Interface Motion and Pinning in Small World Networks

4 Pages·0.19 MB·English

by  Denis Boyer

Interface Motion and Pinning in Small World Networks
Denis Boyer and Octavio Miramontes
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Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 20-364, 01000 M´exico D.F., M´exico
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0 (February 1, 2008)
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Weshow that the nonequilibrium dynamics of systems with many interacting elements located on

n a small-world network can be much slower than on regular networks. As an example, we study
a the phase ordering dynamics of the Ising model on a Watts-Strogatz network, after a quench in
J
the ferromagnetic phase at zero temperature. In one and two dimensions, small-world features

0 produce dynamically frozen configurations, disordered at large length scales, analogous of random
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field models. This picture differs from the common knowledge (supported by equilibrium results)

that ferromagnetic short-cuts connections favor order and uniformity. We briefly discuss some
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n implications of these results regarding thedynamics of social changes.
n
-
s
i
d Small-worldnetworkshave receiveda greatdeal of at- tures are reminiscent of nonequilibrium processes in the
. tention in the past few years,in particular for their real- randomfield Ising model (RFIM) [9], in binarymixtures
t
a istic description of the topology of the interactions that with fixedimpurities [10],as wellasin a few socialmod-
m take place among populations in various biological, so- els onregularlattices like the voter model [11]. This has
- cial or economical systems [1]. An important feature of to be contrasted with the much more efficient phase or-
d
small worlds, that is not shared neither by regular lat- dering kinetics of the Ising model on regular lattices (or
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tices nor random networks, is the interplay that exists ModelA in the lexiconof HohenbergandHalperin [12]),
o
between local (or “physical”) interactions, e.g. between wherethe meansize ofordereddomainsgrowswithtime
c
[ nearest-neighbors, and non-local ones, involving nodes as t1/2 [13]. Our present analysis focuses on the motion

(or agents) separated by large distances but connected ofdomainwallsbetween“up”and“down”domains,and
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through short-cuts. Among other outstanding topologi- showsevidenceofcompetingeffects betweensurfaceten-
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cal properties, the effective space dimension of such net- sion and pinning (or localizing) effects.
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5 works grows linearly with their size [2], even if the frac- We use a standard model of small-world network [14]
3 tion of sites with short cuts is very small. consisting of a regular square lattice (or a chain in 1D)
0 The strong connectivity of small worlds usually en- composed of N nodes connected to their nearest neigh-
1 hances dramatically cooperative effects, as predicted by bors. Foreachsite,wethenestablishwithaprobabilityp
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epidemic models of spreading of diseases [4], or of prop- anadditionalconnection,orshort-cut,linkingtheconsid-
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agation of conventions or rumors in social systems [5]. eredsite toanothersite chosenatrandominthe lattice.
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t Naturally, many models of social dynamics have been (We do not remove the nearest neighbors connections.)
a
m inspired from the Ising model [6]. The Ising model on For p=0, the lattice is regular,while for p=1, the net-
a small world exhibits ferromagnetic order at low tem- work is strongly disordered. Here, we will consider only
-
d peratures even in one dimension [7], while, in higher di- theso-called“smallworld”limit,thatcorrespondstothe
n mensions,thecriticaltemperatureisincreasedcompared case p≪1, where connections are mainly localand only
o with that of the regular lattice [8]. In addition, the fact long-rangedfor a small fraction of nodes.
