U. of Toronto Dept. of Comp. Science Dept. of Comp. Science

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1 On Power-Law Reationships of the Internet Topoogy Michais Faoutsos Petros Faoutsos U.C. Riverside U. of Toronto Dept. of Comp. Science Dept. of Comp. Science Christos Faoutsos * Carnegie Meon Univ. Dept. of Comp. Science christosqcs.cmu.edu Abstract Despite the apparent randomness of the Internet, we discover some surprisingy simpe power-aws of the Internet topoogy. These power-aws hod for three snapshots of the Internet, between November 1997 and December 1998, despite a 45% growth of its size during that period. We show that our power-aws fit the rea data very we resuting in correation coefficients of 96% or higher. Our observations provide a nove perspective of the structure of the Internet. The power-aws describe concisey skewed distributions of graph properties such as the node outdegree. In addition, these power-aws can be used to estimate important parameters such as the average neighborhood size, and faciitate the design and the performance anaysis of protocos. Furthermore, we can use them to generate and seect reaistic topoogies for simuation purposes. 1 Introduction What does the Internet ook ike? Are there any topoogica properties that don t change in time? ~ How wi it ook ike a year from now? How can I generate Internet-ike graphs for my simuations? These are some of the questions motivating this work. In this paper, we study the topoogy of the Internet and we identify severa power-aws. Furthermore, we discuss mutipe benefits from understanding the topoogy of the Internet. First, we can design more efficient protocos that take advantage of its topoogica properties. Second, we can create more accurate artificia modes for simuation purposes. And third, we can derive estimates for topoogica parameters (e.g. the average number of neighbors within h *This research was partiay funded by the Nationa Science Foundation under Grants No. IRI and DMS Aso, by the Nationa Science Foundation, ARPA and NASA under NSF Cooperative Agreement No. IRI , and by DARPA/ITO through Order F463, issued by ESC/ENS under contract N C-851. Additiona funding was provided by dooations from NEC and Inte. Views and concusions contained in this document are those of the authors and shoud not be interpreted as representing officia poicies, either expressed or impied, of the Defense Advanced Research Projects Agency or of the United States Government. Permission to make digita or hard copies of a or part of this work for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. To copy otherwise, to repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. SIGCOMM Cambridge, MA, USA Q 1999 ACM /99/ $5.00 hops) that are usefu for the anaysis of protocos and for specuations of the Internet topoog in the future. Modeing the Internet topoogy P. is an important open probem despite the attention it has attracted recenty. Paxson and Foyd consider this probem as a major reason Why We Don t Know How To Simuate The Internet [16]. Severa graph-generator modes have been proposed [23] [5] [27], but the probem of creating reaistic topoogies is not yet soved; the seection of severa parameter vaues are eft to the intuition and the experience of each researcher. As our primary contribution, we identify three poweraws for the topoogy of the Internet over the duration of a year in Power-aws are expressions of the form y 0: zq, where a is a constant, z and y are the measures of interest, and o( stands for proportiona to. Some of those exponents do not change significanty over time, whie some exponents change by approximatey 10%. However, the important observation is the existence of power-aws, i.e., the fact that there is some exponent for each graph instance. During 1998, these power-aws hod in three Internet instances with good inear fits in og-og pots; the correation coefficient of the fit is at east 96% and usuay higher than 98%. In addition, we introduce a graph metric to quantify the density of a graph and propose a rough power-aw approximation of that metric. Furthermore, we show how to use our poweraws and our approximation to estimate usefu parameters of the Internet, such as the average number of neighbors within h hops. Finay, we focus on the generation of reaistic graphs. Our power-aws can hep verify the reaism of synthetic topoogies. In addition, we measure severa crucia parameters for the most recent graph generator [27]. Our work in perspective. Our work is based on three Internet instances over a one-year period. During this time, the size of the network increased substantiay (45%). Despite this, the sampe space is rather imited, and making any generaizations woud be premature unti additiona studies are conducted. However, the authors beieve that these power-aws characterize the dynamic equiibrium of the Internet growth in the same way power-aws appear to describe various natura networks such as the the human respiratory system [12], and automobie networks [6]. At a more practica eve, the reguarities characterize the topoogy concisey during If this time period turns out to be a transition phase for the Internet, our observations wi obviousy be vaid ony for In absence of revoutionary In this paper, we use the expression the topoogy of the Internet, athough the topoogy changes and it woud be more accurate to tak about Internet topoogies. We hope that this does not misead or confuse the reader. 251

