Sandpile model on an optimized scale-free network on Euclidean space

0.22 MB·English

by  R. Karmakar

Sandpile model on an optimized scale-free network on Euclidean space
R. Karmakar and S. S. Manna
Satyendra Nath Bose National Centre for Basic Sciences Block-JD, Sector-III, Salt Lake, Kolkata-700098, India
Deterministic sandpile models are studied on a cost optimized Barab´asi-Albert (BA) scale-free
5 network whose nodes are the sites of a square lattice. For the optimized BA network, the sandpile
0 model has the same critical behaviour as the BTW sandpile, whereas for the un-optimized BA
0 network thecritical behaviouris mean-field like.
2
PACSnumbers: 05.65.+b05.70.Jk,45.70.Ht05.45.Df
n
a
J
VariousmodelsofStatisticalMechanicswhichareusu- sandcolumnmusttoppleandinatopplingitloosessome
5
allystudiedonregularlatticesarebeingstudiedinrecent grainswhicharedistributedamongthe neighboringsites

] years on graphs or networks of very complex structures. [13]. This creates anavalancheofsandcolumn topplings
h
For example, the Ising model has been studied both on andthestrengthofsuchactivitymeasuresthesizeofthe
c
the Small-World Networks (SWN) [1, 2] and on Scale- avalanche. There must be some ‘sinks’, i.e., a set of ver-
e
m Free Networks (SFN) [3]. The phenomenon of Percola- ticesthroughwhichgrainsflowoutofthe systemsothat
tionhasalsobeenstudiedonsuchnetworks[4,5]. Inad- in the steady state the balance of fluxes of inflow and
-
t ditiondiseasespreadingmodels likesusceptible-infected- outflow currents is maintained.
a
t susceptible (SIS) [6] have been studied on networks for In this paper, we have studied the deterministic sand-
s
. spreading diseases in the society or spreading of viruses pilemodelonascale-freenetworkplacedonanEuclidean
t
a in the Internet. substrate,namelyasquarelattice. Themotivationofthis
m study is to acquire support for the validity of our recent
Over the last few years it has been observed that the
- conjecture [18] that in a sandpile model the precise bal-
nodal degree distributions of many real-world networks,
d
anceatalllatticesites(exceptontheboundary)between
n e.g., World Wide Web [7] and the Internet [8] are char-
o acterized by power law tails: P(k) ∼ k−γ (degree k of thenumberofoutflowinggrainsHi whicharedistributed
c a node being the number of links attached to it). These among the neighbouring sites in a toppling at the site i
[ and the number of inflowing grains H′ received by the
networks are called ‘scale-free networks’ [9, 10, 11, 12] i
site i when its all neighbouring sites topple once ensures
1 due to the absence ofa characteristicvalue for nodal de-
v that the sandpile model behaves like the BTW model
grees. Theoretically a number of graphs are generated
2 [13] with a multiscaling avalanche size distribution. The
to model SFNs. One of them is by Baraba´si and Albert
7 absence of the site-to-site balance of H = H′ leads to
0 (BA) which has the following ingredients, namely: (i) A i i
the behaviour of Manna sandpile [14]. Below we define
1 network growsfrom an initial set of mo nodes with links
and study the sandpile model on the SFN where H is
0 connecting all m nodes. At every time step a new node i
o
5 isintroducedandisrandomlylinkedtom(<m )distinct equalto the degree ki ofthe SFN andtherefore is an ex-
o
0 tremelyfluctuatingquantity. Inspiteofthattheequality
previous nodes. (ii) Any of these m links introduced at
/ H = H′ is maintained by construction. We see below
t time t connects a previous node i with an attachment i i
a that the sandpile model on the optimized SFN indeed
m probabilityπi(t)whichislinearlyproportionaltothede-
behaves like the BTW model.
gree k (t) of the i-th node at time t: π (t) ∝ k (t). For
- i i i Recently, BTW sandpile model has been studied on a
d BA model γ =3 [9].
n static model of SFN [19]. In contrast to the usual sand-
o Sandpile models are the prototype models of Self- pile models there are no specific sinks at fixed positions.
c organizedCriticality (SOC) [13, 14, 15, 16, 17]. In these Instead, during a toppling any grain can evaporate from
v: models, long ranged correlations both in space as well the system from any arbitrary node with a small prob-
i as in time spontaneously emerge under a self-organizing ability f. The distribution of avalanche sizes (s) which
X
dynamics, in absence of a fine tuning parameter. In its do not dissipate (i.e., grains do not evaporate in these
r
a very general form, a sandpile model can be defined on avalanches) is:
an arbitrary connected graph, having a set of vertices
Prob(s)∼s−τexp(−s/s ) (1)
connected by another set of edges. An integer height c
variableh representingthenumberofgrainsinthe sand where the cut-off of the avalanche size s ∼ 1/f. It is
i c
column is associated with every vertex i of the graph. to be noted that the cut-off size does not depend on the
Starting from an arbitrary initial sand height distribu- network size N but only on the dissipation rate.
tion the system is driven by adding unit grains of sand Toclaimthatadynamicalprocessactiveinasystemis
at the randomly selected vertices h → h + 1. This self-organizedcritical,itisimportanttoensurethatboth
i i