c
thatthetransitionisofmean-fieldnatureagreeswiththe On a fixed network, we then assign to each node a
:
v intuitive argument that each site is effectively close to a spin-like variable S = ±1: it represents a social con-
i
i largenumber ofsites due to the short-cuts ofthe lattice. vention, initially chosen at random for each node. At
X
However, because of their inherent random topology, each time step, each node updates its convention in or-
r
a onemayaskwhetherinsomesituationssmall-worldnet- der to reach a better consensus with the nodes it is con-
works would not rather exhibit features characteristic of nected to. In other words, the system follows a zero
disorderedsystems. Inthiswork,westudy asabasicex- temperature Glauber dynamics with the Hamiltonian
ample the nonequilibrium dynamics of the Ising model, H = − J S S , where the sum is performed over
hi,ji ij i j
as observedafter a rapidquench from the high tempera- all possPible pairs of nodes. Jij = 1 if sites i and j are
turephasetotheferromagneticphase. Weshowthatthe connected,J =0otherwise. Ateachstep,aspinisthus
ij
random(all ferromagnetic)connectionsthat enhance or- chosen at random and flipped. If H decreases, does not
dered states at thermodynamic equilibrium, are respon- change, or increases, the new configuration is accepted
sible in the present case for very slow dynamics and sta- with probability 1, 1/2 and 0, respectively.
bilize atlargetimes configurationsthat, insteadofbeing In regular networks (p = 0), the system evolves to-
uniform, are spatially heterogeneous. At zero tempera- ward a minimum of H (all S ’s equal to +1 or −1).
i
ture, systems do not perform long range order dynami- Transient configurations are characterized by the pres-
cally, but remain asymptotically trapped in metastable ence of growingand competing ordereddomains of “up”
states characterized by a finite domain size. These fea- and“down”spins. Thelargetimedynamicsiscontrolled
1
by the motion and annihilation of interfaces (or domain restricted to perform a random walk within the interval
walls) that separate these domains. As for many other [A,B]. Interface I is therefore unable to annihilate with
1
systems ordering in phases with broken symmetries, its interface I , that is localized between C and D. Hence,
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time evolution is self-similar: The two-point correlation the two disjoint black domains can not merge to form
function, C(r,t) = hS (t)S (t)i, obeys a scaling rela- a bigger one, what would happen in the standard Ising
i i+r
tion C(r,t) = f(r/ξ(t)), where f is a scaling function, model. Thedomainsize,orcorrelationlengthξ,doesnot
while ξ, the domain size, grows as t1/2 [13]. exceedthe distance, of order1/p,that separatesinfluent
unlike (antagonist) successive nodes.
2 ξ∞, 1D a)
100 ր 2 L2/L∞ ւ A B YES A B
2D 2 2
ց
2 2 Slope −1
2
ξ∞2 2 I NO I
2
2 −0.642 2 b)
2 2 2
I I I
2 2 1 2 3
10 2
2 A B C D E F
2
2
ξ
0.001 0.01 p 0.1 FIG.2. Up and down domains. Domain walls In become
localized in 1D.
FIG.1. Asymptoticcorrelation lengthξ0 inlatticespacing
units as a function of the reconnection probability p, in 1D Wealsofindthatthestructureoffrozenconfigurations
and 2D. L2/L∞ is the interface characteristic length in 2D. obeys a scaling relation with p, i.e. that they are statis-
tically independent of p via properlength rescaling. The
On the contrary, in small-world networks (p 6= 0, asymptoticcorrelationfunctionC(r,t=∞)is plottedin
p ≪ 1) one observes after some time that the typical Figure 3 as a function of the reduced variable r/ξ (p),
∞
domain size of ordered spins saturates to a finite value, for various values of p. Data collapse rather well on a
which decreases when the density of short-cuts (p) in- single curve. At short times, the kinetics is not affected
creases. Foraonedimensionalchain(oflengthL=105), by the small world structure of the lattice, and that ξ(t)
weplotξ(t=∞)asafunctionofp inFigure1. The cor- starts growing as t1/2. When interfaces become local-
relation length is determined from the half-width of C ized, the structure can be roughly seen as the one given
averaged over 10 networks and initial conditions. A be- bythestandardGlauberdynamicsofthe1DIsingmodel
havior ξ(t = ∞) ∝ 1/p can be observed. 1/p represents stopped at a time (1/p)2. Since that problem obeys dy-
the characteristic size of the one dimensional network, namical scaling, frozen configuration should scale with
i.e. the average distance between two nodes that have parameter p. However, this picture is not quantitatively
long range connections (or “influent” nodes). correct,asthescalingfunctionsinbothproblemsslightly
Influent sites strongly affect the motion of interfaces. differ.
