Influence of the link weight structure on the shortest path

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Influence of the link weight structure on the shortest path

Piet Van Mieghem and Stijn van Langen

Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands 共Received 28 October 2004; published 20 May 2005兲

The shortest path tree rooted at a source to all other nodes is investigated in a graph with polynomial link weights tunable by the power exponent␣. By varying ␣, different types of shortest path trees, in short ␣ trees, appear. Especially, the␣→0 regime that corresponds to heavily fluctuating link weights possesses a peculiar type of tree. The most important properties of this␣→0 tree are derived in the asymptotic limit for large N. The application of the theoretical insights to real networks共such as the Internet兲 are discussed: steering flow by adjusting link weights共traffic engineering兲, sensitivity of link weights and modeling of the network by␣ trees.

DOI: 10.1103/PhysRevE.71.056113 PACS number共s兲: 89.75.Fb

I. INTRODUCTION

The simple shortest path problem asks for the computa-tion of the path from a source to a destinacomputa-tion node that minimizes the sum of the weights of its constituent links. The related shortest path tree共SPT兲 is the union of the short-est paths from a source node to all other nodes in the graph. The SPT belongs to the fundamentals of graph theory and has many applications. Moreover, powerful shortest path al-gorithms like that of Dijkstra exist. Nevertheless, little seems known about the influence of the link weight structure on the properties of the SPT. The motivation to study the impact of the link weights on the path properties arose in multicon-strained routing关1兴. Also from a traffic engineering perspec-tive, a network operator may want to tune the weight of each link such that the resulting shortest paths between a particu-lar set of ingresses and egresses follows the desirable routes in his network. Thus, apart from the topology of the graph, the link weight structure clearly plays an important role. Of-ten as in complex molecules, global social interactions or other large infrastructures as the Internet, both the topology and the link weight structure are not accurately known. This uncertainty about the precise structure leads us to consider both the underlying graph and each of the link weights as random variables.

Since the shortest path tree problem is mainly sensitive to the smaller, non-negative link weights, the probability distri-bution of the link weights around zero will dominantly influ-ence the properties of the resulting shortest path tree. A

regu-lar link weight distribution Fw共x兲=Pr关w艋x兴 has a Taylor series expansion around x = 0,

Fw共x兲 = fw共0兲x + O共x2兲

since Fw共0兲=0 and Fw

⬘

共0兲= fw共0兲 exists. A regular link weight distribution is thus linear around zero. The factor f_w共0兲 only scales all link weights, but it does not influence the shortest path. The simplest distribution of the link weight w with a distinct different behavior for small values is the polynomial distribution

Fw共x兲 = x␣1x苸关0,1兴+ 1x苸关1,⬁兲, ␣⬎ 0, 共1兲 where the indicator function 1_x is one if x is true else it is zero. The corresponding density is fw共x兲=␣x␣−1, 0⬍x⬍1.

The average and variance of the link weight are E关w兴 =␣/共␣+ 1兲 and var关w兴=␣/共␣+ 2兲−关␣/共␣+ 1兲兴2_{, respectively.} The exponent ␣ is called the extreme value index of the probability distribution of w and ␣= 1 for regular distribu-tions. By varying the exponent␣ over all non-negative real values, any extreme value index can be attained and a large class of corresponding shortest path trees, in short ␣-trees, can be generated. Finally, we assume independence of link weights which we deem a reasonable assumption in most large networks.

The main purpose of this paper is to show that, by con-sidering a polynomial link weight structure tunable by one parameter ␣ on, e.g., the complete graph, a broad class of shortest path trees are found as function of ␣. Instead of worrying about the precise topology of the graph, we show that a tunable link weight structure thins out the complete graph to the extent that a specific shortest path tree can be constructed. The approach thus eliminates the precise knowl-edge of the underlying graph by immediately concentrating on the tree properties induced by polynomial link weights.

Secondly, we show that relatively small variations in the link weight structures cause large differences in the proper-ties of the SPT. In particular, the average hop count共i.e., the number of links or number of nodes minus 1 in a path兲 in a graph with N nodes follows a different scaling: E关HN兴 = O共ln N兲 for ␣ around 1 while E关HN兴=O共N1/3兲 if ␣→0. Moreover, for␣→0, all traffic in the graph routed along the SPT traverses precisely N − 1 links and among those links there seems a large difference in the “critical backbone” over which nearly all traffic flows and the other links. The ␣

→0 regime corresponds to a strong disorder regime. In this

article, many properties of the resulting tree for␣→0 will be derived.

The paper is outlined as follows. We first overview three types of␣trees for␣→⬁, 1, and 0 and introduce a critical ␣c⬎0 for which all␣⬍␣care␣→0 trees 共with overwhelm-ing probability兲. The major part 共Sec. III兲 of the paper con-sists of computing properties of ␣→0 trees. The potential applications of␣trees in the setting of Internet are discussed in Sec. IV.

II. SPECIAL␣ TREES

Let us consider a graph G共N,L兲 with N nodes and L links and with independent polynomial link weights specified by

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Eq.共1兲. The shortest path tree rooted at an arbitrary node in the graph to all other nodes is called the␣tree. In this sec-tion, we will meet three special␣ trees for␣=⬁, 1, and 0, respectively.

A. The case␣\ⴥ

If␣→⬁, it follows from Eq. 共1兲 that w=1 for all links.

Since all link have unit weight, the␣→⬁ regime reduces to the computation of the shortest path tree in the underlying graph. The␣→⬁ regime is thus entirely determined by the topology of the graph because the link weight structure does not differentiate between links. In case of the complete graph

and␣→⬁, the␣tree is a star. In other underlying graphs, it

is more difficult共see, e.g., Ref. 关2兴兲 to determine properties of the shortest path tree. In this paper, the␣→⬁ regime is not further considered.

