On the robustness of power-law random graphs in the finite mean, infinite variance region

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1 On the robustness of power-law random graphs in the finite mean, infinite variance region arxiv: v [math.pr] 7 Jan 008 Ilkka Norros and Hannu Reittu VTT Technical Research Centre of Finland Abstract We consider a conditionally Poissonian random graph model where the mean degrees, capacities, follow a power-tailed distribution with finite mean and infinite variance. Such a graph of size N has a giant component which is super-small in the sense that the typical distance between vertices is of the order of log log N. The shortest paths travel through a core consisting of nodes with high mean degrees. In this paper we derive upper bounds of the distance between two random vertices when an upper part of the core is removed, including the case that the whole core is removed. Introduction Power-law random graphs or scale-free random graphs have become popular objects of both applied and theoretical interest, because they are simple to define and generate and yet share some important characteristics with many large and complex real-world networks. The mathematical models help to understand how and on what conditions those characteristics emerge, and this understanding can be valuable also in the design of new, artificial networks. The general characteristic of power-law random graphs is that their degree distribution possesses a regularly varying tail, the most interesting region of tail exponents being that of finite mean and infinite variance. Several models with these features have been studied. In this paper, we work with the conditionally Poissonian random graph [7], a modification of the expected degree sequence model of Chung and Lu [3]. We draw an i.i.d. sequence of mean degrees, capacities, that follow a Pareto distribution with finite mean and infinite variance. Our graph of size N has a giant component which is ultra-small in the sense that the typical distance between vertices is proportional to log log N more precisely, this is an upper bound, like most of our distance results, whereas a similar random graph whose degrees have the same mean but finite variance cannot offer better than log N scalability of distances. Remarkably, the speciality of the infinite variance case is the emergence of a core network consisting of nodes with high degrees, through which the short paths typically travel. It is now natural to ask, what happens to these distances, if vertices with highest degrees are deleted. Our main findings can be summarized as follows:

2 i Any deletion of core vertices leaves the asymptotical size of the largest, giant component intact. ii If vertices with capacity higher than N γ are deleted, γ not depending on N, then paths that would have gone through the deleted vertices can be repaired using back-up paths that increase the distances only by a constant depending on γ but not on N. This increase is negligible on the log log N scaling of distances. iii If the whole core is removed for the exact meaning of this, see Section, the distances still scale slightly better than log N. Because of structural similarity compare [8] with [7], we conjecture that our results are transferable to the configuration model of Newman et al. [6, 8]. Perhaps they can be extended even to preferential attachment models in the finite mean and infinite variance region, since van der Hofstad and Hooghiemstra showed recently that these models share with the previously mentioned models the loglog-scalability of distances [5]. The next section specifies the graph model and reviews the relevant results of [7]. Some of them are slightly sharpened in Section 3. Section 4 applies results on the diameter of classical random graphs to measure the core network in the horizontal direction. The main result on robustness against core losses is stated and proven in Section 5. Model and earlier results Throughout the paper, we work with the following conditionally Poissonian power-law random graph model [7]. There are N vertices that possess, respectively, i.i.d. capacities Λ,..., Λ N with distribution PΛ > x = x τ+, x [, ; τ, 3. Given Λ = Λ,..., Λ N, each pair of vertices {i, j}, is connected with E ij edges, where Λi Λ j E ij Poisson L N, L N = N Λ j, and the E ij s are independent. Loops, i.e. the case i = j, are included for principle, although they have no significance in the present study. Given Λ, the degree of vertex i then has the distribution PoissonΛ i. The resulting random graph is denoted as G N. The following coupling between neighborhood shells and branching processes will play an important role, as it did in [7]. Fix N, and assume that the sequence Λ has been generated and probabilities are now conditioned on it. We shall consider a marked branching process, i.e., a branching process where each individual is associated with some element of the mark space {,..., N}. More specifically, let us define a process Z, J = Z n, J n,i, where Z n is the size of generation n, and J n,i {,..., N} is the mark of member i of generation n. We set Z 0 and take J 0, from the uniform distribution U{,..., N}. The process then proceeds so that an j=

