Compact Routing on Internet-Like Graphs

Transcription

1 Compact Routing on Internet-Like Graphs Dmitri Krioukov Kevin Fall Intel Research, Berkeley Xiaowei Yang Massachusetts Institute of Technology Abstract The Thorup-Zwick (TZ) compact routing scheme is the first generic stretch-3 routing scheme elivering a nearly optimal per-noe memory upper boun. Using both irect analysis an simulation, we erive the stretch istribution of this routing scheme on Internet-like inter-omain topologies. By investigating the TZ scheme on ranom graphs with power-law noe egree istributions, P k k γ, we fin that the average TZ stretch is quite low an virtually inepenent of γ. In particular, for the Internet inter-omain graph with γ., the average TZ stretch is aroun., with up to 70% of all pairwise paths being stretch- (shortest possible). As the network grows, the average stretch slowly ecreases. We fin routing table sizes to be very small (aroun 50 recors for 0 4 -noe networks), well below their theoretical upper bouns. Furthermore, we fin that both the average shortest path length (i.e. istance) an with of the istance istribution σ observe in the real Internet inter- AS graph have values that are very close to the minimums of the average stretch in the - anσ-irections. This leas us to the iscovery of a unique critical point of the average TZ stretch as a function of an σ. The Internet s istance istribution is locate in a close neighborhoo of this point. This is remarkable given the fact that the Internet inter-omain topology has evolve without any irect attention pai to properties of the stretch istribution. It suggests the average stretch function may be an inirect inicator of the optimization criteria influencing the Internet s inter-omain topology evolution. Inex Terms Routing, Internet Topology, Simulations, Graph Theory, Combinatorics, Statistics. I. INTRODUCTION The question as to what rives the evolutionary process of the Internet s topology is of interest to many researchers. While various moels of its topological structure appear to escribe it reasonably well, most neither ai in unerstaning why the Internet s graph has evolve as it has, nor offer the metric which is effectively being optimize by its implementers as it grows. In aition, as the network grows, its global routing scalability is being stresse [], leaing several groups to explore alternatives to the present Internet routing system. We believe that a better unerstaning of the Internet s topological growth process, couple with knowlege of the theoretical unerpinnings of the routing problem on graphs, coul help in evaluating these proposals (or eveloping others). In particular, we are intereste in the performance of the most scalable theoretical routing algorithms on realistic topology graphs. To further investigate this relationship, we focus on the performance of compact routing schemes on scale-free graphs. Several authors are currently pursuing such moels, however. For one of the latest examples, see []. Compact routing schemes comprise a set of algorithms that aim to make a goo trae-off between stretch versus the amount of storage require at each vertex for routing tables. Stretch refers to the (usually worst-case) multiplicative factor increase of path length between a pair of vertices uner a particular routing scheme versus the length of the shortest existing path between the same pair. The most efficient stretch- 3 routing scheme for generic (arbitrary) graphs currently known is ue to Thorup-Zwick [3], which we will simply refer to as the TZ scheme. It is known to be optimal, up to a logarithmic factor, for per-noe memory utilization. We investigate the performance of the TZ scheme on scalefree graphs because these graphs represent our best current unerstaning of the Internet s inter-as topological structure. It is worth mentioning that although we base our analysis on properties of the real-worl Internet, we are not suggesting that the TZ scheme is ripe for use within the Internet to solve its scalability problems. Rather, we employ the TZ scheme as a tool to analyze the funamental limits for average stretch an routing table sizes on realistic graphs. The scheme is generic, so that it can be irectly applie to any graphs to scalefree graphs, in particular. As we are unaware of any routing schemes evelope specifically for scale-free graphs, we must turn to generic schemes to pursue our investigation. One might expect that for scale-free graphs, the majority of known generic routing schemes woul be very inefficient. Inee, many routing schemes (incluing the TZ scheme) incorporate locality by carefully ifferentiating between close an remote noes. This approach can make routing more efficient (in the stretch-versus-space trae-off sense, in particular) by keeping only approximate (non-shortest path) routing information for remote noes, while full (shortest path) routing information is kept for local noes. In scale-free graphs, with very low average istances an istance istribution withs, local noes comprise huge portions of all the noes in a network, so that one might suspect that locality-sensitive approaches might break for such networks. For a goo example emonstrating that this might be quite plausible, see the Appenix, where the stretch factor is foun to be very high for the Kleinrock-Kamoun (KK) routing scheme [4] applie to the scale-free networks. The choice of the KK scheme for such analysis is partially riven by the fact that many relatively recent routing architecture proposals [5], [6], aime at resolving the Internet routing scalability issues, have been base on the ieas of [4].

