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arXiv:1810.01686v2 [cond-mat.stat-mech] 3 Jan 2019

Scaling properties of d-dimensional complex networks

Samura´ı Brito1,∗Thiago C. Nunes2,†Luciano R. da Silva2,3,‡and Constantino Tsallis3,4,5,6§

1International Institute of Physics, Universidade Federal do Rio Grande do Norte,

Campus Universit´ario, Lagoa Nova, Natal-RN 59078-970, Brazil

2Departamento de F´ısica Te´orica e Experimental,

Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-900, Brazil

3National Institute of Science and Technology of Complex Systems, Brazil

4Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil

5Santa Fe Institute, 1399 Hyde Park Road, New Mexico 87501, USA and

6Complexity Science Hub Vienna, Josefstaedter Strasse 39, A 1080 Vienna, Austria.

The area of networks is very interdisciplinary and exhibits many applications in several ﬁelds

of science. Nevertheless, there are few studies focusing on geographically located d-dimensional

networks. In this paper, we study scaling properties of a wide class of d-dimensional geographically

located networks which grow with preferential attachment involving Euclidean distances through

r−αA

ij (αA≥0). We have numerically analyzed the time evolution of the connectivity of sites, the

average shortest path, the degree distribution entropy, and the average clustering coeﬃcient, for

d= 1,2,3,4, and typical values of αA. Remarkably enough, virtually all the curves can be made

to collapse as functions of the scaled variable αA/d. These observations conﬁrm the existence of

three regimes. The ﬁrst one occurs in the interval αA/d ∈[0,1]; it is non-Boltzmannian with very-

long-range interactions in the sense that the degree distribution is a q-exponential with qconstant

and above unity. The critical value αA/d = 1 that emerges in many of these properties is replaced

by αA/d = 1/2 for the β-exponent which characterizes the time evolution of the connectivity of

sites. The second regime is still non-Boltzmannian, now with moderately long-range interactions,

and reﬂects in an index qmonotonically decreasing with αA/d increasing from its critical value to

a characteristic value αA/d ≃5. Finally, the third regime is Boltzmannian-like (with q≃1), and

corresponds to short-range interactions.

I. INTRODUCTION

Networks are everywhere, from the Internet to social

networks. We are living in the network age and the emer-

gence of more and more related researches is natural. The

theory of networks has applications in a diversity of sci-

entiﬁc ﬁelds, such as medicine [1], cosmology [2], quan-

tum information theory [3] and social networks [4]. For a

long time, it was diﬀusely believed that the statistics gov-

erning complex networks was only the Boltzmann-Gibbs

(BG) one. However, in 2005 the connection between net-

works and nonextensive statistical mechanics started to

be explored [5–7], and is presently very active [8–14].

In the literature, systems with long-range interactions

are characterized by paired potentials that decay slowly

with the distance. A potential of the form 1/rαis typi-

cally said to be long-range if 0 ≤α≤d, where dis the

spatial dimension of the system. Some examples of such

potentials are gravitational systems, two-dimensional hy-

drodynamic systems, two-dimensional elastic systems,

charged systems and dipole systems. Unlike the case

of classical systems with short-range interactions (usu-

ally described within BG statistics), where many results

∗E-mail address: samuraigab@gmail.com.br

†E-mail address: thiago.cris@yahoo.com.br

‡E-mail address: luciano@ﬁsica.ufrn.br

§E-mail address: tsallis@cbpf.br

are well understood, there is a lack of complete knowl-

edge about the dynamic and statistical properties of sys-

tems with long-range interactions (for which BG statis-

tics fails). In this sense, many theories have been pro-

posed to understand the systems that interact at long-

range, and q-statistics has shown satisfactory results for

this regime [15–20].

In 2016, we studied a d-dimensional network model

where the interactions are short- or long-ranged depend-

ing on the choice of the parameter αA≥0. The results

that were obtained reinforced the connection between

nonextensive statistical mechanics and the networks the-

ory [13]. In that work, we found some quantities which

present a universal behaviour with respect to the partic-

ular variable αA/d and observed the existence of three

regimes. In the ﬁrst one, namely 0 ≤αA/d ≤1, q

is constant and larger than unity, characterizing a non-

Boltzmannian regime with very-long-range interactions.