3 invidual bearing mark i gives for each j =,..., N birth independently to a PoissonΛ i Λ j /L N distributed number of j-marked members of the next generation. On the other hand, we consider the neighborhood sequence around a random vertex of G N. Take a vertex i 0 from uniform distribution, and define recursively The following coupling was proven in [7]. N 0 i 0 = {i 0 } { k c } N k+ i 0 = j N l i 0 : j N k i 0. l=0 Proposition. Let Z, J be the marked branching process defined above. Define a reduced process by proceeding generation by generation, i.e., in the order J 0, ; J,, J,,..., J,Z ; J,,..., and pruning that is, deleting from Z, J each individual, whose mark has already appeared before, together with all its descendants these are considered as not seen in the procedure. Denote the resulting finite process by Ẑ, Ĵ, and let Jk ˆ be the set of the marks in generation k of the reduced process. Then, the sequence of the sets J ˆ k has the same distribution as the sequence N k. Moreover, we observed that the size of each generation and its marks can be generated independently. Denote by q N the distribution q N j. = Λ j L N, j =,..., N, and by π the mixed Poisson distribution PoissonΓ, where Γ is a random variable with distribution PΓ dx = xpλ dx /E {Λ}. Obviously, the distribution of PoissonΛ J N, J N q N, converges weakly to π in probability. Let J N, J N,... be i.i.d. random variables with distribution q N, and let J N 0 be a random variable with uniform distribution on {,..., N}, independent of the previous ones; we often suppress the superscript N. Start now by Z 0 =, J 0, and proceed generationwise by drawing the size of the n + th generation from the distribution Z n+ Poisson M n i=m n + where M = 0, M n = n 0 Z k, and giving then each individual a fresh mark from the sequence J Mn+, J Mn+,.... Λ Ji, 3

4 Proposition. The marked generation sequence Z n, J n,i is stochastically equivalent to Z n, J Mn +,..., J Mn. We next turn to the definition of the core network mentioned in the introduction. Fix an increasing function l : N R such that and define Note that l =, Define recursively the functions ln log log log N 0, ln log log log log N ɛn := ln log N. as N, ɛn 0, N ɛn = e ln as N. 3 β 0 N = τ + ɛn τ, β j N = τ β j N + ɛn, j =,,..., 4 and denote k. log log N =. logτ For N sufficiently large, we have β 0 N > ɛn/3 τ, and the sequence β k N k=0,,... decreases toward the limit value ɛn/3 τ. With k = k, we are already in the ɛn order of magnitude: β k NN ɛn + τ i = 4 τ ɛn. 5 3 τ i=0 The key of our analysis of G N is the notion of its core C, consisting of all vertices with capacity higher than N β k. Note that the exact boundary of the core depends on ln and is thus somewhat arbitrary. By the definition of ɛn, we can choose a natural number κ = κn such that κn κn, N θɛn k N 0, as N, 6 where θ = τ 4 τ/3 τ. We can now collect the main results of [7] to the following theorem: Theorem.3 i The graph G N has a.a.s. asymptotically almost surely a giant component, whose relative size approaches the value j= PD = j P Z π = 0 j, 7 4

5 where D is distributed as the conditionally Poissonian variable PoissonΛ, and Z π n is a Galton-Watson branching process with offspring distribution π. ii A random vertex of the giant component is connected with the core C in less than κn hops. The distribution of the capacity of the first core vertex found in a random-order breadth-first neighborhood { search } is asymptotically identical to the distribution of J n N C, where n C = inf n : J n N C. iii A random vertex of the core is a.a.s. connected with the vertex i with highest capacity in at most k hops. iv As a consequence of the previous items, the distance between two randomly chosen vertices of the giant component is at most k N + o a.a.s.. Denote T γ,δ = { i {,..., N} : Λ i N γ, N δ ] }. The idea of the proof of claim iii of Theorem.3 is that if the core is divided into tiers V 0 = {i }, V = T β, \ {i }, V k = T βk,β k, k =,..., k, then, for any k > 0, a random vertex of tier k has a.a.s. an edge to V k V 0. 3 Auxiliary results In this section we sharpen the estimate used in [7] for the aggregated capacity of a set of form {i : Λ i > N γ }. In particular, we obtain that the aggregated capacity of a tier T γ,γ+ɛn is asymptotically overwhelmingly larger than all strictly higher vertex capacities together. This holds also when γ depends on N. We use frequently the notation x + I := {x + y : y I}, where x R and I R is an interval. Lemma 3. Let α = αn 0, ᾱ, where ᾱ < /τ. Then, for N sufficiently large, N i= P Λ τ+ i {Λi >N α } E {Λ} N τ α E {Λ} 4, 4 N τ ᾱτ. Proof Denote n n i= pn = P Λ i E {Λ},, N α = n {Λi >N α }. 5