2 In analyzing the TZ scheme s performance for the Internet graph, however, we fin that the situation is quite opposite with respect to the KK schemes: the TZ scheme prouces extremely low average stretch values an succinct routing tables that turn out to be well below their theoretical upper bouns. A. Routing Backgroun The aim of compact routing schemes is to approach the optimal stretch- (shortest-path) routing but with significantly reuce memory requirements. It is shown in [7] that for any stretch- routing scheme, there exists a graph of size n an maximum noe egree, 3 <n, such that Ω(n log ) bits of memory are require at Θ(n) noes. Since the trivial upper boun for shortest path full-table routing is also O(n log ), this result effectively emonstrates incompressibility of generic shortest path routing. That is to say, if we must accommoate both graphs of arbitrary topology an with all paths being stretch-, we must be willing to have large routing tables. Although there are some results, [8], showing that the majority of graphs are better, very little can be sai conclusively regaring the practical implications of these results with respect to real-worl graphs. Thus, in orer to stuy compact routing on the Internet s graph, we must turn to the only existing tool we have for analyzing compact routing performance in all cases: generic routing schemes. Generic shortest path routing is incompressible, so if memory space is to be reuce, then the stretch must be increase. The memory space lower boun epenence on stretch is not continuous. As shown in [7], any generic routing scheme with stretch strictly less than.4 must use at least Ω(n log n) bits of memory on some noes of some graphs. In other wors, the lower boun for generic schemes with stretch s<.4 is the same as in the incompressible case of shortest path routing iscusse above (if one consiers graphs with = Θ(n)). Furthermore, as shown in [9], the lower boun for schemes with stretch strictly less than 3 is nearly the same as for shortest path routing Ω(n) bits of memory on some noes of some graphs. The minimum stretch factor that we must be prepare to accept in orer to significantly ecrease memory requirements below the incompressible limits is therefore 3. Cowen introuces a simple stretch-3 routing scheme with a local memory space upper boun of O(n /3 log 4/3 n) in [0]. Thorup an Zwick improve upon Cowen s result an eliver a per-noe space upper boun of O(n / log / n) in [3]. We call these two schemes the Cowen an TZ schemes, respectively. The local memory space upper boun provie by the TZ scheme is nearly optimal (up to a logarithmic factor) because, as emonstrate in [], any generic routing scheme with stretch strictly less than 5 must use at least Ω(n / ) bits of memory on some noes of some graphs. To the best of our knowlege, the TZ scheme is the only known generic stretch- 3 routing scheme elivering a nearly optimal local memory upper boun. For this reason, we use it as the basis for our analytic work an simulations below. They also show how to implement routing ecisions at constant time per noe. B. Scale-free networks Until fairly recently, most ranom graph analyses have been base on classical Erős-Réni ranom n-noe graphs [], which have links between every pair of vertices with the uniform probability p. The ensemble of such graphs is calle G n,p an their average vertex egree is k np.forlargen,the vertex egree probability istribution for these graphs is the quickly-ecaying Poisson istribution with an exponentially small number of high-egree noes, P k k k e k /k!, an average istance is log n/ log k [3]. These graphs are uncorrelate, 3 an their entire statistical properties can be erive from this vertex egree istribution. Almost all the networks observe in nature iffer rastically from the G n,p graphs. For our work, the most important ifference is an inconsistency between the average istance an average vertex egree preicte by the Erős-Réni moel for the Internet. In the real Internet inter-omain 000-noe graph, k 5.7 an 3.6 [4], while the G 000, graphs have 5.3. TheG n,p graphs of the same size with the right average istance 3.6 woul have to have the average egree k 4. The simultaneously small values of the average istance an average vertex egree in the Internet necessarily imply a larger portion of its noes are high-egree than in a comparably-size G n,p graph. In other wors, the vertex egree istribution must be fat-taile. The power-law istribution, P k k γ, one of such fat-taile istributions, is what has been observe in many real-worl networks, with the exponent γ ranging between an 3. For the Internet interomain graph, γ. [5], [4]. Both the G n,p graphs an graphs with fat-taile egree istributions are often sai to possess the small-worl property, [6], to emphasize that they have extremely low average istances (for networks of such size), even though average istances in G n,p graphs are only slightly higher. Networks with power-law egree istributions are also calle scale-free since their noe egree istribution lacks any characteristic scale, [7], in contrast to the G n,p graphs with the narrow Poisson egree istribution centere aroun the characteristic average value k np. The most popular moel for growing scale-free networks uses linear preferential attachment by Barabási an Albert (BA), [7]. The BA moel is very simple; it oes not have external parameters, an in its pure form, it preicts γ =3. The moel can be easily moifie to prouce other values of γ, but its ability to help explain the evolutionary processes influencing the growth of the Internet has been questione in [8], [9]. In particular, in [8], it is note that the BA moel an its erivatives are capable of reproucing what has been alreay measure but fail to preict correctly anything new about the Internet topology growth. 4 Construction of explanatory moels capturing elements of the actual factors 3 That is, vertex-vertex egree correlations are absent. 4 A goo example is the power-law ecay of the clustering coefficient. Moels reproucing this effect were constructe only after it ha been observe in real networks.

