DEGREE DISTRIBUTION OF SHORTEST PATH TREES AND BIAS OF NETWORK SAMPLING ALGORITHMS

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1 DEGREE DISTRIBUTION OF SHORTEST PATH TREES AND BIAS OF NETWORK SAMPLING ALGORITHMS SHANKAR BHAMIDI 1, JESSE GOODMAN 2, REMCO VAN DER HOFSTAD 3, AND JÚLIA KOMJÁTHY3 Abstract. In this article, we explicitly erive the limiting egree istribution of the shortest path tree from a single source on various ranom network moels with ege weights. We etermine the asymptotics of the egree istribution for large egrees of this tree an compare it to the egree istribution of the original graph. We perform this analysis for the complete graph with ege weights that are powers of exponential ranom variables weak isorer in the stochastic mean-fiel moel of istance), as well as on the configuration moel with ege-weights rawn accoring to any continuous istribution. In the latter, the focus is on settings where the egrees obey a power law, an we show that the shortest path tree again obeys a power law with the same egree power-law exponent. We also consier ranom r-regular graphs for large r, an show that the egree istribution of the shortest path tree is closely relate to the shortest path tree for the stochastic mean-fiel moel of istance. We use our results to she light on an empirically observe bias in network sampling methos. This is part of a general program initiate in previous works by Bhamii, van er Hofsta an Hooghiemstra [7, 8, 6] of analyzing the effect of attaching ranom ege lengths on the geometry of ranom network moels. 1. Introuction In the last few years, there has been an enormous amount of empirical work in unerstaning properties of real-worl networks, especially ata transmission networks such as the Internet. One functional which has witnesse intense stuy an motivate an enormous amount of literature is the egree istribution of the network. Many real-worl networks are observe to have a heavy-taile egree istribution. More precisely, empirical ata suggest that if we look at the empirical proportion p k of noes with egree k, then p k 1/k τ, k. 1.1) The quantity τ is calle the egree exponent of the network an plays an important role in preicting a wie variety of properties, ranging from the typical istance between ifferent noes, robustness an fragility of the network, to iffusion properties of viruses an epiemics, see [24, 18, 16, 32, 17, 33] an the references therein. 1 Department of Statistics an Operations Research, 34 Hanes Hall University of North Carolina, Chapel Hill NC Department of Mathematics, Technion, Haifa 32, Israel. 3 Department of Mathematics an Computer Science, Einhoven University of Technology, P.O. Box 513, 56 MB Einhoven, The Netherlans. aresses: bhamii@ .unc.eu, jgooman@tx.technion.ac.il, rhofsta@win.tue.nl, j.komjathy@tue.nl. Date: April 13, Mathematics Subject Classification. Primary: 6C5, 5C8, 9B15. Key wors an phrases. Flows, ranom graph, ranom network, first passage percolation, hopcount, Bellman-Harris processes, stable-age istribution, bias, network algorithms, power law, mean-fiel moel of istance, weak isorer. 1

2 2 BHAMIDI, GOODMAN, VAN DER HOFSTAD, AND KOMJÁTHY In practice, such network properties often cannot be irectly measure an are estimate via inirect observations. The egree of a given noe, or whether two given noes are linke by an ege, may not be irectly observable. One metho to overcome this issue is to sen probes from a single source noe to every other noe in the network, tracking the paths that these probes follow. This proceure, known as multicast, gives partial information about the unerlying network, from which the true structure of the network must be inferre, see e.g. [1, 15, 21, 22, 29, 34]. Probes sent between noes to explore the structure of such networks are assume to follow shortest paths in the following sense. These networks are escribe not only by their graph structure but also by costs or weights across eges, representing congestion across the ege or economic costs for using it. The total weight of any given path is the sum of ege weights along the path. Given a source noe an a estination noe, a shortest path is a potentially non-unique) path joining these noes with smallest total weight. It is generally believe that the path that ata actually takes is not the shortest path, but that the shortest path is an acceptable approximation of the actual path. For our moels, the shortest paths between vertices will always be unique. For a given source noe, the union of the shortest paths to all other noes of the network efines a subgraph of the unerlying network, representing the part of the network that can be inferre from the multicast proceure. When all shortest paths are unique, which we assume henceforth, this subgraph is a tree, calle the shortest path tree. This will be the main object of stuy in this paper. Given the shortest path tree an its egree istribution, one can then attempt to infer the egree istribution of the whole network. Empirical stuies such as [1, 29] show that this may create a bias, in the sense that the observe egree istribution of the tree might iffer significantly from the egree istribution of the unerlying network. Thus a theoretical unerstaning of the shortest path tree, incluing its egree istribution an the lengths of paths between typical noes, is of paramount interest. By efinition, the unique path in the shortest path tree from the source v s to any given target vertex v t is the shortest path in the weighte network between v s an v t. Thus, the shortest path tree minimizes path lengths, not the total weight of a spanning tree. Hence it is ifferent from the minimal spanning tree, the tree for which the total weight over all eges is the tree is minimal. The last few years have seen a lot of interest in the statistical physics community for the stuy of isorere ranom systems that brige these two regimes, with moels propose to interpolate between the shortest weight regime first passage percolation or weak isorer) an the minimal spanning tree regime strong isorer), see [14]. Consier a connecte graph G n = V n, E n ) on n vertices with ege lengths L n := {l e : e E n }. Now fix the isorer parameter s R +, change the ege weights to L n s) := {l s e : e G n }, an consier the shortest paths corresponing to the weights L n s). For finite s, this is calle the weak isorer regime. As s, it is easy to check that the optimal path between any two vertices converges to the path between these two vertices in the minimal spanning tree where one uses the original ege weights Ln) to construct the minimal spanning tree. This is calle the strong isorer regime. The parameter s allows one to interpolate between these two regimes. Unerstaning properties of the shortest path tree an its epenence on the parameter s is then of relevance. The aim of this paper is to stuy the egree istribution of shortest path trees, motivate by these questions from network sampling an statistical physics Mathematical moel. In orer to gain insight into these properties, we nee to moel a) the unerlying networks an b) the ege weights. We shall stuy two main settings in

