Topological phase transition in a network model with preferential attachment and node removal

0.34 MB·English

by  Heiko Bauke

EPJ manuscript No.
(will be inserted by the editor)
Topological phase transition in a network model with preferential
attachment and node removal
Heiko Bauke1, Cristopher Moore2,3a, Jean-Baptiste Rouquier3,4, and David Sherrington1,3
1 Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, United Kingdom
2 2 Computer Science Department and Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM
1
87131, USA
0 3 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
2 4 RhoˆneAlpesComplexSystemsInstitute,E´coleNormaleSup´erieuredeLyon,Universit´edeLyon1,15parvisRen´eDescartes,

n BP 7000 69342 Lyon Cedex 07, France
a
J
DOI: 10.1140/epjb/e2011-20346-0

9
1
Abstract. Preferential attachment is a popular model of growing networks. We consider a generalized

model with random node removal, and a combination of preferential and random attachment. Using a
]
h high-degreeexpansionofthemasterequation,weidentifyatopologicalphasetransitiondependingonthe
c rate of node removal and the relative strength of preferential vs. random attachment, where the degree
e distribution goes from a power law to one with an exponential tail.
m
-
t
a 1 Introduction alistic model has to take into account node removal, edge
t rewiring[8]orremoval[9],andotherdynamicalprocesses,
s
. Complex networks are found in nature, social and eco- as well as deviations from linear preferential attachment.
t
a nomic systems, technical infrastructures, and countless Hereweconsiderageneralizedpreferentialattachment
m other fields. The macroscopic properties of such networks model with asymptotically linear attractiveness function
- emerge from the microscopic interaction of many individ- and random node removal.
d ualconstituents.Variousmodelsofcomplexnetworkshave The degree distribution of generalized preferential at-
n
been proposed and the statistical mechanics of networks tachment models is very sensitive to model-specific fea-
o
c has become an established branch of statistical physics tures.Varyingtheattractiveness function [10,11,12,13,14]
[ [1,2,3,4,5,6]. Since complex networks are non-equilibrium or including node removal [15,16,17] can shift the expo-
systems, they do not have to obey detailed balance, and nent of the power-law degree distribution to 2 < γ < ∞.
1 may show many fascinating features not found in equilib- For networks of constant size [17] or a sublinear attrac-
v
rium systems. tiveness function [11], the degree distribution can become
4
4 In order to explain the power-law degree distribution a stretched exponential. We will demonstrate that gener-
0 observed in many complex networks, Barab´asi and Al- alizations of the Barab´asi-Albert model can dramatically
4 bert [7] introduced a preferential attachment model for affect the degree distribution even in the case of growing
. growing networks. When new nodes enter the network, networksandasymptoticallylinearattractiveness,leading
1
0 theyprefertoattachtonodeswithhighdegree.Inagener- to a topological phase transition from a power-law degree
2 alizationofthismodel,theprobabilitypadd(v)thatanew distribution to an exponential degree distribution, with a
1 node u establishes an edge to an existing node v with de- stretched exponential at the critical point.
: greek(v)isproportionaltoanattractivenessfunctionA ;
v k
(cid:80)
normalizing, p (v) = A / A . Barab´asi and
Xi Albert consideraeddd the casekw(vh)ere wtheka(wtt)achment is pro- 2 The Model
r portional to degree, A = k. This results in a power-law
k The topological phase transition in generalized preferen-
a degree distribution p ∼k−γ with exponent γ =3, which
k tial attachment networks can be illustrated by consider-
isclosetotheobservedexponentsofmanyrealworldnet-
ing the following model. Vertices arrive at rate 1, each
works [3].