B. The case␣=1

If ␣= 1, the link weights are iid uniformly distributed. Earlier 关3兴, it was shown that the shortest path tree in the complete graph with uniform共or exponential兲 link weights is precisely a uniform recursive tree 共URT兲 while a URT is asymptotically the shortest path tree in the Erdös-Rényi ran-dom graph Gp共N兲关4兴 with link density p above the discon-nectivity threshold pc⬃ln N/N. The interest of the URT is that analytic modeling is possible such as the computation of the degree, the hop count 关3,5兴, the number of links in the URT or the multicast problem关6,7兴 and the hop count to the most nearby server or the anycast problem关8兴. Comparison with Internet measurements共see Sec. IV兲 shows that proper-ties of the shortest path tree in the regime around␣= 1 agree reasonably well with those measured, but they remain first order estimates.

In an earlier analysis关3兴, it was shown that, for N large and fixed␣, the hop count H_Nsatisfies

E关HN兴 ⬃ ln N ␣ , 共2兲 var关HN兴 ⬃ ln N ␣2 , 共3兲

and that the hop count is共at least asymptotically兲 indepen-dent of the link density p. That analysis has assumed inde-pendence of the links in the shortest path. Also, for large N, it follows from Eqs.共2兲 and 共3兲 that

␣⬃ E关HN兴

var关H_N兴共4兲

which gives a method to relate␣ to Internet measurements 共see, e.g., Fig. 9兲 provided that␣is around 1.

C. The case␣\0 If␣→0,

冑

var关w兴 E关w兴 ⬃ 1

冑

␣

which means that, in this limit, the link weights possess strong fluctuations and that the above ratio diverges. This observation inspired by Braunstein et al.关9兴 is crucial in the analysis of the behavior of the shortest path for small␣. If the set of L link weights兵wk其1艋k艋Lis ordered as

w_共1兲⬍ w_共2兲⬍ ¯ ⬍ w_共L兲

it is of interest to know if there exist a critical value␣c⬎0 such that, for all ␣⬍␣c, the following inequalities are obeyed共with high probability兲:

w_共2兲⬎ w_共1兲, w_共3兲⬎ w_共2兲+ w_共1兲, w_共4兲⬎ w_共3兲+ w_共2兲+ w_共1兲, . . . , w_共k+1兲⬎

兺

j=1 k w_共j兲, . . . , w_共L兲⬎

兺

j=1 L−1 w_共j兲. 共5兲

In any graph with this link weight structure, the weight of a path w共P兲=兺i→j苸Pw共i→ j兲 consists of one link that domi-nates all others. Indeed, by ordering the set 兵w共i→ j兲其i_→j苸P and assuming that maxi_→j苸P兵w共i→ j兲其=w_共k兲, then the in-equalities above indicate that w共P兲⬍w_共k+1兲. Hence, the set of inequalities assure that the weight of a path can be upper bounded by the next in order link weight higher than its own maximum link weight. Any path in the graph that does not contain the link with weight w_共L兲possesses a path weight that is always smaller than w_共L兲.

Theorem 1. If the link weights obey the inequalities 共5兲,

the union of all shortest path trees rooted at each node in the graph is a tree.

Proof. Suppose that the shortest path graph G_艛consisting of the union of all shortest paths would contain a loop, then the largest link weight of that loop is larger than the sum of all other links of that path. Since subsections of the shortest paths are again shortest paths, that largest link weight in the loop will not appear in a shortest path because it is bypassed by the other part of the loop with smaller link weight. Hence, loops do not occur in the graph G_艛, and a loop-free graph is a tree.

This curious property implies that all shortest path trees rooted at different nodes are precisely the same and equal to the minimum spanning tree 共MST兲. The coincidence of all SPT’s with the MST for a general graph with strongly fluc-tuating weights has been reported by Dobrin and Duxbury 关10兴.

We will now show that the inequalities共5兲 can be satisfied provided ␣c= O共L−2_{兲. It is convenient to make the} transfor-mation w = e−V/␣_{. Since w}_{苸关0,1兴, the random variable V}

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ranges over all positive real numbers and V is distributed as Pr关e−V/␣艋 x兴 = x␣1_x_苸关0,1兴

or, since the event兵w=e−V/␣艋x其 is equivalent to the event 兵V艌−␣ln x其, and denoting y=−␣ln x,

Pr关V 艌 y兴 = e−y,

which shows that the random variable V is exponentially distributed with mean 1. The set of inequalities generated by

w_共k+1兲⬎2w_共k兲 for each link k苸兵L其 is more severe than the original set of inequalities共5兲. In other words, if there exists an␣csuch that w_共k+1兲⬎2w_共k兲for each link k苸兵L其 is obeyed, then also the original set of inequalities is satisfied. Since

w_共k+1兲 w_共k兲 = exp

冉

1

␣共V共k兲− V共k+1兲兲

冊

⬎ 2

the inequalities are all satisfied if the minimum spacing ⌬min⬅min兩V共k兲− V共k+1兲兩⬎␣ln 2. We further replace the expo-nential distribution Pr关V艋y兴=1−e−y _{with the}_{共qualitatively} similar兲 uniform distribution on 关0, 1兴. The L random vari-ables V_共k兲 constitute a Poisson process and the spacings ⌬ between two consecutive V’s are independent and identically distributed as Pr关⌬⬎t兴=共1−t兲L _{with average spacing 1 /}_共L + 1兲⬇1/L 共see, e.g., Ref. 关11兴, Chap. 7兲. For large L, we have Pr关L⌬⬎t兴=共1−t/L兲L_→e−t _{and the spacing}_{⌬ tends to} the exponential distribution f_⌬共x兲=L exp共−Lx兲. The distribu-tion of the minimum spacing⌬_minis given by

Pr关⌬min⬎␣ln 2兴 = 关Pr共⌬ ⬎␣ln 2兲兴L= exp共−␣L2ln 2兲. In conclusion, by choosing␣c⬍L−2_{/ ln 2, the probability of a} violation of the inequalities w_共k+1兲⬎2w_共k兲 for each link k 苸兵L其 is less than one. Although this probability rapidly de-cays, only if␣→0, the inequalities are surely satisfied. The argument has shown that there is indeed a range of 0⬍␣ ⬍␣cthat obeys the inequalities with high probability. How-ever, it merely demonstrates the existence of␣c⬎0. More-over, the resulting estimate␣c= O共L−2兲 is much too conser-vative because共a兲 the inequalities w_共k+1兲⬎2w_共k兲for each link

k苸兵L其 are more stringent than the set of inequalities 共5兲, 共b兲

even the set of inequalities共5兲 is likely to impose too many restrictions, in particular on the links with larger link weights

w_共k兲 that are already unlikely to be part of a shortest path. Consequently, the extreme ␣→0 regime is expected to be entered more rapidly than for␣c= O共L−2_兲.

