Apr 11, 2008 - Boston University, Boston, Massachusetts 02215, USA ... 3Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel. 4Minerva ...
Fractal Boundaries of Complex Networks Jia Shao1 , Sergey V. Buldyrev2,1 , Reuven Cohen3 , Maksim Kitsak1 , Shlomo Havlin4 , and H. Eugene Stanley11 11
Center for Polymer Studies and Department of Physics,
arXiv:0804.1968v1 [math-ph] 11 Apr 2008
Boston University, Boston, Massachusetts 02215, USA 2
Department of Physics, Yeshiva University,
500 West 185th Street, New York, New York 10033, USA 3
Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel 4
Minerva Center and Department of Physics,
Bar-Ilan University, 52900 Ramat-Gan, Israel (Dated: April 11, 2008)
Abstract We introduce the concept of boundaries of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundaries nodes of complex networks. We find that for both Erd¨os-R´enyi and scale-free model networks, as well as for several real networks, the boundaries have fractal properties. In particular, the number of boundaries nodes B follows a power-law probability density function which scales as B −2 . The clusters formed by the boundary nodes are fractals with a fractal dimension df ≈ 2. We present analytical and numerical evidence supporting these results for a broad class of networks. Our findings imply potential applications for epidemic spreading.
1
Many complex networks are “small world” due to the very small average distance d between two randomly chosen nodes. Usually d ∼ ln N, where N is the number of nodes [1, 2, 3, 4, 5, 6]. Thus, starting from a randomly chosen node following the shortest path, one can reach any other node in a very small number of steps. This phenomenon is called “six degrees of separation” in social networks [4]. That is, for most pairs of randomly chosen people, the shortest “distance” between them is not more than six. Many random network models, such as Erd¨os-R´enyi network (ER) [1], Watts-Strogatz network (WS) [5] and random scale-free network (SF) [3, 6, 7, 8], as well as many real networks, have been shown to possess this small-world property. Much attention has been devoted to the structural properties of networks within the average distance d from a given node. However, almost no attention has been given to nodes which are at distances greater than d from a given node. We define these nodes as the boundaries of the network and study the ensemble of boundaries. An interesting question is how many “friends of friends of friends etc...” one has at distance greater than the average distance d? What is their probability distribution and what is the structure of the boundaries? The boundaries have an important role in several scenarios, such as in the spread of viruses or information in a human social network. If the virus (information) spreads from one node to all its nearest neighbors, and from them to all next nearest neighbors and further on until d, how many nodes do not get the virus (information), and what is their distribution with respect to the origin of the infection. Our results may explain why epidemics such as “black death” in medieval Europe stopped before reaching the entire population. In this Letter, we find theoretically and numerically that the nodes at the boundaries, which are of order N, exhibit similar fractal features for many types of networks, including ER and SF models as well as several real networks. Song et al. [9] found that some networks have fractal properties while others do not. Here we show that almost all model and real networks including non-fractal networks have fractal features at their boundaries which are different from Song et al. Fig. 1 demonstrates our approach and analysis. For each node, we identify the nodes at distance ℓ from it as nodes in shell ℓ. We chose a random origin node and count the number of nodes Bℓ at shell ℓ. We see that B1 =10, B2 =11, B3 =13, etc... We estimate the average distance d ≈ 2.9 by averaging the distances between all pairs of nodes. After 2
