0, d 5 n A 3 hen. 6 The amplitude A cannot be found by solving the linear equation 5; its determination requires analysis of the full problem 4. Note also that for the deterministic link eight distribution = 1, the eight of the node is equal to the degree, so the model becomes equivalent to the basic groing netork model ith preferential attachment hich also leads to a node eight distribution ith 3 tail. In general, the large eight tail of the node eight distribution is alays given by Eq. 6 if the link eight distribution has a cutoff. To prove this assertion, suppose that =0 for 1. For 1, Eq. 2 becomes

3 WEIGHT-DRIVEN GROWING NETWORKS + n = 7 ith = 1 0 dxx xn x; for the uniform distribution =F F 1 and Eq. 7 turns into Eq. 4. Expanding the integrand in, yields xn x = n xn + = n n +, 2 9 here relations d=1 and d=/2 ere used. In conjunction ith Eq. 9, Eq. 7 reduces to Eq. 5 leading to the tail given by Eq. 6. This derivation, as ell as the earlier one that led to Eq. 5, ignores the higher terms in the expansion 8. Keeping such terms, one ould obtain dn d +3n = 2n 1 3 3n + 10 ith 2 = 1 d 2, 3 = 1 d 3, etc. Solving Eq. 10 gives higher-order corrections to the tail asymptotics n = A but it does not affect the leading 3 tail. The exponential link eight distribution = e 12 is particularly appealing as the node eight distribution in this case is remarkably simple. The governing equation 2 becomes 2+e n dxxe =0 x nx Writing G=2+ 0 dxxe x nx, e recast integral equations 13 into a simple differential equation 2 dg +1 d = G. 14 We find G=8+2 2 e, from hich n = Thus for the exponential link eight distribution, the emerging node eight distribution is scale-free that is, purely algebraic over the entire eight range. We no outline the behavior for link distributions ith heavy tails. Consider particularly distributions ith a poerla tail ; the exponent must obey the inequality 1 to ensure normalization d=1. It is easy to check that for 3 the leading asymptotic of the node eight distribution is n 3 ; moreover, expansion 11 holds up to the order of n, here n is the largest integer smaller than i.e., nn For 23, the leading term shos even sloer decay n3 1. The situation is very different for 12 hen the total eight gros ith size faster than linearly: N 1/ 1. Using the usual definition N =Nn, e observe that the square bracket term in Eq. 1 is negligible for large N values. Hence n, suggesting that the number of nodes ith more than one links gros sloer than linearly in N, that is, only the dangling nodes give contribution to n. B. Weight-degree distribution The node eight distribution is perhaps the most natural, and therefore readily tractable, local characteristic for netorks ith eight-driven groth. Of course, the degree distribution remains geometrically the simplest local characteristic, yet for eighted netorks it is generally impossible to rite don a closed equation for the degree distribution n k. To compute n k, one must determine the joint eight-degree distribution n k ; the degree distribution is then found by integration, n k =0 dn k. 16 The eight-degree distribution obeys a set of equations similar to Eq. 2. The density of dangling nodes is given by + n 1 = hile for k2 the eight-degree distribution satisfies 17 + n k dx xxn k 1 x. 18 =0 One can treat Eqs. 17 and 18 recursively, yet even for the simplest link eight distributions like the uniform distribution 3 the exact expressions for n k become very unieldy as the degree gros. Exceptionally neat results emerge again for the exponential link eight distribution 12. In this case, 2+e n k dxe =0 x xn k 1 x. 19 Starting ith n 1 =2/2+e e explicitly computed a fe more n k hich led us to the hypothetical solution (e use the shorthand notation = 2 ln1+/2) n k = PHYSICAL REVIEW E 71, k 1 e 3 k 1!. 20 Of course, the above ansatz agrees ith the sum rule n = k1 n k. Having guessed the solution, it is then straightforard to verify its validity. Note that asymptotically k 1, the joint eight-degree distribution approaches a Gaussian centered around =k 4, or k k+2 lnk, ith idth k. Interestingly, the average eight of the nodes of degree k, viz., k k+2 lnk, slightly exceeds 27 the expected value k =k

4 T. ANTAL AND P. L. KRAPIVSKY PHYSICAL REVIEW E 71, i 0 n 1 0 = The density i 0, satisfies i 0, =0 dx x 0 +2xi 0,x FIG. 2. In- and out-components of node x. In this example, the out-component has size 3 and the in-component has size 5 node x itself belongs to its in- and out-components. The degree distribution does not admit a simple closed form even for the exponential link eight distribution. The exact expression n k =0 d k 1 e 2 k 1! 21 simplifies for k1. Indeed, utilizing to properties of the quantity k e /k!, viz., i it has a sharp maximum at =k and ii 0 d k e /k!