2 changes, it is reasonabe to expect that our power-aws wi continue to hod in the future. The rest of this paper is structured as foows. In Section 2, we present some definitions and previous work on measurements and modes for the Internet. In Section 3, we present our Internet instances and provide usefu measurements. In Section 4, we present our three observed poweraws and our power-aw approximation. In Section 5, we expain the intuition behind our power-aws, discuss their use, and show how we can use them to predict the growth of the Internet. In Section 6, we concude our work and discuss future directions. 2 Background and Previous Work The Internet can be decomposed into connected subnetworks that are under separate administrative authorities, as shown in Figure 1. These subnetworks are caed domains or autonomous systems. This way, the topoogy of the Internet can be studied at two different granuarities. At the router eve, we represent each router by a node [14]. At the inter-domain eve, each domain is represented by a singe node [o] and each edge is an inter-domain interconnection. The study of the topoogy at both eves is equay important. The Internet community deveops and empoys different protocos inside a domain and between domains. An intra-domain protoco is imited within a domain, whie an inter-domain protoco runs between domains treating each domain as one entity. Symbo Definition G An undirected graph. N E Number of nodes in a graph. Number of edges in a graph. 6 The diameter of the graph. dv Outdegree of node u. 2 The average outdegree of the nodes of a 1 graph: 6=2 E/N - Tabe 1: Definitions and symbos. Metrics. The metrics that have been used so far to describe graphs are mainy the node outdegree, and the distances between nodes. Given a graph, the outdegree of a node is defined as the number of edges incident to the node (see Tabe 1). The distance between two nodes is the number of edges of the shortest path between the two nodes. Most studies report minimum, maximum, and average vaues and pot the outdegree and distance distribution. We denote the number of nodes of a graph by N, t.he number of edges by E, and the diameter of the graph by 6. Rea network studies. Govindan and Reddy [o] study the growth of the inter-domain topoogy of the Internet between 1994 and The graph is sparse with 75% of the nodes having outdegrees ess or equa to two. They distinguish four groups of nodes according to their outdegree. The authors observe an increase in the connectivity over time. Pansiot and Grad [14] study the topoogy of the Internet in The definition of an autonomous system can vary in the iterature, but it usuay coincides with that of the domain [IO] at the router eve. The distances they report are approximatey two times arger compared to those of Govindan and Reddy. This eads to the interesting observation that, on average, one hop at the inter-domain eve corresponded to two hops at the router eve in Generating Internet Modes. Regarding the creation of reaistic graphs, Waxman introduced what seems to be one of the most popuar network modes [23]. These graphs are created probabiisticay considering the distance between nodes in a Eucidean sense. This mode was successfu in representing sma eary networks such as the ARPANET. As the size and the compexity of the network increased more detaied modes were needed [5] [27]. In the most recent work, Zegura et a. [27] introduce a comprehensive mode that incudes severa previous modes 3. They ca their mode transit-stub, which combines simpe topoogies (e.g. Waxman graphs and trees) in a hierarchica structure. There are severa parameters that contro the structure of the graph. For exampe, parameters define the tota number and the size of the stubs. An advantage of this mode ies in its abiity to describe a number of topoogies. At the same time, a researcher needs experimenta estimates to set vaues to the parameters of the mode. Power-aws in communication networks. Power-aws have been used to describe the traffic in communications networks, but not their topoogy. Actuay, both sef-simiarity, and heavy tais appear in network traffic and are both reated to power-aws. A variabe X foows a heavy tai distribution if P[X > z] = kaea L(z), where k E 8?