new vertex makes c connections to existing vertices, and
However, the linear preferential attachment function
we remove vertices randomly at a rate r. Each new edge
of the Barab´asi-Albert model was introduced as an ad
attaches to a given pre-existing vertex of degree k with
hoc ansatz without fundamental justification, and many
probability proportional to its attractiveness A . We as-
k
generalizations are conceivable. For many networks, a re-
sume that A is of the form
k
a [email protected] Ak =k+k∗
2 Heiko Bauke et al.: Topological phase transition in a network model with preferential attachment and node removal
for some constant k∗. Thus the choice of a link endpoint 3.2 Master Equation
is somewhere between preferential attachment (the case
k∗ =0) and uniform attachment (the limit k∗ →∞). We Let n be the number of vertices at time t. The expected
canalsotreatk∗ asatheinitialdegreeofthevertexwhen number of vertices with degree k is n = np . One time
k k
it is added to the network (e.g. by adding k∗ self-loops) step later this is n(cid:48) = (n+1−r)p(cid:48) where p(cid:48) is the new
k k k
and then run the “pure” preferential attachment model. value of p . Thus
k
The network starts to grow at time t = 0 with N
0
c
nodes and M0 links. We are interested in systems where (n+1−r)p(cid:48) =np +δ + (A p −A p )
the number of nodes is much larger than unity. Since we k k kc (cid:104)A(cid:105) k−1 k−1 k k
want to study the influence on the topology of the dy- +r(k+1)p −rkp −rp . (3)
k+1 k k
namics of our model, and not the influence of the initial
network, we let the system evolve until we have added a Thetermδ correspondstoaddingofavertexofdegreec
kc
numberofnodesmuchlargerthanN ,andthedegreedis- to the network. The term cA p /(cid:104)A(cid:105) is the probabil-
0 k−1 k−1
tribution has reached equilibrium. For r < 1 the number ity that a vertex of degree k−1 gains an extra edge from
of nodes grows with time, so the initial value N is unim- the new vertex and becomes of degree k, and similarly
0
portant. For r = 1, nodes are added and removed at the cA p /(cid:104)A(cid:105) is the flow from degree k to degree k+1.
k−1 k−1
same rate; in that case the expected number of nodes is The terms r(k+1)p and rkp are the flows from k+1
k+1 k
constant, so we start with N (cid:29)1. tok andfromk tok−1respectively,asverticesloseedges
0
when one of their neighbors is removed from the network.
Finally, rp is the probability that a vertex of degree k is
k
removed. Contributions from processes in which a vertex
3 Analytic Solution
gains or loses two or more edges in a single unit of time
vanish in the limit of large n and have been neglected.
In this section, we derive the average degree (cid:104)k(cid:105) and the Weareinterestedintheasymptoticformofthedegree
averageattractiveness(cid:104)A(cid:105).Wethenwritethemasterequa- distribution p in the limit of large t. Setting p(cid:48) = p
k k k
tionforthedegreedistributionp andsolveforitsasymp- in (3) gives
k
totic behavior for large k in terms of the parameters r, c,
and k∗. c
δ + (A p −A p )
k,c (cid:104)A(cid:105) k−1 k−1 k k
+r(k+1)p −rkp −p =0. (4)
k+1 k k
3.1 Mean Degree and Attractiveness
as previously appeared in [17]. Naively, this equation ap-
pears linear in the p . But since it involves the mean at-
k
Let pk be the expected fraction of vertices in the network tractiveness (cid:104)A(cid:105), the combination of (2) and (4) gives a
atagiventimethathavedegreek.Asin[17],theexpected nonlinear system of equations.
meandegreeofavertex(cid:104)k(cid:105)=(cid:80)∞k=0kpk canbederivedas When the attractiveness is proportional to the degree,
follows. The expected increase in the number of vertices A = k, the system (2) and (4) separates, and has been
k
per unit time is 1−r. The expected number of edges re- solved analytically in [17]. The authors showed that in
moved when a randomly chosen vertex is removed is (cid:104)k(cid:105), this case the degree distribution exhibits a power-law tail
so the expected increase in the number of edges per unit in the case 0 ≤ r < 1 of growing networks, and follows
timeisc−r(cid:104)k(cid:105).Attimettheexpectednumberofvertices a stretched exponential in the constant-size case r = 1.
andedgesaren=(1−r)t+N0 andm=(c−r(cid:104)k(cid:105))t+M0, For more general attractiveness functions like the one in
so in the limit t→∞ the mean degree obeys this paper, a fully analytic solution seems more difficult.