In the second one, qmonotonically decreases as αA/d

increased from its critical value αA/d = 1 to a charac-

teristic value αA/d ≃5. The third regime, above this

characteristic value [21], is Boltzmannian-like (q≃1)

and corresponds to short-range interactions. For the β

exponent (deﬁned here below) the behaviour is somewhat

diﬀerent: a ﬁrst regime is exhibited for 0 ≤αA/d ≤1/2,

a second regime appears between αA/d = 1/2 and a char-

acteristic value once again close to 5, and a third regime,

Boltzmannian-like with q≃1, between this value and

inﬁnity; it cannot be excluded that the purely Boltzman-

nian behaviour only occurs for αA/d → ∞.

Our model was constructed through two stages: the

2

number of the sites increases at time and the connections

between the sites follow a preferential attachment rule,

given by:

Πi∝kiri−αA.(1)

Each newly created site can connect to mothers. In

the present work, all results were obtained for m= 1.

The growth of the network starts with one site at the

origin, and then, we stochastically locate a second site

(and then a third, a fourth, and so on up to N) through

the d-dimensional isotropic distribution

p(r)∝1

rd+αG(αG>0; d= 1,2,3,4),(2)

where r≥1 is the Euclidean distance from the newly ar-

rived site to the center of mass of the pre-existing system.

For more details see [13]. This network is characterized

by three parameters αA,αGand d, where αAcontrols the

importance of the distance in the preferential attachment

rule, αGis associated with the geographical distribution

of the sites, and dis the dimension of the system.

The connectivity distribution was the only property

studied in the previous work. Our results showed that the

degree distribution of this model is very well described by

the q-exponential functions that emerges from nonexten-

sive statistical mechanics [22–24], more precisely P(k)∼

eq−k/κ ∀(αA, αG, d), with ez

q≡[1 + (1 −q)z]1

1−q. The re-

lation between qand γ(the exponent of the asymptotic

power law) is given by γ≡1/(q−1) (see [5] for more

details). When αA= 0 we recover the Barab´asi-Albert

(BA) model [25] with q= 4/3 (γ= 3). Remarkably

enough, our previous results showed that the index qand

κexhibit universal behaviours with respect to the scaled

variable αA/d (∀d).

Motivated by the results in [13], in the present work

we are interested in investigating, for the same network

model, other possible universal behaviors with respect

to the same scaled variable αA/d. Besides that, we also

are interested in verifying the existence of the same three

regimes that we have previously observed. We have ana-

lyzed the exponent β, which is associated with the time

evolution of the connectivity of sites, the average shortest

path hli, the degree distribution entropy Sqand the aver-

age clustering coeﬃcient hCi. Along the lines of [13], in

order to analyse these properties, we choose the typical

value αG= 2 and vary the parameters (αA, d).

II. RESULTS

A. Time evolution of the connectivity of sites

One of the most common analyses that are done in net-

works theory is to verify how the degree of a site changes

at time. This property is usually referred to as connec-

tivity time evolution and it usually follows the behaviour

ki(t)∝t

tiβ

where ti≤t. (3)

We analyzed the time evolution of the connectivity of

sites for typical values of αAand d= 1,2,3,4 (see Fig. 1).

In order to do that, we choose the site i= 10 (the result

is independent of i), and then we compute the time evo-

lution of its connectivity. All simulations were made for

105sites and 103samples.

We observe that the dynamic exponent βis not con-

stant, in discrepancy with its value for the BA model:

βdecreases with αAand increases with d(see Fig. 2a).

Moreover, we notice that βexhibits universal curves with

respect to the scaled variable αA/d. When αA/d ≥0 up

to the critical value αA/d = 1/2, the system is in the

same universality class of the BA model with β= 1/2

and it is in the non-Boltzmannian very-long-range in-

teractions regime (qis constant above unity). From

αA/d > 1/2 on, β < 1/2 and decreases nearly exponen-

tially with αA/d down to the value 0.11 for αA/d ≈5.

For αA/d above this value up to inﬁnity, βremains prac-

tically constant, indicating a Boltzmannian-like regime

(with q≃1). It cannot be excluded that the terminal

value of βis achieved only at the limit αA/d → ∞ (see

Fig. 2b).

