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However, previous studies were performed numerically, and there are no analytical arguments for long-range degree correlations in the fractal networks. In this study, we focus on the giant component of an uncorrelated random network. By characterizing its long-range degree correlation as a function of degrees of a pair of nodes and their distance, we analytically derive the emergence of the negative long-range degree correlation in the giant component at a critical state. Let us consider an infinitely-large uncorrelated random network with a pk which has a locally tree-like structure. The probability, u, that an edge does not lead to the giant component is given as the solution of k−1 u = G1(u), where G1(u)= k qku , and it is less than 1 in the presence of the giant component. Assuming that a given network contains the giant component, i.e., u < 1, we extract it from this network. We start our analytical P ′ analysis by introducing probability PGC(k, k |l), stating that two ends of a randomly selected l-chain have degrees k and k′, given that the chain belongs to the giant component. Using the expectation number of nodes with degree k′ at distance l from a degree-k node and the expectation number of nodes at distance l from a random node [see Eqs. (18) and (21)], we obtain

′ l−1 k+k −2 ′ 1 − v u P (k, k |l)= q q ′ , (1) GC 1 − vl−1u2 k k where v is the probability that a node in between the l-chain is not connected to the giant component, which can be ′ ′ ′ k−2 expressed as a function of u, as v = G1(u)/G1(1) with G1(u)= dG1(u)/du = (k − 1)qku /hki. For l = 1, it has ′ k been reported that P (k, k′|l = 1)= (1 − uk+k −2)q q ′ /(1 − u2), which depicts the joint probability that an edge GC k k P selected randomly from the giant component is connected to degree-k and -k′ nodes [28, 29]. We introduce the characteristic lengths associated with the distance and the degrees of a node pair as

ξl = −1/ log v (2) and

ξk = −1/ log u, (3) respectively. Consequently, Eq. (1) is rewritten as

′ −(l−1)/ξl −(k+k −2)/ξk ′ 1 − e e PGC(k, k |l)= qkqk′ . (4) 1 − e−(l−1)/ξl e−2/ξk

′ For a finite ξl, PGC(k, k |l) decays exponentially to qkqk′ with increasing chain length l. Thus, a pair of l-distant nodes selected from the giant component is degree-correlated for l . ξl and degree-uncorrelated for l ≫ ξl. Notably, ′ relation P (k, k |l)= qkqk′ holds for l-distant node pairs selected from the entire network. ′ ′ We further introduce the probability, PGC(k |k,l), that one end of a chain has degree k , given that the chain has a degree-k node at the other end, has length l, and belongs to the giant component. From Bayes’ rule, this probability is given as

′ ′ PGC(k, k |l) PGC(k |k,l)= ′ k′ PGC(k, k |l) ′ −(l−1)/ξl −(k+k −2)/ξk P1 − e e = qk′ , (5) 1 − e−(l−1)/ξl e−k/ξk

GC and the average degree, kl (k), of l-distant nodes from degree-k nodes on the giant component is given as

GC ′ ′ kl (k)= k PGC(k |k,l) ′ Xk hk2i h(u)e−(l−1)/ξl e−k/ξk = + , (6) hki 1 − e−(l−1)/ξl e−k/ξk

k−2 GC where h(u) = k kqk(1 − u ) ≥ 0. Note that kl (k) of l = 1 corresponds to the average degree of the nearest- GC neighbor degree-k nodes on the giant component [28, 29]. Equation (6) shows that kl (k) is a decreasing function P GC of k for a fixed value of l, indicating that the giant component is negatively degree-correlated. Moreover, kl (k) is a decreasing function of l for any k, indicating that any degree correlation gradually disappears with increasing l (as in 3