3 DEGREE DISTRIBUTION OF SHORTEST PATH TREES 3 this paper, the first motivate by network sampling issues an the secon to unerstan weak isorer moels. a) Configuration moel with arbitrary ege weights: An array of moels have been propose to capture the structure of empirical networks, incluing preferential attachment-type moels [5, 12, 13] an, what is relevant to this stuy, the configuration moel CM n ) [11, 31]) on n vertices given a egree sequence = n = 1,..., n ) which is constructe as follows. Let [n] := {1, 2,... n} enote the vertex set of the graph. To vertex i [n], attach i half-eges an write l n = i [n] i for the total egree, assume to be even. For i rawn inepenently from a common egree istribution D, l n may be o; if so, select one of the i uniformly at ranom an increase it by 1). Number the half-eges in any arbitrary orer from 1 to l n, an sequentially pair them uniformly at ranom to form complete eges. More precisely, at each stage pick an arbitrary unpaire half-ege an pair it to another uniformly chosen unpaire half-ege to form an ege. Once paire, remove the two half-eges from the set of unpaire half-eges an continue the proceure until all half-eges are paire. Call the resulting multi-graph CM n ). Although self-loops an multiple eges may occur, uner mil conitions on the egree sequence, these become rare as n see for example [28] or [11] for more precise results in this irection). For the ege weight istribution, we will assume any continuous istribution with a ensity. In the case of infinite-variance egrees, we nee to make stronger assumptions an only work with exponential ege weights an inepenent an ientically istribute i.i..) egrees having a power-law istribution. b) Weak isorer an the stochastic mean-fiel moel: The complete graph can serve as an easy mean-fiel moel for ata transmission, an for many observables it gives a reasonably goo approximation to the empirical ata, see [37]. The complete graph with ranom exponential mean one ege weights is often refere to as the stochastic mean-fiel moel of istance an has been one of the stanar workhorses of probabilistic combinatorial optimization, see [27, 2, 3, 38] an the references therein. In this context, we consier the weak isorer moel where, with s > fixe, the ege lengths are i.i.. copies of E s, where E has an exponential istribution with mean one. In [27], the optimal paths were analyze when s = 1, an in [1], the case of general s was stuie as a mathematically tractable moel of weak isorer Our contribution. We rigorously analyze the asymptotic egree istribution of the shortest path tree in the two settings escribe above. We give an explicit probabilistic escription of the limiting egree istribution that is intimately connecte to the ranom fluctuations of the length of the optimal path. These in turn are intimately connecte to Bellman-Harris-Jagers continuous-time branching processes CTBP) escribing local neighborhoos in these graphs. By analyzing these ranom fluctuations, we prove that the limiting egree istribution has markely ifferent behaviour epening on the unerlying graph: i) Configuration moel: The shortest path tree has the same egree exponent τ as the unerlying graph for any continuous ege weight istribution when τ > 3, an for exponential mean one ege weights when τ 2, 3). This reflects the fact that, for a vertex of unusually high egree in the unerlying graph, almost all of its ajoining eges if τ > 3) or a positive fraction of its ajoining eges if 2 < τ < 3) are likely to belong to the shortest path tree. See Figure 1. ii) Weak isorer: Here the limiting egree istribution of the shortest path tree has an exponential or stretche exponential tail epening on the temperature s. Furthermore,

4 4 BHAMIDI, GOODMAN, VAN DER HOFSTAD, AND KOMJÁTHY 5 5 logq k) Tree Truth logq k) Tree Truth logk) 2 4 logk) τ = 2.5, n = 1 million τ = 4.5, n = 1 million 5 5 logq k) Tree Truth logq k) Tree Truth logk) logk) τ = 1.5, n = 1 million τ = 15.5, n = 1 million Figure 1. Empirical istributions of unerlying egrees truth ) in the full graph an observe egrees tree ) in the shortest path tree, shown in log-log scale. The vertical axis measures the tail proportion q k = j k pn) j of vertices having egree at least k. The unerlying graphs are realizations of the configuration moel on n vertices with power-law egree istributions having exponent τ an minimal egree 5 so as to ensure connectivity). Ege weights are i.i.. exponential variables. this limiting egree istribution arises as the limit r of the limiting egree istribution for the r-regular graph when the ege weights are exponential variables raise to the power s; see Figure 2 for the case s = Notation. In stating our results, we shall write v s an v t for two vertices the source an the target ) chosen uniformly an inepenently from a graph G n on vertex set [n] =

5 DEGREE DISTRIBUTION OF SHORTEST PATH TREES 5 logq k) logq k) logk) k r = 1 r = 1 Figure 2. Empirical istributions of observe egrees in the shortest path tree. At left, both the egree k an the tail proportion q k = j k pn) j of vertices having egree at least k are shown in logarithmic scale; at right, only q k is shown in logarithmic scale. The unerlying graph is a ranom r-regular graph, r = 1, on n = 1 vertices. The blue line in the right-han graph is the curve q = 2 k, corresponing to the Geometric1/2) istribution. Ege weights are i.i.. exponential variables. {1,..., n}, which will either be the complete graph or a realization of the configuration moel. For the configuration moel, we write v for the egree of vertex v [n]. On the eges of G n, we place i.i.. positive ege weights Y e rawn from a continuous istribution. We enote by T n the shortest path tree from vertex v s, i.e., the union over all vertices v v s of the a.s. unique) optimal path from v s to v. We write eg Tn v) for the egree of vertex v in the shortest path tree an p n) k for the proportion of vertices having egree k in the shortest path tree. We write E for an exponential variable of mean 1 an Λ = log1/e) for a stanar Gumbel variable, i.e., PΛ x) = exp e x ) Organization of the paper. We escribe our results in Section 2 an set up the necessary mathematical constructs for the proof in Section 3. Theorems about convergence of the egree istribution have three parts: part a) escribes the limiting egree istribution of a uniformly chosen vertex in the shortest path tree; this is prove in Section 4, part b) states the convergence of the empirical egree istribution in the shortest path tree to the asserte limit from part a); this is prove in Section 5, part c) ientifies the limiting expecte egree in the shortest path tree; this is prove in Section 6. Section 2 also contains results about the tail behaviour of the egrees in the shortest path tree, prove in Section 7, an a link between the limiting egree istributions an those for breath-first tree setting, prove in Section 8.