3 governing the Internet evolution is a hot topic of Internetrelate research these ays []. II. COMPACT ROUTING SCHEMES In this section, we briefly review the Cowen an TZ schemes an establish the terminology an notation we will require. Both schemes are very simple. They involve four separate components: the lanmark set (LS) construction proceure, routing table construction, labeling, an routing (message forwaring) function. The TZ scheme iffers from the Cowen scheme by improving only the LS construction proceure. Both schemes operate on any unirecte connecte graph G =(V,E) with positive ege weights. Let n = V be the graph size, δ(u, v) be the istance (in hops) between a pair of noes u, v V, L be the LS, L(v) be a lanmark noe closest to noe v V, an C(v) be v s cluster, efine for v V as the set of all noes c that are closer to v than to their closest lanmarks, C(v) = { c V } δ(c, v) <δ(c, L(c)). () Clusters are similar to Voronoi iagrams but they can intersect. If l L, then L(l) =l an C(l) = by efinition. If L is empty, then for v V, L(v) = an C(v) =V. The TZ LS construction algorithm iteratively selects lanmarks from the set of large-cluster noes T. At the first iteration, T = V an every noe t T is selecte to be a lanmark with a specific uniform probability q/n with q =(n/ log n) /. The expecte LS size after the first iteration is q. During subsequent iterations, T is reefine to be a set of noes that have clusters of size greater than a specific threshol q =4n/q, T = { t V } C(t) > q, () an aitional portions of lanmarks are selecte from T with a uniform probability q/ T. The iterations procee until T is empty. Every noe v V then calculates its outgoing port for the shortest path to every l L an every c C(v). This is the routing information that is store locally at v. As one can see, the essence of the LS construction proceure is the right balance between the LS an cluster sizes (or, effectively, between q an q). The cluster sizes are upper-boune by efinition (), an the involve part of the proof in [3] is to emonstrate that the algorithm terminates with a proper limit for the expecte LS size, which turns out to be q log n. This guarantees the overall local memory upper boun of O(n / log / n). The label of noe v (use as its estination aress in packet heaers) is then a triple of its ID, the ID of its closest lanmark L(v), an the local ID of the port at L(v) on the shortest path from L(v) to v. With these labels, routing of a packet estine to v at some (intermeiate) noe u occurs as follows: if v = u, one; if v L C(u), the outgoing port can be foun in the local routing table at u; ifu = L(v), the outgoing port is in the estination label in the packet; otherwise, the outgoing port for the packet is the outgoing port to L(v) the L(v) ID is in the label an the outgoing port for it can be foun in the local routing table. The emonstrations of correctness of the algorithm an that the maximum stretch is 3 are straightforwar ([0], [3]). III. ANALYTICAL RESULTS In this section, we provie analytical expressions for the TZ stretch istribution on a small-worl graph with a given istance istribution. To obtain our results we make a simplifying assumption: we consier only the first iteration of the TZ LS construction algorithm. There are two justifications making this assumption reasonable. First, as shown in [], the first iteration guarantees that the average cluster size is n/q; the subsequent iterations guarantee that all cluster sizes are no larger than 4n/q. Therefore, the error introuce by this assumption for the average stretch is small as we see in the next section. Seconly, we are concerne with small-worl graphs which have very short average istances an narrow istance istributions. Inee, if there are no long istances in a graph, then even after just the first iteration, the majority of clusters are small. For the rest of this section, we let q enote the actual size ofthels(q = L ) an D be the graph iameter (i.e. the maximum shortest path length in the graph). We also enote the istance p.m.f. an c..f. by f() an F (), respectively. With D being the graph iameter, {...D}. Let w an v be inepenently selecte ranom vertices from the ranom graph G =(V,E) corresponing to a source an estination noe, respectively. We introuce the following three ranom variables X, Y, an Z: X = δ(w, L(v)), p.m.f. p X (x), (3) Y = δ(v, L(v)), p.m.f. p Y (y), (4) Z = δ(w, v), p.m.f. p Z (z) =f(z). (5) These ranom variables correspon to the istances from some ranom noe w to the lanmark nearest another ranom noe v, the istance from that lanmark to v, an the actual shortest path between the two ranom noes. From these, we may construct another ranom variable, S, efine when w v to escribe the stretch value S = X + Y Z. (6) This expression for stretch is approximate for two reasons. First, it oes not account for stretch- paths to estinations in the local cluster of W. Secon, it oes not incorporate the shortcut effect. Recall that the Cowen routing algorithm is such that if estination v L an if a message on its way to L(v) passes some noe u v C(u), then the message never reaches L(v) but instea goes along the shortest path from u to v. To refine our approximation, we correct the above efinition of S to form S as follows: if Z<Y, if Z<X, S = S(X, Y, Z) = (7) if Z =0, X+Y Z otherwise.