A better estimate for␣cshould be derived from the joint density function f of order statistics兵w_共k兲其 of the set of L link weights which is

f共w_共1兲= x1, . . . ,w共L兲= xL兲 = L!

兿

j=1 L

fw共xj兲 ⫻ 1x₁⬍x₂⬍¯⬍x_L.

共x1+x2兲1兵x1+x2艋1其 1 dx3x3␣−1 ⫻ ¯

冕

共兺j=1 L−1_x j兲1兵兺_j=1L−1x_j艋1其 1 dxLxL␣−1. 共7兲

The largest value of␣ that solves the equation p_␣= 1 −⑀for an arbitrarily small ⑀⬎0 can be obtained. Although we be-lieve that the integral in共7兲 can be solved exactly, the result seems too cumbersome to be useful here. If all link weights are iid exponential random variables with a same mean, we find that Pr

冋

w_共2兲⬎ w_共1兲,w_共3兲⬎ w_共2兲+ w_共1兲, . . . ,w_共L兲⬎

兺

j=1 L−1 w_共j兲

册

= L! 2L共L−1兲/2 共8兲

which shows that for independent exponential link weights in any graph the inequalities共5兲 are almost never satisfied for large L. Since the exponential has the same extremal index as the uniform distribution␣= 1, we see that to satisfy the in-equalities共5兲, the link weight structure must be highly fluc-tuating. If operators in the Internet assign link weights in-versely proportional to the bandwidth关12兴, it is a priori not unlikely that the inequalities共5兲 are almost all satisfied be-cause of the larger heterogeneity in links 共roughly from 10 kbit/ s up to 10 Gbit/ s兲 in the Internet.

In summary, three distinct␣regimes have been identified. Values of ␣ around either ⬁, 1, and 0 are likely to yield ␣-trees with properties that are essentially the same as those of the shortest hop count tree, the URT or the MST. After a study of properties of the MST共␣→0 tree兲, we will motivate in Sec. III D a slightly better order estimate for ␣c = O共N−2ln−2N兲 rather than␣c= O共N−4兲 in worst case where

L = O共N2兲.

D. Simulations

Figure 1 shows the average hop countE关H_N兴 as a function of the number of nodes N for various values of the exponent ␣苸关0,1兲. For ␣ near to 1, the average hop count E关HN兴 scales linearly in ln N, in agreement with Eq.共2兲. For smaller ␣, the average hop countE关HN兴 increases faster than ln N. If

␣→0, the strong overlap 共dependence兲 of paths 共all shortest

path trees are equal to the MST兲 radically excludes assump-tions of independence made in the earlier analysis 关3兴 that lead to a logarithmic scaling in N of the hop countE关HN兴.

The simulations seem to indicate that ␣c may be much larger than estimated in previous section which implies a higher probability that they really appear in practice. Braun-stein et al.关9兴 investigated the lengths of minimum-weight

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paths in the regime of large link weight fluctuations 共␣

→0兲 in random graphs, small-world and scale-free graphs.

Using arguments from the theory of critical phenomena and numerical simulations they showed that for Erdös-Rényi ran-dom graphs the hop count scales as lim_␣→0E关HN共␣兲兴 = O共N1/3兲.

Figure 2 visualizes the different structure of a typical MST 共a兲 and a typical URT 共b兲 of the same size N=100. Figure 3 shows the probability that the union of all shortest

paths G_{艛spt共␣兲}between all node pairs in the complete graph with polynomial link weights is a minimum spanning tree. In real networks where almost all flows follow shortest paths through the network, the union of all shortest paths is the observable part of a network. For example, the union of all trace routes between all node pairs in the Internet, would represent the observable graph of the Internet. The real In-ternet is larger because it also contains dark links for backup paths needed in case of failures. We observe in Fig. 3 a phase FIG. 1. The average hop count E关HN兴 in the random graph

G_p共N兲 with link density p=2 ln N/N as a function of the number of nodes N.

FIG. 2. An example in the graph with N = 100 nodes of共a兲 the MST which is the SPT for␣=0 and 共b兲 the URT which is the SPT for ␣=1. Both trees are structured per level set where each level shows the number of nodes at different hop count from the root 共here node with label 7兲.

FIG. 3. The probability that the union of all shortest paths G_艛spt共_␣_兲 in the complete graph with polynomial link weights is a minimum spanning tree as a function of␣.

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transition around ␣c, which we here define as Pr关G_艛spt共␣

c兲

= MST兴=₂1. For ␣⬍␣c, most graphs G_{艛spt共␣兲} are trees with high probability while for␣⬎␣chardly any graph G_{艛spt共␣兲}is a tree. The width⌬␣of the phase transition共e.g., defined as Pr关G_艛spt共␣

c−⌬␣兲= MST兴−Pr关G艛spt共␣c+⌬␣兲= MST兴=0.9兲

de-creases with N. The simulations are limited to relatively small graphs because for small␣the large relative variations in the link weights require a Dijkstra shortest path algorithm that runs with arbitrarily long real numbers 共mantisse and exponent兲. Indeed, a polynomial random variable is gener-ated as U1/␣_{= exp}_{共ln U/}_␣_{兲 where U is a uniform random} variable that is easy to generate with computers. For a small ␣of about 10−3and for a typical value of U around1₂, we see that link weights appear of the order of 2−1000_{and smaller.}

III. PROPERTIES OF MINIMUM SPANNING TREES

As mentioned in Sec. II B, the properties of the URT共␣ = 1兲 have been investigated earlier in detail. We devote this section to study some properties of MSTs corresponding to

the␣→0 regime. Although some results on MSTs have been

found earlier in a different setting with different methods, we present here a unified and elegant approach in the asymptotic regime for N→⬁.