0.14 0.14 0.14 (a) (b) (c) 0.12 0.12 0.12 0.1 0.1 0.1 0.08 0.08 0.08 (k,k'|l=1) (k,k'|l=5) 0.06 0.06 (k,k'|l=10) 0.06 GC GC 0.04 0.04 GC 0.04 P P P 0.02 0.02 0.02 0 0 0 1 1 1 2 2 2 1 1 1 3 2 3 2 3 2 4 3 4 3 4 3 4 5 4 k’ 5 4 k’ 5 5 k’ 5 5 6 6 6 6 6 6 7 7 k 7 7 k 7 7 k 8 8 8 8 8 8

0.14 0.14 0.14 (d) (e) (f) 0.12 0.12 0.12 0.1 0.1 0.1 0.08 0.08 0.08 (k,k'|l=1) (k,k'|l=5) 0.06 0.06 (k,k'|l=10) 0.06 GC GC 0.04 0.04 GC 0.04 P P P 0.02 0.02 0.02 0 0 0 1 1 1 2 1 2 1 2 1 3 2 3 2 3 2 4 3 4 3 k’ 4 3 k’ 4 k’ 4 4 5 5 5 5 k 5 5 k 6 6 k 6 6 6 6

0.07 0.07 0.07 (g) (h) (i) 0.06 0.06 0.06 0.05 0.05 0.05 0.04 0.04 0.04 (k,k'|l=1) (k,k'|l=5) 0.03 0.03 (k,k'|l=10) 0.03 GC GC 0.02 0.02 GC 0.02 P P P 0.01 0.01 0.01 0 0 0 1 1 1 2 2 2 3 3 3 4 1 4 1 4 1 5 2 5 2 5 2 6 4 3 6 4 3 6 4 3 k’ 7 5 k’ 7 5 k’ 7 5 8 7 6 8 7 6 8 7 6 9 8 k 9 8 k 9 8 k 10 10 9 10 10 9 10 10 9

′ ′ FIG. 1: Probability distribution PGC(k,k |l) for Erd˝os-R´enyi random graphs as a function of k and k for several distances. Wireframes depict the analytical calculation (1), and symbols represent the corresponding simulation results. Top panels represent the results for the average degree, λ = 1, and distances (a) l = 1, (b) l = 5, and (c) l = 10; middle panels represent the results for λ = 1.1 and distance (d) l = 1, (e) l = 5, and (f) l = 10; bottom panels depict the average degree, λ = 2.5, and distance (g) l = 1, (h) l = 5, and (i) l = 10. The results for each average degree are obtained from one sampled network of 107 nodes.

Eq. (4)). In summary, the giant component in a random network has a negative degree correlation for l < ξl, whereas GC 2 it has no correlations for l ≫ ξl: kl (k) → hk i/hki. To further discuss the emergence of the long-range degree correlation of the giant component in detail, we employ k −λ the Erd˝os-R´enyi , whose degree distribution is pk = λ e /k!, where λ = hki. Prior to a detailed analysis, we test the validity of the theoretical analysis (1) by comparing it with the simulation results. In Fig. 1, ′ we plot the theoretical predictions (wireframes) of probability PGC(k, k |l) for the Erd˝os-R´enyi random graphs with λ = 1(= λc), λ = 1.1, and λ = 2.5, where λc is the critical average degree above which the giant component exists. The wireframes in all cases match the corresponding Monte-Carlo simulations (symbols) perfectly. We assume that λ = λc + δ and u = v =1 − ǫ, where both δ and ǫ are infinitely small values. For λ & λc, we have ǫ ∼ λ − λc and two characteristic lengths, ξl and ξk, as

−1 −1 ξl = ξk ∼ ǫ ∼ (λ − λc) . (7)

The critical exponents for ξl and ξk are unity, which corresponds to the critical exponent of the correlation (chemical) GC length for the mean size of the finite cluster in the problem [35]. For λ & λc, the second term of kl (k) becomes a power-law with an exponential cutoff of both l and k within ξl and ξk as 1 kGC(k)=2+ e−(l−1)/ξl e−(k−1)/ξk . (8) l (l − 1) + k

GC The two characteristic lengths in Eq. (8) diverge asymptotically in a critical state (λ → λc). At λ = λc, kl (k) 4