100

101

102

103

hk10i

(a) (b)

d= 1 d= 2 d= 3 d= 4

(c)

101102103104

t/t10

100

101

102

103

hk10i

(d)

101102103104

t/t10

(e)

101102103104

t/t10

(f)

FIG. 1. Time evolution of the connectivity of the site i= 10

in log −log plot for diﬀerent values of αAand d= 1,2,3,4.

The ﬁgure sublabels refer to (a) αA= 0, (b) αA= 2, (c)

αA= 3, (d) αA= 5, (e) αA= 6 and (f ) αA= 8. We can see

that ki∝(t/ti)β(αA,d)where β(αA, d) is the asymptotic slope

of the curves. For αA= 0, independent of the dimension, we

recover the BA model with β= 1/2, and when αA→ ∞ the

dimension does not matter either.

B. Average shortest path length

The average shortest path length is a concept, in net-

work theory, deﬁned as the average number of steps along

the shortest paths for all possible pairs of sites of the net-

work. In real networks, a short path makes it easier to

3

0246810

αA

0.0

0.1

0.2

0.3

0.4

0.5

0.6

β

(a) d= 1

d= 2

d= 3

d= 4

01/2 12345678910

αA/d

0.0

0.1

0.2

0.3

0.4

0.5

0.6

β

β(αA/d) =

1

2if 0 ≤αA/d ≤1/2

0.41e0.45−αA

d+ 0.11 if αA/d > 1/2

(b) d= 1

d= 2

d= 3

d= 4

FIG. 2. (a) βdecreases with αAand increases with d.(b)

As we can see, from the rescaling αA→αA/d, all the curves

of βcollapse, and three regions clearly emerge. The ﬁrst one

is from αA/d = 0 up to the critical value αA/d = 1/2, β=

1/2. The second regime, from the critical value αA/d = 1/2

up to the characteristic value αA/d ≃5, βdecreases nearly

exponentiallly. When αA/d &5, βreaches a terminal value

β≃0.11, and the Bolzmannian-like regime is achieved. The

simulations have been run for 103samples and N= 105.

transfer information and can reduce costs. Mathemati-

cally, the average shortest path length is deﬁned by

hli=2

N(N−1) X

i<j

dij ,(4)

where dij is the shortest path (smaller number of edges)

between the sites iand j. We have computed the aver-

age shortest path length hlifor typical values of αAand

∀d. When αA= 0 the results are the same as the BA

model where hli ∼ ln N(for m= 1), independent of the

dimension of the system. We have numerically veriﬁed

that hlidepends on (αA,d), increasing with αAand de-

creasing with d(see Fig. 3a). Remarkably enough, all

the curves can be made to collapse through the scalings

αA→αA/d and hli → hli(1 + αA/d)−1(see Fig. 3b).

Again, we can see the existence of three regimes. The

non-Boltzmannian very-long-range interactions go up to

the critical value αA/d = 1, as we can see in the inset

plot (Fig. 3b). The non-Boltzmannian moderate long-

range interactions go to up to the characteristic value

αA/d ≃5, as can be seen from the derivative of the

collapse curve. And ﬁnally, from αA/d &5 on, the

Boltzmannian-like limit is reached.

0 1 2 3 4 5 6 7 8 9 10

αA

8

10

12

14

16

hli

(a)

d= 1

d= 2

d= 3

d= 4

012345678910

αA/d

0

2

4

6

8

10

hli(1 + αA/d)−1

(b)

d= 1 d= 2 d= 3 d= 4

0 1 2 4 6 8 10

αA/d

−8

−6

−4

−2

0

∆(hli(1 + αA/d)−1)/∆(αA/d)

FIG. 3. Shortest path length. (a) We can see that the

chemical distance hliincreases with αAand decreases with

d.(b) We can observe that the curves exhibit universal-

ity when we re-scale the axis replacing αA→αA/d and

hli → hli(1 + αA/d)−1. In the inset plot we show the deriva-

tive collapsed curve in order to see more precisely the exis-

tence of the three regimes. The critical value αA/d = 1, show

us the end of the non-Boltzmanninan very-long-range inter-

actions. The second regime, the non-Boltzmannian moder-

ate long-range interactions, go to up the characteristic value

αA/d ≃5. Finally, the third regime, from the characteristic

value up to αA/d → ∞, we see the Boltzmannian-like behav-

ior characterizing short-range interactions. This results are

for N= 104and 103samples.