A. Earlier work

Frieze 关13兴 computed the average weight of an MST on the complete graph KNfor a general weight distribution with finite fw共0兲. He also showed that the variance vanishes as-ymptotically. Aldous 关14兴 has computed the distribution of the degree in a 共single sample兲 MST. He has generalized Frieze’s result for the MST weight to a more general class of polynomial link weight distributions. Janson关15兴 shows that the distribution of the MST weight on KN with uniform weights is asymptotically normal, and he gives an expression for the variance. Penrose关16兴 proves that the degree distri-bution of the Euclidean MST on a d-dimensional hypercube converges to Aldous’ result, Eq.共24兲 below, in the mean-field limit d→⬁.

Barabasi 关17兴 clarified the equivalence between invasion percolation and Prim’s algorithm for the MST 共or strong-disorder SPT兲. Invasion percolation has been widely studied by physicists working in the area of phase transitions usually in two or three dimensions 共rather than in the infinite-dimensional case that we study兲. It models the penetration of fluid in a porous medium saturated by another fluid. The link weight plays the role of a potential barrier for fluid to invade into a pore. Dobrin关10兴 shows that the MST geometry on a random network is universal 共i.e., does not depend on the energy or weight distribution兲, which yields a simple way to compute the MST weight for general energy distributions.

B. The Kruskal growth process of the MST

Since the link weights in the underlying complete graph are chosen independently and assigned randomly to links in the complete graph, the resulting graph is probabilistically the same if we first order the set of link weights and assign

them in increasing order randomly to links in the complete graph. In the latter construction process, only the order sta-tistics or the ranking of the link weights suffice to construct the graph because the precise link weight can be unambigu-ously associated共later兲 to the rank of a link. Hence, assume the existence of a set of L =

共

N₂

兲

iid polynomially link weights

with␣→0 that are ordered, then we need only to take the

rank of each link weight into account to construct the MST which coincides with the shortest path tree in the case ␣

→0. This observation immediately favors the Kruskal

algo-rithm关18兴 for the MST over Prim’s algorithm. Although the Prim algorithm leads to the same MST, it gives a more com-plicated, long-memory growth process, where the attachment of each new node depends stochastically on the whole growth history so far. Pietronero and Schneider关19兴 illustrate that in our approach Prim, in contrast with Kruskal, leads to a very complicated stochastic process for the construction of the MST.

The Kruskal growth process described here is closely re-lated to a growth process of the random graph G共N,L兲 with

N nodes and L links. The construction or growth of G共N,L兲

starts from N individual nodes and in each step an arbitrary, not yet connected random pairs is connected. The only dif-ference with Kruskal’s algorithm for the MST is that, in Kruskal, links generating loops are forbidden. Those forbid-den links are the links that connect nodes within the same connected component or “cluster.” As a result, the internal wiring of the clusters differs, but the cluster size statistics 共counted in nodes, not edges兲 is exactly the same as in the corresponding random graph. The metacode of the Kruskal growth process for the construction of the random ␣→0 trees is

We now relate the link density p in the random graph

Gp共N兲 to the link density in the corresponding stage of the Kruskal growth process. We first compute the size of the giant cluster in the forest as a function of the number of links added. Let S = Pr关n苸C兴 denote the probability that a node n belongs to the giant component C. If n苸C, then none of the neighbors of node n belongs to the giant component. The number of neighbors of a node n is the degree d共n兲 of a node such that Pr关n 苸 C兴 = Pr关all neighbor of n 苸 C兴 =

兺

k艌0 Pr关all k neighbors of n 苸 C兩d共n兲 = k兴 ⫻Pr关d共n兲 = k兴.

Since in G_p共N兲 all neighbors of n are independent, the con-ditional probability becomes with 1 − S = Pr关n苸C兴,

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Pr关all k neighbors of n 苸 C兩d共n兲 = k兴 = 共Pr关n 苸 C兴兲k =共1 − S兲k.

Moreover, this probability holds for any node in n苸G_p共N兲 such that, writing the random variable D_RG= d共n兲,

1 − S =

兺

k=0 ⬁ 共1 − S兲k_Pr_{关d共n兲 = k兴 =}_␸D RG共1 − S兲, where␸D RG共u兲=E关u

D_RG_{兴 is the generating function of the} de-gree DRG in Gp共N兲. For large N, the degree distribution in

Gp共N兲 is Poisson distributed with mean degree ␮RG= p共N − 1兲 and␸D

RG共u兲=e

␮RG共u−1兲_{. Hence, for large N, the fraction S}

of nodes in the giant component in the random graph is given by

S = 1 − e−␮RGS 共9兲

and the average size of the giant component is NS. For ␮RG⬍1 the only solution is S=0 whereas for␮RG⬎1 there is a nonzero solution for the size of the giant component. The solution can be expressed as a Lagrange series共Ref. 关20兴, p. 94兲,

S共␮RG兲 = 1 − e−␮RG

兺

n=0

⬁ _{共n + 1兲}n

共n + 1兲!共␮RGe−␮RG兲n. 共10兲 By reversing Eq.共9兲, the average degree in the random graph can be expressed in terms of the fraction S of nodes in the giant component

␮RG共S兲 = −ln共1 − S兲

S . 共11兲

We will now transform the mean degree␮RG in the ran-dom graph to the mean degree ␮MST in the corresponding stage of the MST. In early stages of the growth each selected link will be added with high probability such that ␮MST =␮RG almost surely. After some time the probability that a selected link is forbidden increases, and thus ␮RG exceeds ␮MST. In the end, when connectivity of all N nodes is reached,␮MST= 2共since it is a tree兲 while␮RG= O共ln N兲.