0 10 λ =1.001 -1 λ =1.1 10 λ =1.2 -2 λ =1.5 10 λ =2.5

10-3

10-4 |C(l)|

10-5 y~x-2 10-6

10-7

10-8 100 101 102 l

FIG. 2: |C(l)| of Erd˝os-R´enyi random graphs as a function of l. Lines from right to left correspond to |C(l)| for λ = 1.001, 1.1, 1.2, 1.5, and 2.5, respectively. Dashed line with slope −2 is plotted as a guide to the eye.

decreases with increasing degree k in a power-law for any l(< ∞):

GC −1 kl (k)=2+(l + k − 1) . (9) Hence, the negative long-range correlation in the giant component stretches entirely at criticality. Furthermore, we propose a degree-degree correlation function C(l), which characterizes the critical behavior of the ′ networks. Probability PGC(k, k |l) has full information on the structure of the giant component. We define the ′ ′ ′ correlation function for degrees of l-distant node pairs on the GC, as C(l) = hkk il − hkilhk il, where hf(k, k )il = ′ ′ k,k′ f(k, k )PGC(k, k |l). Combined with Eq.(4), C(l) is expressed as

P −(l−1)/ξl 2 k−2 2 −e u k kqk 1 − u C(l)= 2 . (10) 1 − e−P(l−1)/ξlu2 

We observe that the degree correlation of l-distant node pairs on the giant component disappears for l > ξl, as |C(l)| is an exponentially decreasing function of l. The correlation function exhibits critical behavior when the giant component exists but infinitely small, i.e., ξl ≫ 1: C(l) in the critical region, and drops according to a power-law with an exponential cutoff,

a2 C(l) ∼− e−(l−1)/ξl , (11) (b(l − 1)+2)2

2 −2 where a = hk (k − 2)i/hki and b = hk(k − 1)(k − 2)i/hki [36]. At the critical point, ξl diverges, and C(l) ∼−l for l ≫ 1. Figure 2 shows the absolute value |C(l)| of the correlation function for Erd˝os-R´enyi random graphs for several values of λ. We observe that the correlation function decays exponentially in the off-critical region (λ ≥ 1.1), and a power-law with exponent −2 exists near criticality (λ =1.001 ≈ λc). All our analyses conclude that the long-range degree correlation in the giant component of an uncorrelated ran- ′ GC dom network emerges at the critical point. Both PGC(k, k |l) and kl (k) indicate that the giant component of an uncorrelated random network exhibits a long-range degree correlation. The giant component is negatively correlated for l < ξl, whereas it becomes neutral for l ≫ ξl. At criticality, where ξl diverges and the giant component is fractal, the negative degree correlation is observed at any distance. Moreover, the correlation function C(l) for degrees of l-distant node pairs decays exponentially in the off-critical region. In contrast, it obeys a power-law with a cutoff, C(l) ∼ −l−2el/ξl for l ≫ 1 near criticality and becomes a power-law, C(l) ∼ −l−2 at criticality. In summary, the negative long-range degree correlation spontaneously emerges in the fractal networks. The long-range degree correlation for a given network is intrinsic or extrinsic. Extrinsic ones are correlations arisen form nearest-neighbor degree correlations. We note that even when a given network has only a strong negative nearest-neighbor degree correlation, it will exhibit an extrinsic long-range degree correlation, such that the degrees of node pairs are positively correlated at l = 2, 4, ··· . Extrinsic correlations can be described by the products of 5 probability P (k′|k) that a random neighbor of a degree-k node has k′ edges. In an extrinsic case, for example, a triplet probability, P (k′, k′′|k), that a degree-k node is connected to a degree-k′ node and a degree-k′′ node satisfies P (k′, k′′|k) = P (k′|k)P (k′′|k). As stated in [33], most empirical networks have intrinsic correlations which cannot be explained by only nearest-neighbor degree correlations. With regard to the giant component in an uncorrelated random network, the long-range degree correlation is considered intrinsic. The triplet probability P (k′, k′′|k) ′ ′′ GC ′ ′′ k+k +k −4 k ′ ′′ indicates PGC(k , k |k)=(1 − u )qk′ qk′′ /(1 − u ) 6= PGC(k |k)PGC(k |k). Bialas and Ole´shave investigated the correlation function for generic trees [27]. Their correlation function (Eq. (12) in [27]) behaves as a power-law, which is similar to C(l) in the present study. Interestingly, the long-range degree correlation of the giant component is attributed to the divergence of the correlation length in the , while the behavior in the generic trees is not associated with criticality. (However, we note that ξl is directly connected to the correlation length through Eq. (2.29) in [20].) The tree structures found in both generic trees and the giant component at criticality may result in a power-law behavior of C(l), although further studies are required to gain an understanding of this common mechanism. Previous works have captured nearest-neighbor degree correlations of fractal networks which are renormalized at several length scales, implying a correlation between small- and large-scale degree correlations [21, 23]. We did not attain the problem how the long-range degree correlation treated in this study associates with the nearest-neighbor degree correlations of renormalized networks. This study deepens our understanding of the relation between the emergence of the long-range degree correlation and criticality/fractality. Acknowledgement. The authors would like to thank K. Yakubo for fruitful discussions. S.M. and T.H. acknowledge the financial support from JSPS (Japan) KAKENHI Grant Number JP18KT0059. S.M. was supported by a grant- in-aid for Early-Career Scientists (No. 18K13473) and a grant-in-aid for JSPS Research Fellow (No. 18J00527) from the Japan Society for the Promotion of Science (JSPS). T.H. acknowledges the financial support from JSPS (Japan) KAKENHI, grant number JP19K03648.