C. Degree distribution entropy

The computation of the entropy in complex networks

is important to verify the heterogeneity and structure of

the network [26]. The degree distribution entropy mea-

sures the quantity of randomness present in the connec-

tivity distribution. In our simulations, it is possible to

realize the change of the topology of the network. When

αA= 0 the network is a scale-free with an asymptotically

power law connectivity distribution. As we increase the

value of αAthe randomness of the degree distribution

also increases. For αA→ ∞ (q→1) the network is not

scale-free anymore since it presents an exponential degree

distribution, in agreement with some results available in

the literature [27].

We have computed the degree distribution entropy S

for each value of αAand d= 1,2,3,4. We computed the

4

q-entropy (Sq), from nonextensive statistical mechanics,

and the Boltzmann-Gibbs (BG) (SBG) entropy (alterna-

tively referred to as Shannon entropy) for the same con-

nectivity distributions studied in [13] (see Fig. 4). The

BG entropy was calculated from SBG =Pkpkln(1/pk),

where pkis the probability to ﬁnd sites with kdegree

and the sum is over k= 1 up to kmax under the con-

straint Pkpk= 1. Since we have P(k)∼eq−k/κ, to

each value αA/d, a pair of parameters (q, κ) is associ-

ated. This enables, in particular, the computation of

the q-entropy Sq≡Pkpklnq(1/pk) for the same data,

where lnqz≡z1−q−1

1−qis the inverse of the q-exponential

function. When αA→ ∞ (q→1), both entropies con-

verge to the same asymptotic limit. This result was of

course expected since, for q= 1, the q-entropy recovers

the standard entropy SBG .

Our results show that there is a region where the two

entropies are diﬀerent. It is known that the BG entropy

is not appropriate for systems where long-range interac-

tions are allowed. So, this result provides evidence that

Sqis adequate to describe the interactions in this nonex-

tensive domain. Besides this result, we also studied the

dependence of Sqwith both (αA, d) and αA/d. We ver-

iﬁed that, although Sqdepends on αAand dseparately

(see Fig. 5a), the curves exhibit universal behavior with

regard to the scaled variable αA/d (see Fig. 5b). Once

again, we clearly see the existence of three regimes. In

the ﬁrst one, Sqhas a constant value up to the critical

value αA/d = 1. From that value on, the characteristic

value αA/d ≃5, Sqincreases nearly exponentially and

then, from αA/d &5 on, the Boltzmannian-like limit is

achieved.

D. Average clustering coeﬃcient

The average clustering coeﬃcient is an important mea-

sure in the theory of networks and it is associated with

how the neighbours of a given node are connected to each

other. This coeﬃcient is deﬁned as follows:

hCi=1

NX

i

2ni

ki(ki−1),(5)

where kiis the degree of the site i,niis the number

of connections between the neighbours of the site iand

ki(k1−1)/2 is the total number of possible links between

them.

In order to compute it, we run our network model for

m= 2 (because hCi= 0 when m= 1) and analyzed

how hCichanges with both (αA, d) and αA/d. We see

that hCiincreases with αAand decreases with d(see

Fig. 6a). The larger αAthe more aggregated the network

is. In the standard Barab´asi-Albert model (αA= 0),

the clustering coeﬃcient is inﬂuenced by the size Nof

the network, such that hCican be numerically approxi-

mated by hCi ∼ N−0.75 (in later works, Barab´asi analyt-

ically claimed that hCi ∼ (log N)2/N; for further details

0 1 2 3 4 5 6 7 8 9

0.8

1.0

1.2

1.4

entropy

(a)

SBG

Sq

0 5 10 15

(b)

SBG

Sq

0 10 20

αA

0.8

1.0

1.2

1.4

entropy

(c)

SBG

Sq

0 10 20 30

αA

(d)

SBG

Sq

FIG. 4. Measure of entropy in complex networks. Com-

parison between q−entropy (Sq) and the standard entropy

(SBG ). The BG entropy was calculated in the network us-

ing SBG =−kPkpkln pk(we used k= 1), whereas the q-

entropy was calculated using Sq=−Pkpq

klnqpk, where pk

is the probability of ﬁnding a site with connectivity kand

Pkpk= 1. In the region of long-range interactions we can

see that Sqis very diﬀerent from SBG, exhibiting that Sq

is more sensitive for describing this model in this domain.