+ and L(z) is a sowy varying function: Zimt,,[L(tx)/L(x)] = 1 [20] [24]. A Pareto distribution is a specia case of a heavy tai distribution with P[X > z] = k x-=. It is easy to see that power-aws, Pareto and heavy-taied distributions are intimatey reated. In a pioneering work, Leand et a. [] show the sef-simiar nature of Loca Area Network (LAN) traffic. Second, Paxson and Foyd [15] provide evidence of sef simiarity in Wide Area Network (WAN) traffic. In modeing the traffic, Wiinger et a. [25] provide structura modes that describe LAN traffic as a coective effect of simpe heavytaied ON-OFF sources. Finay, Wiinger et a. [24] bring a of the above together by describing LAN and WAN traffic through structura modes and showing the reation of the sef-simiarity at the macroscopic eve of WANs with the heavy-taied behavior at the microscopic eve of individua sources. In addition, Crovea and Bestavros use power-aws to describe traffic patterns in the Word Wide Web [3]. At an intuitive eve, the previous works seem to attribute the heavy-taied behavior of the traffic to the heavy-taied distribution of the size of the transmitted data fies, and to the heavy-taied characteristics of the human-computer interaction. Recenty, Chuang and Sirbu [2] use a power-aw to estimate the size of muticast distribution trees. Note that in a foow-up work, Phiips et a. [17] verify the reasonabe accuracy of the Chuang-Sirbu scaing aw for practica purposes, but they aso propose an estimate that does not foow a power-aw. 3 Internet Instances In this section, we present the Internet instances we acquired and we study their evoution in time. We examine the inter-domain topoogy of the Internet from the end of 1997 unti the end of We use three rea graphs that correspond to six-month intervas approximatey. The data 3The graph generator software is pubicy avaiabe [27]. 252

3 Domain 2 Domain 3 Domain 2 Domain 3 (4 1: Host : 0 Router i 0 Domain (b) Figure 1: The structure of Internet at a) the router eve and b) the inter-domain eve. The hosts connect to routers in LANs. E 4600 L "r 4wov) : E s 2 's E z' Nov "%?i:%s Internet Growth - Dec1996 Figure 2: The growth of the Internet: the number of domains versus time between the end of 1997 unti the end of is provided by the Nationa Laboratory for Appied Network Research [9 The information is coected by a route server from BGP a. routing tabes of mutipe geographicay distributed routers with BGP connections to the server. We ist the three dataset.s that we use in our paper, and we present more information in Appendix A. Int-11-97: the inter-domain topoogy of the Internet in November of 1997 with 3015 nodes, 5156 edges, and 3.42 avg. outdegree. Int-04-98: the inter-domain topoogy of the Internet in Apri of 1998 with 3530 nodes, 6432 edges, and 3.65 avg. outdegree. Int-12-98: the inter-domain topoogy of the Internet in December of 1998 with 4389 nodes, 8256 edges, and 3.76 avg. outdegree. Note that the growth of the Internet in the time period we study is 45% (see Figure 2). The change is significant, and it ensures that the three graphs refect different instances of an evoving network. Athough we focus on the Internet topoogy at the interdomain eve, we aso examine an instance at the router 4BGP stands for the Border Gateway Protoco [19], and it is the inter-domain routing protoco. 1 eve. The graph represents the topoogy of the routers of the Internet in 1995, and was tediousy coected by Pansiot and Grad [14]. Rout-95: the routers of the Internet in 1995 with 3888 nodes, 5012 edges, and an average outdegree of Ceary, the above graph is consideraby different from the first three graphs. First of a, the graphs mode the topoogy at different eves. Second, the Rout-95 graph comes from a different time period, in which Internet was in a fairy eary phase. To faciitate the graph generation procedures, we anayze the Internet in a way that suits the graph generator modes [27]. Namey, we decompose each graph in two components: the tree component that contains a nodes that beong excusivey to trees and the core component that contains the rest of the nodes incuding the roots of the trees. We report severa interesting measurements in Appendix A. For exampe, we find that 40-50% of the nodes beong to trees. Aso, 80% the trees have a depth of one, whie the maximum tree depth is three. 4 Power-Laws of the Internet In this section, we observe three of power-aws of the Internet topoogy. Namey, we propose and measure graph properties, which demonstrate a reguarity that is unikey to be a coincidence. The exponents of the power-aws can be used to characterize graphs. In addition, we introduce a graph metric that is taiored to the needs of the compexity anaysis of protocos. The metric refects the density or the connectivity of nodes, and we offer a rough approximation of its vaue through a power-aw. Finay, using our observations and metrics, we identify a number of interesting reationships between important graph parameters. In our work, we want to find metrics or properties that quantify topoogica properties and describe concisey skewed data distributions. Previous metrics, such as the average outdegree, fai to do so. First, metrics that are based on minimum, maximum and average vaues are not good descriptors of skewed distributions; they miss a ot of information and probaby the interesting part that we woud want to capture. Second, the pots of the previous metrics are difficut to quantify, and this makes difficut the comparison of graphs. Ideay, we want to describe a pot or a distribution with one number. 253