Thuswefocusonthebehaviorofp forlargek.Depending
k
2m 2(c−r(cid:104)k(cid:105)) on the model parameters we find either a degree distribu-
(cid:104)k(cid:105)= = ,
n 1−r tion with a power-law tail (Figure 2) or an exponential
tail (Figure 3). We confirm our calculations with numeri-
and solving for (cid:104)k(cid:105) gives cal simulation of the master equation, and through direct
simulation of the network dynamics.
2c
(cid:104)k(cid:105)= . (1)
1+r
3.3 High-Degree Expansion
The average attractiveness is then
Wenowspecializethemasterequationtoourmodel.Sub-
(cid:88)∞ 2c stituting (2) into (4) gives, for k ∈/ {c−1,c,c+1},
(cid:104)A(cid:105)= A p =(cid:104)k(cid:105)+k∗ = +k∗. (2)
k k 1+r
k=0 c (cid:0)(k−1+k∗)p −(k+k∗)p (cid:1)
(cid:104)A(cid:105) k−1 k
In the case r =1 where the network has constant size, we
have (cid:104)k(cid:105)=c and (cid:104)A(cid:105)=c+k∗. +r(k+1)pk+1−rkpk−pk =0. (5)
Heiko Bauke et al.: Topological phase transition in a network model with preferential attachment and node removal 3
We will determine the asymptotic behavior of pk with 100
a “high-degree expansion”, by approximating the ratio
p /p as a Taylor series in 1/k. We find a phase tran-
k k−1
sition between power-law and exponential behavior, and 80
determine the phase diagram as a function of the param-
eters c, r, and k∗. 60 exponential
degreedistribution
As an ansatz, assume that p is a power-law times an
k
∗
exponential: k
p =Ckαβk. (6) 40
k
We can determine α and β by taking k (cid:29)1, and expand- powerlaw
ing the ratio p /p to leading orders in 1/k. This gives 20 degreedistribution
k k−1
(cid:18) (cid:18) (cid:19)(cid:19)
p α 1
k =β 1+ +O , (7) 0
p k k2 0 0.2 0.4 0.6 0.8 1
k−1
(cid:18) (cid:18) (cid:19)(cid:19) r
p 2α 1
k+1 =β2 1+ +O . (8)
p k k2 Fig. 1. Phase diagram of the degree distribution with
k−1
c=20.Thetwoblackdotsindicatelocationsinthephase
Substituting this into (5), multiplying by (cid:104)A(cid:105)/p , and space for data shown on Figures 2, 3 and 4.
k−1
ignoring O(1/k) terms yields the equation
k(β−1)(r(cid:104)A(cid:105)β−c)+α(cid:0)r(cid:104)A(cid:105)β(2β−1)−cβ(cid:1) In the special case k∗ =0, this recovers the result of [17]
+(cid:104)A(cid:105)β(rβ−1)+c(cid:0)k∗(1−β)−1(cid:1)=0. (9) 3−r
α=− ,
1−r
Since (9) must be true for all k, we can set the coefficient
of k to zero. This gives two solutions for β, namely β =1
whileinthespecialcaser =0,itincludestheresultof[10]
and
c
β = . (10) k∗
r(cid:104)A(cid:105) α=−3− .
c
If r(cid:104)A(cid:105) > c, the solution β < 1 of (10) is physically
relevantandp decaysexponentially.However,ifr(cid:104)A(cid:105)<c Finally, since k∗ ≥0 and r ∈[0,1], we have
k
then (10) would give β >1, which does not correspond to
anormalizableprobabilitydistribution.Inthatcaseβ =1 −∞<α≤−3,
istherelevantsolution,andp ∼kα isapower-law.Thus
k
a phase transition occurs at r(cid:104)A(cid:105) = c. Applying (2), we sothatthedegreedistributionhasafiniteaverageaswell
can write this in terms of a critical value of k∗, as (except when k∗ =r =0) a finite variance.