Atlowp,mostofthesenodesarecharacterizedbyonead- Finite, low temperature effects are quite subtle in one
ditional connection. On Figure 2a, two nodes far apart, dimensional small-worlds since they do not destroy the
A and B, are connected, and SA = −SB. Any inter- ferromagnetic order observed at T = 0, unlike in usual
face I passing through node A leftward can not jump Ising chains. One can interpret here the order/disorder
back toward the right, since it is energetically unfavor- transition temperature T as the temperature where or-
c
able. Therefore,atlargetimes,throughinterfacemotion, dering via interface jumps over localizing barriers (that
influent nodes will tend to be (irreversibly)connected to enable further domain merging) no longer overcomes
nodes that have the same spin (a situation analogous to disordering happening within domains (subdomain cre-
assortativemixing,asobservedinsomereallifenetworks ation). In Figure 2b, interface I (or I ) can jump in
2 1
[3]). interval [B,C] at a rate r = exp(−2/T). Besides, the
a
This argument can be extended to a succession of do- rate at which any spin among the p−1 spins of interval
mains. Figure 2b illustrates a typical large time configu- [D,E]wouldflipisr =p−1exp(−4/T): itisroughlythe
b
ration: In this example, interface I1 stands between two rate at which a white domain is createdandcan startto
influent nodes A and B with opposite spins: for the rea- grow. Qualitatively,the order/disordertransitionoccurs
son mentioned above, I1 can not jump to the left of A, whenra =rb. Thisgives: Tc ≃−2/lnp,aexpressionde-
norto the rightofB. The interfaceis thenlocalized,i.e. rived(with a∝sign)inref.[7]usingthe replicamethod.
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Theaboverelationmaybeexactasp→0(asthenumer- in white, those which are connected to an other influent
ical prefactors in the different rates become irrelevant). spin of same sign (“like” pairs, of number density n ),
l
Simulation results (not shown) give T = −2.3/lnp for in black those connected to a spin of opposite sign (“un-
c
p=0.03. like” pairs, number density n ). Initially, n ≃ n , but
u l u
1 ascoarseningproceeds,“unlike”dotsturnmoreeasilyto
“like” than the contrary, as in one dimension (Fig. 2a).
Onceagain,mixingtendstobeassortative(n >n ),but
0.8 p=0.2;0.1;0.07;0.05;0.03; l u
with the increaseof n , atsome point, there are no more
l
0.02;0.01;0.005
possible moves toward better consensus. We find nu-
0.6 mericallythatcoarseningstopsandinterfacesgetpinned
C(r,t=∞) when nl ≃1.86 nu.
The finite domainsize canbe interpretedas the result
0.4 ofcompetingeffectsbetweensurfacetension(the driving
force for domain growth) and energy barriers created by
the multiplicationofinfluent“like”sites. We picture the
0.2
systemasacollectionofL2/R2 domainsofradiusR,and
estimate its energy change when domains coarsen from
00 1 2 3 4 5 6 R to R+ dR (dR > 0). The usual contribution from
r/ξ(t=∞) surface tension is δE ∝ −2L2dR/R2. Meanwhile, the
I
numberofinfluent nodes thatflipspinis proportionalto
FIG.3. Twopointcorrelationfunctionin1D(dottedlines) 2pRdR(L2/R2). “Like” nodes turn to “unlike” (with an
and 2D (solid lines) as a function of r/ξ∞, for various p. energy costper spin of 2), and reversely(with anenergy
decrease of -2). The total energy difference thus reads
2 n −n 4p
δE ∝ − + l u L2dR. (1)
(cid:20) R2 n +n R(cid:21)
l u
The second term is positive and dominate at large R.