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DERIVATION OF PGC(ks,kt)

Let us consider an infinitely-large uncorrelated random network with a locally tree-like structure and focus on l-distant nodes, which are nodes at distance l from a randomly chosen node (seed). We suppose that for a randomly chosen seed the numbers of l-distant nodes with degrees 1, 2, ··· , k, ··· , and K are n1,n2, ··· ,nk, ··· , and nK , respectively, where K is the maximum degree. We denote the sequence of the number l-distant nodes with each degree by n = (n1,n2, ··· ,nK ). Let Pl(n, GC|Nl) be the probability that the sequence for the l-distant nodes is n and the focal component belongs to the giant component under the condition that the total number of l-distant nodes K n Nl(= k=1 nk) is given. Using the multinomial distribution, Pl( , GC|Nl) is given as

P K K N ! n l k−1 nk nk Pl( , GC|Nl)= 1 − (u ) qk , (12) n1!n2! ··· nk! ··· nK ! ! kY=1 kY=1 where u is the probability that an edge does not lead to the giant component, pk is the degree distribution, and K k−1 nk qk = k kpk/hki is the neighbor’s degree distribution. The term 1 − k=1(u ) corresponds with the probability that at least one of outgoing edges from l-distant nodes connects to the giant component. By combining Pl(n, GC|Nl) P Q and Pl(Nl|ks), which is the probability that the number of l-distant nodes is Nl given that the seed has ks edges, we construct the probability P (n, GC|ks) that the sequence for the l-distant nodes is n and the focal component belongs to the giant component given that the seed has ks edges as

Pl(n, GC|ks)= Pl(Nl|ks)Pl(n, GC|Nl) XNl K K Nl! k−1 nk nk = Pl(Nl|ks) 1 − (u ) qk . (13) n1!n2! ··· nk! ··· nK ! ! XNl kY=1 kY=1

We introduce the generating function F˜GC(x|ks) for Pl(n, GC|ks) as

K ˜ x n nk x FGC( |ks)= Pl( , GC|ks) xk , where = (x1, x2, ··· , xK ). (14) n X kY=1 This function is calculated as follows:

N ! K K K ˜ x l nk nk k−1 nk nk nk FGC( |ks)= Pl(Nl|ks) qk xk 1 − (u ) qk xk n n1!n2! ··· nk! ··· nK ! ! X XNl kY=1 kY=1 kY=1 K K Nl! nk k−1 nk = Pl(Nl|ks) (qkxk) − (qkxku ) n n1!n2! ··· nk! ··· nK ! ! XNl X kY=1 kY=1 K Nl K Nl k−1 = Pl(Nl|ks) qkxk − qkxku  ! !  XNl Xk=1 kX=1 ˜  ˜ k−1  = Gl ( kqkxk|ks) − Gl kqkxku |ks . (15) P P  7

Here G˜l(x|ks) is the generating function for the probability distribution Pl(Nl|ks) and is given as

ks G˜l(x|ks) = (G1(G1(··· (G1(x)) ··· ))) , (16) which is known as the generating function for the distribution of the number of children in l generation un- der a given offspring distribution qk and initial population ks [37]. The probability distribution Pl(n, GC, ks) that the sequence for the l-distant nodes is n, the seed has ks edges, and the focal component belongs to the n n ˜ x giant component is Pl( , GC, ks) = Pl( , GC|ks)pks , and the corresponding generating function FGC( , ks)(= n K nk ˜ x n Pl( , GC, ks) k=1 xk ) is thus given by the product of FGC( |ks) and pks , P Q x ˜ x FGC( , ks)= pks FGC( |ks) ˜ ˜ k−1 = pks Gl ( kqkxk|ks) − pks Gl kqkxku |ks . (17) P x P1  Differentiating Eq. (17) with respect to xkt and substituting = into it, we have the expectation number ks,kt hN (l, GC)i of l-distant nodes with degree kt where a randomly chosen seed has ks edges and its component belongs to the giant component:

∂F (x, k ) hN ks,kt (l, GC)i = GC s ∂x t k x=1 ′l−1 l−1 ks+kt−2 = kspks qkt G (1) 1 − v u , (18) 1 ′ ′  where v = G1(u)/G1(1). In a similar way, we easily find the generating function FGC(x) for the probability distribution Pl(n, GC) that for a randomly chosen node, the sequence of the number of l-distant nodes with each degree is n and these nodes are member of the giant component as

K x x k−1 FGC( )= FGC( , ks)= Gl ( kqkxk) − Gl kqkxku , (19) ks=1 X P P  where

Gl(x)= pkG˜l(x|k) Xk = G0(G1(G1(··· (G1(x)) ··· ))) (20)

From Eq. (19), the expectation number hN(l, GC)i of l-distant nodes from a random seed which belong to the giant component is calculated as

K ∂F (x) hN(l, GC)i = GC ∂xkt x 1 kt=1 = X ′l−1 l−1 2 = hkiG1 (1) 1 − v u (21)  By dividing Eq. (18) by Eq. (21), we obtain the probability PGC(ks, kt|l) that two ends of a randomly chosen l-chain from the giant component have degree ks and kt as

hN ks,kt (l, GC)i P (k , k |l) ≡ GC s t hN(l, GC)i ′ l−1 l−1 ks+kt−2 G1(1) 1 − v u = ′ l−1 l−1 2 qks qkt , G1(1) (1 − v u )  1 − vl−1uks+kt−2 = q s q t , (22) 1 − vl−1u2 k k where we call a connected path with length l as an l-chain. Here, the denominator is proportional to the number ′ l−1 ′ l−1 l−1 2 of l-chains in the giant component in that hkiG1(1) (hkiG1(1) v u ) represents the average number of nodes at distance l from a randomly chosen node in the whole network (finite components). When the networks are singly 8

connected, i.e., u = 0, Eq. (22) reduces to P (ks, kt|l) = qks qkt which is a known result for uncorrelated random networks [32]. Taking the limit u → 1 (v → 1), we obtain PGC(ks, kt|l) at criticality as ′′ G1 (1) ′ (l − 1)+(ks + kt − 2) G1(1) lim PGC(ks, kt|l, GC) = ′′ qks qkt . (23) u→1,v→1 G1 (1) ′ (l − 1)+2 G1(1)