6 6 BHAMIDI, GOODMAN, VAN DER HOFSTAD, AND KOMJÁTHY We now set out to state our main results. 2. Main results an iscussion 2.1. Weak isorer in the stochastic mean-fiel moel. Let G n s) enote the complete graph with each ege e equippe with an i.i.. ege weight l e = E s where E exp1) an s >. Here we escribe our results for the shortest path tree T n := T n s) from a ranomly selecte vertex. Let E i, i = 1, 2,..., enote inepenent copies of E. Define X 1 < X 2 < by X i = E E i ) s ; 2.1) equivalently, X i ) i 1 are the orere points of a Poisson point process with intensity measure Let Γ ) be the Gamma function an set µ s x) = 1 s x1/s 1 x. 2.2) λ s = Γ1 + 1/s) s ; 2.3) a short calculation verifies that e λsx µ s x) = 1. Then there exists a ranom variable W with W > an EW ) = 1 whose law is uniquely efine by the recursive istributional equation W = e λsxi W i, 2.4) i 1 where W 1, W 2,... are i.i.. copies of W. This ientity will arise from the basic ecomposition of a certain continuous-time branching process, an the uniqueness in law of W follows from stanar arguments; see Section Our first theorem escribes the egrees in the shortest path tree for the weak-isorer regime from Section 1: Theorem 2.1. Let s > an place i.i.. positive ege weights with istribution E s on the eges of the complete graph K n. Let X i ) i 1 be as in 2.1), let Λ i ) i 1 be i.i.. stanar Gumbel variables, an let W i ) i 1 be an i.i.. sequence of copies of W. Then: a) The egree eg Tn V n ) of a uniformly chosen vertex in the shortest path tree converges in istribution to the ranom variable D efine by D = 1 + i 1 1 {Λi+log W i+λ sx i<m}, with M = max i N Λ i + log W i λ s X i ). 2.5) b) The empirical egree istribution in the shortest path tree converges in probability as n, p n) k = 1 P 1 {egtn v)=k} P n D = k). 2.6) v [n] c) The expecte limiting egree is 2, i.e., as n, E[eg Tn V n )] E[ D] = ) We remark that D an M take finite values: the law of large numbers implies that i 1 X i 1 a.s., whereas Λ i + log W i = oi) a.s. by Markov s inequality an the Borel-Cantelli lemma. Since T n is a tree on n vertices an V n is a uniformly chosen vertex, E[eg Tn V n )] = 21 1/n) 2 as n. The convergence in 2.7) is in this sense a triviality. However, when combine with the convergence in istribution of eg Tn V n ) to D from part a), the assertion of 2.7) is that no mass is lost in the limit, i.e., the variables eg Tn V n ), n N, are

7 DEGREE DISTRIBUTION OF SHORTEST PATH TREES 7 uniformly integrable. In practical terms, this means that a small number of vertices cannot carry a positive proportion of the egrees in the shortest path tree. The following theorem escribes the tail of the egree istribution in the tree in terms of the exponent s on the exponential weights: Theorem 2.2. Let s > an place i.i.. positive ege weights with istribution E s on the eges of the complete graph K n. a) For s = 1, the variable D efine by 2.5) is a Geometric ranom variable with parameter 1 2. Then: b) For s < 1 an k, c) For s > 1 an k, log P D = k ) λs k s. log P D = k ) 1 1/s)k log k. Theorem 2.2 shows that the tail asymptotics of D ecay less rapily when s becomes small. Note that the bounary case s = correspons to constant ege weights. However, λ s as s, an Theorem 2.2 is not uniform over s. Inee, the limit s is surprisingly subtle, see [19] The configuration moel with finite-variance egrees. We next consier the configuration moel for rather general egree sequences n, which may be either eterministic or ranom, subject to the following convergence an integrability conitions. To formulate these, we think of n as fixe an choose a vertex V n uniformly from [n]. We write v for the egree of v in the original graph. Then the istribution of Vn is the istribution of the egree of a uniformly chosen vertex V n, conitional on the egree sequence n. We assume throughout that v 2 for each v [n]. Conition 2.3 Degree regularity). The egrees Vn satisfy Vn 2 a.s. an, for some ranom variable D with PD > 2) > an ED 2 ) <, Vn D, E 2 V n ) ED 2 ). 2.8) Furthermore, the sequence 2 V n log Vn ) is uniformly integrable. That is, for any sequence a n, lim sup E 2 V n log + Vn /a n )) =. 2.9) n In the case where n is itself ranom, we require that the convergences in Conition 2.3 hol in probability. In particular, Conition 2.3 is satisfie when 1,..., n are i.i.. copies of D an ED 2 log D) <. Define the istribution of the ranom variable D the size-biase version of D by PD = k) = kpd = k). 2.1) ED) We efine ν = ED 1); it is easily checke that ν = E[DD 1)]/E[D]. The assumptions Vn 2 an PD > 2) > imply that ν > 1. We take the ege weights to be i.i.. copies of a ranom variable Y > with a continuous istribution. Since ν > 1, we may efine the Malthusian parameter λ, ) by the requirement that νee λy ) = )