Hence, coarsening is arrested when δE = 0, or R ∼
∞
p−1. This argument is somehow similar to the (equilib-
rium) Imry-Ma argument for the RFIM [15]. Yet, an
important difference is that here the average magnetic
field felt on influent nodes (or “impurities”) is not zero,
but has been biased(n 6=n ) due to previous spin flips.
l u
The above continuous Imry-Ma-like argument quali-
FIG.4. Frozen domains (in gray) for p = 0.05 (6% of the tatively explains frozen states, but over-estimates R∞
system total area). The white and black dots represent the (∼ p−1 instead of p−2/3). The exponent −2/3 can be
“like” and “unlike” influentnodes, respectively. explained as an effect of the square lattice. As shown
on Figure 5a, a single influent “like” node located at a
In two dimensions, one also observes that random ini- domain corner can disappear through the diffusive mo-
tialconfigurationsfreezeatlargetimes. Figure1displays tion of a step. Figure 5b represents then the simplest
as a function of p the asymptotic correlation length (de- distribution of “like” nodes such that the hatched do-
terminedfromC(r,t=∞)averagedover8networkswith main can not shrink. It is composed of two right-angle
15002 spins), as well as the length associated with inter- corners {A,A ,A } and {B,B ,B } defining a square
1 2 1 2
face density. The latter is defined as L2/L, where L is r×r. Iftheotherwhitenodes{D ,..D }containedinthe
1 n
the system linear extend and L the total length of all square do not form any right-anglecorners,then this re-
boundaries. Both length scales remain proportional to gion encloses the smallest (or “critical”) pinned domain:
eachotherwhenvaryingp,suggestingthatfrozenconfig- any bubble of hatched region comprised in the square
urationscanbecharacterizedbyonecharacteristiclength and that does not contain both corners {A,A ,A } and
1 2
scale,referredtoasthe“domainsize”,R (p). Numerical {B,B ,B } will shrink. Any larger bubble will not. We
∞ 1 2
resultssuggestthatR variesasaninversepower-lawof now calculate the probability P (r) that a configu-
∞ freeze
p,withannon-trivialexponentcloseto−2/3overnearly ration such as represented in Fig. 5b has a size r, and
two decades. Surprisingly, R does not scale as p−1/2, thenidentifyr∗ suchthatP (r∗)ismaximalwiththe
∞ freeze
the characteristic length scale of the network [2]. Once asymptotic domain size R in the disordered medium.
∞
again, the spin-spin correlation function at t=∞ scales GiventhenodeAlocatedattheorigin,theprobability
rather well as C(r)=f(r/ξ (p)), see Figure 3. that there is at least one white dot (A ) on the same
∞ 1
Figure 4 shows a typical frozen pattern at p = 0.05. line within a distance r is P (r) = 1−(1−p )r, with
1 l
Thepositionsofthe“influent”spinsaremarkedbydots: p /p=n /(n +n )thefractionofinfluentnodesthatare
l l l u
3
“like” (here, the numerical value of this ratio − close to heterogeneities (or segregation), despite that these con-
0.65−is unimportantandcouldbe setto 1). Therefore, figurations are not the most desired ones by individual
P (r)=p [P (r)]4P (r), withP the probabilitythat agents. A similar picture, in agreement with some em-
freeze l 1 2 2
the D ’s do not form right-angle corners, i.e. that each pirical observations, was drawn recently from antiferro-
n
node D is at least located on a line or a column not magnetic models on scale-free networks [16]. Right after
i
occupied by an other D (see the dotted lines in Fig. strong political changes (in Eastern European countries
j
5b). P can be approximated as in 1989, in Mexico in 2000) the evolution of reforms can
2
be fast, but rapidly social inertia takes over and renders
(r−2)2 further adjustments difficult or null. Physically speak-
P2(r)≃ (1−pl)(r−2)2−npnlC(nr−2)2[1−P1(r)]n, (2) ing, the response of social systems to external forcings
nX=0 (i.e. large-scalepolicies)issusceptibletoexhibitsomeof
theinterestingfeaturesknownfordisorderedsystems[9].