Replacing u and v by ξk = −1/ log u and ξl = −1/ log v, respectively, we can rewrite Eq. (22) as 1 − e−(l−1)/ξl e−(ks+kt−2)/ξk PGC(ks, kt|l)= qks qkt . (24) 1 − e−(l−1)/ξl e−2/ξk

Let us introduce the probability P (kt|ks, l, GC) that one end of a chain has degree kt given that the chain has length l and has a degree-ks node as a starting node, and belongs to the giant component. The probability is given from Eq. (22) as 1 − vl−1uks+kt−2 P (kt|ks, l, GC) = qkt . (25) 1 − vl−1uks GC We get the average degree kl (ks) of l-distant nodes from a degree-ks node on giant component as GC kl (ks)= ktP (kt|ks, l, GC) Xkt hk2i h(u)vl−1uks = + (26) hki 1 − vl−1uks

k−2 where h(u)= k kqk(1 − u ). We consider the situation that the giant component exists but infinitely small i.e., u ∼ 1 − ǫ and v ∼ 1 − hk(k − 1)(k − 2)iǫ/hk(k − 1)i. In the situation, Eq. (26) approximates P hk2i hk3i− 2hk2i 1 kGC(k ) ∼ 1+ . (27) l s hki hk2i (l − 1)hk(k − 1)(k − 2)i/hk(k − 1)i + k  s 

CORRELATION FUNCTION C(l)

We consider the correlation function C(l) defined as ′ ′ C(l)= hkk il − hkilhk il, (28) ′ ′ ′ where hf(k, k )il = k,k′ f(k, k )PGC(k, k |l). From Eqs. (22) and (28), we have 2 P ′ ′ 1 − vl−1uk+k −2 1 − vl−1uk+k −2 ′ ′ ′ C(l)= kk l−1 2 qkqk − k l−1 2 qkqk ′ 1 − v u  ′ 1 − v u  Xk,k Xk,k 2 2 2  2  2 hk i − vl−1 kq uk−1 hk i l−1 k−1 hki k k hki − v k kqku u = l−1 2 − l−1 2 . (29)   1 − vPu   1 −vP u     The first term of the right hand side in Eq. (29) is

2 2 hk i l−1 k−1 2 hki − v k kqku   1 − vl−P1u2  2 2 l−1 2 hk i l−1 k−1 2 1 − v u hki − v k kqku =     (1 − vl−1u2)2P  2 2 2 2 hk i hk i l−1 2 2(l−1) k−1 2 2 l−1 k−1 2 hki − hki v u + v k kqku u − v k kqku = 2 , (30)     (1 −Pvl−1u2)  P  9 and the second term is

2 2 hk i l−1 k−1 hki − v k kqku u l−1 2  1 −vP u    2 2 2  hk i hk i l−1 k−1 2(l−1) k−1 2 2 hki − 2 hki v k kqku u + v k kqku u = . (31)   P (1 − vl−1u2)2 P  Then, we have

2 2 2 l−1 k−1 2 hk i l−1 2 hk i l−1 k −v k kqku − hki v u +2 hki v k kqku C(l)= 2 P  (1 − vl−1u2) P  2 2 −(l−1)/ξl k−1 hk i −e k kqku − u hki = 2 , (32) 1 −Pe−(l−1)/ξl u2 

−1/ξl −(l−1)/ξl  −(l−1)/ξl where we used v = e . Expanding u and e in the denominator as u ∼ 1 − ǫ and e ∼ 1 − (l − 1)/ξl, we drive the correlation function in the critical region as

a2 C(l) ∼− e−(l−1)/ξl , (33) (b(l − 1)+2)2 where a = hk2(k − 2)i/hki and b = hk(k − 1)(k − 2)i/hki.