Abovethetransition,β isgivenby (10).Againsetting
c(1−r) the constant term of (9) to zero gives
k∗ = . (11)
c r(1+r)
c(k∗−1)+(cid:104)A(cid:105)(1+r(1−k∗))
α=− . (13)
We illustrate the resulting phase diagram in Figure 1. (cid:104)A(cid:105)r−c
Thus we have a power-law correction to the exponential
3.4 The Power-Law decay, p ∼kαβk. In terms of our parameters,
k
To solve for the power-law exponent α, we again use (9), c(1+r)
β =
butnowsettheconstantterm(withrespecttok)tozero. r(2c+k∗(1+r))
If β =1, this gives pk ∼kα where (1+r)(c+k∗(1+r))
α=k∗− .
(cid:104)A(cid:105)(1−r)+c k∗r(1+r)−c(1−r)
α=− <0. (12)
c−(cid:104)A(cid:105)r
Note that in this regime α may be positive.
Note that α approaches −∞ as we approach the transi-
tion. As we show in Section 3.6, at criticality p takes a
k
stretched-exponential form. 3.5 Finite-Degree Corrections
Substituting (1)and(2)into(12),wecanexpressαin
terms of c, r, and k∗ as We have derived the leading behavior of pk for large k,
namely a power-law times an exponential. In this section,
c(3−r)+k∗(1−r2) we obtain the next-order correction, by taking the Tay-
α=− .
c(1−r)−k∗r(1+r) lor series to second order in 1/k. This correction becomes
4 Heiko Bauke et al.: Topological phase transition in a network model with preferential attachment and node removal
important in the exponential regime, where the exponen- α and β as before. Setting the coefficient of 1/k to zero
tialdecayofthedegreedistributionmakessmallerdegrees and applying (17) gives
more relevant.
Ratherthanstartingwithanansatzforthecorrection αc(1+α+k∗)+(cid:104)A(cid:105)(2+r(1+α−6β−4αβ))
δ = .
term, we derive it by expanding p /p to second order 2 c+r(cid:104)A(cid:105)(1−2β)
k k−1
in 1/k. Write
Multiplicativelyspeaking,thecorrectiontermeδ/k be-
(cid:18) (cid:18) (cid:19)(cid:19)
p α κ 1 comes negligible as k → ∞. However, it makes a signifi-
k =β 1+ + +O . (14)
p k k2 k3 cant difference for small values of k, and greatly improves
k−1
agreement with the simulations in the next section. We
For k >c, we then have notethatthesametechnique,expandingtheratiobetween
p , p , and p to higher degree in 1/k, can give us as
k−1 k k+1
k (cid:18) (cid:18) (cid:19)(cid:19) many correction terms as we wish.
(cid:89) α κ 1
p ∼βk 1+ + +O
k i i2 i3
i=1
(cid:32) k (cid:18) (cid:18) (cid:19)(cid:19)(cid:33) 3.6 The Stretched Exponential At Criticality
(cid:88) α κ 1
=βkexp ln 1+ + +O
i i2 i3 We saw above that as we approach the critical point, the
i=1
=βkexp(cid:32)(cid:88)k (cid:18)α + κ−α2/2 +O(cid:18)1(cid:19)(cid:19)(cid:33) tpioawl efar-cltaowr eβxpaopnpernotacαhedsiv1e.rgInestthois−s∞ec,tiaonndwtheeshexowpontheant-
i i2 i3 the degree distribution in fact becomes a stretched expo-
i=1
(cid:18) (α+α2)/2−κ (cid:18) 1 (cid:19)(cid:19) nentialatthispoint,duetotheappearanceofhalf-integer
∼βkexp αlnk+ +O , (15) powers of 1/k in the high-degree expansion of pk/pk−1.
k k2
We start with the ansatz
√
where ∼ hides multiplicative constants. Here we used the p =Ckαβkeζ k.