When αA→ ∞ (q→1) both entropies converges to the same

asymptotic behaviour. The sublabels refer to (a) d= 1, (b)

d= 2, (c) d= 3, and (d) d= 4.

see [28]). From this behaviour we can see that, in the

thermodynamical limit (N→ ∞), hCi → 0. So, we have

numerically veriﬁed that when N→ ∞ hCi → 0 not only

for αA/d = 0, but for 0 ≤αA/d ≤1 (see the inset plot in

Fig. 6b). We also analyzed how the clustering coeﬃcient

changes with Nand we found that hCi ∼ N−ǫ(αA,d).

This power-law form was in fact expected since it agrees

with the numerical result previously found for the par-

ticular case αA= 0. However, surprisingly enough, when

αA&2dthe clustering coeﬃcient does not change with

Nanymore (see Fig. 7). Analyzing how ǫ(αA, d) changes

with both (αA, d) and αA/d, we see that this exponent

decreases with αA, but increases with d(see Fig. 8a).

Although we did not get collapse for these curves, by

rescaling αA→αA/d we clearly can see that all curves

perfectly intersect in αA/d = 1, strongly indicating a

change of regime (see Fig. 8b). The results found for

ǫ(αA, d) are somewhat reminiscent of the κ(αA, d) expo-

nent associated with the maximal Lyapunov exponent for

the generalized Fermi-Pasta-Ulam (FPU) model [29].

CONCLUSION

Our present results reveal an intriguing ubiquity of the

variable αA/d for the class of networks focused on here,

where both topological and metric aspects exist. Sur-

prisingly, the use of this variable indeed provides col-

5

0123456789

αA

0.8

1.0

1.2

1.4

Sq

(a)

d= 1

d= 2

d= 3

d= 4

0123456789

αA/d

0.8

1.0

1.2

1.4

Sq

Sq(αA/d) =

0.82 if 0 ≤αA/d ≤1

−0.56e1−αA

d+ 1.38 if αA/d > 1

(b) d= 1

d= 2

d= 3

d= 4

FIG. 5. Entropy dependence of αAand d= 1,2,3,4. (a)

Sqincreases with αAand decreases with d.(b) Once again,

we obtain the collapse of Sqwhen rescaling αA→αA/d.

The entropy Sqhas a constant value up to the critical value

αA= 1 and a nearly exponential behavior emerges up to the

characteristic value αA≃5.

lapses or quasi collapses for all the properties studied

here. Another interesting point is the existence of three,

and not only two, regimes. The ﬁrst one is a non-

Boltzmannian regime characterized by very-long-range

interactions and it goes from αA/d = 0 up to the critical

value αA/d = 1, except for βwhose critical value turns

out to be αA/d = 1/2, curiously enough. The second

one, from the critical value up to the characteristic value

αA/d ≃5, is still non-Boltzmannian and corresponds

to moderate long-range interactions. The third and last

regime, from αA/d ≃5 on, is Boltzmannian-like and is

characterized by short-range interactions. The existence

of the intermediate regime has also been observed in clas-

sical many-body Hamiltonians, namely the α-generalized

XY [16, 17] and Heisenberg [18] rotator models as well

as the Fermi-Pasta-Ulam model [19, 20, 29–31].

The present work neatly illustrates a fact which is not

always obvious to the community working with com-

plex networks, more precisely those exhibiting asymp-

totic scale-free behavior. Such networks frequently be-

long to the realm of applicability of nonextensive statis-

tical mechanics based on nonadditive entropies, and to

its superstatistical extensions [32]. Such connections be-

tween thermal and geometrical systems are by no means

rare in statistical mechanics since the pioneering and en-

lightening Kasteleyn and Fortuin theorem [33]. In the

present scenario, such connection can be naturally un-

derstood if we associate half of each two-body interac-

tion energy between any two sites of the Hamiltonian

0 1 2 3 4 5 6 7 8 9 10

αA

0.0

0.1

0.2

0.3

0.4

0.5

0.6

hCi

(a)