6 Since P[Λ > y Λ > x] = PxΛ > y, we have the distribution equality N N α D Λ i {Λi >N α } = N α Λ k, 8 i= where the Λ k s are fresh independent copies of Λ. Then k= Nα Λ N P α k= k E {Λ} P N α < E {N α} [, ] = E {pn α } } + E {pn α {Nα E{Nα}}. 9 The distribution of N α is BinN, N τ α. The well-known bounds for a binomial random variable X, yield PX < E {X} x exp x E {X}, PX > E {X} + x exp x E {X} + x3 E {X} 3, Nα P E {N α } < e E{Nα}/8, Nα P E {N α } > e E{Nα}/8+. 0 By the well-known result on subexponential variables with finite mean see, e.g., [4], n P Λ i > E {Λ} PmaxΛ,..., Λ n > ne {Λ} E {Λ} τ+ n τ+. n i= For the large deviation to the opposite direction, the classical Cramér theorem applies and the probability goes to zero at exponential speed. Thus, for N sufficiently large, E {pn α } e E{Nα}/8 + 3 τ+ E E {Nα } {Λ} τ+. Combining 8-, replacing α by the worse case ᾱ and upperbounding the exponential terms by replacing the factor 3/ in by we get the assertion. Lemma 3. Let α 0 N, α N 0, ᾱ, where ᾱ < /τ, and α 0 N α N ɛn. Then, for N sufficiently large, N i= P Λ i {Λi N α0,n α ]} τ+ E {Λ} N τ α 0 E {Λ} 5, 5 N τ ᾱτ. Proof This follows by applying Lemma 3. to both α 0 and α and noting that N τ α α

7 Lemma 3.3 For any fixed b >, N i= Λ i {Λi >N bɛn } N i= Λ i {Λi >N ɛn } 0 in probability. Proof By Lemma 3., N i= Λ i {Λi >N bɛn } N i= Λ N b τ ɛn [, 6] a.a.s., i {Λi 6 >N ɛn } and the claim follows since N ɛn. Lemma 3.3, combined with our earlier results in [7], entails the important observation that the first contact to the core is a.a.s. close to its bottom: Proposition 3.4 Fix b > 0. Let I N be a random node of the giant component of G N. Asymptotically almost surely, I N is connected to T βk,bβ k with a path that avoids the set { i : Λ i > N bβ k }, and whose length coincides with the distance between I N and the core. Proof Since the proportional size of the core shrinks to zero, it suffices to consider the case that I N is picked from outside the core. By the results in [7], conditional that I N is connected with the core, one of the minimal length paths to the core has its core endpoint distributed as the first core element in an i.i.d. sequence J n {,..., N} with common distribution. Now, β k is proportional to ɛn, and Lemma 3.3 can be applied with β k in the role of ɛn. We conclude that the first core element in the sequence J n belongs a.a.s. to the set T βk,bβ k, and the statement of the proposition follows. 4 Horizontal paths in the core In this section, we show that quite thin tiers of the form T γ,γ are internally as induced subgraphs almost connected in the sense that the proportional size of the largest component approaches one and, moreover, the diameter of that largest component is obtained with high accuracy from remarkable results on classical random graphs. For deterministic γ, the tiers T γ ɛn,γ are a.a.s. connected, and their diameters can be picked almost unequivocally from the following classical theorem []. Theorem 4. Consider the Erdös-Rényi random graph G n,p and let p = pn and d = dn > satisfy Then G n,p has diameter d a.a.s. log n 3 log log n d p d n d log n p d n d log n 7

8 To include also lower parts of the core, we apply the following, rather recent result by Chung and Lu []. Theorem 4. If np and log n/log np, then almost surely diamg n,p = + o log n log np, where diamg denotes the diameter of the largest giant component of G. Denote τ γ wγ :=. 3 τγ Proposition 4.3 Let γn be a non-increasing function such that γn [ɛn, for all N. If lim γn > 0, assume also that γ does not correspond to a jump of the ceiling function in. Then, there exists another function γ N such that and γ N < γn, γ N/γN, T γ N,γN, diamt γ,γ = + owγn a.a.s., 3 where diam is generally interpreted as in Theorem 4.. When γn is bounded away from zero, T γ,γ is a.a.s. connected and its diameter is a.a.s. equal to wγn. Proof We shall suppress the N s in γn etc. when it improves the clarity of presentation. Considered as an induced subgraph, T γ,γ is denser resp. sparser than the classical random graph G n,p resp. G n,p with n = nn = T γ,γ, p = p N = N γ L N, p = p N = N γ L N. Denote γ δ N = γ δɛn, where δ 0,. We fix δ for a while and choose γ = γ δ. By Lemma 3., T γ,γ, and On the other hand, Thus, a.a.s., T γ,γ N [, 5], a.a.s. τ γ 5 L N /N E {Λ} [, ] a.a.s. log n τ γ log N + [ log 4, log 4], lognp 3 τγ log N log E {Λ} + [ log 8, log 8], lognp γ τ γ log N log E {Λ} + [ log 8, log 8]. 8