removing nodes with ℓ < d = 2.9, the network is fragmented into 12 clusters, with sizes s3 ={1, 1, 2, 5, 1, 3, 1, 1, 8, 1, 2, 3}. 000 111 000 111 000 111 000 111 000 111 000 111 000 111 111 000 000 111 000 111 000 111 000 111 000 111 000 111 000000 111111 0 1 000 111 00 11 000 111 000 111 0000 1111 00000011 111111 00 0 1 00 11 000 111 000 111 0000 1111 00 11 0 1 00 11 00 11 0000 1111 000 111 0000 1111 00 11 000000 111111 0000000 1 00 11 0000 1111 000 111 0 1 000 111 0111111 1 00 11 0000 1111 0 1 000 111 00 11 0000001 111111 0111111 1 00 11 0000 1111 0 000 111 00 11 0000000 1 00 11 00 11 00 11 0 1 000 111 00 11 0000000 111111 0 1 00 11 00 11 00 11 000 111 0000001 0111111 1 00 11 00 00 11 0000011 11111 0111111 1 0000001 111111 00 11 0 1 00 11 000000 00 0000011 11111 0 00 11 000000 111111 00 11 0000011 11111 0111111 1 00 11 000 111 000000 00 00000 11111 0 1 0 1 000 111 000 111 000000 111111 000 111 00 11 00000 11111 0 11111 1 000 111 000 111 00000 000 111 00 11 00000 11111 0 1 000 111 00000 11111 000 111 00000 11111 000 111 0 11111 1 000 111 00000 11111 000 111 0 00000 1 00000 11111 0000 1111 000 111 00000 11111 0 1 00000 11111 0000 1111 000 111 00000 0 11111 1 00000 11111 0000 1111 000 111 00000 11111 00000 11111 0 111 1 00 11 000 00 11 0000 1111 000 111 00000 11111 00 11 00111 11 000 00 11 0000 1111 00000 11111 11111111 000000 000 111 00 11 00 11 000 111 00 000000 111111 0000 1111 00 11 000 111 00000 11111 000 111 00 0 1 00 111 11 00000011 111111 0000 1111 00 11 000 00000 11111 000 111 0 1 00 11 1111111 0000000 00 11 000 111 000 111 000 111 00 11 000 111 0 1 00 11 00 11 0 1 00 11 000 111 000 111 000 111 00 11 0 1 00 11 00 11 0 1 000 111 000 111 00 11 0 1 00 11 00 11 011 1 000 111 111 000 111 00 000 000 111 0 1 00 11 00 11 0 1 000 111 000 111 00 11 000 111 000 111 1111 0000 1111 0000 011 000 111 000 111 00 000 111 000 1 111 011 000 111 000 111 000 111 000 111 000 1 111 00 11 000 111 011 1 00 000 1 000 111 000 111 000 111 00 11 000 111 011 1 00 0 000 111 000 111 000 1 111 00 11 0000 1111 000 111 011 1 00 11 0 1 00 11 000 111 000 111 00 11 0000 1111 000 111 000 111 0 1 00 0 1 00 11 000 111 0000 1111 00 11 00 11 000 111 000 111 011 1 00 00 00000011 111111 00 11 000 111 0000 1111 00 11 00 11 000 111 000000 111111 000 111 0 1 00 11 00 11 000000 111111 00 111 11 000 111111 0000 1111 00 11 00 11 00 11 000000 00 11 00000 11111 00 11 000 111 00 11 00 11 00 11 000000 111111 00 111111 00000000 11111111 000000 111111 0 000 1 0000 1111 00000 11111 0000000 111 00 11 00 11 000000 111111 00 111111 00000000 11111111 000000 111111 0 111 1 0000 1111 0000000 111 00 11 000000 111111 00000000 11111111 0000001 111111 0shell 1 0000 1111 000 111 00 11 1 0000 11 000000 111111 000000 111111 0 0000 1111 00000000 11111111 00 11 00000 11111 0011 11 000000 111111 00 11 00000000 11111111 0000001 111111 0 0000 1111 00 11 00000000000 11111111111 00000 11111 00 11 0000 1111 0 1 00000000 11111111 000000 111111 0 1 0000 1111 00 11 00 11 00000000000 11111111111 00000 11111 000 111 00 11 0000 1 1111 0 00000000 11111111 0 1 0000 1111 1111 00 11 00000000000 11111111111 000 111 0000 0 1 0 1 0000 1111 00 11 00000000000 11111111111 000 111 11111 00000 0000 1 1111 0 1 0 1 0000 1111 1111 00000000000 11111111111 00 11 1111111 00000 00 11 0000 0 0 1 0000 1111 00000000000 11111111111 00 00 11 0000 1 1111 0 0 1 0000 1111 00000000000 11111111111 00 11 00 11 00 11 00 11 0000 1111 0 1 00 11 00 11 00 11 0 1 11111111 000000 00 11 00 11 00 0 1 00 11 0 1 00 11 0 1 shell 2 00 11 000 111 0 1 00 11 000 111 00 11 000 111 0 1 000 111 0 1 00 11 0 1 00 11 0 1 00 11 00 11 0 1 00 11 00 11 00 11 111111 0000 000 111 00 1111 0000 000 111 00 0000 1111 000 111 000 111 0011 11 0 1 0000 1111 000 111 00 11 0 1 0000 1111 000 000 cluster of 8 nodes 111 00111 11 0 1 000 111 0000 1111 000 111 000 111 00 11 0 1 000 111 000 111 00 11 0 1 000 111 000 111 000 111 000 111 000 111 000 111 000 111
FIG. 1:
(Color on line) Illustration of shells and clusters originating from a randomly chosen
node, which is shown in the center (red). Its neighboring nodes are defined as shell 1, the nodes at distance ℓ are defined as shell ℓ. When removing all nodes with ℓ < 3, the remaining network becomes fragmented into 12 clusters.