Consider now an intermediate stage of the growth as il-lustrated in Fig. 4. Assume there is a giant component of average size NS and nc= N共1−S兲/sc small components of average size sceach. Then we can distinguish six types of links labeled a–f in the figure. Types a and b are links that have been chosen earlier in the giant component共a兲 and in the small components 共b兲, respectively. Types c and d are eligible links between the giant component and a small com-ponent共c兲 and between small components 共d兲, respectively. Types e and f are forbidden links connecting nodes within the giant component共e兲, respectively, within a small compo-nent 共f兲. For large N, we can enumerate how many links there are of each type Lkwith k =兵a,b,c,d,e,f其:

L_a+ L_b=1 2␮MSTN, Lc= SN共1 − S兲N, Ld= 1 2nc 2 s_c2, Le= 1 2共SN兲 2_{− SN,} Lf= 1 2ncsc共sc− 1兲 − nc共sc− 1兲. To highest order in O共N2兲, we have

L_c= N2S共1 − S兲, L_d=1 2N 2_{共1 − S兲}2_, Le= 1 2N 2_S2_.

The probability that a randomly selected link is eligible is

q =共c+d兲/共c+d+e+f兲 or

共Lc+ Ld兲/共Lc+ Ld+ Le+ Lf兲 q = 1 − S2. 共12兲 In contrast with the growth of the random graph Gp共N兲 where at each stage a link is added with probability p, in the Kruskal growth of the MST we are only successful to add one link共with probability 1兲 per 1/q stages on average. Thus the average number of links added in the random graph cor-responding to one link in the MST is 1 / q = 1 /共1−S2_{兲. This} provides an asymptotic mapping between ␮RG and␮MSTin the form of a differential equation

d␮RG d␮MST=

1 1 − S2. By using Eq.共11兲, we find

FIG. 4. Component structure during the Kruskal growth process.

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d␮MST dS = d␮MST d␮RG d␮RG dS = 共1 + S兲关S + 共1 − S兲ln共1 − S兲兴 S2 .

Integration with the initial condition␮MST= 2 at S = 1, finally gives the average degree␮MSTin the MST as function of the fraction S of nodes in the giant component

␮MST共S兲 = 2S −共1 − S兲 2

S ln共1 − S兲. 共13兲

As shown in Fig. 5, this result agrees well with the simula-tion 共even for a single sample兲, except in a small region around the transition␮MST= 1 and for relatively small N.

The key observation is that all transition probabilities in the Kruskal growth process asymptotically depend on only one parameter, the fraction of nodes in the giant component

S. This quantity S is called an order parameter in statistical

physics. In general, the expectation of an order parameter distinguishes the qualitatively different regimes 共states兲 be-low and above the phase transition. In higher dimensions fluctuations of the order parameter around the mean can be neglected and the mean value can be computed from a self-consistent mean-field theory. In our problem, the underlying complete 共or random兲 graph topology makes the problem effectively infinite dimensional. The argument leading to Eq. 共9兲 is essentially a mean-field argument.

C. The average weight of the minimum spanning tree

The crucial observation关10兴 is that in any graph, the to-pology of a MST depends only on the ranks of the link weights. By definition, the weight of the MST is

WMST=

兺

j=1 L

w_共j兲1j苸MST, 共14兲

where w_共j兲is the jth smallest link weight. The average weight of the MST is

E关WMST兴 =

兺

j=1 L

E关w共j兲1j苸MST兴.

The random variables w_共j兲 and 1j_苸MST are independent be-cause the value w_共j兲of jth smallest link only depends on the link weight distribution and the number of edges L while the appearance 1j_苸MSTof the jth edge in the MST only depends on the graph’s topology. Hence,

E关w共j兲1j苸MST兴 = E关w共j兲兴E关1j苸MST兴 = E关w共j兲兴Pr关j 苸 MST兴 such that the average weight of the MST is

E关WMST兴 =

兺

j=1 L

E关w共j兲兴Pr关j 苸 MST兴. 共15兲

In general for independent link weights with probability density function f_w共x兲 and distribution function F_w共x兲 = Pr关w艋x兴, the density function of the jth order statistic 共Ref. 关11兴, Chap. 3兲 is fw_共j兲共x兲 = jfw共x兲 Fw共x兲

冉

of the mean degree␮MST. Each simulation for different number of

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E关WMST兴 ⯝ N 2

冕

_2/N N−1 F_w−1

冉

x N

冊

共

1 − S共N/2兲x 2

兲

_dx ⯝N 2

冕

₀ N F_w−1

冉

x N

冊

关1 − S 2_{共x兲兴dx.}

It is known 关22兴 that if the number of links 共edges兲 in the growth process of the random graph is below N / 2, with high probability 共and ignoring a small onset region just below

N / 2兲, there is no giant component such that S共x兲=0 for x

苸关0,1兴. Thus, we arrive at the general formula valid for large N, E关WMST兴 ⯝ N 2

冕

₀ 1 F_w−1

冉

x N

冊

dx + N 2

冕

₁ N F_w−1

冉

x N

冊

关1 − S 2_{共x兲兴dx.} 共18兲 The first term is the contribution from the smallest N / 2 links in the graph, which are included in the MST almost surely. The remaining part comes from the more expensive links in the graph, which are included with diminishing probability since 1 − S2共x兲 decreases exponentially for large x as can be deduced from Eq.共10兲. The rapid decrease of 1−S2_{共x兲 makes} only relatively small values of the argument F_w−1共x/N兲 con-tribute to the second integral.

At this point, the specifics of the link weight distribution needs to be introduced. The Taylor expansion of 共N/2兲Fw

−1_{共x/N兲 for large N to first order is}

N 2Fw −1

冉

0 ⬁ x1/␣+1 e −x 共1 − e−x_兲1/␣dx. Finally, we end up with

E关WMST共␣兲兴 ⯝ N1−1/␣

冉

1 1/␣+ 1

冕

₀ ⬁ x1/␣+1 e −x 共1 − e−x_兲1/␣dx

冊

. 共19兲 If ␣⬍1, then E关WMST共␣兲兴→0 for N→⬁, while for␣⬎1, E关WMST共␣兲兴→⬁. In particular, lim␣→⬁ E关WMST共␣兲兴=N−1. Only for ␣= 1, E关WMST共1兲兴 is finite for large N. More pre-cisely,

E关WMST共1兲兴 =␨共3兲 = 1.202, ... , 共20兲 where we have used关Ref. 关24兴, Eq. 共23.2.7兲兴 the integral of the Riemann Zeta function

⌫共s兲␨共s兲 =

冕

0

⬁ _us−1

eu− 1du

convergent for Re共s兲⬎1. This particular case for ␣= 1 has been proved earlier by Frieze 关13兴 based on a different method. Asymptotically, as shown in Ref. 关6兴, the average weight of a shortest path tree is␨共2兲=␲2_{/ 6, while here the} average weight of the MST is␨共3兲⬍␨共2兲.