9 Note that p N 0 and nnp N, so Theorem 4. applies to G n,p and G n,p. We distinguish between two cases. First, if γn is bounded away from zero, we are in fact in the regime of Theorem 4., and a computation of the diameter d leads to the expression of wγ. Thus, Theorem 4. yields the inequalities τ γ diamt γ,γ a.a.s., 3 τγ τ γ diamt γ,γ a.a.s. γ + τγ Now, the right hand sides are with sufficiently large N both equal to the right hand side of wγ, and the claim follows. The value of δ did not matter. The second case is that γn 0. Since nnp N, the subgraph still has a giant component whose proportional size approaches one, although it is not necessarily connected. Choose any sequence δ k 0, and denote { 3 τγ η k = max, γ + τ γ } δ k. 3 τγ δk 3 τγ Note that η k. Now define a sequence N k as { N k = inf M : P T γδk,γ N γ δ k k /k N M, L N P diamt γ,γ η k, } η k /k N M. 3 τγ Thanks to Theorem 4., the N k s are finite numbers, and obviously they grow unboundedly. Finally, denote kn := max {k : N k N} and choose γ N = γ δkn. Then obviously diamt γ,γ = + o 3 τγn a.a.s. In a similar way, we can prove a diameter result for the tiers V k = V N k, defined at the end of Section : Proposition 4.4 The tiers V 0,..., V k, considered as induced subgraphs of G N, are almost connected in the sense that the relative sizes of their largest components approach as N, and their diameters satisfy, a.a.s., diamv 0 = 0, diamv =, diamv k + o[wβ k, wβ k ], k =,..., k. 9

10 By Proposition 4.3, we can say that the width of the core at height N γ, measured by hop distance, is wγ. Moreover, this holds also for varying γ = γn downto the bottom of the core. When γ 0,, we have wτ γ > wγ +. This has the following heuristic consequence. For connecting two vertices with capacities N γ, a horizontal path staying at about the same height in the core is longer than a path that jumps at both ends with one hop to the height N τ γ and finds the horizontal connection at that level. On the other hand, this is how high in the core the neighborhood of a vertex typically reaches. Thus, a horizontal move should be made at as high level as possible. 5 The main result We have now essentially collected all elements needed to understand the situation where a top part of the core is deleted. First, since the aggregated capacity of any top part is concentrated at its bottom, the remains of the core are found from outside almost as easily as the original core was found. Second, if a lower part of the core is alive, we can find vertical paths from the bottom of the core to a top tier of its remaining part, as we did with the undamaged core in [7] a complete proof of this step requires a slight modification of the proof of Proposition 4.3 of [7] and is presented below in detail. Third, Proposition 4.3 guarantees the existence of a horizontal path between almost any pair of vertices within this top tier, and provides also an explicit upper bound for its length. It remains to couple these pieces together. Theorem 5. Let γ := γn ɛn, be such that γn is non-increasing and γn/ɛn non-decreasing w.r.t. N. Denote by H γ = H N the graph obtained from G N by deleting all vertices with capacity higher than N γ, together with the edges attached to them. The following hold: i The graph H γ has a.a.s. a giant component whose relative size approaches the relative size of the giant component of G N. ii The distance of two randomly chosen vertices of the giant component of H γ is a.a.s. less than + o log log N log + wγ, 4 logτ γ where wγ is given in. Proof o. Denote γ 0 = γ, γ = γ 0 ɛn, γ k+ = τ γ k + ɛn, k =,,..., 0