4 , P(h) The frequency of an outdegree, d, is the num- The rank of a node, v, is its index in the order The number of pairs of nodes is the tota number of pairs of nodes within ess or equa to h hops, incuding sef-pairs, and counting a other nairs twice. NN(h) The average number of nodes in a neighborhood of h hops. x The eigen vaue of a square matrix A: 3x E RN and Ax = Xx. i The order of A; in xi 2 AZ... 1 AN Tabe 2: Nove definitions and their symbos. To express our power-aws, we introduce severa graph metrics that we show in Tabe 2. We define frequency, fd, of some outdegree, d, to be the number of nodes that have this outdegree. If we sort the nodes in decreasing outdegree sequence, we define rank, rv, to be the index of the node in the sequence, whie ties in sorting are broken arbitrariy. We define the number of pairs of nodes P(h) to be the tota number of pairs of nodes within ess or equa to h hops, incuding sef-pairs, and counting a other pairs twice. The use of this metric wi become apparent ater. We aso define NN(h) to be the average number of nodes in a neighborhood of h hops. Finay, we reca the definition of the eigenvaues of a graph, which are the eigenvaues of its adjacency matrix. In this section, we use inear regression to fit a ine in a set of two-dimensiona points [18]. The technique is based on the east-square errors method. The vaidity of the approximation is indicated by the correation coefficient which is a number between -1.0 and 1.0. For the rest of this paper, we use the absoute vaue of the correation coefficient, ACC. An ACC vaue of 1.0 indicates perfect inear correation, i.e., the data points are exacty on a ine. 4.1 The rank exponent R In this section, we study the outdegrees of the nodes. We sort the nodes in decreasing order of outdegree, d,, and pot the (ru, d,) pairs in og-og scae. The resuts are shown in Figures 3 and 4. The measured data is represented by diamonds, whie the soid ine represents the Ieaat-squares approximation. A striking observation is that the pots are approximated we by the inear regression. The correation coefficient is higher than for the inter-domain graphs and for the Rout-95 graph. This eads us to the foowing power-aw and definition. Power-Law 1 (rank exponent) The outdegree, d,, of a node v, is proportiona to the rank of the node, rv, to the power of a constant, R: d,, K r? Definition 1 Let us sort the nodes of a graph in decreasing order of outdegree. We define the rank exponent, R, to be 4 the sope of the pot of the outdegrees of the nodes versus the rank of the nodes in og-og scae. For the three inter-domain instances, the rank exponent, R, is -0.81, and in chronoogica order as we see in Appendix B. The rank exponent of the Rout-95 graph, -0.48, is different compared to that of the first three graphs. This is something that we expected, given the differences in the nature of the graphs. On the other hand, this difference suggests that the rank exponent can distinguish graphs of different nature, athough they both foow Power-Law 1. This property can make the rank exponent a powerfu metric for characterizing famiies of graphs, see Section 5. Intuitivey, Power-Law 1 most ikey refects a principe of the way domains connect; the inear property observed in our four graph instances is unikey to be a coincidence. The power-aw seems to capture the equiibrium of the tradeoff between the gain and the cost of adding an edge from a financia and functiona point of view, as we discuss in Section 5. Extended Discussion - Appications. We can estimate the proportionaity constant for Power-Law 1, if we require that the minimum outdegree of the graph is one (dn = 1). This way, we can refine the power-aw as foows. Lemma 1 The outdegree, d,, of a node v, is a function of the rank of the node, rv and the rank exponent, I?., as foows 1 d, = ~a rc Proof. The proof can be found in Appendix C. Finay, using emma 1, we reate the number of edges with the number of nodes and the rank exponent. Lemma 2 The number of edges, E, of a graph can be estimated as a function of the number of nodes, N, and the rank exponent, R, as foows: EC 2(7Z+1) (1 - Proof. The proof can be found in Appendix C. Note that Lemma 2 can give us the number of edges as a function of the number of nodes for a given rank exponent. We tried the emma in our datasets and the estimated number of edges differed by 9% to 20% from the actua number of edges. More specificay for the Int-12-98, the emma underestimates the number of edges by 10%. We can get a coser estimate (3.6%) by using a simpe inear interpoation in the number of edges given the number of nodes. Note that the two prediction mechanisms are different: our emma does not need previous network instances, but it needs to know the rank exponent. However, given previous network instances, we seem to be better off using the inear interpoation according to the above anaysis. We examined the sensitivity of our emma with respect to the vaue of rank exponent. A 5% increase (decrease) in the absoute vaue of the rank exponent increases (decreases) the number of edges by 10% for the number of nodes in Int The outdegree exponent 0 In this section, we study the distribution of the outdegree of the graphs, and we manage to describe it concisey by a singe number. Reca that the frequency, fd, of an outdegree, d, is the number of nodes with outdegree d. We pot the frequency fd versus the outdegree d in og-og scae in 254