8 8 BHAMIDI, GOODMAN, VAN DER HOFSTAD, AND KOMJÁTHY Then there is a ranom variable W whose law is uniquely efine by the requirements that W >, EW ) = 1, an W = D 1 i=1 e λyi W i, 2.12) where W 1, W 2,... are i.i.. copies of W. Again, this ientity is erive from the basic ecomposition of a certain branching process; see Section The next theorem, the counterpart of Theorem 2.1, is about the egrees in the shortest path tree in the configuration moel: Theorem 2.4. On the eges of the configuration moel where the egree sequences n satisfy Conition 2.3 with limiting egree istribution D, place as ege weights i.i.. copies of a ranom variable Y > with a continuous istribution. Let Λ i ) i 1, W i ) i 1, an Y i ) i 1 be i.i.. copies of Λ, W, an Y, respectively. Then, a) the egree eg Tn V n ) of a uniformly chosen vertex in the shortest path tree converges in istribution to the ranom variable D efine by D = 1 + D i=1 1 {Λi+log W i+λy i<m}, with M = max 1 i D Λ i + log W i λy i ) ; 2.13) b) the empirical egree istribution in the shortest path tree converges in probability: p n) k = 1 P 1 {egtn v)=k} P n D = k), as n ; 2.14) v [n] c) the expecte limiting egree is 2, i.e., as n, E[eg Tn V n )] E[ D] = ) As in Theorem 2.1 c), part c) implies that the egrees eg Tn V n ), n N, are uniformly integrable. Since eg Tn V n ) Vn, part c) follows from Conition 2.3 using ominate convergence, but for completeness we will give a proof that uses the stochastic representation 2.13) irectly. In 2.13), the behaviour of D epens strongly on the value of D, an in particular D D a.s. This boun is clear in the original egree problem; to see it from 2.13), note that the summan for which M = Λ i + log W i λy i must vanish.) Thus very large observe egrees D must arise from even larger original egrees D. To unerstan this relationship, we efine a family of ranom variables D k ) k=1 by D k = 1 + k i=1 1 {Λi+log W i+λy i<m k }, with M k = max 1 i k Λ i + log W i λy i ). 2.16) The istribution of D k correspons to the limiting istribution of eg Tn V n ) when, instea of being selecte uniformly, V n is conitione to have egree k. The limiting istribution D from 2.13) is then the composition D = D D, 2.17) where D has the asymptotic egree istribution from Conition 2.3. As well as epening on k, the istribution of D k epens on λ > an on the istributions of Λ i ) i 1, W i ) i 1 an Y i ) i 1, which we always assume to be i.i.. copies of Λ, W an Y, respectively. We omit this epenence from the notation.

9 DEGREE DISTRIBUTION OF SHORTEST PATH TREES 9 The asymptotic behaviour of D is establishe by large values of D, hence we stuy D k in the limit k. The following theorem shows that the form of 2.13) an 2.16) etermines the asymptotic behaviour uner very general conitions. Theorem 2.5. Define D k accoring to 2.16), where the variables Λ i ) i 1, W i ) i 1 an Y i ) i 1 are i.i.. copies of arbitrary ranom variables Λ, W, Y, inepenently for each i N, with Y > a.s. If PΛ > x) > for each x R, or if PW > x) > for each x R, then D k = k1 o P 1)) as k. Theorem 2.5 shows that the proportion of summans in 2.16) that o not contribute to D k tens to. In wors, if the vertex has large egree in the original graph, then it is likely that almost all of the outgoing eges will be reveale by the shortest path tree. On the contrary, the next result shows that uner certain circumstances the orer of magnitue of the error is not necessarily small, i.e., finite behaviour might moify the empirical ata significantly compare to the true limit behaviour. We pay particular attention to the case when the ege weights Y i ) i 1 are i.i.. exponential or uniform variables. In these cases we can etermine the precise asymptotic orer of magnitue of the ifference between the egrees in the original graph an in the shortest path tree. Theorem 2.6. Define D k, M k accoring to 2.16), where the variables Λ i ) i 1, W i ) i 1 an Y i ) i 1 are i.i.. copies of a Gumbel variable Λ, a positive ranom variable W with EW ) <, an a positive ranom variable Y. Then, a) M k = log k + O P 1) as k ; b) if Ee λy ) <, then k D k is tight; c) if Y is a stanar exponential variable an the Malthusian parameter λ satisfies λ > 1, then k D k = Θ P k 1 1/λ ); 2.18) ) If Y is a stanar exponential variable an λ = 1, then k D k = Θ P log k). 2.19) Theorem 2.6 b) applies to the setting where Y is a stanar exponential variable an < λ < 1. Interestingly, for the CM with exponential ege weights, one has λ = ν 1, where we recall that ν = E[DD 1)]/E[D] enotes the expecte forwar egree. Thus, λ, 1) precisely when ν 1, 2). The other cases are treate in Theorem 2.6c) an 2.6), where the behaviour is really ifferent. Further, Theorem 2.6b) applies to the setting where Y is a uniform ranom variable, regarless of the value of λ. An immeiate consequence is the following corollary, hanling the case of i.i.. egrees with power-law exponent τ > 3. Here we shall assume that the istribution function F x) = PD x) of the unerlying egrees satisfies where x Lx) is a slowly varying function as x. 1 F x) = x 1 τ Lx), 2.2) Corollary 2.7. Suppose that the configuration moel egrees are i.i.. copies of a ranom variable D whose istribution function satisfies 2.2) with τ > 3. Then a) conitional on {D = k}, we have D = D1 o P 1)) in the limit k ; an b) the istribution function of D satisfies 2.2) also, for the same τ. If in aition ν > 2 an the ege weights are exponentially istribute, then c) conitional on {D = k}, we have D D = Θ P k 1 1/ν 1) ) in the limit k.