2. Generalizations

We now return to the Taylor series valid for link weights where 0⬍ f_w共0兲⬍⬁. The above result for␣= 1 immediately yields

E关WMST兴 = ␨共3兲

fw共0兲

. 共21兲

This result is for the complete graph K_N. A random graph

Gp共N兲 with p⬍1 and weight density fw共x兲 is equivalent to

KN with a fraction 1 − p of infinite link weights. Thus the effective link weight distribution is pfw共x兲+共1−p兲␦w,⬁, and we can simply replace fw共0兲 by pfw共0兲 in the expression 共21兲 for the MST weight on KN.

D. The largest link weight in a group

Consider a group 共e.g., in multicasting兲 consisting of m members and one source node on the MST. Both the source and the m receiving group members are chosen uniformly among the N nodes of the MST. For large N, the m paths from source to each multicast group member have a same bottleneck almost surely关25兴 if and only if the source joins

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the giant component later than the m destinations which hap-pens with probability 1 /共m+1兲. We will now compute the rank of the largest link weight in the tree that spans the m members.

During the growth of the MST as explained in Sec. III B, at each step links are added and the last link that connects the

m members and the source will be determined. When link

weights are ranked in increasing order, the rank R of that last link corresponds to the Rth order statistics w_共R兲whose distri-bution is specified in Eq.共16兲. We first determine the rank R which equals the step T in the Kruskal growth process for which T = max共t1, t2, . . . , tm, tm+1兲, where tjis the time in the growth process at which the jth multicast member is swal-lowed by the giant component and where tm+1is the time at which the source joins the giant component. Since all m nodes and the source are uniformly selected during the growth process measured in terms of S苸关0,1兴, the random variables tjare iid uniform on关0, 1兴 and

Pr关T 艋 x兴 = xm+1

for x苸关0,1兴. From the relation 共11兲 of the average degree and the fraction of nodes in giant component S, the normal-ized rank R* _{of the largest link weight follows from R}* =共2/N兲R=兩␮RG共S兲兩S=T and the event兵T艋x其 is equivalent to 兵␮RG共T兲艋␮RG共x兲其 because␮RG共x兲 is monotonically increas-ing in x such that

Pr关R*艋 y兴 = 关␮RG−1共y兲兴m+1,

where x =␮RG−1共y兲 is the inverse function of y=␮RG共x兲 and explicitly given in Eq.共10兲. The probability density function

fR*共y兲=共d/dy兲Pr关R*艋y兴 is

fR*共y兲 = 共m + 1兲 关␮RG

−1_共y兲兴m

␮RG

⬘

关␮RG−1_共y兲兴共22兲 and is shown in Fig. 6 for m = 1, 2, 5, and 10. The mean is best computed from the tail probability formula

E关R*_{兴 =}

冕

0 N−1 共1 − Pr关R*_{艋 y兴兲dy =}

冕

0 1 共1 − xm+1_兲d_␮RG_共x兲 = −共m + 1兲

冕

0 1 xm−1ln共1 − x兲dx =共m + 1兲

兺

k=1 ⬁ 1 k共k + m兲.

After partial fraction decomposition, the average normalized rank is E关R*_{兴 =}m + 1 m

兺

_k=1 m 1 k 共23兲

which shows that for large m,E关R*兴⯝ln m. In case m=1, the rank R =共N/2兲R*is the rank of the largest link weight in the shortest path from the source to the only destination and has E关R*_{兴=2, thus E关R兴=N. For large N, the probability that R} ⬍N/2 共i.e., R*_{⬍1兲 is zero asymptotically because the first}

N / 2 are almost surely not connected. Connectivity occurs

asymptotically when the mean degree ␮RG exceeds 1. The probability distribution of the largest weight follows from Eqs.共16兲 and 共22兲 using the law of total probability

fw_共R*_兲共x兲 =

冕

0 N fw共R*_兲共x兩R*= y兲dPr关R*艋 y兴 = N 2 f_w共x兲 Fw共x兲

冕

1 N y

冉

L 共N/2兲y

冊

关Fw共x兲兴共N/2兲y ⫻关1 − Fw共x兲兴L−共N/2兲yfR*共y兲dy.

Observe that w_共R*_兲 in contrast with R* does depend on link

probability p of the random graph. For large N, fw_共j兲共x兲 tends to a Gaussian共as explained in Sec. III C兲 resulting in

fw_共R*_兲共x兲 ⯝ 1 ␴

冑

2␲

冕

0

N

e−关共N/2兲y −␮兴2/2␴2fR*共y兲dy ⬇ f_R*关NF_w共x兲兴.

This expression shows that the probability density function 共PDF兲 of the highest link weight in the multicast group is asymptotically distributed as the normalized rank with the rank parameter y in Eq.共22兲 replaced by NFw共x兲.

Formula 共23兲 shows that the rank of the largest link weight in the MST共m=N兲 is about E关Rmax兴⯝共N/2兲ln N. Re-turning to the determination of␣cin Sec. II C, this observa-tion suggests that the number of relevant equaobserva-tions in Eq.共5兲 is L = O共N ln N兲. In fact, if N→⬁, the claim is very likely correct which leads to the critical regime␣c= O共N−2ln−2N兲

in stead of␣c= O共N−4兲 determined in Sec. II C.

E. The degree distribution

Aldous关14兴 has shown that the probability distribution of the degree DMSTof nodes in the MST equals

FIG. 6. The PDF of the normalized rank R*for various sizes m of the multicast group.