5 " rank exp( ) x "( ) " rank" exp( ) 'x*'( ) (a) Int (b) Int Figure 3: The rank pots. Log-og pot of the outdegree d, versus the rank ru in the sequence of decreasing outdegree... ' rank" : exp( ) 'x"( ) - : ooOe oo (a) Int (b) Rout-95 Figure 4: The rank pots. Log-og pot of the outdegree d, versus the rank r,, in the sequence of decreasing outdegree. figures 5 and 6. In these pots, we excude a sma percent- The second striking observation is that the vaue of the age of nodes of higher outdegree that have frequency of one. outdegree exponent is practicay constant in our graphs of Specificay, we pot the outdegrees starting from one unti the inter-domain topoogy. The exponents are -2.15, we reach an outdegree that has frequency of one. As we saw and -2.2, as shown in Appendix B. It is interesting to note earier, the higher outdegrees are described and captured by that even the Rout-95 graph obeys the same power-aw (Figthe rank exponent. In any case, we pot more than 98% of ure 6.b) with an outdegree exponent of These facts the tota number of nodes. The soid ines are the resut of suggest that Power-Law 2 describes a fundamenba property the inear regression. The major observation is that the pots are approximatey inear (see Tabe 8). The correation coefficients are between for the inter-domain graphs and for the Rout-95. This eads us to the foowing power-aw and definition. Power-Law 2 (outdegree exponent) The frequency, fd, of an outdegree, d, is proportiona to the outdegree to the power of a constant, C? of the network. The intuition behind this power-aw is.that the distribution of the outdegree of Internet nodes is not arbitrary. The quaitative observation is that ower degrees are more frequent. Our power-aw manages to quantify this observa- tion by a singe number, the outdegree exponent. This way, we can test the reaism of a graph with a simpe numerica comparison. If a graph does not foow Power-Law 2, or if its outdegree exponent is consideraby different from the rea exponents, it probaby does not represent a reaistic topoogy. Definition 2 We define the outdegree exponent, 0, to be the sope of the pot of the frequency of the outdegrees versus the outdegrees in og-og scae. 4.3 The hop-pot exponent Y In this section, we quantify the connectivity and distances between the Internet nodes in a nove way. We choose to 255

6 ut ~~( ) * x * ( ) Ut exp( ) * x ** ( ) (a) Id-:10-W (b) Int-ii Figure 5: The outdegree pots: Log-og pot of frequency fd versus the outdegree d ur exp( ) x ** ( ) - -r0utes.out~ exp( ) * x ( ) - : (a) Int-1198 oo (b) Rou:95 oo Figure 6: The outdegree pots: Log-og pot of frequency fd versus the outdegree d. study the size of the neighborhood within some distance, instead of the distance itsef. Namey, we use the tota number of pairs of nodes P(h) within h hops, which we define as the tota number of pairs of nodes within ess or equa to h hops, incuding sef-pairs, and counting a other pairs twice. Let us see the intuition behind the number of pairs of nodes P(h). For h = 0, we ony have the sef-pairs: P(0) = N. For the diameter of the graph 6, h = 6, we have the sefpairs pus a the other possibe pairs: P(b) = N*, which is the maximum possibe number of pairs. For a hypothetica ring topoogy, we have P(h) oc h, and, for a 2-dimensiona grid, we have P(h) 0: h*, for h < b. We examine whether the number of pairs P(h) for the Internet foows a simiar power-aw. In figures 7 and 8, we pot the number of pairs P(h) as a function of the number of hops h in og-og scae. The data is represented by diamonds, and the dotted horizonta ine represents the maximum number of pairs, which is iv*. We want to describe the pot by a ine in east-squares fit, for h < 6, shown as a soid ine in the pots. We approximate the first 4 hops in the inter-domain graphs, and the first 12 hops in the Rout-95. The correation coefficients are is 0.98 for inter-domain graphs and 0.96, for the Rout-95, as we see in Appendix B. Unfortunatey, four points is a rather sma number to verify or disprove a inearity hypothesis experimentay. However, even this rough approximation has severa usefu appications as we show ater in this section. Approximation 1 (hop-pot exponent) The tota number of pairs of nodes, P(h), within h hops, is proportiona to the number of hops to the power of Definition 3 Let us pot the number of pairs of nodes, P(h), within h hops uersus the number of hops in og-og scae. For h < 6, we define the sope of this pot to be the hop-pot exponent,%. Observe that the three inter-domain datasets have practicay equa hop-pot exponents; 4.6,4.7, and 4.86 in chronoogica order, as we see in Appendix B. This shows that the hop-pot exponent describes an aspect of the connectivity of the graph in a singe number. The Rout-95 pot, in fig. 8.b, 256