10 1 BHAMIDI, GOODMAN, VAN DER HOFSTAD, AND KOMJÁTHY Corollary 2.7 a) an b) show that large egrees are asymptotically fully etecte in the shortest path tree. Corollary 2.7 c) provies a counterpart by showing that D, though asymptotically of the same orer as D, may nevertheless be substantially smaller when D is of moerate size. Furthermore, this effect is accentuate when ν is large. Note that Theorems an thus Corollary 2.7 rely heavily on the fact that the unerlying egree istribution an the Malthusian parameter λ stay fixe whereas k is large. In other wors, these results pertain to a single vertex of unusually large egree. In particular, Theorems o not hol for the ranom k-regular graph in the limit k. In that case every vertex not just the target vertex has egree k an hence the Malthusian parameter λ = k 1 tens to infinity together with the egree k. In the context of an r-regular graph, Theorems apply instea to the asymptotic egree behaviour of a vertex of egree k ae artificially to the ranom r-regular graph on n vertices, with r fixe, k r an n The configuration moel with infinite-variance egrees. Section 2.2 treats the configuration moel with egree istribution having a finite limiting variance. However, in many real-life networks, this is not the case. Quite often, the available empirical work suggests that the egrees in the network follow a power-law istribution with exponent τ 2, 3). Thus, throughout this section we shall have in min that the egrees 1,..., n of the configuration moel are i.i.. copies of D, where D 2 a.s. an the istribution function F x) = PD x) satisfies 2.2) for 2 < τ < 3 an x Lx) a slowly varying function as x. We further assume that the ege weights are stanar exponential ranom variables. In the parameter range 2 < τ < 3, the egree istribution has finite mean but infinite variance. Hence the size-biase istribution in 2.1) is well efine, but has infinite mean, an the Malthusian parameter in 2.11) oes not exist. Instea, let V be the positive non-trivial) ranom variable that satisfies V = i= ) i j=1 D j 2), where E i is an i.i.. collection of exponential ranom variables, an inepenently, Dj are i.i.. copies of the size-biase istribution efine in 2.1), now having infinite mean. It is not har to see that V also satisfies V = E i min E i=1,...,d i + V i ), 2.22) 1 where E i an V i are i.i.. copies of E an V, respectively. This recursive characterisation can be erive again from the basic ecomposition of Markov chains. Our next theorem escribes the behaviour of egrees in the shortest path tree on the configuration moel with i.i.. infinite-variance egrees, an exponential ege weights: Theorem 2.8. On the eges of the configuration moel whose egree sequence n is given by inepenent copies of D, where the istribution function of D satisfies 2.2) with τ 2, 3), place i.i.. ege weights istribute as E, an exponential ranom variable of mean 1. Let V i ) i 1 an E i ) i 1 be i.i.. copies of V an E, respectively. Then, a) the egree eg Tn V n ) of a uniformly chosen vertex in the shortest path tree converges in istribution to the ranom variable D efine by D = 1 + D i=1 1 {Vi E i>ξ}, with ξ = min 1 i D V i + E i ); 2.23)

11 DEGREE DISTRIBUTION OF SHORTEST PATH TREES 11 b) the empirical egree istribution in the shortest path tree converges in probability, i.e., p n) k = 1 P 1 {egtn v)=k} P n D = k); 2.24) v [n] c) the expecte limiting egree is 2, i.e., as n, E[eg Tn V n )] E[ D] = ) As with Theorem 2.4 c), part c) of Theorem 2.8 asserts that the egrees eg Tn V n ), n N, are uniformly integrable, an we will give both a ominate convergence proof an a proof using the stochastic representation 2.23). As in Section 2.2, we wish to unerstan the asymptotic behaviour of the egrees by looking at vertices with large original egree. Thus, we efine a family of ranom variables D k ) k=1 by D k = 1 + k i=1 1 {Vi E i>ξ k }, with ξ k = min 1 i k V i + E i ). 2.26) Then the following theorem escribes the egree in the shortest path tree of a vertex conitione to have a large original egree: Theorem 2.9. Define D k accoring to 2.26), where the variables V i ) i 1 an E i ) i 1 inepenent i.i.. copies of arbitrary continuous positive ranom variables V an E, respectively. If PV < x) an PE < x) are positive for each x >, then for V, E inepenent, p = PV > E) satisfies < p < 1 an, as k, D k = p k 1 + o P 1)). 2.27) Theorem 2.9 asserts that an asymptotic fraction p neither nor 1) of the summans in 2.26) contribute to D k. Compare to Theorem 2.5, where p = 1, the ifference stems from the fact that V is supporte on, ) whereas Λ + log W is supporte on, ). Corollary 2.1. If the istribution function of the configuration moel egrees D satisfies 2.2) with τ 2, 3), then a) conitional on {D = k}, we have D = p D 1 + o P 1)) in the limit k ; an b) the istribution function of D satisfies 2.2) also, for the same τ Discussion. In this section, we iscuss our results an compare them to existing literature Convergence to the limiting egree istribution. Part a) of Theorems 2.1, 2.4 an 2.8 states that the egree istribution of a single uniformly selecte vertex converges to the istribution of D. Part b) strengthens this to state that the empirical egree istribution converges in probability, i.e., the ranom) proportion of vertices of egree k in the shortest path tree T n is with high probability close to the limiting value P D = k), for all k. Finally, part c) states that the convergence of the egree istribution from part a) also happens in expectation. Note that these convergences are not uniform over the choice of original egree istribution or ege weight istribution: see the remarks following Theorem 2.2 an Corollary 2.7.