4 Note that the first case on the r.h.s. of (7) accounts exactly for the stretch- paths to the estinations in the local cluster (cf. efinition ()), while the secon case accounts approximately for the shortcut effect as shown in []. With the above notations, the p.m.f. for the istance Y i between a ranom noe an its i th closest lanmark is very similar to the p..f. for orer statistics (see, for example, [0]). One can show (cf. []) that ( ) q p Yi () =i F () i f()( F ()) q i. (8) i The p.m.f. for the average istance to all lanmarks from a ranomly-selecte noe is p X () = q q p Yi (). (9) i= Since lanmarks are just q ranom noes, p X () is equivalent to f(), p X () =f() q = f(). q ( ) q i F () i ( F ()) q i i i= (0) Our problem now is to fin the p.m.f. p S (s) for the ranom variable S. If X, Y, an Z were inepenent an unconstraine then p S (s) woul be given by a simple sum over the joint istribution where p XY Z (x, y, z) =p X (x)p Y (y)p Z (z). They are not inepenent, however, because they are efine in equations (3)-(5) on the same ranom events. Furthermore, the form of their efinition results in the triangle inequality, X Y Z X + Y, () which causes some portions of the joint p.m.f to be zero. As such, the complete joint p.m.f. we require is erive in [] to be p XY Z (x, y, z) = p X(x)p Y (y)p Z (z)i t (x, y, z), () F (x + y) F ( x y ) where the enominator is positive an the triangle inicator function is efine as follows { if x y z x + y, I t (x, y, z) = (3) 0 otherwise. The intuition behin equation () is as follows. We are selecting three ranom noes on a graph. These noes form a triangle. The istance istribution of one of the triangle s sies p Y (y) is special (given by the LS construction proceure). The probability mass is therefore concentrate only in the feasible combinations of x, y an z (i.e. those for which the triangle inequality hols). The enominator normalizes the prouct to form a proper p.m.f. Using equation (), the stretch istribution p S (s) an the average stretch s are compute as follows: p S (s) = D D x= y= z= D p XY Z (x, y, z) S(x, y, z) =s (4) s = s p S (s) (5) s For the average stretch, the summation in equation (5) is over the finite set of rational stretch values s boune above by, as follows from equation (7). Equations (4) an (5) are our final analytical results that we require for the numerical evaluations of the next section. Of particular note is that the stretch istribution an average epen only on f() an q. IV. SIMULATION AND NUMERIC RESULTS We are now reay to substitute the Internet inter-as istance istribution into the analytical expressions of the previous section. Because the Internet is an evolving network, it contains vertex-vertex correlations [], an so to fully achieve our immeiate goal, we woul nee to know an analytic result for the istance istribution in correlate scale-free networks. Unfortunately, this problem has not yet been solve. 5 Surprisingly, the eterministic scale-free graph moel by Dorogovtsev, Goltsev, an Menes (the DGM moel) [4] analytically prouces the Gaussian istance istribution, which is very close to the istance istribution observe in the real Internet inter-omain graph []. Given this observation, we choose to parameterize istance istributions in small-worl graphs we consier in this section by Gaussian istributions. Using a iscrete form of the Gaussian istribution as the istance p.m.f. f() from the previous section transforms equations (4) an (5) into expressions that cannot be evaluate analytically, so that we retreat to numeric evaluations with f() taken to be an explicitly normalize Gaussian, ( ) f() =ce σ, (6) where c is s.t. D = f() =, an an σ are respectively the average istance an the with (stanar eviation) of the istance istribution. Distributions p X (x) an p Y (y) are also explicitly normalize. Variables X, Y, an Z, efine in (3)- (5), take integer values within the following ranges: D = [ ] + 0σ, (7) X, Y =...D, (8) Z = max (, X Y )...min (D, X + Y ), (9) where [ ] roun ( ) an iameter D becomes a istance istribution cutoff parameter, f(), > D since f(d)/f() e 00. The TZ LS size q (cf. Section II) is roune, q = [ (n/ log n) /]. 5 Although, there are some recent analytical results on istance istributions in uncorrelate scale-free graphs, [3].

5 For the following simulation results, we evelope our own TZ scheme simulator an use it on graphs prouce by the PLRG generator [5]. For a given parameter set, all the ata is average over 0 ranom graphs. All average graph sizes n are between 0, 000 an, 000 unless mentione otherwise. A. Distance istribution While the PLRG generator has been foun to prouce topologies largely consistent with those observe in the Internet inter-as graph [8], it outputs uncorrelate graphs, an, hence, there are some concerns regaring its capability of reproucing all the features of strongly correlate nets, such as the Internet. However, since the stretch istribution is a function of the istance istribution an the graph size only (Section III), all we nee from a graph generator for our purposes is that istance istributions in graphs prouce by it be close to istance istributions observe in real-worl graphs. Base on the experiments performe in [8], one can expect that the istance istribution in PLRG-generate graphs with the noe egree istribution exponent γ =. shoul be close to the one in the Internet. We fin that it is inee so. See Fig. (a) for etails. We then procee as follows. Paying special attention to the value of the noe egree istribution exponent γ equal to., which is observe in the Internet, we generate a series of graphs with γ ranging from to 3, an calculate their istance istributions. We fit these istributions by explicitly normalize Gaussians (6) yieling values of an σ that we use in numerical evaluations of our analytical results. For fitting, we use the stanar non-linear least squares metho. All fits are very goo: the maximum SSE we observe in our fits is an the minimum R-square is The values of an σ in fitte Gaussians are slightly off from the means an stanar eviations of istance istributions in generate graphs as epicte in Fig. (b). In fact, Fig. (b) is a parametric plot of σ() with γ being a parameter. We observe an almost linear relationship between an σ with such parametrization. Note that the almost linear relationship between the istance c..f. center an with parameterize by γ is analytically obtaine in [3] as well. We further iscuss this subject in Section V. In Fig. (c,), we show fitte an σ as functions of γ (cf. with the results in [3], [9]). Average graph sizes for ifferent values of γ are slightly ifferent, but the epenence of an σ on n (not shown) is negligible compare to their epenence on γ. Thisisin agreement with [3], [9]. B. Stretch istribution We obtain a very close match between the simulations an analysis of the average TZ stretch an stretch istribution. The average stretch as a function of γ isshowninfig.