11 and let { k = min k : γ k 4 τ } 3 τ ɛn. It is easy to check see 4 o below that k = Olog log N. Consider the tiers T γ,γ 0, T γ,γ,..., T γ k,γ k. The idea of the proof is to apply the diameter result, Theorem 4., on T γ,γ 0, and, if k, to imitate our proof of Theorem.3 to show that vertices of tier T γ k,γ k are with high probability connected to T γ,γ 0 with a path jumping from tier to tier. o. Assume that k. Let I 0 = I 0 N be a stochastic not necessarily uniformly random vertex of T γ k,γ k such that, conditioned on Λ, I 0 is independent of the edges within k T γ j,γ j. Define the sequence of vertices I, I,... recursively as follows. If I n T γ k n, then I,γ k n n+ is a neighbor say, with smallest index of I n belonging to T γ k n if such a neighbor exists.,γ k n Otherwise I n+ = I n, and the rest of the sequence repeats I n as well. Denote A n = { } I n T γ k n,γ k n, n = 0,,..., k, { } B LN = E {Λ}, N B k = {LT γk+,γk E {Λ} N τ γ k 5 }, 5, k = 0,..., k, where LS := i S Λ i, B = B B 0 B k. We have A k A k A 0. Proposition 3. yields τ+ E {Λ} PB 0 B k k N τ γτ as N. On B in turn, we have for each n by Proposition 3. that P [ [ ] A c n+ Λ, An = E exp Λ ] I n LT γ k n,γ k n Λ, A n L N exp E {Λ} N N γ k n 5 N τ γ k n exp eln. 0E {Λ} It follows that on B P[A k Λ, A 0 ] = k P [ ] A c n+ Λ, An n=0 k exp eln 0E {Λ}

12 k exp ce ln = k log N exp log 3 N ce log4 N ln/ log4 N = k log N exp log 3 N as N by the assumption that ln/ log 4 N. clog 3 N ln/ log4 N 3 o. Now let I and I be two independent uniformly randomly chosen vertices of G N. By Proposition 3.4, I is a.a.s. either outside the largest component of G N, or it is connected to T γ k,γ k with a path that avoids the set {i N : Λ i > N γ k } and whose length is at most κn. Denote the endpoint of that path by I 0. In the case that k, step o above showed that I 0 is a.a.s. connected to a vertex I k T γ,γ 0 over a path of length k. The same claims holds for I, with vertices I 0 and I k respectively. Now, I k and I k belong a.a.s. to the largest component of the induced subgraph T γ,γ 0, and Theorem 4. yields that their distance is at most + owγ. 4 o. We have now shown that if I and I both belong to the largest component of G N, their distance is a.a.s. at most + o k + wγ. It remains to show that k can in this expression be replaced by First, if γn CɛN for all N and some finite C >, then log log N log/γ. 5 logτ log γ log log N log ln + [ log C, 0], and 5 is in this region negligible compared with wγ. On the other hand, k is in this region limited by a constant, whereas wγn. Thus, the assertion of the theorem holds. Assume then that γn/ɛn is unbounded. Then γ > 4 τ/3 τɛn for large N, so that k. Since k γ k+ = τ k γ + ɛn τ i, a short computation using the geometric series formula yields that the condition γ k 4 τ/3 τɛn is equivalent to the condition τ k γ ɛn ɛn, 3 τ and further to k logτ = + o i=0 log log N + log γn 4 τ log log N log/γn. logτ 3 τ ɛn log ln

13 Thus, k equals the last, asymptotic expression. This finishes the proof. Theorem 5. tells that removing a top part of the core downto N γ with fixed γ > 0 has no effect to the asymptotic upper bound 4. On the other hand, with γn ɛn we obtain wγn 3 τɛn = log N 3 τln. Combining this with Proposition 3.4, we find that if the whole core is removed, the remaining graph still has a giant component with same proportional size as the original and diameter proportional to logn/ln slightly smaller than that of a supercritical power-law graph with τ > 3. Asymptotics of intermediate cases between these two extremes can be computed from 3.4 as well. References [] B. Bollobás. Random Graphs. Cambridge University Press, 00. nd ed. [] F. Chung and L. Lu. The diameter of sparse random graphs. Adv. Appl. Math., 6:57 79, 00. [3] F. Chung and L. Lu. The average distance in a random graph with given expected degrees full version. Internet Mathematics, :9 4, 003. [4] C.M. Goldie and C. Klüppelberg. Subexponential distributions. In R.J. Adler, R.E. Feldman, and M.S. Taqqu, editors, A Practical Guide to Heavy Tails. Birkhäuser, 998. [5] R. van der Hofstad and G. Hooghiemstra. Diameters in preferential attachment models. In The Fourteenth Applied Probability Society of INFORMS Conference, July 007. Available at [6] M.E.J. Newman, S.H. Strogatz, and D.J. Watts. Random graphs with arbitary degree distribution and their applications. Phys. Rev. E, 64:068, 00. [7] I. Norros and H. Reittu. On a conditionally Poissonian graph process. Adv. Appl. Prob., 38:59 75, 006. [8] H. Reittu and I. Norros. On the power-law random graph model of massive data networks. Performance Evaluation, 55:3 3,