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Pr关DMST= k兴 =

冕

0 1 e−⌽共s兲⌽共s兲 k−1 共k − 1兲!ds, 共24兲 where ⌽共s兲=兰₀sdt关ln t/共t−1兲兴. Remarkably, this expression

for the probability of the degree D_MSTis close共but not iden-tical兲 to a Poisson共␭兲 distribution conditioned to positive in-tegers k艌1, Pr关DMST= k兴 ⯝ ␭k k! e−␭ 1 − e−␭, 共25兲 where␭ is chosen such that the mean degree equals ␮MST =E关D_MST兴=2. It is straight forward to show that ␭ is the positive solution of e−␭_{= 1 −}_{␭/2, ␭Ⲙ1.593624}_{¯ and that}

var关D_MST兴=2共␭−1兲. Table I compares the two expressions 共24兲 and 共25兲.

During the growth process, we observed that the degree distribution of the forest is found to be Poisson共x兲共␮MST兲 below the transition共␮MST⬍1兲. Above the transition 共␮MST ⬎1兲, the average degree of the giant component equals the final value of 2. At the transition the distribution of degree minus one in the giant component is found to be Poisson共1兲. Thus Pr关DMST− 1 = k兴=1/共ek!兲. For larger values of␮MSTit evolves gradually to the asymptotic form at ␮MST= 2. The degree distribution over all nodes not in the large component is Poisson for all 1⬍␮MST⬍2, with average degree follow-ing from 2S +共1−S兲␮sc=␮MST:

␮sc=␮MST− 2S 1 − S = −

1 − S

S ln共1 − S兲共26兲

F. The hop count

The hop count between two arbitrary nodes in the MST

共␣→0 regime兲 has been simulated and is shown in Fig. 7.

The simulation also indicated that per MST, there is a large variation in the hop count which heavily contrasts with the almost sure behavior for the hop count in the ␣= 1 regime 关26兴. The large variation between samples is indirectly de-duced from the fact that var关HN兴⬎E关HN兴 in the inset of Fig. 7.

The PDF fh共x兲 of the scaled hop count h=HN/E关HN兴 ver-sus the scaled number of hops x = j /E关HN兴 is plotted in Fig. 8 as a full line. With the dotted line, we have added the PDF of the limit random variable W of a Poisson branching process 共BP兲 with mean ␮BP. Two exact expressions for f_W共x兲 are TABLE I. Comparison of the degree distribution in the MST

k Exact Eq.共24兲 Conditioned Poisson Eq.共25兲

1 0.40658 0.40637 2 0.32429 0.32380 3 0.17112 0.17201 4 0.068353 0.068529 5 0.022006 0.021842 6 0.0059347 0.0058013 7 0.0013768 0.0013207 8 0.00028022 0.00026309 9 0.000050790 0.000046586 10 8.2970⫻10−6 _7.4240_⫻10−6 var关D_MST兴 1.1917 2共␭−1兲⯝1.1872

FIG. 7. The probability density function of the hop count simulated for various sizes of the␣=0 tree. The inset plots the average and variance of the hop count versus the number of nodes N.

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presented in Ref. 关27兴 and fW共x兲 is shown in the inset for various 2艋␮BP艋10. We observe that the PDF of the scaled hop count for N = 50 up to N = 6400 lies between those of the BP corresponding to the Poisson degree distribution with mean between␮BP= 4 and ␮BP= 6. We have also plotted the fit of a Maxwellian fh共x兲=4/

冑

␲共2x/

冑

␲兲2exp共−共2x/

冑

␲兲2兲

which was proposed by Braunstein et al. in关28兴 purely by fitting the simulation data.

The PDF fW共x兲 of the Poisson BP is clearly superior to the guess of a Maxwellian. The suggestion to compare the scaled hop count共with mean 1兲 to the limit random variable W 共also with mean 1兲 is explained by Aldous 关14兴. However, we still

FIG. 9. Trace routes from CAIDA, May 2004. Each of the 21 sources has several 104destinations. The local internal hops that follow a single path towards the Internet have been subtracted. In the inset,␣ is computed as ␣=E关H兴/var关H兴.

FIG. 8. The scaled hop count h = H_N/ E关H_N兴 versus the scaled number of hops x= j/E关H_N兴 for N=25⫻2k_{with k = 0 , . . . , 8. The arrows} show how this scaled hop count varies with increasing N. The inset shows the PDF fW共x兲 of the limit random variable of a Poisson branching process.

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need to associate the mean ␮BP of the Poisson BP to the number of nodes in the MST which turns out to be a rather difficult theoretical problem.

IV. DISCUSSION AND CONCLUSIONS

Properties of both the␣= 1-tree共URT兲 and the␣→0-tree 共MST兲 have been characterized. We have shown that both types of trees are quite different. Based on trace-route hop count measurements as shown in Fig. 9, the regime of inter-est for the Internet seem关29兴 to lie between 0.5⬍␣⬍1.5.

From a topological view, the Internet trees indeed seem to consist of a critical bearer tree共corresponding to the ␣→0 tree兲 overgrown with URT-like small trees 共influence of ␣ = 1兲. Based on Eq. 共2兲, the latter causes that the hop count in the Internet still scales logarithmically in N, rather than poly-nomially as for the␣→0 tree 共MST兲. This effect is similar to the small world graphs: by adding a few links in a “large average hop count graph,” the hop count may decrease dra-matically.

Our theoretical study inspires some new research. If In-ternet trees can be modeled via ␣ trees, insight about an

effective link weight structure in the Internet may be gained.

That effective link weight structures arises as a combination of intra-domain 共shortest path兲 routing with interdomain 共not-shortest path兲 routing. In spite of the relatively low number of sources, Fig. 9 seems to suggest that the effective link weight is close to a regular distribution. Although con-troversial to adopt link weights, from a modeling perspec-tive, simpler tools as shortest paths, no violation of the tri-angular inequality, etc., can be applied to deduce first order estimates. In a next stage, the modeling of multicast trees in terms of␣trees may be interesting since little about Internet multicast trees is known.