12 12 BHAMIDI, GOODMAN, VAN DER HOFSTAD, AND KOMJÁTHY Degree exponents, bias an the effect of ranomness. If the initial graph is the configuration moel whose original egrees obey a power law with exponent τ, then Theorems 2.5 an 2.9 show that in both cases the power-law exponent τ is preserve via the shortest path tree sampling proceure. 1 In particular, if the egrees from a shortest path tree are use to infer the power-law exponent τ, then asymptotically they will o so correctly. In the literature, several papers consier the question of bias. Namely, o the observe egrees arising from network algorithms accurately reflect the true unerlying egree istribution, or can they exhibit power law behaviour with a moifie or spurious exponent τ? This question has rawn particular attention in the setting of the breath-first search tree BFST), where paths are explore in breath-first orer accoring to their number of eges, instea of accoring to their total ege weight. Exact analysis [1] an numerical simulations [29] have shown that the BFST can prouce an apparent bias, in the sense that observe egree istributions appear to follow a power law, for a relatively wie range of egrees, when the true istribution oes not. Surprisingly, this phenomenon occurs even in the ranom r-regular graph, where all vertices have egree r: efining a r) k the limiting egree istribution = Γr)Γk 1 + 1/r 2)) r 2)Γr + 1/r 2))Γk), 2.28) D BFST satisfies P D BFST = k) = a r) k if 1 k r, an a r) k 1 as k. 2.29) r k1 1/r 2) See [1, Section 6.1]; note that the requirement k r is not mentione in their iscussion.) In this case, since the unerlying egrees are boune, the power law in 2.29) is of course truncate, an is therefore not a power law in the sense of 1.1) or 2.2). The breath-first search tree correspons in our setup to the non-ranom case where all ege weights are 1. Although our proof of Theorem 2.4 relies on a continuous ege weight istribution, we may nevertheless set Y = 1 in the efinition 2.13) of D. In this case, we recover the limiting egree istribution arising from the breath-first search tree: Theorem Let D be any egree istribution with D 3 a.s. an ED 2 log D) <, an set Y = 1. Then with λ an W as in Section 2.2, the limiting egree istribution D from 2.13) is equal to the limiting egree istribution for the breath-first search tree ientifie in [1, Theorem 2]. In particular, Theorem 2.6 which makes no assumptions on the ege weights except positivity) applies to the breath-first search tree egrees. Consequently, Theorem 2.6 an Corollary 2.7 must be unerstoo with the caveat that they pertain to true power laws, but not truncate power laws such as 2.29). For the truncate power law in 2.29) to look convincingly like a true power law, r must be relatively large. It is worth noting, however, that the limiting egree istribution is ill-behave in the limit r : we have a r) 1 1 an a r) k for k 2, so that the egree of a typical vertex converges to 1 an most vertices are leaves. In particular, the truncate power law in 2.29) isappears in this limit. Furthermore, the expecte limiting egree E D BFST ) which continues to be 2 for each finite r) is reuce to 1 after taking r, so BFST that D is not uniformly integrable in this limit. By way of comparison, the limiting egree istribution for the ranom r-regular graph with i.i.. exponential ege weights perhaps raise to some power s > ) is well-behave in 1 To be precise, this is prove only for τ > 3, for certain parts of the regime τ = 3, an for 2 < τ < 3 with exponential ege weights.

13 DEGREE DISTRIBUTION OF SHORTEST PATH TREES 13 the limit r, an inee converges 2 to the limiting egree istributions for the complete graph efine in Section 2.1. By Theorem 2.2, the tails of this istribution ecay faster than a power law, for any s >. Figure 2 shows a simulation of the case r = 1, s = 1, with n = 1. The observe egree istribution oes not resemble a power law at all, an in fact it agrees very closely with the Geometric1/2) istribution which, by Theorem 2.2 a) an the preceing iscussion, correspons to the case r. While not a proof, this strongly suggests that the truncate power laws foun in [1, 29] are anomalous an reflect specific choices in the breath-first search moel. It woul be of great interest to unerstan uner what conitions truncate power laws can be expecte to appear in general. It is tempting to conjecture that spurious power laws o not arise whenever the ege weights are ranom with support reaching all the way to Special cases. The statement of Theorem 2.2 for s = 1 is well-known, since in this case the shortest path tree is the uniform recursive tree, an the egrees in the uniform recursive tree can be unerstoo via martingale methos: see for instance [24, Exercise 8.15, Theorem 8.2]. The proof we give here is ifferent, with the main avantage that it is easier to generalize to the case s 1. It is base on the representation 2.5) for D together with the observation that the martingale limit W is a stanar exponential variable: see for instance [24] or [27], or verify irectly that E satisfies 2.4). The r-regular graph on n vertices correspons to the choice D = r in Theorem 2.4. If in aition the ege weights are exponential, the martingale limit W can be ientifie as a Gamma r 1 r 2, r 2 r 1 ) ranom variable, i.e., the variable with Laplace transform φ W u) = 1+ r 2 r 1 u) r 1)/r 2). Even though we can characterize W, obtaining an explicit escription of the law of D for example, through its generating function) appears ifficult Branching processes: limit ranom variables W an V. In analyzing the shortest path tree T n, it is natural to consier the exploration process, or first passage percolation, that iscovers T n graually accoring to the istance from the source vertex v s. Starting from the subgraph consisting of v s alone, reveal the original egree vs. Reveal whether any of the vs half-eges associate to v s form self-loops; if any o, remove them from consieration. This step is unnecessary in the complete graph case.) For each remaining half-ege, there is an i.i.. copy of the ege weight Y. Set t =. Iteratively, having constructe the subgraph with i vertices an i 1 eges, wait until the first time t i > t i 1 when some new vertex v i can be reache from v s by a path of length t i. Thus t 1 will be equal to the smallest ege weight incient to v s, apart from self-loops.) Reveal the egree vi an a the unique new ege in the path between v i an v s, using one of the vi half-eges associate to v i. For the remaining vi 1 half-eges, remove any that form self-loops or that connect to alreay explore vertices, an iterate this proceure as long as possible. The subgraph so constructe will be T n. When n, no half-ege will form a self-loop or connect to a previously explore vertex by any fixe stage i of the exploration, for any fixe i. It follows that the exploration process is well approximate at least initially) by a continuous-time branching process CTBP) that we now escribe. 2 This follows from the convergence of the collection r s Y i ) i [r] of rescale ege weights towars the Poisson point process X i ) i=1, cf. 4.14) an the surrouning material, an the consequent convergence of the corresponing martingale limits W. Problems relate to the unboune number of terms in 2.5) an 2.13) can be hanle by the observation that the collection r s Y i ) i [r] is stochastically ominate by X i ) i 1 for each r.