7 1e+11 \ 1e+O - e+09 r hoppot exp( ) * x * ( ) - : maximum number of pairs... 1e+11 t e+o - 1e+o9 I hoppot exp( ) * x** ( ) - : maximum number of pairs... e+ob e+o6 7 e+07 - e+07 - e+o6 r e+o (a) Int (b) Int Figure 7: The hop-pots: Log-og pots of the number of pairs of nodes P(h) within h hops versus the number of hops h. ;., [:I maxmum number of paws... ;.f i/ maxmum number of pairs ,e--::;... ; =J$-/+ I :I: 1 10 (a) Int t 1 (b) Rou::95 J 100 Figure 8: The hop-pots: Log-og pots of the number of pairs of nodes P(h) within h hops versus the number of hops h. ham more points, and thus, we can argue for its inearity with more confidence. The hop-pot exponent of Rout-95 is 2.8, which is much different compared to those of the inter-domain graphs. This is expected, since the Rout-95 is a sparser graph. Reca that for a ring topoogy, we have 3t = 1, and, for a 2-dimensiona grid, we have H = 2. The above observations suggest that the hop-pot exponent can distinguish famiies of graphs efficienty, and thus, it is a good metric for characterizing the topoogy. Extended Discussion - Appications. We can refine Approximation 1 by cacuating its proportionaity constant. Let us reca the definition of the number of pairs, P(h). For h = 1, we consider each edge twice and we have the sef-pairs, therefore: P(1) = N + 2 E. We demand that Approximation 1 satisfies the previous equation as an initia condition. Lemma 3 The number of pairs within h hops is h < 6 h>b where c = N + 2 E to satisfy initia conditions. In networks, we often need to reach a target without knowing its exact position [7] []. In these cases, seecting the extent of our broadcast or search is an issue. On the one hand, a sma broadcast wi not reach our target. On the other hand, an extended broadcast creates too many messages and takes a ong time to compete. Ideay, we want to know how many hops are required to reach a sufficienty arge part of the network. In our hop-pots, a promising soution is the intersection of the two asymptote ines: the horizonta one at eve N2 and the asymptote with sope?y. We cacuate the intersection point using Lemma 3, and we define: Definition 4 (effective diameter) Given a graph with iv nodes, E edges, and 7- hop-pot exponent, we define the effective diameter, 6,f, as: /N N2 &f= - ( N+2E > Intuitivey, the effective diameter can be understood as foows: any two nodes are within 6,s hops from each other with high probabiity. We verified the above statement experimentay. The effective diameters of our inter-domain 257

8 4500 I Actua using hoppot using avg OutdeQree a../ wo t /(I/ :, -_ --._ Number of Hops Figure 9: Average neighborhood size versus number of hops the actua, and estimated size a) using hop-pot exponent, b) using the average outdegree for Int Hops hop-pot avg. outdegree I 2 I I I I 3 I I I I 4 I 0.17 I I Tabe 3: The reative error of the two estimates for the average neighborhood size with respect to the rea vaue. Negative error means under-estimate. graphs was sighty over four. Rounding the effective diameter to four, approximatey 80% of the pairs of nodes are within this distance. The ceiing of the effective diameter is five, which covers more than 95% of the pairs of nodes. An advantage of the effective diameter is that it can be cacuated easiy, when we know N, and U. Reca that we can cacuate the number of edges from Lemma 2. Given that the hop-pot exponent is practicay constant, we can estimate the effective diameter of future Internet instances as we do in Section 5. Furthermore, we can estimate the average size of the neighborhood, NN(h), within h hops using the number of pairs P(h). Reca that P(h) - N is the number of pairs without the sef-pairs. NN(h) = 7-1 Using Equation 1 and Lemma 3, we can estimate the average neighborhood size. Lemma 4 The average size of the neighborhood, NIV(h), within h hops as a function of the hop-pot exponent, f, for h << 6, is NIV(h) = + hs - 1 where c = N + 2 E to satisfy initia conditions. The average neighborhood is a commony used parameter in the performance of network protocos. Our estimate is an improvement over the commony used estimate that uses the average outdegree outdegree estimate: [26] [7] which we ca average- NN (h) = d (6- )h- In figure 9, we pot the actua and both estimates of the average neighborhood size versus the number of hops for the Int graph. In Tabe 3, we show the normaized error of each estimate: we cacuate the quantity: (p - r)/r where p the prediction and T the rea vaue. The resuts for the other inter-domain graphs are simiar. The superiority of the hop-pot exponent estimate is apparent compared to the average-outdegree estimate. The discrepancy of the averageoutdegree estimate can be expained if we consider that the estimate does not compy with the rea data; it impicity assumes that the outdegree distribution is uniform. In more detai, it assumes that each node in the periphery of the neighborhood adds d - 1 new nodes at the next hop. Our data shows that the outdegree distribution is highy skewed, which expains why the use of the hop-pot estimate gives a better approximation. The most interesting difference between the two estimates is quaitative. The previous estimate considers the neighborhood size exponentia in the number of hops. Our estimate considers the neighborhood as an?