In networks where the link weights can be varied, con-trolled or determined independent of the topology, we have shown that if the extreme value index of the link weight distribution is larger than␣c, transport in a network is spread out over more paths, while if the extreme value index is below␣c, transports starts concentrating on very few “back-bone links.” Hence, by tuning␣in a same underlying topol-ogy, we may create two very different types of transport in the network. The analogy with a normal conduction共above a critical temperature Tc兲 and a superconducting transport 共be-low T_c兲 of an electrical current in some solids 共network of atoms兲 comes to mind.

From a control point of view in networks, operators may wish to steer flows by tuning link weights. Our study indi-cates that large variations in the link weights共␣ small兲 will result in overall properties close to the ␣→0 tree 共MST兲: many flows will traverse over a same set of links and the overall hop count will increase. From a robustness point of view, choosing␣around 1 will lead to the use of more paths and, hence, a more balanced overall network load. A next step in understanding the influence of the link weight struc-ture is to find out what the maximum amount ⌬w in link weight change can be in order not to modify the set of short-est paths in a network. This insight is important to short-estimate the topology update overhead in networks共e.g., in road traf-fic where the link weights may be associated with the traftraf-fic load兲. An accurate view of the updated topology is crucial for route planner in cars.

ACKNOWLEDGMENTS

We would like to thank Serena Magdalena for providing Figs. 1–3 and Xiaoming Zhou for Fig. 9.

关1兴 F. A. Kuipers and P. Van Mieghem, IEEE/ACM Trans. Netw. 共to be published兲.

关2兴 R. van der Hofstad, G. Hooghiemstra, and P. Van Mieghem, Random Struct. Algorithms 26, 598共2005兲.

关3兴 P. Van Mieghem, G. Hooghiemstra, and R. van der Hofstad, Delft University of Technology Report No. 2000125, 2000. 关4兴 B. Bollobas, Random Graphs, 2nd ed. 共Cambridge University

Press, Cambridge, 2001兲.

关5兴 R. van der Hofstad, G. Hooghiemstra, and P. Van Mieghem, Prob. Eng. Inf. Sci .共 PEIS兲 15, 225 共2001兲.

关6兴 R. van der Hofstad, G. Hooghiemstra, and P. Van Mieghem, Combinatorics, Probab. Comput. 14, 795共2005兲.

关7兴 P. Van Mieghem, G. Hooghiemstra, and R. van der Hofstad, IEEE/ACM Trans. Netw. 9, 719共2001兲.

关8兴 P. Van Mieghem, Int. J. Commun. Syst. 17, 269 共2004兲. 关9兴 L. A. Braunstein, S. V. Buldyrev, R. Cohen, S. Havlin, and H.

E. Stanley, Phys. Rev. Lett. 91, 168701共2003兲.

关10兴 R. Dobrin and P. M. Duxbury, Phys. Rev. Lett. 86, 5076 共2001兲.

关11兴 P. Van Mieghem, Performance Analysis of Communications Systems and Networks 共Cambridge University Press, Cam-bridge, 2005兲.

关12兴 In Cisco’s OSPF implementation, it is suggested to use w共i

→ j兲=108_{/ B}_{共i→ j兲, where B共i→ j兲 denotes the bandwidth 共in}

bit/s兲 of the link between nodes i and j.

关13兴 A. M. Frieze, Discrete Appl. Math. 10, 47 共1985兲. 关14兴 D. Aldous, Random Struct. Algorithms 1, 383 共1990兲. 关15兴 S. Janson, Random Struct. Algorithms 7, 337 共1995兲. 关16兴 M. D. Penrose, Ann. Prob. 24, 1903 共1996兲. 关17兴 A.-L. Barabasi, Phys. Rev. Lett. 76, 3750 共1996兲.

关18兴 T. H. Cormen, C. E. Leiserson, and R. L. Rivest, An Introduc-tion to Algorithms共MIT Press, Boston, 1991兲.

关19兴 L. Pietronero and W. Schneider, Physica A 170, 81 共1990兲. 关20兴 A. I. Markushevich, Theory of Functions of a Complex

Vari-able 共Chelsea Publishing Company, New York, 1985兲, Vols. I–III.

关21兴 In general holds that w共k兲= Fw

−1_共U 共k兲兲 and E关w共k兲兴 = E关Fw −1 共U共k兲兲兴 ⫽ Fw −1 共E关U共k兲兴兲.

but, for a large number of order statistics L, the central limit theorem leads to E关w共k兲兴 ⯝ Fw −1

冉

j L

冊

⯝ Fw −1_共E关U 共k兲兴兲

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weight of the jth smallest link is exactly E关w共i兲兴 = j L + 1⯝ j L .

关22兴 S. Janson, D. E. Knuth, T. Luczak, and B. Pittel, Random Struct. Algorithms 4, 233共1993兲.

关23兴 Since the average of the kth smallest link weight can be com-puted from Eq.共16兲 as

E关w共k兲兴 = L! ⌫

冉

L + 1 +1 ␣

integral, substituting x = 2u / N and using Eq.共6.1.47兲 of Ref. 关24兴, for large z, that ⌫共z+1/␣兲/⌫共z兲=共z兲1/␣_{关1+O共1/z兲兴, we}

arrive at the same formula.

关24兴 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions共Dover Publications, New York, 1968兲.

关25兴 The probability that m nodes become connected in a cluster different from the giant component tends to zero for large N. 关26兴 P. Van Mieghem, G. Hooghiemstra, and R. W. van der Hofstad,

Proceeding of Passive and Active Measurement (PAM 2001), April 23–24, RIPE NCC, Amsterdam共2001兲.

关27兴 P. Van Mieghem 共unpublished兲.

关28兴 L. A. Braunstein, S. V. Buldyrev, S. Sreenivasan, R. Cohen, S. Havlin, and H. E. Stanley, The Optimal Path in an Erdos-Renyi Random Graph, Proceedings of the 23rd LANL-CALS Conference on Complex Networks 共Springer Verlag, Berlin 2004兲.

关29兴 NLANR traces 共Aug. 2001兲 with 112 sources give E关␣兴 = 1.34, while RIPE traces共Feb. 2004兲 with about 70 sources giveE关␣兴=0.7.