We begin our study by simulating ER and SF networks, and later present analytical proofs. Fig. 2a shows simulation results for the number of nodes Bℓ reached from a randomly chosen origin node for an ER network. The results shown are for a single network realization of size N = 106 , with average degree hki = 6 and d ≈ 7.9 [11]. For ℓ < d, the cumulative distribution function, P (Bl ), which is the probability that shell ℓ has more than Bℓ nodes, decays exponentially for Bℓ > Bℓ∗ , where Bℓ∗ is the maximum typical size of shell ℓ [12]. However, for ℓ > d, we observe a clear transition to a power law decay behavior, where P (Bℓ ) ∼ Bℓ−β , with β ≈ 1 and the pdf of Bℓ is P˜ (Bℓ ) ≡ dP (Bℓ )/dBℓ ∼ Bℓ−2 . Thus, our results suggest a broad “scale-free” distribution for the number of nodes at distances larger than d. This power law behavior demonstrates the fractal nature of the boundaries of network, suggesting that there is no characteristic size and a broad range of sizes can appear in a shell at the boundaries. Further fractal features of the boundaries structure will be shown below. In SF networks, the degrees of the nodes, k, follow a power law distribution function q(k) ∼ k −λ , where the minimum degree of the network is chosen to be 2. Fig. 2b shows, for SF networks with λ = 2.5, similar power law results, P (Bℓ ) ∼ Bℓ−β for ℓ > d as for ER, with 3
a similar power β ≈ 1. We find similar results also for λ > 3 (not shown). 0
0
10
10
l =11 -1
β=1.0
-2
P(Bl )
P(Bl )
-1
10
l =12
10
10
l =13
10
β=1.0 -2
-3
10
-3
10
l =8
l =11 l =10
-4
10
-4
(a) ER
10
0
2
10
10
0
4
Bl
6
10
l =9
(b) SF 0
10
2
10
10
0
4
Bl
6
10
10
10
10
l =9 l =10
β=1.0
P(Bl )
10
l =11 l =12
-2
0
10
-2
l =7
-3
(d) AS
10
(c) HEP
-3
1
10
2
10
Bl
3
10
4
0
10
l =4
l =5 β=1.0
10
10
10
l =6
-1
10
P(Bl )
-1
10
1
10
2
10
Bl
3
10
4
10
FIG. 2: The cumulative distribution function, P (Bℓ ), for two random network models: (a) ER network with N = 106 nodes and hki = 6, and (b) SF network with N = 106 nodes and λ = 2.5, and two real networks: (c) the High Energy Particle (HEP) physics citations network and (d) the Autonomous System (AS) Internet network. The shells with ℓ > d are marked with their shell number. The thin lines from left to right represent shells ℓ =1, 2... respectively, with ℓ < d. For ℓ > d, P (Bℓ ) follows a power-law distribution P (Bℓ ) ∼ Bℓ−β , with β ≈ 1 (corresponding to P˜ (Bℓ ) ∼ Bℓ−2 for the pdf). The appearance of a power law decay only happens for ℓ larger than d ≈ 7.9 for ER and d ≈ 4.7 for the SF network. The straight lines represent a slope of −1.