k
Taylor series for the logarithm,
Expanding to order k−3/2, we have
ln(1+(cid:15))=(cid:15)−(cid:15)2/2+O((cid:15)3),
p (cid:18) ζ α+ζ2/8 ζ3/48+αζ/2+ζ/8(cid:19)
k =β 1+ √ + +
the approximation for the kth harmonic number pk−1 2 k k k3/2
p (cid:18) ζ 2α+ζ2/2 ζ3/6+2αζ(cid:19)
(cid:88)k 1 1 k+1 =β2 1+ √ + + ,
=lnk+γ+ +O(1/k2), pk−1 k k k3/2
i 2k
i=1
with error terms of order 1/k2. Substituting this into the
where γ is Euler’s constant, and master equation (5) as before, if ζ (cid:54)= 0 then the terms of
√
order k and k force b=1 and c=r(cid:104)A(cid:105). In other words,
(cid:88)k 1 π2 1 ζ can be nonzero only at the critical point. The term of
= − +O(1/k2). order 1 then gives
i2 6 k
i=1 √
ζ =−2/ r,
Eq. (15) gives a multiplicative correction of the form
eδ/k to our earlier form (6) for p , andthetermoforderk−1/2givesthepower-lawcorrection
k
p =Cβkkαeδ/k(cid:0)1+O(1/k2)(cid:1) , (16) 3 k∗
k α=− + .
4 2
where
This recovers the results of [17] for the special case
α+α2
δ = −κ. (17) k∗ =0 and r =1, where ζ =−2 and α=−3/4. However,
2
these calculations are significantly more technical, evalu-
To determine κ and therefore δ, using (14) we write ating generating functions and their derivatives in terms
ofspecialfunctions.Ourhigh-degreeexpansioniscloserin
p (cid:18) 2α 2κ+α2 (cid:18) 1 (cid:19)(cid:19) spirit to [13], where p is written as a telescoping product
k+1 =β2 1+ + +O . (18) k
pk−1 k k2 k3 of ratios pk/pk−1.
Substituting (14) and (18) into the master equation (5)
and multiplying by (cid:104)A(cid:105)/p , we obtain an equation akin 4 Simulations
k−1
to(9),withtermsoforderk,1,1/k,andnegligibletermsof
order O(1/k2). The coefficient of k and the constant term To check that our asymptotic calculations are correct, we
are identical to those in (9), giving the same solutions for conducted two kinds of simulations: direct simulation of
Heiko Bauke et al.: Topological phase transition in a network model with preferential attachment and node removal 5
1
asymptoticanalyticsolution asymptoticanalyticsolution
1 masterequationsimulation 0.1 masterequationsimulation
networksimulation networksimulation
0.01
0.01
k k 0.001
p p
y y
t 1e-04 t 1e-04
bili bili
a a 1e-05
b b
o 1e-06 o
r r 1e-06
P P
1e-07
1e-08
1e-08
1e-10 1e-09
1 10 100 1000 10000 20 40 60 80 100 120
Degreek Degreek
Fig. 2. Comparison between simulations for k∗ = 10, Fig. 3. Comparisonbetweensimulationsfork∗ =10,r =
r = 0.10, c = 20 and our solution, which gives a power- 0.80, c=20 and our solution, which gives p ∼βkkαeδ/k
k
lawp ∼kα withα=−4.02.Thereisgoodagreementfor with β = 0.776, α = 3.423 and δ = 80.89. There is good
k
k >200.Thissetofparametersisinthepower-lawregime, agreement for k > 30. This set of parameters is in the
below the phase transition at k∗ ≈ 163.6. Note the log- exponential regime, above the phase transition at k∗ ≈
c c
log scale. The integrated master equation dips down at 2.78. Note the semi-logarithmic scale.
k ≈ 3000 because it has not yet reached equilibrium at
the highest degrees.
We show a network simulation for a single network. For
1
large degrees, we use logarithmic binning, so that each asymptoticanalyticsolution
point represents the average over an interval of degrees of 1e-05 masterequationsimulation
width proportional to logk.
