Evolution of reference networks with aging

5 Pages·0.18 MB·English

by  S. N. Dorogovtsev

Evolution of reference networks with aging
S.N. Dorogovtsev1,2,∗ and J.F.F. Mendes1,†
1 Departamento de F´ısica and Centro de F´ısica do Porto, Faculdade de Ciˆencias, Universidade do Porto
Rua do Campo Alegre 687, 4169-007 Porto – Portugal
2 A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
We study the growth of a reference network with aging of sites defined in the following way.
Each new site of the network is connected to some old site with probability proportional (i) to the
connectivity of theold site as in theBarab´asi-Albert’s model and (ii) to τ−α, where τ is theage of
the old site. We consider α of any sign although reasonable values are 0 ≤ α ≤ ∞. We find both
from simulation and analytically that thenetwork shows scaling behavior only in theregion α<1.
Whenαincreasesfrom−∞to0,theexponentγofthedistributionofconnectivities(P(k)∝k−γ for
0 largek)growsfrom 2tothevalueforthenetworkwithoutaging, i.e. to3fortheBarab´asi-Albert’s
0 model. The following increase of α to 1 makes γ to grow to ∞. For α>1 the distribution P(k) is
0 exponentional, and thenetwork hasa chain structure.
2
PACS numbers: 05.10.-a, 05.40.-a, 05.50.+q, 87.18.Sn
n
a
J
The explosion of the general interest on the problem low both numerically and analytically that this change
8
ofthe structureandevolutionofmostdifferentnetworks is dramatic: the scaling disappears when α>1, and the
2
[1–7] is connected not only with the sudden understand- exponents of the scaling laws depend strongly on α in

1 ingthatourworldisinfactahugesetofvariousnetworks the range α < 1. This result could not be foreseen be-
v (withthe moststrikingexamplesofWebandneuralnet- forehand: the scaling is very sensitive to changes of the
9 works – see the papers [4,5,8,9] and references therein) model. For example, as it is noted in [6], the scaling dis-
1
butalsowiththerecentfindingthatmanynetworksobey appearsiftheprobabilityoftheconnectiontoanoldsite
4
scaling laws [2,6]. That was the reason to renew old is proportional to kǫ,ǫ6=1.
1
0 studies [10–13]. One may note also that the growth of Here, we consider exclusively the network, in which
0 networks is only a particular kind of fascinating growth only one extra link appears when the new site is added,
0 processes [14–16]. since the results for the exponents do not depend on the
/ The simplestkindofgrowingnetworksisthe reference number of links which are added each time [6].
t
a networks [17] in which new links appear only between Letusstartfromthesimulationwhichturnstobeeasy
m
new sites and the old ones but they never appear be- for the problem under consideration because we study
- tweenoldsites. Therefore,inthe referencenetworks,the only the characteristics dependent on the connectivity.
d
average connectivity of old sites is always higher than They are: (i) the distribution of connectivities in the
n
o that one of younger sites. The well-known example of network P(k,t) in the instant t (only one site is in the
c the reference networks is the network of citations of sci- network at t=0, and one more site is added in each in-
: entific papers [2] in whicheachpaper is a site of the cor- stant) and (ii) the mean connectivity k(s,t) of the site s
v
i responding net and links are the references to the cited (0<s<t) in the instant t, i. e. the local density of the
X papers. connectivities. If one is interested only in these quanti-
r A clear beautiful model of a reference network ties, there is no need to keep a matrix of connections in
a
that shows scaling behavior was recently proposed by memory, and the simulation is very fast.