--dimensiona sphere with radius equa to the number of hops, which is a nove way to ook at the topoogy of a network5. Our data suggests that the hop-pot exponent-based estimate gives a coser approximation compared to the average-outdegreebased metric. 4.4 The eigen exponent & In this section, we identify properties of the eigenvaues of our Internet graphs. There is a rich iterature that proves that the eigenvaues of a graph are cosey reated to many basic topoogica properties such as the diameter, the number of edges, the number of spanning trees, the number of connected components, and the number of waks of a certain ength between vertices, as we can see in [8] and [4]. A of the above suggest that the eigenvaues intimatey reate to topoogica properties of graphs. We pot the eigenvaue Xi versus i in og-og scae for the first 20 eigenvaues. Reca that i is the order of X; in the decreasing sequence of eigenvaues. The resuts are shown in Figure 10 and Figure 11. The eigenvaues are shown as diamonds in the figures, and the soid ines are approximations using a east-squares fit. Observe that in a graphs, the pots are practicay inear with a correation coefficient of 0.99, as we see in Appendix B. It is rather unikey that such a canonica form of the eigenvaues is purey coincidenta, and we therefore conjecture that it constitutes an empirica power-aw of the Internet topoogy. Power-Law 3 (eigen exponent) TILe eigenvaues, A;, of a graph are proportiona to the order, i, to the power of a constant, E: Xi o( i Definition 5 We define the eigen exponent, E, to be the sope of the pot of the sorted eigenvaues versus their order in og-og scae. 5Note that our resuts focus on reativey sma neighborhoods compared to the diameter h <( 6. Other experimenta studies focus on neighborhoods of arger radius [17]. 258

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11 ber of nodes, the number neighborhood size. of edges, and the average We propose power-aw exponents, instead of averages, as an efficient way to describe the highy-skewed graph metrics which we examined..4part from their theoretica interest, we showed a number of practica appications of our power-aws. First, our power-aws can assess the reaism of synthetic graphs, and enhance the vaidity of our simuations. Second, they can hep anayze the average-case behavior of network protocos. For exampe, we can estimate the message compexity of protocos using our estimate for the neighborhood size. Third, the power-aws can hep answer what-if scenarios ike what wi be the diameter of the Internet, when the number of nodes doubes? ( what wi be the number of edges then? In addition, we decompose and measure the Internet in a way that reates to the state-of-the-art graph generation modes. This decomposition provides measurements that faciitate the seection of parameters for the graph generators. For the future, we beieve that our suggestion to ook for power-aws wi open the foodgates to discovering many additiona power-aws of the Internet topoogy. Our optimism is based on two facts: (a) power-aws are intimatey reated to fractas, chaos and sef-simiarity [21] and (b) there is overwheming evidence that sef-simiarity appears in a arge number of settings, ranging from traffic patterns in networks [24], to bioogica and economica systems [12]. ACKNOWLEDGMENTS. We woud ike to thank Mark Craven, Danie Zappaa, and Adrian Perrig for their hep in earier phases of this work. The authors are gratefu to Pansiot and Grad for providing the Rout-95 routers data. We woud aso ike to thank Vern Paxson, and Een Zegura for the thorough review and vauabe feedback. Finay, we woud ike to thank our mother Sofia Faoutsou-Kaamara and dedicate this work to her. References [31 [41 [51 [61 Christos Faoutsos and Ibrahim Kame. Beyond uniformity and independence: Anaysis of R-trees using the concept of fractai dimension. In- Proc. ACM SIGACT-SIGMOD- SIGART PODS, oages Minneanois. MN. Mav Aso avaiide-% CS-Tk-3198, GMIkCS-i R M. Faoutsos, A. Banerjea, and R. Pankaj. QoSMIC: a &OS Muticast Internet protoco. ACM SIGCOMM. Computer Communication Review., Sep 2-4, Vancouver BC K. Carberg and J. Crowcroft. Buiding shared trees using a one-to-many joining mechanism. ACM Computer Communication Review, pages 5-11, January J. Chuang and M. Sirbu. Pricing muticast communications: A cost based approach. In Proc. of the INET 98, M. Crovea and A. Bestavros. Sef-simiarity in Word Wide Web traffic, evidence and possibe causes. SIGMETRICS. pages 160-i69, D. M. Cvetkovit, M. Boob, and H. Sachs. Spectra of Graphs. Academic press, M. Doar. A better mode for generating test networks. Proc. Goba Internet, IEEE, Nov M. Faoutsos, P. Faoutsos, and C. Faoutsos. Power-aws of the Internet topoogy. Technica Report UCR-CS-99-01, University of Caifornia Riverside, Computer Science, Nationa Laboratory for Appied Network Research. Routing data. Supported by NSF, [ B. Mandebrot. I+acta Geometry of Nature. W.H. Freeman, New York, [I31 [I41 P51 WI [I71 P-4 PO1 [ =I [231 P Water