To test how general is our finding, we also study several real networks (Figs. 2c, 2d), including the High Energy Particle (HEP) physics citations network [14] and the Autonomous System (AS) Internet network [10, 15]. Our results suggest that the fractal properties of the boundaries appear also in both networks, with similar values of β ≈ 1 for ℓ > d [16]. Next we ask how many nodes are on average at the boundaries? Are they a finite fraction 4
4
0
-1
-2
10
ER N=128000 6 ER N=10 SF N=128000 6 SF N=10
3
10
~
(a) ER
10
/N
10
N=200 N=400 N=800 N=8000 N=32000 N=128000 6 N=10
kl + 1
10
-3
2
10
10
10
-4
10
(b) 0
-5
10
-6
-4
-2
0
2
4
l-(lnN/ln)
6
10 1 2 3 4 5 6 7 8 9 10 11 12 13
8
l
0
0
10
µ=1.5
10
10
l =1 l =2 l =3 l =4 l =5 l =6 l =7 l =8
-4
10
(d) ER -1
m=1
10
rl+m
-2
PDF
1
Eq. (6)
m=2
-2
10
-6
m=3
10
_ (c) ER (l=6. For different N , the curves collapse. (b) k˜ℓ +1, which is hkℓ2 i/hkℓ i, as function of ℓ shown for both ER and SF network. (c) The probability distribution function P˜ (Bℓ ) in shells ℓ ≤ d for ER network. For small values of Bℓ , P˜ (Bℓ ) ∼ Bℓµ , where µ depends on the hki of the network (Eq. (4)). (d) The fraction of nodes outside shell ℓ + m, rℓ+m , as a function of rℓ for ER network, where rℓ is calculated for any possible ℓ. The (red) lines represent the theoretical iteration function (Eq. (6)).
of N, or less? In Fig. 3a, we study the mean number hBℓ i in shell ℓ, and plot hBℓ i/N as function of ℓ − ln N/ lnhki for different values of N for ER network. The term ln N/ lnhki represents the average distance d of the network [2]. We find that, for different values of N, the curves collapse, supporting a relation independent of network size N. Since hBℓ i/N is apparently constant and independent of N, it follows that hBℓ i ∼ N, i.e., a finite fraction of N nodes appear at each shell including shells with ℓ > d. We find similar behavior for SF network with λ = 3.5 (not shown). The branching factor [13] of the network is
5
k˜ = hk 2 i/hki − 1, where the averages are calculated for the entire network. Similarly, we define k˜ℓ = hkℓ2 i/hkℓ i − 1, where the averages are calculated only for nodes in shell ℓ. Above the average distance, k˜ℓ + 1 decreases with ℓ for both ER and SF networks (Fig. 3b). Thus, at the shells where power law behavior of P (Bℓ ) appears (Fig. 2), the nodes have much lower k˜ℓ + 1 compared with the entire network. The approach of k˜ℓ + 1 to 1 (ER network) and 2 (SF network) is consistent with a critical behavior at the boundaries of the network [13]. Fig. 3c shows that P˜ (Bℓ ) for ℓ < d and small values of Bℓ increase as a power law, P˜ (Bℓ ) ∼ B µ , for ER network, where µ depends on k˜ (supporting the theory developed ℓ
below). We define the fraction of nodes outside shell m as rm = 1 − (
Pm
ℓ=1
Bℓ )/N. There
exists a functional relation Eq. (6), which is independent of ℓ, between any two rℓ and rℓ+m (m = 1, 2, 3...), for ER network in Fig. 3d. Figs. 3c, 3d provide empirical evidences for the theory developed below. Next, we study the structural properties of the boundaries. Removing all nodes that are within a distance ℓ > d (not including shell ℓ), the network will become fragmented into several clusters (see Fig. 1). We denote the size of those clusters as sℓ , the number of clusters of size sℓ as n(sl ), and the average distance in the clusters as dℓ [17]. We find n(s) ∼ s−θ , with θ ≈ 3.0 (Figs. 4a and 4b). Similar relations are also found for ER and other real networks. The relation between the size of the clusters sℓ and their mean distance dℓ is shown in Figs. 4c and 4d, for SF (λ = 2.5) and HEP citations networks respectively. These plots suggest a power law relation, sℓ ∼ dϕℓ , with ϕ ≈ 2. It indicates that the clusters at the boundaries are fractals with fractal dimension df = 2 as percolation clusters at criticality [18]. Note that, for very large clusters their average distances dℓ decrease with size, suggesting that the largest clusters are not fractals. We find that the fractal dimension is df = ϕ ≈ 2 also for ER, SF with λ = 3.5 and several other real networks. Next we present analytical derivations supporting the above numerical results. We denote the degree distribution of a network as q(k). In infinitely large network we can neglect loops for ℓ < d and approximate the behavior of forming a network as a branching process [19, 20, 21]. The probability of reaching a node with k outgoing links through a link is q˜(k) = (k + 1)q(k + 1)/hki. The probability of number of branches equals to q˜(k). We define the generating function of q(k) as G0 (x) ≡ G1 (x) =
P∞
˜(k)xk k=0 q
P∞
k=0
′
q(k)xk , the generating function of q˜(k) as
= G0 (x)/hki. For ER networks we have G0 (x) = G1 (x) = ehki(x−1) .