14 14 BHAMIDI, GOODMAN, VAN DER HOFSTAD, AND KOMJÁTHY Consier first the configuration moel. The vertex v s is uniformly chosen by assumption. The vertex v 1, however, is generally not uniformly chosen. Conitional on v s we have, P v 1 = v v s ) = v1 {v vs} w v s w. 2.3) Note for instance that v1 can never be ). Owing to the finite mean assumption on the CM egrees, it follows that w v s w ned) an P v1 = k v s ) kpd = k)/ed) in the limit n. This size-biasing effect means that the number vi 1 of new half-eges will asymptotically have the istribution D 1, where D is efine in 2.1). The CTBP approximation for the CM is therefore the following: An iniviual v born at time T v has a ranom finite number N v of offspring, born at times T v + Y v,1,..., T v + Y v,nv. The Y v,i are i.i.. copies of Y ; the initial iniviual v s has family size N vs = vs ; an all other iniviuals have family size N v = D 1. For the complete graph, the egrees are eterministic but large, an it is necessary to rescale the ege weights: the collection of ege weights incient to a vertex, multiplie by n s, converges towars the Poisson point process X i ) i 1 efine in 2.1), for a formal version of this statement, see 4.14) below. The corresponing CTBP is as follows: Every iniviual v born at time T v has an infinite number of offspring, born at times T v + X v,1, T v + X v,2,..., where X v,i ) i 1 are i.i.. copies of the Poisson point process efine in 2.1). The ranom variables W an V from Sections arise naturally from these CTBPs. In the complete graph context from Section , the CTBPs grow exponentially in time, with asymptotic population size cw e λt for λ = λ s efine by 2.3) an c > a constant, an inee W arises as a suitable martingale limit; see [4]. For the CM contexts from Sections , we must take the initial iniviual v s to have egree istribution D 1 in orer to obtain the variables W an V instea of Ŵ an V from Section 3 below). When the family sizes D 1 have finite mean, as in Section 2.2, the population size again grows asymptotically as cw e λt for λ given by 2.11). In the setting of Section 2.3, the CTBP exploes in finite time, i.e., there is an a.s. finite time V = lim k t k at which the population size iverges; see [23]. The recursive relations 2.4), 2.12) an 2.22) result from conitioning on the size an birth times of the first generation in the CTBP. For the uniqueness in law of W, see [3, Theorem 4.1, Page 111]. We note that in all cases, the value of W or V is etermine from the initial growth of the branching process approximations: we can obtain an arbitrarily accurate guess, with probability arbitrarily close to 1, by examining the CTBP until it reaches a sufficiently large but finite size. In terms of the exploration process, it is sufficient to examine a large but finite neighbourhoo of the initial vertex. Large values of W, an small values of V, correspon to faster than usual growth uring this initial perio, an thereafter the growth is essentially eterministic. In Theorems 2.1 an 2.4, a large value of M might be expecte to correspon to one large value of W i, an a large value of D might be expecte to arise from having many vertices j with small values of W j. As we shall see in the proofs, however, this intuition is incorrect, an it is the variables Λ i, an seconarily the ege weights Y i, whose eviations are most relevant to the sizes of M an D Shortest path trees an giant components. In Theorems 2.4 an 2.8, the hypothesis D 2 implies that v s an v t are connecte with high probability. If egrees 1 or are possible, we must impose the aitional assumption that ν > 1 in Theorem 2.4. Having mae this assumption, the CM will have a giant component, i.e., asymptotically, the largest component will contain a fixe positive fraction of all vertices, an the next largest component will contain on) vertices. The variable W from Section 2.2 has a positive

15 DEGREE DISTRIBUTION OF SHORTEST PATH TREES 15 probability of being, in which case we set log W =, an the variable V from Section 2.3 has a positive probability of being. Furthermore, there is a positive probability that T n contains only a fixe finite number of vertices, corresponing to the case where the branching process approximations from Section go extinct. This possibility will be reflecte mathematically in the possibilities that Ŵs = in Proposition 3.2 or V s = in Proposition 3.3.) If we conition v s to lie in the giant component corresponing to non-extinction of the branching process starte from v s ), then in the resulting shortest path tree, the outegree of v t has the same limiting conitional istribution as D 1 in Theorems 2.4 an 2.8. The variable M respectively, ξ) equals respectively, ) whenever W i = respectively, V i = ) for each i = 1,..., D, corresponing to the case that v t oes not belong to the giant component, an in this case the outegree an the egree of v t are both Open problems. There are several interesting questions that serve as extensions of our results. First, as iscusse in Section 2.4.2, our results reveal the existence or nonexistence of true power laws, but not truncate power laws. A precise characterization of when truncate power laws arise woul be of great interest. Secon, many real-life networks have power-law behaviour with egree exponent τ 2, 3). In this regime where the egrees have infinite variance as well as part of the regime τ = 3 when Conition 2.3 is not satisfie), it is natural to exten beyon the exponential ege weights that we consier. We expect that Theorems 2.8 an 2.9 remain vali with slight moification if the corresponing CTBP is explosive, i.e., if the CTBP reaches an infinite population in finite time. When the corresponing CTBP is not explosive, even the probabilistic form of the limiting istribution D is unknown. Such a representation woul in particular be expecte to give rise to the limiting BFST egree istribution, as in Theorem Finally, real-worl traceroute sampling typically uses more than just a single source. It is natural to exten our moel to several shortest path trees from ifferent sources. In this setup, the resulting behaviour might epen on whether we observe, for a given target vertex, either the egree in each shortest path tree; or the egree in the union of all shortest path trees; or the entire collection of incient eges in each shortest path tree. In any of these formulations, we may ask how accurately the observe egree reflects the true egree when the number of sources is large, an whether this accuracy varies when both the true egree an the number of sources are large. 3. Limit theorems for shortest paths The proofs of Theorems 2.1, 2.4 an 2.8 are base on Propositions 3.1, 3.2 an 3.3 respectively which in turn follow from [1, Theorem 1.1], [9, Theorems ] an [7, Theorem 3.2] respectively. These theorems etermine the istribution of the shortest paths between two uniformly chosen vertices in the complete graph, an in the configuration moel. Since we nee the results about shortest paths jointly across a collection of several target points i.e., not just between two vertices), we state only the neee versions here. These moifications easily follow from the results mentione earlier, combine with an iea about marginal convergence from the work of Salez [36] who prove the joint convergence of typical istances between several points for the particular case of the ranom r-regular graph with exponential mean one ege lengths. His argument extens however to the more general situation as well. We give an iea of how these results were proven in Section 3.1 but omit full proofs. Our first proposition is about the joint convergence of shortest weight paths on the complete graph. Recall the notation for W from Section 2.1.