Wiinger, Murad Taqqu, Robert Sherman, and Danie V. Wison. Sef-simiarity through high variabiity: statistica anaysis of ethernet LAX traffic at the source eve. ACM SIGCOMM 95. Computer Communication Review, 25: , [261 D. Zappaa, D. Estrin, and S. Shenker. Aternate path routing and pinning for interdomain muticast routing. Technica Report USC CS TR , U. of South Caifornia, P71 E. W. Zegura, K. L. Cavert, and M. J. Donahoo. A quantitative comparison of graph-based modes for internetworks. IEEE/ACM Transactions on Networking, 5(6): , December R. Govindan and A. Reddy. An anaysis of internet interdomain topoogy and route stabiity. Proc. IEEE INFO- COM, Kobe, Japan, Apri W.E. Leand, M.S. Taqqu, W. Wiinger, and D.V. Wison. On the sef-simiar nature of ethernet traffic. IEEE Transactions on Networking, 2(1):1-15, February (earier version in SIGCOMM 93, pp ). J. Moy. Muticast routing extensions for OSPF. ACM Connunications, 37(8):61-66, J.-J. Pansiot and D Grad. On routes and muticast trees in the Internet. ACM Computer Communication Review, 28(1):41-50, January V. Paxson and S. Foyd. Wide-area traffic: The faiure of Poisson modeing. IEEE/ACM Transactions on Networking, 3(3): , June (earier version in SIGCOMM SI, pp ). V. Paxson and S. Foyd. Why we don t know how to simuate the internet. Proceedings of the 1997 Winter Simuation Conference, December G. Phiips, S. Shenker, and H. Tangmunarunkit. Scaing of muticast trees: Comments on the chuang-sirbu scaing aw. ACM SIGCOMM. Computer Communication Review., Sep Wiiam H. Press, Sau A. Teukosky, Wiiam T. Vettering, and Brian P. Fannery. Numerica Recipes in C. Cambridge University Press, 2nd edition, Y. Rekhter and T. Li (Eds). A Border Gateway Protoco 4 (BGP-4). Internet-Draft.:draft-ietf-idr-bgp4-08.txt avaiabe from ftp://ftp.ietf.org/internet-drafts/, S. R. Resnick. Heavy tai modeing and teetraffic data. Annas of Statistics, 25(5): , Manfred Schroeder. +actas, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman and Company, New York, D. Waitzman, C. Partridge, and S. Deering. Distance vector muticast routing protoco. IETF RFC 1075, B. M. Waxman. Routing of mutipoint connections. IEEE Journa of Seected Areas in Communications, pages , W. Wiinger, V. Paxson, and M.S. Taqqu. Sef-simiarity and heavy tais: Structura modeing of network traffic. In A Practica Guide to Heavy Tais: Statistica Techniques and Appications, Ader, R., Fedman, R., and Taqqu, M.S., editors, Birkhauser. G.K. Zipf. Humun Behauior and Principe of Least Efort: An Introduction to Human Ecoogy. Addison Wesey, Cambridge, Massachusetts,

12 Int Int Int nodes edges avg outdegree max outdegree diameter avg. distance Tabe 6: The evoution of the Internet at the inter-domain eve. Int Int Int #nodes in trees (%) #trees over #nodes (%) max depth avg. tree size core outdegree Tabe 7: The evoution of the Internet considering the core and the trees. A Decomposing the Internet We anayze the Internet in a way that suits the graph generator modes [27]. The measurements we present can faciitate the seection of parameters for these generators. We study the graphs through their decomposition into two components: the tree component that contains a nodes that beong excusivey to trees and the core component that contains the rest of the nodes incuding the roots of the trees. We measure severa parameters from this decomposition that are shown in Tabe 7. These resuts eads to the foowing observations. Approximatey haf of the nodes are in trees 40-50% The number of nodes in trees decreased with time by 10% means that the Internet becomes more connected a around. The maximum tree depth is 3, however more than 80% of the trees have depth one. More than 95% of the tree-nodes have a degree of one. This eads to the foowing interesting observation: if we remoue the nodes with outdegree one from the origina graph, we practicay get the core component. These observations can hep users seect appropriate vaues for the parameters used in various graph generation techniques [27]. 1 Exponent 1 Int Int Int Rout-95 1 rank ACC outdeeree ACC um I s hop-pot ACC eigen ACC Tabe 8: An overview of a the exponents for a our graphs. Note that ACC is the absoute vaue of the correation coefficient. C The Proofs Here we prove the Lemmas we present in our paper. Lemma 1. The outdegree, d,, of a node v, is a function of the rank of the node, rv and the rank exponent, R, as foows d, = & rr Proof. We can estimate the proportionaity constant,c, for Power-Law 1, if we require that the outdegree of the N-th node is one, dn = 1. dn = CN R * C = /N (2) We combine Power-Law 2 with Equation 2, and concude the proof. n Lemma 2. The number of edges, E, of a graph can be estimated as a function of the number of nodes, N, and the rank exponent, R, as foows: E= 1 2 (R+) (1 - j&i) N Proof: The sum of a the outdegrees for a the ranks is equa to two times the number of edges, since we count each edge twice. 2E = ed, In the ast step, above we approximate the summation with an integra. Cacuating the integra concudes the proof. n B The Exponents of Our Power-Laws We present the exponents of our power-aws in Tabe 8.