6
10
10 10
l =5 l =6 l =7
θ=3.0
6
10
n(sl )
n(sl )
10
10
8
4
10
2
10
0 10 (a) SF
10
sl
10 10 10
10
2
sl
10
4
10
6
6
l =5 l =7 l =8
θ=3.0
4
2
0 10 (b) HEP 0 1 10 10
10
2
sl
10
3
10
4
8
6
4
l =6 l =6.5 l =7
10 10
ϕ=1.9
10
2
(c) SF
10
0
10 0 10
10
6
4
l =5 l =6 l =7 l =8 ϕ=2.0
sl
10
0
8
1
dl
10
2
1
2
(d) HEP
0
dl
10
FIG. 4: The number of clusters of sizes sℓ , n(sℓ ), as function of sℓ after removing nodes within shell ℓ for: (a) SF network with N = 106 and λ = 2.5, (b) HEP citations network, and sℓ as function of average distance dℓ of the clusters for (c) SF network with N=106 and λ = 2.5, (d) HEP citations network. The relation between n(sℓ ) and sℓ is characterized by a power law, n(sℓ ) ∼ s−θ ℓ , with θ ≈ 3. Also, sℓ scales with dℓ as sℓ ∼ dϕ ℓ , with ϕ ≈ 2.
The generating function for the number of nodes, Bm , at the shell m is [22]: ˜ m (x) = G0 (G1 (...(G1 (x)))) = G0 (Gm−1 (x)), G 1
(1)
where G1 (G1 (...)) ≡ Gm−1 (x) is the result of applying G1 (x), m − 1 times. P˜ (Bm ), which 1 is the probability distribution of Bm , is the coefficient of xBm in the Taylor expansion of ˜ m (x). G For shells with large m but still much smaller than d, we expect [22] that the number of ˜ It is possible to show [20] that Gm−1 nodes will increase by a factor of k. (x) converges to 1 a function of the form f ((1 − x)k˜m ) for large m (m m (a detailed proof will be given elsewhere): rn = G0 (G1n−m (G−1 0 (rm ))),
(5)
where rn is the fraction of nodes outside shell n. It can also be shown that the branching ˜ n ) = uG′′ (u)/G′ (u), where u = G−1 factor in the rn fraction of nodes is k(r 0 (rn ). 0
8
0
For ER networks, Eq. (5) yields: k rℓ+1 = ehki(rℓ −1) = Σ∞ ℓ=0 q(k)rℓ ,
(6)
which is valid for all possible ℓ. We test it in Fig. 3d. When m > d, using the same considerations as before it can be shown that: ˜ − rm )]−µ−1 + r∞ , rn = [ak(1
(7)
where r∞ = G0 (f∞ ) is the fraction of nodes not belonging to the giant component of the network, a is a constant. Based on Eqs. (4) and (7), expressing rm and rn in terms of Bm and Bn , we find that for m > d, Bn ∼ B −µ−1 . Using P˜ (Bn )dBn = P˜ (Bm )dBm , we obtain m
P˜ (Bn ) ∼ Bn−1−µ/(µ+1)−1/(µ+1) = Bn−2 ,
(8)
supporting the numerical findings in Fig. 2. These results are rigorous when k˜ exists and when the minimum degree km ≤ 2. For SF networks with λ < 3, k˜ diverges for N → ∞. But for finite N, k˜ still exists. Thus the above results can also be applied to the case of λ < 3. For both ER and SF networks with km ≥ 3, the power law of P (Bn ) with n >> d cannot be observed, as we indeed confirm by simulations. Relating our problem with percolation theory, we can explain the simulation results of probability distribution of cluster size sℓ . The cluster size distribution in percolation at some concentration p close to pc is determined by the formula [13]: Pp (s > S) ∼ S −τ +1 exp(−S|p − pc |1/σ ) .