d= 1

d= 2

d= 3

d= 4

012345678910

αA/d

0.0

0.5

1.0

1.5

2.0

[hCi(1 + αA)(d+ 2)]/αA

(b)

d= 1

d= 2

d= 3

d= 4

0.025 0.050 0.075

(1/N)0.55

0.00

0.05

0.10

0.15

hCi

d= 1 αA= 0.75

FIG. 6. Clustering coeﬃcient for typical values of αAand

d= 1,2,3,4. (a) hCiincreases with αAbut it decreases with

d.(b) All curves collapse with the rescaling hCi → hCi[(αA+

1)(d+2)]/αAand αA→αA/d. In the thermodynamical limit

hCi → 0 from αA= 0 up to the critical value αA/d = 1.

In the inset plot we show an example for d= 1 and αA=

0.75. From the characteristic value αA/d ≃5 on we reach the

Boltzmannian-like regime.

10−3

10−2

10−1

hCi

(a)

αA= 0.0

αA= 0.2

αA= 0.4

αA= 0.6

αA= 0.8

αA= 1.0

αA= 1.5

αA= 2.0

αA= 4.0(b)

αA= 0.0

αA= 0.5

αA= 1.0

αA= 1.5

αA= 2.0

αA= 2.5

αA= 3.0

αA= 4.0

αA= 6.0

102103104105106

N

10−3

10−2

10−1

hCi

(c)

αA= 0.0

αA= 1.0

αA= 1.5

αA= 2.0

αA= 2.5

αA= 3.0

αA= 4.0

αA= 6.0

αA= 8.0

102103104105106

N

(d)

αA= 0.0

αA= 1.0

αA= 2.0

αA= 3.0

αA= 4.0

αA= 5.0

αA= 6.0

αA= 8.0

αA= 10.0

FIG. 7. Clustering coeﬃcient as a function of Nfor (a) d= 1,

(b) d= 2, (c) d= 3, and (d) d= 4. hCidecreases with N,

but increases with αA. Interestingly, from αA∼2don, hCi

does not change any more with N.

to each of the connected sites, thus generating, for each

node, a degree (number of links) in the sense of networks.

Through this perspective, it is no surprise that the degree

distribution corresponds to the q-exponential function

which generalizes the Boltzmann-Gibbs weight within

thermal statistics. The fact that in both of these geo-

metrical and thermal systems, the scaled variable αA/d

6

0246810

αA

0.0

0.2

0.4

0.6

0.8

1.0

ǫ(αA, d)

(a) d= 1

d= 2

d= 3

d= 4

01234

αA/d

0.0

0.2

0.4

0.6

0.8

1.0

ǫ(αA, d)

(b) d= 1

d= 2

d= 3

d= 4

FIG. 8. Analysis of the exponent ǫ(αA, d). (a) This graph

reminds us the behavior of the maximum Lyapunov exponent

[29]. The exponent ǫ(αA, d) decreases with αAand does so

faster for smaller d.(b) Although we did not get collapse for

these curves through the rescaling αA→αA/d, we can clearly

see that all curves appear to perfectly intersect at αA/d =

1. This result strongly indicates a change of regime, with

something special occurring at the value αA/d = 1. When

αA/d &2 we observe that ǫ→0 agrees with the result showed

in Fig. 7.

plays a preponderant role becomes essentially one and the

same feature. Mathematically-based contributions along

such lines would be more than welcome. Last but not

least, it would surely be interesting to understand how

come the critical point of the βexponent diﬀers from

the all the critical points that we studied here. This is

somewhat reminiscent of the two-dimensional XY ferro-

magnetic model with short-range interactions for which

nearly all properties exhibit a singularity at the positive

temperature of Kosterlitz and Thouless [34, 35], whereas

the order parameter critical point occurs at zero temper-

ature.

ACKNOWLEDGMENTS

We gratefully acknowledge partial ﬁnancial support

from CAPES, CNPq, Funpec, Faperj (Brazilian agencies)

and the Brazilian ministries MEC and MCTIC. We thank

the High Performance Computing Center at UFRN for

providing the computational facilities to run the simula-

tions.

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