16 16 BHAMIDI, GOODMAN, VAN DER HOFSTAD, AND KOMJÁTHY Proposition 3.1. Consier the complete graph with ege weights istribute as E s, s >. Let v 1,..., v k be istinct vertices, all istinct from v s the source vertex), an enote the length of the shortest path between v s, v i by C n v s, v i ). Then λ s n s C n v s, v i ) log n) k i=1 Λ i log W s log W i ) k i=1, 3.1) where Λ 1,..., Λ k are i.i.. copies of Λ an W s, W 1,..., W k are i.i.. copies of the ranom variable W from Section 2.1. Note that, ue to the presence of the term log W s, the limiting variables in Proposition 3.1 are exchangeable but not inepenent for ifferent i. When k = 1, the case s = 1 is ue to [27] an the case s 1 is ue to [1]. For the configuration moel with finite-variance egrees, we will nee to apply a similar result to the neighbours of the uniformly chosen vertex v t. Proposition 3.2. Consier the configuration moel with egrees satisfying Conition 2.3. Let v 1,..., v k be istinct vertices, all istinct from v s, which may be ranomly chosen but whose choice is inepenent of the configuration moel an of the ege weights. If the egrees v1,..., vk ) converge jointly in istribution to inepenent copies of D 1, then there is a constant λ > an a sequence λ n λ such that λ n C n v s, v i ) log n) k i=1 Λ i log Ŵs log W i + c ) k i=1, 3.2) jointly in i = 1,..., k, where c is a constant, Λ i are i.i.. copies of Λ, W 1,..., W k are i.i.. copies of the variable W from Section 2.2, an Ŵs is a positive ranom variable, all inepenent of one another. As iscusse in Section 2.4.4, each time we connect a half ege of v t to another vertex, the probability of picking a vertex of egree k is approximately proportional to k PD = k). Thus, for each neighbour, the egree converges in istribution to the size-biase variable D efine in 2.1), an the number of half-eges not connecte to v t converges in istribution to D 1. This motivates the assumptions on the egrees in Proposition 3.2. The constant c arises as a function of the stable age-istribution of the associate branching process [9]. Since it oes not play a role in the proof, we omit a full escription of this constant. Finally, we state the corresponing result for the infinite-variance case: Proposition 3.3. Consier the configuration moel with i.i.. egrees satisfying 2.2) with τ 2, 3). Let v 1,..., v k be istinct vertices, all istinct from v s, which may be ranomly chosen but whose choice is inepenent of the configuration moel an of the ege weights. If the egrees v1,..., vk ) converge jointly in istribution to inepenent copies of the size-biase istribution D 1, then ) C n v s, v i )) k k i=1 Vs + V i, i=1 where V i ) i 1 are i.i.. copies of the ranom variable V from Section 2.3 an V s is a ranom variable inepenent of V 1,..., V k Iea of the proof. We give the iea behin the proof of Proposition 3.2. The proofs of the other propositions are similar, using the corresponing branching process approximations of local neighborhoos as escribe in Section Let n ) n 1 be a egree sequence satisfying Conition 2.3 an fix a continuous positive ranom variable Y. Let G n = [n], E n ) be the configuration moel constructe from this

17 DEGREE DISTRIBUTION OF SHORTEST PATH TREES 17 egree sequence, with E n enoting the ege set of the graph, an let the ege weights {Y e : e E n } be i.i.. copies of Y. As in 2.1) 2.11), we efine PDn = k) = kp Vn = k)/e Vn ) the size-biasing of Vn ) an the corresponing size-biase expectations ν n = EDn 1), Malthusian parameters λ n satisfying ν n Ee λny ) = 1, an martingale limit W n) satisfying W n) = D n 1 i=1 e λnyi W n) i. Assuming Conition 2.3, we have ν n ν so that ν n > 1 an λ n, W n) are well-efine for n sufficiently large), λ n λ an W n) W One target vertex: the case k = 1. Let us first summarize the ieas behin [9, Theorems ], which erive the asymptotics for the length of the optimal path between two selecte vertices v, v 1 G n. To unerstan this optimal path, think of a flui flowing at rate one through the network using the ege lengths, starte simultaneously from the two vertices v, v 1 at time t =. When the two flows collie, say at time Ξ 1) n, there exists one vertex in both flow clusters. This implies that the optimal path is create an the length of the optimal path is essentially 2Ξ 1) n. Write F i t)) t for the flow process emanating from vertex v i. As escribe in Section 2.4.4, these flow processes can be approximate by inepenent Bellman-Harris processes where each vertex has offspring istribution Dn 1 an lifetime istribution Y. By [26], n) the size of both flow processes grow like F i t) W i expλ n t) as t, where λ n is the n) Malthusian rate of growth of the branching process an W i > ue to the fact that by assumption our branching processes survive with probability 1) are associate martingale limits. Furthermore, an analysis of the two exploration processes suggests that for t >, the rate at which one flow cluster picks a vertex from the other flow cluster thus creating a collision in a small time interval [t, t + t)) is approximately γ n t) κ F n) t) F 1 t) κ W W n) 1 exp2λ n t) t, 3.3) n n where the constant κ arises ue to a subtle interaction of the stable-age istribution of the associate continuous time branching process with the exploration processes. This suggests that times of creation of collision ege scales like 2λ) 1 log n, an further the time of birth of the first collision ege, re-centere by 2λ) 1 log n, converges to the first point Ξ of a Cox process with rate γ x) := κ W W1 exp2λx), x R. It is easy to check that 1 Ξ = Λ log 2λ W log W ) 1 + c 3.4) where c is a constant epening on λ an κ an Λ has Gumbel istribution inepenent of W, W 1. In [9], both v an v 1 are chosen uniformly an therefore have a egree istribution ifferent from the offspring istribution Dn 1 associate to the rest of the branching n) n) process. Consequently, W an W 1 are not istribute as the martingale limit W n) but as a certain sum Ŵ s n) of such variables with Ŵ s n) Ŵs as n ). By contrast, in the setting of Proposition 3.2 for k = 1, the vertex v 1 has istribution close to D 1 by n) assumption, so that this replacement is not necessary an W 1 = W n) W. Since the length of the optimal path scales like 2Ξ 1) n, rearranging 3.4) gives Proposition 3.2 with k = 1. The actual rigorous proof in [9] is a lot more subtle albeit following the above unerlying iea. The optimal path is forme not quite at time 2Ξ n, one has to keep track of resiual