(a).forthe Internet-like graphs, γ =., the average stretch we observe in simulations is.09 an the average stretch given by (5) with f() in (6), with =3.4 an σ =0.9, is.4. Thus, we fin that the average stretch is very low. TABLE I THE TOP TEN STRETCH VALUES AND PERCENTAGE OF PATHS ASSOCIATED WITH THEM. Stretch Analysis (%) Simulations (%) / / / / / / / / Furthermore, while both the average istance an istance istribution with in power-law graphs o epen on γ (cf. Fig. (c,)), the average stretch oes not. We elay the iscussion of this topic until Section V. The stretch istributions obtaine both analytically, (4), an in simulations are shown in Fig. (b). The sets of significant stretch values (that is, stretch values having noticeable probabilities) match between the analysis an simulations. The top ten stretch values corresponing to virtually 00% of paths are presente in Table I. We notice that a majority of paths (up to 7% accoring to the simulations) are shortest. There are only a very few significant stretch values for the rest of paths. All the significant stretch values are below. The small amount of stretch values with noticeable probabilities is ue to the narrow with of the istance istribution. Inee, in 86% cases, two ranom noes are either 3 or 4 hops away from each other. That is, the probability for X or Z to be either 3 or 4 is 0.86, see Fig. (a). In 8% cases, a ranom noe is just one hop away from its closest lanmark, p Y () 0.8. This explains why stretch-4/3 (X =3, Y =, an Z =3) an stretch-5/4 (X =4, Y =, an Z =4) paths are most probable among stretch s> paths in Table I. In Fig. (c), the analytical results for the average stretch as a function of the graph size are shown. Note that epenence on n in (5) is only via the LS size q. We present ata for the case when an σ are fixe at their values observe in the Internet, an the case when they are allowe to scale as in the DGM moel. In both cases, the average stretch slowly ecreases as the network grows, although this ecrease is sprea over multiple orers of magnitue of n an the stretch change is confine to a narrow region between.3 an.. Wealso notice that after a certain point, the stretch stops ecreasing. Although it becomes very small, it oes not reach its minimal value. Finally, in Fig. (), we report the simulation ata on the average cluster an LS sizes. (Recall that the sum of the cluster an LS sizes in the TZ scheme is the number of recors in the local routing tables.) We notice that they are well below their theoretical bouns. Inee, for the Internet-like graphs we stuie, n 0 4, γ., this sum is 5, while the theoretical upper boun, 6(n log n) /,is 00.

6 AS ( = 3.7, σ = 0.9) PLRG with γ =. ( = 3.6, σ = 0.9) Gaussian fit of PLRG ( = 3.4, σ = 0.9) 3 PLRG graphs Gaussian fits Internet (γ =.) f() 0.5 σ a) b) σ c) 0.8 ) γ γ Fig.. a) The istance istributions. The circles represent the istance istribution from a typical AS (AS #) average over a perio of approximately March-May, 003. (The source of ata is [6]; for other measurements, see [4], [7].) The mean an stanar eviation is 3.7 an 0.9 respectively. The istance istribution in PLRG-generate graphs with γ =. is shown by squares. The stanar eviation is the same as before, the mean is 3.6. The soli line is the Gaussian fit of the PLRG istribution, =3.4 an σ =0.9. b) The means an stanar eviations (squares) of istance istributions in PLRG-generate graphs with γ =.0,.,..., 3.0 (from left to right), an the corresponing values of an σ (crosses) in their Gaussian fits. The fitte values of an σ as functions of γ are shown in (c) an () respectively. The Internet value of γ =. is circle in (b)-(). C. G n,p graphs Looking at Figs. (a,c), one may be tempte to assume that the average stretch just moerately epens on n an oes not epen on either or σ for a wie class of ranom graphs. To emonstrate that this is incorrect, we consier the most common class of ranom graphs, G n,p.wetaken 0 4 an choose p to match approximately the Internet average istance (p ) an average noe egree (p ). The analytical an simulation results for the average stretch in these two cases are presente in Table II. We fin that the average stretch is substantially higher than in the case of ranom graphs with power-law noe egree istributions. V. MINIMUM STRETCH AND THE INTERNET GRAPH Our investigation so far suggests that the average TZ stretch epens strongly on the characteristics of the graph istance istribution its average istance an with, in particular. Recall that now we are taking the istance istribution in a graph to be Gaussian, (6), an, hence, the average TZ stretch s in (5) is a function of the average istance an the with of the istance istribution σ, s s(, σ). At this point, we wish to explore the analytical structure of s(, σ) in more etail. The natural starting point is to fix either or σ to their observe values in the Internet, (3.4 an 0.9 respectively), an vary the other. This results of this exercise are illustrate in Figures 3(a,b). The left graph shows the stretch values when σ is fixe at 0.9 an is allowe to vary between 0 an 7. The right graph shows the stretch values when is fixe at 3.4 an the with σ is allowe to vary between 0 an 7. To our great surprise, we iscover that these two functions have unique minimums an that the point corresponing to the Internet istance istribution (the large ots, which we will call the Internet point ) are very close to them. In other wors, one may get an impression that the Internet topology has been carefully crafte to have a istance istribution that woul

7 3.8 simulations analysis Internet 0.9 simulations analysis s a) ρ(s) 0.5 b) γ s.3 = 3.4, σ = 0.9 ~ log n, σ ~ log / n s(n). c) q, C 30 ) n LS size cluster size γ Fig.. a) The analytical results (circles) an simulation ata (squares) for the average TZ stretch as a function of γ. b) The same for the TZ stretch istribution with γ =.. c) The analytical ata for the average stretch as a function of the graph size. The ashe line correspons to the case when the istance istribution parameters an σ are fixe to the values observe in the Internet. The soli line presents the ata when an σ scale accoring to the DGM moel. ) The simulation ata for the LS (circles) an cluster (squares) sizes. In the Internet case, γ =., the average graph size in simulations is 0,687, the average LS size is 50.0, an the average cluster size is.43. TABLE II THE AVERAGE TZ STRETCH ON THE G n,p GRAPHS. n p Avg. egree k (, σ) in graphs (, σ) in Gaussian fits s (analysis) s (simulations) (3.9, 0.6) (3.9, 0.5) (5.5, 0.9) (5.6, 0.8) (nearly) minimize the average TZ stretch. Of course, this can be only an impression an not an explanation since the Internet evolution, as we know it toay, has ha nothing to o with stretch. The next question we have to ask is if the minimums we observe in Fig. 3(a,b) correspon to a true local minimum of the stretch function. Our analytical results allow us to construct Fig. 3(c), where the stretch function is plotte against both an σ. Note that not all combinations of (, σ) correspon to Gaussian-like istance istributions. Inee, when σ>, f() from (6) looks more like an exponential ecay since it is cut off from the left by conition. Also, when σ is very small, (corresponing to highly regular graphs like complete graphs, stars, etc.), the peculiar peak formation in the σ 0 area in the picture occurs. In this region, accurate computation of the stretch requires etaile knowlege of the particular graph topology. 6 The region of primary interest to us, an which correspons to real-worl networks where our Gaussian moel is likely to be most accurate, is when σ is somewhat greater than 0 an somewhat less then. Here, we observe a concave area in a form of a channel between the other regions escribe above. This area, which we shall term the minimal stretch 6 Note, however, that for the complete network case, =, σ =0,we obtain the correct answer for the average stretch,. For etaile explanation of the peak structure, see [].