=1, e estimate the integral in Eq. 21 as the value of the sloly varying part of the integrand /2 1 1+/2 2 near the maximum, i.e., at k k+2 lnk. Thus n k = 8 k 3 48ln k k 4 + Ok C. In-component eight distribution The emerging netork has the natural structure of a directed graph since each ne link starts at the ne node and ends up at some previous node. Taking into account the orientation of each link allos us to define an in-component and an out-component ith respect to each node. For instance, the in-component of node x is the set of all nodes from hich node x can be reached folloing a path of directed links Fig. 2. The computation of in- and out-component size distributions is quite complicated as one must first determine joint distributions that involve both size and eight. In contrast, in- and out-component eight distributions can be determined directly. Here e compute the in-component eight distribution. Let 0 be the eight of the link emanating from node x and the total eight of all other links in the in-component of node x; then the total eight of that in-component, that is, the sum of eights of all nodes in the in-component of node x, is Denote by i 0, the density of such incomponents. Here e tacitly assume that 0, that is, the size of the in-components is larger than 1. The density i 0 of in-components of size 1 and eight 0 is found by noting that such in-components are just dangling nodes, so 0 +2i 0, + i 0 0. For the exponential link eight distribution 12, this equation can be reritten in a simple form 2 G = G, 24 here G=G 0, is the auxiliary variable G =0 dxe x 0 +2xi 0,x e Solving Eq. 24, e obtain 2+ 0 G 0, = G 0, e. Equation 25 gives the initial condition, so the auxiliary variable reads 2 0 G = e 0. This result leads to the in-component density 2 0 i 0, = e 26 No the total eight of the in-component is s= 0 +2, and the respective eight distribution Is is s Is = is + 1 d 0 i 0,s 0 / Plugging Eqs. 23 and 26 into Eq. 27, e arrive at 1 Is = 2+s s s 2e s. 28 Therefore, up to an exponentially small correction, the incomponent eight distribution is algebraic ith exponent 2 the same exponent characterizes the in-component size distribution 28, hich is not so surprising as this exponent is found to be very robust. III. GENERALIZATIONS The model of the previous section alays results in a tree structure by construction. In this section, e consider to generalizations of that model hich allo the formation of loops in the evolving netork. The simplest generalization of the minimal model leading to a netork ith many loops is to connect a nely created node to m1 target nodes. The attachment probability is still

5 WEIGHT-DRIVEN GROWING NETWORKS proportional to the eight of the target node as before, and the eights of the m ne links are chosen independently from the link eight distribution. The eight of the nely introduced node becomes the sum of the m independent link eights, hich is then distributed according to m, the m-fold convolution of. The governing equation thus remains similar to Eq. 1 except that the last term becomes m, and the square bracketed term gains a factor m due to the m attached links. As the average node eight is also m times larger than previously, that is, =2m, and the node density satisfies equation 2 + n =0 dx xxnx +2 m, hich is almost identical to Eq. 2, the only difference is the second term on the right-hand side, hich no contains m instead of. The modified model leads to the same 3 tail for the node eight distribution for any ith a cutoff since then m also has a cutoff and the same argument applies as before. For the exponential link eight distribution =e the convoluted distribution is m =e m 1 /m 1!, and the resulting node eight distribution is algebraic up to an exponentially small correction, 2mm +3 n 2+ 3 for A more substantial generalization of the minimal model can be obtained by combining to processes: In addition to the original process of creating and linking ne nodes to the netork Sec. II, links beteen already existing nodes are created as ell 29. We again choose the simplest eightdriven rule, namely e assume that a ne link beteen to existing nodes is created ith a rate proportional to the product of their eights. Let r be the average number of link creations beteen already existing nodes per node creation, so in the netork ith N nodes the average number of links is 1+rN. This can be modeled by to independent processes: ith probability 1/1+r, a ne node is attached to the netork, and ith probability r/1+r, a ne link is added beteen to already existing nodes. The eights of the ne links created in both processes are dran independently from the same distribution. The node eight distribution N N satisfies an equation similar to Eq. 1, ith the term in the square brackets multiplied by 1 +2r/. The governing equation for the normalized eight distribution n is therefore almost identical to Eq. 2; the only change is the replacement =/1+2r, PHYSICAL REVIEW E 71, n dx xxnx =0 The average node eight = 0 dn is no equal to 21+r since the number of links 1+r times exceeds the number of nodes and each link contributes