1e-10
k
p 1e-15
y
tohfethdeynmaamsticesroefqfiunaittieonn.etworks,andnumericalintegration abilit 1e-20
b
Inourdirectsimulations,wegrewthenetworkstochas- ro 1e-25
P
ticallyaccordingtothemodel,uptosizen=5×107.How- 1e-30
ever,itishardtoexplorethetailofthedegreedistribution
1e-35
in the exponential regime, since p falls off exponentially.
k
For instance, to measure a probability p ≈ 10−10 we 1e-40
k
would need a network of size more than n = 1010, unless 50 100 150 200 250 300 350 400
weuselargebinsizes.Inthiscase,numericallyintegrating Degreek
themasterequation(3)untilitreachesequilibriumletsus Fig. 4. Comparison of our analytic solution and the in-
explore the asymptotics of pk far more efficiently. tegrated master equation for k∗ = 10, r = 0.80, c = 20
Since the normalization constant C depends on the (the same parameters as Figure 3) at larger degrees than
values of pk for small k, which our analysis does not try direct simulations can reach.
to predict, we adjust C to fit the simulations. We do not
tune any other parameters. In particular, α and β are
determined by our analysis, rather than fit to the data. The pseudo-code and source code for the simula-
Figure 2 shows results in the power-law regime below tions can be found at http://www.rouquier.org/jb/
the transition. There is good agreement between our so- research/papers/2010_growing_network/.
lution and both types of simulations above k >200 or so.
The direct simulation differs somewhat from the master
equation at large k due to finite-size effects. 5 Conclusion
Figure3showsresultsintheexponentialregime,above
thephasetransition.Hereaswell,thereisgoodagreement We have studied dynamical networks that are generated
betweensimulationsandourasymptoticsolutionforlarge by a model where growth takes place through a combi-
enough k. Figure 4 shows the same parameters at larger nationofpreferentialanduniformattachment,andwhere
degrees; as discussed above, we reach these larger degrees nodesareremovedrandomlyatacertainrate.Bothgrowth
byabandoningdirectsimulationandintegratingthemas- andnoderemovalarekeyfeaturesofmanyreal-worldnet-
terequation.Theagreementwithourasymptoticsolution works. Nodes in peer-to-peer networks may be added or
is excellent. removed,peoplejoinandleavesocialnetworks,nodesina
6 Heiko Bauke et al.: Topological phase transition in a network model with preferential attachment and node removal
communication network can be attacked or degrade with DOI: 10.1103/RevModPhys.74.47
time, and so on. Uniformly random attachment appears, 2. S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks
for instance, if people choose random seats and strike up From Biological Nets to the Internet and WWW (Oxford
conversationswiththeirneighbors,orareassignedtoran- University Press, 2003)
dom classrooms; random attachment also occurs, by de- 3. M.E.J. Newman. The structure and function of complex
sign, in some peer-to-peer protocols. networks.SIAMReview45(2),167(2003).DOI:10.1137/
Wehavesolvedfortheasymptoticdegreedistribution, S003614450342480
and found a phase transition between power-law and ex- 4. R. Pastor-Satorras, A. Vespignani, Evolution and Struc-
tureoftheInternet:AStatisticalPhysicsApproach (Cam-
ponential behavior, with a stretched exponential at the
bridge University Press, Cambridge, 2004)
critical point. Thus, in contrast to pure growth models,
5. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.U.
an asymptotically linear attractiveness function is not a
Hwang. Complex networks: Structure and dynamics.
sufficient condition for a power-law degree distribution. If
Physics Reports 424(4–5), 175 (2006). DOI: 10.1016/j.
the growth rate is too small, or the node removal rate is
physrep.2005.10.009
too high, the degree distribution is exponential.