8 s(,0.9) Internet (s(3.4,0.9)).8.7 s(3.4,σ) Internet (s(3.4,0.9)) s(,0.9).5.4 s(3.4,σ).4.3 b).3.. a) σ irection mins σ irection mins Internet average stretch s c) σ σ S = { (,σ) s(,σ) ~ s(3.4,0.9) } I minimums in irection (ata) minimums in σ irection (ata) minimums in irection (fit) minimums in σ irection (fit) PLRG graphs (γ =.0,.,,.8) PLRG graphs (linear fit) Internet (γ =.) Internet with DGM σ =. G n,p with ~ 3.9 G n,p with k ~ 5.7 s minimums in irection minimums in σ irection PLRG graphs (γ =.0,.,,.8) Internet (γ =.) Internet with DGM σ =. G n,p with ~ 3.9 G n,p with k ~ 5.7 e) ) Fig. 3. a),b) The average stretch as functions of with σ =0.9 an of σ with =3.4 respectively. The Internet is represente by the ots. c) The average stretch as a function of an σ. The Internet is represente by the ot. The stretch minimums along the - anσ-axes, M an M σ, are the light-grey an black lines respectively. ) The projection of (c) onto the -σ plane. The soli bottom an top lines represent respectively M an M σ (the light-grey an black lines from (c)). The two ashe lines are their linear fits in the MSR. The crosses are the same as in Fig. (b), the bottom-most ashe line being their linear fit. The Internet, γ =., is circle. The shae area is M I from the text. The plus is the point with the average istance observe in the Internet an the Gaussian with preicte by the DGM moel, =3.4, σ =.. The iamon an square are the istance istributions of the G n,p graphs from Table II matching the Internet average istance an noe egree. e) The projection of (c) onto the -s plane. The notations are the same as in (). The graph sizes n 0 4 everywhere.

9 region (MSR), is characterize by particularly low stretch values. The with an epth of the MSR slowly increase as (, σ) grow. For small (, σ), the MSR has a unique critical point, which we call the MSR apex. The Internet point is locate very close to the apex, which is characterize by the shortest istance between the sets of minimums of the average stretch function s(, σ) along the - an σ-axes. We enote these two sets by M an M σ respectively. We fin that M an M σ almost touch each other at the apex. The MSR apex can be more easily observe in Fig. 3() showing a projection of Fig. 3(c) on the -σ plane. The soli lines represent the above two sets of minimums forming the MSR, an the Internet point is very near their closest segment. An opportunity to look at the apex from yet another angle is presente in Fig. 3(e) showing a projection of Fig. 3(c) on the -s plane. We see that starting from the apex, as increases, the minimum stretch values along the - an σ- irections become virtually equal an slowly ecrease as grows. We also note that G n,p graphs are far away from the apex an that they have average stretch values that are far from minimal. We can see now that the apex is inee a critical or phase transition point since it is locate at the bounary of the two regions of the average stretch function. The first region, the MSR, is characterize by lowest possible stretch values corresponing to istance istributions observe in real-worl graphs. The secon region, with substantially higher average stretch values, correspons to istance istributions in more regular graphs. To illustrate this point in more etail, we turn our attention back to Fig. 3(). We observe that the two sets of minimums, M an M σ, are linear when σ>. The ashe lines represent the linear fits of M an M σ in the area with σ>. The exact location of the intersection of these fits is (,σ )=(3.6, 0.97). If the linear form of M an M σ sustaine for σ < as well, then M an M σ woul intersect at (,σ ), where we woul observe a stationary 7 point of s(, σ), which we coul then test for the presence of an extremum of the stretch function. This oes not happen, however. Instea, as an σ become small, the linear behavior breaks near the apex ue to increasingly more iscrete structure of the istance istribution ([]). In the extene version of this paper [], we show that linearity of M an M σ can be analytically erive from the fact that the istance istribution is taken to be Gaussian. Of course, this oes not explain why the Internet is so close either to the MSR or to its apex. The linear form of M an M σ in the MSR shes some light on a closely relate issue of why the average stretch is virtually inepenent of γ. In Fig.3(),the shae area represents a set of (, σ), for which the average stretch is approximately the same as for the Internet, M I = { (, σ) s(, σ) s(3.4, 0.9) }. We see that in the 7 Recall that a function has a stationary point where all its first-orer partial erivatives are zero. Thus, we can call the apex a quasi- stationary point emphasizing that s/ an s/ σ are both nearly zero at the apex. MSR, the M I bounaries are almost parallel straight lines. Therefore, if the average stretch is to be inepenent of γ, which is observe in Section IV-B, then the points representing istance istributions in power-law graphs, ( γ,σ γ ) from Fig. (b), shoul lie