(9)
˜ In the In the case of random networks the percolation threshold is given by pc = 1/k. ˜ n ) − 1)/k, ˜ where k(r ˜ n) exterior of the shell n (n >> d), we can estimate |p − pc | ∼ (k(r decreases and reaches the critical percolation value of 1. The cluster size distribution can be estimated by introducing a sharp exponential cutoff ˜ n ) − 1|− σ1 , so that Pn (s > S) ∼ S −τ +1 P (S ∗ > S), where P (S ∗ > S) is the at s = Sn∗ ∼ |k(r n n probability for a given shell to have Sn∗ > S. ˜ ∞ ) < 1, the probability that Since rn − r∞ has a smooth power law distribution and k(r ˜ n )−1| < S −σ = ε is proportional to ε. Thus P (S ∗ > S) ∼ S −σ and Pn (s > S) = S −τ +1−σ |k(r n [25]. Therefore the cluster size distribution follows n(s) ∼ s−(τ +σ) . 9
For ER networks and SF networks with λ > 4, τ = 2.5 and σ = 0.5, the above derivations lead to n(s) ∼ s−3 . For SF networks with 2 < λ < 4, τ = (2λ − 3)/(λ − 2) and σ = |λ − 3|/(λ − 2) [18]. Thus, for SF network with λ > 3, there will be ns ∼ s−3 . We conjecture ˜ n ) does not exist and the above ns ∼ s−3 even for 2 < λ < 3, although in this case k(r derivations are not valid. Our numerical simulations support these results in Fig. 4a, b. In summary, we find empirically and analytically that the boundaries of a broad class of complex networks including non-fractal networks [9] have fractal features. Our findings can be applied to the study of epidemics. It implies that a strong decay of the epidemic will happen in the boundaries of human network, due to the low degree of nodes. The fractal clusters at the boundaries, which are connected sparsely to the bulk, may explain why big breakout of epidemical disease (such as the “black death” in medieval Europe) would suddenly stop after affecting a large percentage of population. We thank ONR and Israel Science Foundation for financial support.
[1] P. Erd¨os and A. R´enyi, Publ. Math. 6, 290 (1959); Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960). [2] B. Bollob´ as, Random Graphs (Academic, London, 1985). [3] R. Albert and A.-L.Barab´ asi, Rev. Mod. Phys. 74, 47(2002). [4] S. Milgram, Psychol. Today 2, 60-67 (1967). [5] D. J. Watts and S. H. Strogatz, Nature (London) 393, 440 (1998). [6] R. Cohen and S. Havlin, Phys. Rev. Lett. 90, 058701 (2003). [7] S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: from Biological nets to the Internet and WWW (Oxford University Press, New York, 2003). [8] R. Pastor-Satorras and A. Vespignani, Evolution and Structure of the Internet: a statistical physics approach (Cambridge University Press, 2004). [9] C. Song et al., Nature (London) 433, 392 (2005); Nature Physics 2, 275 (2006). [10] S. Carmi et al., PNAS 104, 11150 (2007). [11] Different realizations yield similar results. In one realization, a certain fraction of nodes are randomly taken to be origin. The histogram is obtained from Bℓ belonging to different origin nodes.
10
[12] The behavior of the pdf of Bℓ for ℓ < d will be discussed later and is shown in Fig. 3c. [13] R. Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). [14] Derived from the HEP section of arxiv.org; http://vlado.fmf.uni-lj.si/pub/networks/data/hep-th/hep-th.htm (website of Pajek). [15] Y.
Shavitt
and
E.
Shir,
DIMES
-
Letting
the
Internet
Measure
Itself,
http://www.arxiv.org/abs/cs.NI/0506099 [16] We also find similar results (not shown here) for other real networks. [17] Fractional shells with ℓ + a (0 < a < 1) are used here by removing nodes within shell ℓ and a fraction a of nodes at shell ℓ. [18] R. Cohen et al., Phys. Rev. E 66, 036113 (2002); Chap. 4 in Handbook of graphs and networks, Eds. S. Bornholdt and H. G. Schuster (Wiley-VCH, 2002) [19] T. E. Harris, Ann. Math. Statist. 41, 474 (1948); The Theory of Branching Processes (Springer-Verlag, Berlin, 1963). [20] N. H. Bingham, J. Appl. Probab. 25A, 215 (1988). [21] L. A. Braunstein et al., Int. J. Bifurcation and Chaos 17, 2215 (2007). [22] M. E. J. Newman et al., Phys. Rev. E 64, 026118 (2001). [23] G. H. Weiss, Aspects and Applications of the Random Walk (North Holland Press, Amsterdam, 1994). [24] S. Dubuc, Ann. Inst. Fourier 21, 171 (1971). [25] A.L.Barab´ asi et al., Phys. Rev. Lett. 76, 2192 (1996).
11