6. M.E.J.Newman,A.L.Baraba´si,D.J.Watts,TheStructure
Ourfindingsarerelevantforreal-worldnetworkswhere
and Dynamics of Networks (Princeton University Press,
both growth and node removal are important. They also
Princeton, 2006)
imply further potentially interesting consequences for the 7. A.L.Baraba´si,R.Albert. Emergenceofscalinginrandom
evolution of networks whose effective growth rates vary networks. Science 286(5439), 509 (1999). DOI: 10.1126/
with time; for instance, for networks that start with a science.286.5439.509
high growth rate, but that reach a state where nodes are 8. S. Johnson, J. Torres, J. Marro. Nonlinear preferen-
added and removed at about the same rate, e.g. due to tial rewiring in fixed-size networks as a diffusion pro-
limits on the network’s overall size or population. cess. Physical Review E 79(5), 050104 (2009). DOI:
Similarly, it would be interesting to understand how 10.1103/PhysRevE.79.050104
the macro-dynamics of a network change as the growth 9. C. Schneider, L. de Arcangelis, H. Herrmann. Scale-free
and/or removal rates are varied so that we approach or networks by preferential depletion. EPL(EurophysicsLet-
cross the topological phase transition. This includes the ters) 95, 16005 (2011). DOI: 10.1209/0295-5075/95/
approach to the asymptotic degree distribution from a 16005
nonequilibrium initial state; whether this approach shows 10. S.N. Dorogovtsev, J.F.F. Mendes, A.N. Samukhin. Struc-
ture of growing networks with preferential linking. Phys-
critical slowing down near the transition; and the dynam-
ical Review Letters 85(21), 4633 (2000). DOI: 10.1103/
ics after a sudden change of the growth and/or removal
PhysRevLett.85.4633
rates, say from a region from the power-law regime to the
11. P.L. Krapivsky, S. Redner. Organization of growing ran-
exponential one.
dom networks. Physical Review E 63(6), 066123 (2001).
Abroaderquestionishowthetransitionaffectsvarious
DOI: 10.1103/PhysRevE.63.066123
types of dynamics taking place on the network, such as
12. P.L.Krapivsky,G.J.Rodgers,S.Redner. Degree distribu-
search [18], congestion, and robustness to attack.
tionsofgrowingnetworks.PhysicalReviewLetters86(23),
Finally, another direction for future work is to intro-
5401 (2001). DOI: 10.1103/PhysRevLett.86.5401
duce some kind of quenched disorder into the network 13. P.L. Krapivsky, S. Redner, F. Leyvraz. Connectivity of
model. This could include allowing k∗ to vary from node growingrandomnetworks.PhysicalReviewLetters85(21),
tonode,basedonthenode’sintrinsic“fitness”or“attrac- 4629 (2000). DOI: 10.1103/PhysRevLett.85.4629
tiveness”[14,19].Webelievethattheasymptoticbehavior 14. S.N. Dorogovtsev, J.F.F. Mendes. Scaling properties of
of the degree distribution and its phase diagram will be scale-free evolving networks: Continuous approach. Phys-
similar to our results here as long as k∗ has bounded ex- ical Review E 63(5), 056125 (2001). DOI: 10.1103/
pectation and variance. PhysRevE.63.056125
15. S.N. Dorogovtsev, J.F.F. Mendes. Scaling behaviour of
developing and decaying networks. Europhysics Letters
Acknowledgments 52(1), 33 (2000). DOI: 10.1209/epl/i2000-00400-0
16. N.Sarshar,V.Roychowdhury. Scale-free and stable struc-
This work has been partly sponsored by the European
tures in complex ad hoc networks. Physical Review E
Community’s FP6 Information Society Technologies pro- 69(2),026101(2004).DOI:10.1103/PhysRevE.69.026101
grammeundercontractIST-001935,EVERGROW.C.M. 17. C. Moore, G. Ghoshal, M.E.J. Newman. Exact solutions
issupportedbytheMcDonnellFoundation.D.S.acknowl- for models of evolving networks with addition and deletion
edges support from the Leverhulme Trust in the form of of nodes. Physical Review E 74(3), 036121 (2006). DOI:
anEmeritusFellowship.WearealsogratefultotheSanta 10.1103/PhysRevE.74.036121
Fe Institute who hosted the authors and fostered our col- 18. L.Adamic,R.Lukose,A.Puniyani,B.Huberman. Search
laboration. in power-law networks. Physical review E 64(4), 46135
(2001). DOI: 10.1103/PhysRevE.64.046135
19. G. Bianconi, A.L. Baraba´si. Competition and multiscal-
References ing in evolving networks. Europhysics Letters 54(4), 436
(2001). DOI: 10.1209/epl/i2001-00260-6
1. R.Albert,A.L.Barab´asi. Statisticalmechanicsofcomplex
networks. Reviews of Modern Physics 74(1), 47 (2002).