or P2(r) ≃ (1−plP1(r))(r−2)2. In the sum (2) we have While revising the manuscript, we became aware of a
multiplied the probability of having n white dots inside similar study on the voter model on small-worlds [17].
the square by the probability [1−P1(r)]n that n inde- We acknowledge fruitful discussions with G. Cocho, J.
pendent dots have no neighbors on the same line within Vin˜als and R. Boyer.
r. For n small, no or few corners can be formed any-
way, so that relation (2) slightly under-estimated P ,
2
since a small fraction of empty sites are counted twice
([1−P1(r)]n <∼ 1). For n large, on the contrary, rela-
tion(2)over-estimatesP ,sinceitisimpossibletolocate
2
manydots without formingcorners(while [1−P (r)]n is
1
small but 6= 0). We suppose that both errors compen- [1] S.H. Strogatz, Nature410, 268 (2001).
sate. This factorization enable us to compute the most [2] M.E.J. Newman and D.J. Watts, Phys. Rev.E 60, 7332
probablesquaresizer∗ analytically. P (r)increaseswith (1999).
1
r, P (r) decreases with r, and P (r) has one single [3] M.E.J. Newman, Phys. Rev.Lett. 89, 208701 (2002).
2 freeze
maximum. Assuming p ≪ 1, r ≫ 1, rp ≪ 1, we find [4] R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 63
that r∗ =p−2/3 ∝p−2/3, in agreement with the numeri- 066117 (2001); O. Miramontes and B. Luque, Physica
l D 168-169, 379 (2002)
cal results.
[5] D.H. Zanette, Phys.Rev. E 64, R050901 (2001).
[6] Social Dynamics, S.M. Durlauf and P. Young, Eds.,
a) (Brookings Institution, Cambridge, 2001).
[7] A.BarratandM.Weigt,Eur.Phys.J.B13,547(2000).
[8] C.P.Herrero,Phys.Rev.E65,066110(2002);H.Hong,
B.J. Kim, M.Y. Choi, Phys.Rev.E 66, 018101 (2002).
A A [9] T. Nattermann, in Spin Glasses and Random Fields
1
b) D (World Scientific, Singapore, 1998), p.277.
1 [10] F. Qiu, G. Peng, V. Ginzburg, A. Balazc, H.-Y. Chen
and D.Jasnow, J. Chem. Phys. 115, 3779 (2001).
[11] I. Dornic, H. Chat´e, J. Chave and H. Hinrichsen, Phys.
r A2 D2 B1 Ranedv.SL.eRtte.d8n7e,r,04p5r7ep01rin(2t0c0o1n)d;-Fm.aVta/z0q2u0e9z4,4P5..L.Krapivsky
[12] P.HohenbergandB.Halperin,Rev.Mod.Phys.49,435
D
n (1977).
[13] A. Bray, Adv. Phys. 43, 357 (1994); J. Gunton, M. San
Miguel and P, Sahni, in Phase Transitions and Critical
Phenomena, edited by C. Domb and J. Lebowitz (Aca-
B2 B demic, London,1983), Vol. 8.
r [14] D.J. Wattsand S.H.Strogatz, Nature393, 440 (1998).
[15] Y.Imry and S.K.Ma, Phys.Rev.Lett. 35, 1399 (1975).
FIG.5. a)Freeandb)pinneddomain inpresenceof“like” [16] D. Stauffer and K. Kulakowski, preprint cond-
influentnodes. mat/0210225.
[17] C. Castellano, D. Vilone and A. Vespignani, preprint
To summarize, we have shown on an example that as- cond-mat/0210465.
sortative mixing in small world networks can dynami-
cally generate frozen metastable states : At large times,
some influent nodes have simply no immediate interest
to evolve. These results suggest that long term dynam-
icsinhighlyconnectedsocialsystemscanproducespatial
4