Baraba´si and Albert [6]. In their network, each new site The results of the numerics are shown in Figs. 2-5.
is connected with some old site with probability propor- Although only the region α ≥ 0 seems to be of real
tional to its connectivity k. In this case, the distribu- significance, we consider also negative values of α since
tionoftheconnectivitiesinthelargenetwork(i.e. oneof they do not lead to any contradiction. One may see
themostconsiderablecharacteristicsofthestructureand clearly from the figures that P(k,t) ∝ k−γ for large k
evolution of networks) shows a power-law dependence and k(s,t) ∝ s−β for small s, where β is the other scal-
P(k)∝k−γ with the exponent γ =3. ing exponent, only for α < 1. As it should be, we get
However, in real reference networks aging usually oc- the values γ = 3 and β = 1/2 for α = 0 [6]. Note that,
curs: alas, we rarely cite old papers! One may ask, how at 0 <α <1, the deviations of the dependence logP(k)
do the structure of the network change if the aging of vs. logk for small k are stronger as in the real reference
sitesisintroduced,i.e. ifthe probabilityofconnectionof network [2], than those ones at α = 0. At α > 1, P(k)
the new site with some old one is proportional not only turnstobeexponential,andthemeanconnectivitytends
to the connectivity of the old site but also to the power to be constant at large s. Thus, while α changes from 0
of its age, τ−α, for example? This question is quite rea- to 1, γ grows from 3 to ∞, and β decreases from 1/2 to
sonable: indeed, Fig. 1 demonstrates that the structure 0. One seesfromFig. 3,that, forpositiveα the logP(k)
of the network depends obviously on α. We show be- vs. logk dependence looks more curved than those ones
1
for α=0 like in the real reference network [2]. dξ
κ(ξ)=Bexp −β , (7)
The simulations demonstrate also that γ has a ten- (cid:20) Z ξ(1−ξ)α(cid:21)
dency to decrease from 3 to 2, and β to grow from 1/2
to 1 when α decreases from 0 to −∞. where B is a constant. The indefinite integralin Eq. (7)
Let us show now how these results may be described. may be taken:
We use an effective medium approach, inspired by [6].
dξ
We shall explain why it gives so good results for the ex-
=
ponents later but first we outline the applied theory for Z ξ(1−ξ)α
the problem under consideration. ∞
1
The main ansatz of the effective medium approach in lnξ+ α(α+1)...(α+k)ξk+1 =
k!(k+1)2
ourcaseistheapproximationoftheprobabilityP(k,s,t) Xk=0
thattheconnectivityofthesitesatthemomenttisequal lnξ+α F (1,1,1+α;2,2;ξ), (8)
3 2
to k by the δ-function:
where F (, , ; , ; )isthehypergeometricfunction[18].
3 2
P(k,s,t)=δ(k−k(s,t)). (1) Recalling the boundary condition κ(1) = 1, we find the
constant B. Thus the solution is
Thus, the exact function P(k,s,t) is assumed to be
enough sharp. We suppose also that we may use a con-
κ(ξ)=
tinuous approximation. These two strong assumptions
lead immediately to the following equation: e−β(C+ψ(1−α))ξ−βexp[−βαξ3F2(1,1,1+α;2,2;ξ)], (9)
∂k(s,t) k(s,t)(t−s)−α where C = 0.5772... is the Euler’s constant and ψ( ) is
= , k(t,t)=1. (2)
∂t t the ψ-function. Now we see that the constant β indeed
duk(u,t)(t−u)−α is the exponentofmeanconnectivity,since κ(ξ)∼ξ−β if
R0 ξ → 0. The transcendental equation for β may be writ-
ten if one substitutes Eq. (9) into the right side of Eq.
Here, the boundary condition means that only one link
(6):
is added each time.
OnemaycheckthatEq. (2)isconsistent. Letusapply
t β−1 =e−β(C+ψ(1−α))×
ds to it. Then,
0
R 1 dζ
t ∂k(s,t) ∂ t Z ζβ(1−ζ)α exp[−βαζ3F2(1,1,1+α;2,2;ζ)], (10)
ds = dsk(s,t)−k(t,t)=1, (3) 0
Z ∂t ∂tZ
0 0
that is our main equation.