along the M I bounaries, an this is what inee happens. Yet again, the linear relation between γ an σ γ in the power-law graphs, an the fact that this relation is just as require for the average TZ stretch being virtually inepenent of γ, come from two seemingly isjoint omains. To finish the list of various coinciences, we construct a linear fit of ( γ,σ γ ) (the bottom-most ashe line in Fig. 3()). This line is locate exactly at the MSR ege. Inee, in the area of higher values of σ (that is, in the MSR), the stretch function is completely concave, while for smaller σ, we observe multiple convex an concave regions cause by more iscrete structure of the istance istribution. In other wors, the line corresponing to scale-free graphs is right at the bounary between more ranom an more regular graphs. Furthermore, the Internet point, γ =., lies on this line, an our numeric analysis shows that the Internet value of γ =. minimizes the istance between the linear fit of ( γ,σ γ ) an (,σ ), which is the intersection of the linear fits of M an M σ. In other wors, the Internet istance istribution is the point that is closest to the MSR apex, compare to istance istributions in all other scale-free graphs with powerlaw noe egree istributions. VI. CONCLUSIONS We fin that the TZ routing scheme applie to the Internet inter-as graph results in a very low average stretch an succinct routing tables that are well below their upper bouns. The primary reason why the average stretch is of a great concern is that the TZ scheme is not a stretch- scheme, while Internet inter-omain routing is essentially shortest path routing. 8 Thus, any stretch s>routing scheme applie to the Internet woul involve augmentation, in one form or another, of the routing information provie by the scheme with the shortest path routing information for some (or all) non-shortest paths. Our principal fining, that the TZ stretch on the Internet graph is reasonably low, opens a well-efine path for future work in the area of applying relevant theoretical results obtaine for routing to realistic scale-free networks. One of the immeiate next problems on this path is the performance analysis of ynamic low-stretch routing schemes on scalefree graphs. The TZ scheme is not optimal for the ynamic case since it labels noes with topology-sensitive information. In other wors, it is not name-inepenent. As soon as the topology changes, noes nee to be relabele. Significant progress in construction of name-inepenent low-stretch routing schemes has been recently mae by Arias, Cowen, et al. in [30]. 8 A routing scheme that woul prevent, for example, a pair of ASs from utilizing a peering link between them is not realistic, of course.

10 More importantly, however, we fin that the Internet shortest path length istribution is at the minimal istance from the unique critical point of the average stretch function. At present we lack sufficient information to show cause for this effect, but we o believe it strongly suggests the average stretch function may be an inirect (or even irect) inicator of some yet-to-be iscovere process that has influence the Internet s topological evolution. In other wors, a rigorous explanation of this phenomenon woul probably require much eeper unerstaning of the Internet evolution principles an emonstration of a link between them an the TZ scheme. This question is of great interest, as the funamental laws governing the Internet evolution remain unclear. Therefore, a proper explanation of this effect may help us in our intent to move, perhaps along the lines of [], from purely escriptive Internet evolution moels to more explanatory ones, in the terminology of the program outline in [8]. REFERENCES [] H. Chang, S. Jamin, an W. Willinger, Internet connectivity at the AS level: An optimization riven moeling approach, in Proc. of MoMeTools, 003. [] G. Huston, Commentary on Inter-Domain Routing in the Internet, IETF, RFC 3, 00. [3] M. Thorup an U. Zwick, Compact routing schemes, in Proc. of the 3 th SPAA. ACM, 00. [4] L. Kleinrock an F. Kamoun, Hierarchical routing for large networks: Performance evaluation an optimization, Computer Networks, vol., 977. [5] I. Castineyra, N. Chiappa, an M. Steenstrup, The Nimro Routing Architecture, IETF, RFC 99, 996. [6] F. Kastenholz, ISLAY: A New Routing an Aressing Architecture, IETF, Internet Draft, 00. [7] C. Gavoille an S. Pérennès, Memory requirement for routing in istribute networks, in Proc. of the 5 th PODC. ACM, 996. [8] H. Buhrman, J.-H. Hoepman, an P. Vitány, Space-efficient routing tables for almost all networks an the incompressibility metho, SIAM J. on Comp., vol. 8, no. 4, 999. [9] C. Gavoille an M. Genegler, Space-efficiency for routing schemes of stretch factor three, J. of Parallel an Distribute Comp., vol. 6, no. 5, 00. [0] L. Cowen, Compact routing with minimum stretch, J. of Algorithms, vol. 38, no., 00. [] M. Thorup an U. Zwick, Approximate istance oracles, in Proc. of the 33 r STOC. ACM, 00. [] P. Erős an A. Réni, On ranom graphs, Publicationes Mathematicae, vol. 6, 959. [3] B. Bollobás, Ranom Graphs, Acaemic Press, New York, 985. [4] A. Vázquez, R. Pastor-Satorras, an A. Vespignani, Internet topology at the router an Autonomous System level, con-mat/ [5] M. Faloutsos, P. Faloutsos, an C. Faloutsos, On power-law relationships of the Internet topology, in Proc. of the ACM SIGCOMM, 999. [6] S. Milgram, The small worl problem, Psychology Toay, vol. 6, 967. [7] A.