tice. Multiplying Eq. 30 by and integrating over, e indeed obtain =21+r 0 d, thereby providing a useful check of self-consistency. For the exponential link eight distribution 12, the node eight distribution satisfies r 2+2r e n =1+ 1+2r dxxe x nx. 2+2r0 Solving this equation, e again obtain the purely algebraic node eight distribution, n =1+ 1+2r, = 3+4r 2+2r 1+2r. The exponent monotonously decreases from 3 to 2 as r increases from 0 to. IV. CONCLUSIONS We examined a minimal model for the eight-driven netork groth. The virtue of the minimal model is that its many features, such as the node eight distribution, the joint node eight-degree distribution, the in-component eight distribution, etc., are tractable analytically. In particular, e shoed that the node eight distribution exhibits a universal 3 tail independently of the link eight distribution as long as the tail of the latter is sharper than 3, and the incomponent eight distribution displays a robust s 2 tail. Remarkably simple behaviors characterize the exponential link eight distribution: The emerging node eight distribution is purely algebraic, hile the joint node eightdegree distribution and the in-component eight distribution are also given by neat closed formulas. We also studied generalizations of the minimal model for the eight-driven netork groth. When a ne node is connected to several target nodes and/or links are additionally created beteen already existing nodes, the netork acquires loops yet it remains tractable. One generic feature of this class of models is that the node eight distribution remains algebraic, n, ith the exponent varying from 3 to 2 as the average node degree increases from 1 to. ACKNOWLEDGMENT T.A. thanks the Siss NSF for financial support. 1 S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Netorks: From Biological Nets to the Internet and WWW Oxford University Press, Oxford, 2003; R. Pastor-Satorras and A. Vespignani, Evolution and Structure of the Internet: A Statistical Physics Approach Cambridge University Press, Cambridge, P. Erdős and P. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5,

6 T. ANTAL AND P. L. KRAPIVSKY PHYSICAL REVIEW E 71, B. Bollobás, Random Graphs Academic, London, R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, P. L. Krapivsky and S. Redner, Comput. Net. 39, M. E. J. Neman, SIAM Rev. 45, M. E. J. Neman, Phys. Rev. E 64, ; 64, A.-L. Barabási, H. Jeong, Z. Neda, E. Ravasz, A. Schubert, and T. Vicsek, Physica A 311, R. Guimera, M. Sales-Pardo, and L. A. N. Amaral, e-print cond-mat/ A. Barrat, M. Barthélemy, and A. Vespignani, Proc. Natl. Acad. Sci. U.S.A. 101, L. R. Ford and D. R. Fulkerson, Flos in Netorks Princeton University Press, Princeton, NJ, R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Netork Flos: Theory, Algorithms, and Applications Prentice Hall, Engleood Cliffs, NJ, P. G. Doyle and J. L. Snell, Random Walks and Electric Netorks Math. Assoc. Amer., Washington, D.C., L. de Arcangelis, S. Redner, and A. Coniglio, Phys. Rev. B 31, ; 34, R. Rammal, C. Tannous, P. Breton, and A.-M. S. Tremblay, Phys. Rev. Lett. 54, M. E. J. Neman, e-print cond-mat/ E. Almaas, P. L. Krapivsky, and S. Redner, e-print cond-mat/ S. H. Yook, H. Jeong, A.-L. Barabási, and Y. Tu, Phys. Rev. Lett. 86, J. D. Noh and H. Rieger, Phys. Rev. E 66, D. Zheng, S. Trimper, B. Zheng, and P. M. Hui, Phys. Rev. E 67, R P. J. Macdonald, E. Almaas, and A.-L. Barabási, e-print condmat/ A. Barrat, M. Barthélemy, and A. Vespignani, Phys. Rev. Lett. 92, ; Phys. Rev. E 70, Negative eights are occasionally appropriate, e.g., they can represent animosity beteen individuals in a social netork. 24 The analyticity of the node eight distribution n breaks don at integer values as is obvious from Eq. 4. Differentiating Eq. 4, one can express F k 1 via F, F 1,,F k+1. Since F is continuous but not differentiable at =1, the cumulative distribution is continuously differentiable k 1 times at =k implying that the eight distribution is continuously differentiable k 2 times. 25 N is a discrete variable and N N are random variables. Treating N as a continuous variable and N N as the average values of the corresponding random variables is asymptotically exact hen the eight is sufficiently small; see, e.g., P. L. Krapivsky and S. Redner, J. Phys. A 35, for the detailed analysis of these issues in the model here groth is governed by preferential attachment. 26 For integer 4, the th term in expansion 11 acquires a logarithmic correction; for =3, even the leading-order term has a logarithmic correction n 3 ln. 27 The expected value for the sum of k independent identically distributed random variables taken from the exponential distribution is =k; in the present case, the average eight of the node of large degree is slightly higher since the groth is eight-driven. 28 P. L. Krapivsky and S. Redner, Phys. Rev. E 63, P. L. Krapivsky, G. J. Rodgers, and S. Redner, Phys. Rev. Lett. 86,