and we get immediately the proper relation Before we shall find the solution of Eq. (10), let us
showhowtheexponentsβ andγarerelatedifoneapplies
t Eq. (1) of the effective medium approach. k(s,t)∝s−β,
dsk(s,t)=2t, (4)
Z so s∝k−1/β. Hence, k−γ ∝P(k,t)∝∂s/∂k∝k−1−1/β,
0
and one obtains that
i. e. thesumofconnectivitiesequalsthedoublednumber
of links in the network. γ =1+1/β. (11)
We search for the solution of Eq. (2) in the scaling
form The solution of Eq. (10) exists in the range −∞ <
α < 1. The results of the numerical solution are shown
k(s,t)≡κ(s/t) , s/t≡ξ, (5) in Fig. 5. One may also find β(α) and γ(α) at α→0:
which is consistent also with Eq. (4). Then Eq. (2) 1
β ∼= −(1−ln2)α , γ ∼=3+4(1−ln2)α, (12)
becomes 2
dlnκ(ξ) 1 −1 where the numerical values of the coefficients are 1 −
−ξ(1−ξ)α = dζκ(ζ)(1−ζ)−α ≡β,
dξ (cid:20)Z (cid:21) ln2=0.3069...and4(1−ln2)=1.2274.... Weusedthe
0 relation F (1,1,1;2,2;ζ)=Li (ζ)/ζ ≡( ∞ ζk/k2)/ζ
κ(1)=1, (6) 3 2 2 k=1
whilederivingEq. (12). Here,Li2()isthePpolylogarithm
function of order 2 [18].
where β is a constant, which is unknown yet. We
In the limit of α → 1, using the relation
shall see soon that this constant is the exponent of the
F (1,1,2;2,2;ζ)=−ln(1−ζ)/ζ, we find
mean connectivity. Eqs. (4) and (5) give the relation 3 2
1
dζκ(ζ)=2. 1 1
0 β ∼=c (1−α) , γ ∼= . (13)
R The solution of Eq. (6) is 1 c 1−α
1
2
Here, c1 = 0.8065...,c−11 = 1.2400...: the constant c1 [1] J. Lahererre and D. Sornette, Eur. Phys. J. B 2, 525
is the solution of the equation 1+1/c = exp(c ). Note (1998).
1 1
also that β →1 and γ →2 in the limit α→−∞. [2] S. Redner,Eur. Phys. J. B 4, 131 (1998).
One sees from Fig. 5 that the results of the simu- [3] D.J.WattsandS.H.Strogatz, Nature393,440(1998).
lation and of the analytical calculations are in qualita- [4] R. Albert, H. Jeong, and A.-L. Barab´asi, Nature 401,
130 (1999).
tive correspondence. Deviations may be noticed only
[5] B. A. Huberman, P. L. T. Pirolli, J. E. Pitkow, and R.
in regions where simulations can not provide sufficient
J. Lukose, Science 280, 95 (1998).
precision. Why does such a coarse theory demonstrate
[6] A.-L. Barab´asi and R. Albert, Science 286, 509 (1999);
so close agreement with the simulation? One may show
A.-L.Barab´asi, R.AlbertandH.Jeong, PhysicaA272,
[19] that the function P(k,s,t) of k is functionally very
173 (1999).
sharp as compared with the studied long-tailed distribu-
[7] D. J. Watts, Small Worlds (Princeton University Press,
tion P(k,s). Therefore, the main anzats Eq. (1) gives
Princeton, New Jersey, 1999).
excellent results. In fact, Eqs. (1) and (2) provides us [8] G. Parisi, J. Phys. A 19, L675 (1986).
easily with all known results on reference networks. [9] R.MonassonandR.Zecchina,Phys.Rev.Lett.75,2432
One may imagine from Fig. 1 that the point α = 1 (1995).
(β → 0 and γ → ∞) at which the scaling collapses [10] P. Erd¨os and A. Reny´i, Publ. Math. Inst. Hung. Acad.
marks the transition from the multidimensional network Sci. 5, 17 (1960).
forα<1tothechainstructure. Fig. 1demonstratesalso [11] B. Bollob´as, Random Graphs (Academic Press, London,
that, in the case ofα→−∞(β →1 andγ →2)allsites 1985).
of the network are connected with the oldest one. The [12] B. Derrida and D. Stauffer, Europhys. Lett. 2, 739
behavior of the considered quantities near these points (1987).
evokesassociations with the lower and higher criticaldi- [13] D. Stauffer, Philos. Mag. B 56, 901 (1987).
mensions in the theory of usual phase transitions [20]. [14] A.-L. Barab´asi, H. E. Stanley, Fractal Concepts of Sur-
face Growth (Cambridge University Press, Cambridge,
There are some possibilities to change the exponents
1995).
ofthenetworkwithoutaging[6,19]. Onemaycheckthat,
[15] M.Kardar,G.Parisi, andY.C.Zhang,Phys.Rev.Lett.
in these cases, the range of variation of the exponents β
56, 889 (1986).
and γ when α changes from −∞ to 1 is the same as in
[16] J. M. Kim and J. M. Kosterlitz, Phys. Rev. Lett. 62,
the present Letter.