-L. Barabási an R. Albert, Emergence of scaling in ranom networks, Science, vol. 86, 999. [8] W. Willinger, R. Govinan, S. Jamin, V. Paxson, an S. Shenker, Scaling phenomena in the Internet: Critically examining criticality, PNAS, vol. 99, 00. [9] Q. Chen, H. Chang, R. Govinan, S. Jamin, S. J. Shenker, an W. Willinger, The origin of power laws in Internet topologies revisite, in Proc. of the IEEE INFOCOM, 00. [0] S. Ross, Introuction to Probabilty Moels (7th e), Acaemic Press, San Diego, 000. [] D. Krioukov, K. Fall, an X. Yang, Compact routing on Internet-like graphs, Tech. Report IRB-TR-03-00, Intel Research, 003. [] S. N. Dorogovtsev an J. F. F. Menes, Evolution of Networks: From Biological Nets to the Internet an WWW, Oxfor University Press, Oxfor, 003. [3] S. N. Dorogovtsev, J. F. F. Menes, an A. N. Samukhin, Metric structure of ranom networks, Nucl. Phys. B, vol. 653, no. 3, 003. [4] S. N. Dorogovtsev, A. V. Goltsev, an J. F. F. Menes, Pseuofractal scale-free web, Phys.Rev.E, vol. 65, 066, 00. [5] W. Aiello, F. Chung, an L. Lu, A ranom graph moel for massive graphs, in Proc. of the 3 n STOC. ACM, 000. [6] BGP Table Data, [7] A. Broio, E. Nemeth, an k claffy, Internet expansion, refinement, an churn, European Trans. on Telecommunications, vol. 3, no., 00. [8] H. Tangmunarunkit, R. Govinan, S. Jamin, S. Shenker, an W. Willinger, Network topology generators: Degree-base vs. structural, in Proc. of the ACM SIGCOMM, 00. [9] F. Chung an L. Lu, The average istance in a ranom graph with given expecte egrees, Internet Mathematics, vol., no., 003. [30] M. Arias, L. Cowen, K. A. Laing, R. Rajaraman, an O. Taka, Compact routing with name inepenence, in Proc. of the 5 th SPAA. ACM, 003. APPENDIX In this appenix, we calculate a rough estimate of the stretch factor of the Kleinrock-Kamoun (KK) hierarchical routing scheme, [4], applie to the observe Internet interomain topology. We fin that the stretch is very high, which is consistent with the observation mae in [4] that the approach use there works reasonably well only for sparsely connecte networks. The scale-free networks, on the contrary, are extremely ensely connecte. Recall that [4] assumes the existence of a hierarchical partitioning of a network of size n into m levels of clusters. Each k-level cluster consists of n /m (k )-level clusters, k =...m, 0-level clusters being noes. The optimal clustering is achieve when m log n. There are a few other fairly strong assumptions about the properties of the require partitioning. Neither an algorithm for its construction nor proof of its existence are elivere, but if it oes exist then the stretch factor is shown to be [ ] s =+ m n k m k, (0) n k= where is the network average istance an k is the iameter of a k-level cluster. It is further assume in [4] that both the network iameter an average istance are power-law functions of the network size. This is certainly not true for scale-free networks with power-law noe egree istributions. For recent results on the average istance in such networks, see [9], [3]. In the numerical evaluations in this appenix, we use the value of 3.6 observe in the Internet, [6]. As shown in [9], the iameter of networks with power-law noe egree istribution with exponent γ lying between an 3 scales almost surely as Θ(log n). For the Internet, γ., an since the Internet size n is relatively large, we may write the Internet iameter D as D c log n with some multiplicative coefficient c. The observe value of D, D 3 ([6]), efines c. Thesizeofak-level cluster is obviously n k/m but nothing rigorous can be sai about its egree istribution since there is

11 no proceure for its construction. Thus, it is natural to assume that its egree istribution is also power-law with <γ<3, which gives an estimate of the k-level cluster iameter as k c log n k/m Dk/m. Substituting this in (0) an performing summation gives s + D [ m n n n(n m ) (n )(n m ) + ] () n m. m (n m ) Using the numerical values for n,, D, an optimal m =0, we can see that the KK stretch factor on the Internet interomain topology is s 5. () Note that a 5-times path length increase in the Internet woul lea to AS path lengths of 55 an IP hop path lengths of 50. The stretch factor is a nearly linear function of the number of hierarchical levels m, which follows irectly from equation () since it can be rewritten for large n as s + D (m ). (3) Using D log n, log log n ([9]), an optimal m log n, we obtain the following estimate of the stretch factor as a function of the network size: s log n log log n. (4)