2289 (1989).
Insummary,wehaveconsideredthe referencenetwork
[17] We are not sure that the term “reference network” is
with the power-law (τ−α) aging of sites. We have found already accepted but it seems to be reasonable.
bothfromoursimulationsandusingtheeffectivemedium [18] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,
approach that the network shows scaling behavior only Integrals and Series, vol. 1 (Gordon and Breach Science
in the region α < 1. We have calculated the expo- Publishers, New York,1986).
nents of the power-law dependences of the distribution [19] S.N. Dorogovtsev, J.F.F. Mendes, and A. N. Samukhin,
of connectivities P(k,t)∝k−γ andof mean connectivity to be published.
k(s,t)∝s−β andhave shownthatthey depend crucially [20] D. J. Amit, Field Theory, the Renormalization Group,
on α. and Critical Phenomena (McGraw-Hill, New York,
The following questions remainopen. Is our unproved 1978).
idea about the nature of the threshold point α = 1 rea-
sonable? Do the analogies with the lower and higher
critical dimensions exist indeed? What other quantities
of the network demonstrates the scaling behavior?
SND thanks PRAXIS XXI (Portugal) for
a research grant PRAXIS XXI/BCC/16418/98.
JFFM was partially supported by the projects
PRAXIS/2/2.1/FIS/299/94. We also thank E. J. S.
Lage for reading the manuscript and A. V. Goltsev, S.
Redner,andA.N.Samukhinformanyusefuldiscussions.
∗ Electronic address: [email protected]
† Electronic address: [email protected]
3
FIG. 2. Mean connectivity vs. number s of the site for
severalvaluesoftheagingexponentα: 1)α=0,2)α=0.25,
3) α=0.5, 4) α=2.0. The network size is t=10000.
0 0
a = 1
a = 5
-1 -2 a = 10
a = -10.0 a = 0.0 P(k)
-2 log -4
) -3
k -6
(
P 0 5 10 15

g -4 k
o
l
-5 1
2
a = 0.5 a = 1.0
-6 3
4
-7
0 1 2 3
log k
FIG.3. Distribution of the connectivities for several val-
uesof theaging exponent: 1) α=0, 2) α=0.25, 3) α=0.5,
4) α=0.75. The inset shows logP(k) vs. k for α=1,5, and
a = 2.0 a = 10.0 10. Note that, in the later case, all three curves are nearly
thesame.
FIG. 1. Change of the network structure with increase
of the aging exponent α. The aging is proportional to τ−α,
where τ is the age of the site. The network grows clockwise
starting from the site below on the left. Each time one new
site with one link is added.
1
0.9
0.8 0.6
3 b
0.3
0.6
0
b −5 −4 −3 −2 −1 0 1
2 1 0.4 a
) 2
s
(
_ k 3 0.2

g
o
l1
0
−0.5 0 0.5 1
4 a
FIG. 4. Exponent β of the mean connectivity vs. aging
0 exponent α. Points are obtained from the simulations. The
0 1 2 3 4 line is the solution of Eq. (10). The inset shows the analyt-
log s ical solution in the range −5 < α < 1. Note that β → 1 if
α→−∞.
4
FIG. 5. Exponent γ of the connectivity distribution vs.
aging exponent α. The points show the results of the sim-
ulations. The line is the solution of Eq. (10) with account
for Eq. (11). The inset depicts the analytical solution in the
7 range −5<α<1.
20
6
g
10
5
0
g
−5 −3 −1 1
4 a
3
2
−2 −1 0 1
a
5