sand column is said to be unstable when the height h long ranged spatial and temporal correlations dynami-
i
exceeds a pre-assigned threshold value h . An unstable cally evolve in this system. For the ordinary BTW or
c
2
109
108
107
106
105
3.2D(s)L 110034
102
101
100
10−1
10−6 10−5 10−4 10−3 10−2 10−1 100
sL−2
FIG. 1: Scaling of the probability distribution of avalanche
sizesfortheBTWmodelontheun-optimizedSFNfordiffer-
ent system sizes: L = 256, 512 and 1024. The data collapse
gives the values of the scaling exponents D = 2 and β =3.2 FIG.2: AcostoptimizedBarab´asi-Albertscale-freenetwork
2
giving the avalanchesize exponentτun−opt =1.6. on a square lattice of size L = 64. The SFN of N = L
nodes is generated by usual BA algorithm whose nodes are
randomlyassignedlatticesites. ThecostfunctionC(thetotal
MannasandpilesonsystemsofspatialextensionLthisis wiring length) is then minimized by a large number of trials
asdescribedinthetextkeepingthenodaldegreedistribution
verifiedin the followingways: (i) The avalanchesize dis-
tribution has a power law distribution Prob(s,L)∼ s−τ intact. Large degree nodes are visible.
for some intermediate range and this range should in-
creasewiththesystemsizeasthecut-offoftheavalanche
size distribution increases as s (L)∼ LD. (ii) The aver- Here we study the sandpile model on a SFN con-
c
age size of the avalanches increases with the system size structed on a square lattice of size L×L. We first con-
L, hs(L)i ∼ Lν and ν = 2 since the grains while exe- struct a BA SFN of N = L2 nodes. The network starts
cuting a diffusive motion have to travel distances of the growingwith an initial set of mo =(m+1) nodes. Each
orderofLtogooutofthesystemthroughsinkssituated of these nodes is linked to all other m nodes forming a
on the boundary Non-zero values of ν and D indicate (m+1)−clique. After that new nodes are added to the
that system has avalanches of all length scales and the network one by one and each such node is connected to
process is indeed critical. mrandomlyselecteddistinctnodes ofthe alreadygrown
Ifasandpileisgrowninaclosedsystem(i.e.,asystem network with probability πi(t). This process stops when
whichhasnosinksandgrainsdonotevaporatefromthis the network size has grown to L2 nodes. In our calcula-
system) the system eventually reaches a state when an tion we use m = 2, therefore our network has L2 nodes,
“infinite avalanche” which continues for ever and never 2L2−3links amongthe nodes andhas many loops. The
stops. Now, if a slow dissipation rate is introduced like nodes of the network are then assigned randomly with
every 1/f topplings one grain is dissipated from any ar- uniformprobabilitythe sites ofthe squarelattice. If two
bitrary site of the system, there is no infinite avalanche, nodes are linked, the corresponding lattice sites are con-
the system indeed reaches a stationary state, but the nected by straight lines. Thus we place the BA SFN on
avalanche sizes are no more of all length scales. This the square lattice.
is because the large avalanches loose their strengths by Clearly the degree distribution of such an Euclidean
dissipation of grains. SFNisexactlythesameasthatoftheBASFN.Tostudy
For a network or a graph in general, there is no con- the sandpile model we assume that each site (except for
cept of space, only the connections by links between the sitesontheboundary)hasasitedependentcriticalheight
nodes. One still can define a distance between an arbi- hc ofstability whichis equaltothe degreek ofthe node
i i
trarypairofnodesonanetworkmeasuredbythenumber at that site. Therefore in a toppling, the sand height at
of links on the shortest path connecting the two points. site i is reduced to: h →h −hc and in a deterministic
i i i
The largest of all possible shortest paths is called the topplingdynamicslikeBTWmodel,allthek neighbours
i
diameter of the network. A small world network has receiveonegraineach. Theoutletofthesystemisatthe
the diameter varying logarithmically with the number boundary. Thereforeeveryboundarysite(exceptthecor-
of nodes: D(N) ∝ logN. Since scale-free networks are ner sites) has the threshold heights hc = k +1. This
i i
smallworldnetworks,itisdifficulttoobservelongranged implies thatin a toppling atthe boundary site one grain
spatial correlations in sandpile model on SFNs. goes out of the system and never comes back. Similarly
3
105 15
104 (a)
103
10
102
q)
2.4L 101 σ(
b(s,L) 100 5
Pro10−1
10−2
0
10−3 0 1 2 3 4 5
q
10−4 3
10−5 (b)
10−5 10−4 10−3 10−2 10−1 100 101
sL−2
2
q
d
q)/
FIG. 3: Scaling of data for the probability distribution of σ(
d
avalanche sizes for the BTW model on optimized SFN and 1
for different system sizes: L = 32, 64 and 128. The data
collapse gives the values of the scaling exponents D =2 and
β =2.4 giving theavalanche size exponent τopt≈1.2. 0
0 1 2 q 3 4 5
at the corner sites the threshold heights are hc =k +2.
i i
Suchmechanismofoutflowofgrainsthroughthebound- FIG. 4: Comparison of the BTW model on ordinary square
ary sites guarantees that the sandpile dynamics on the lattice(solidline)andtheBTWmodelontheoptimizedSFN
on square lattice (dotted line): (a) the moment exponents
Euclidean SFN must reach a stationary state.
σ(q) vs. q and (b) dσ(q)/dq vs. q.
In this Euclidean SFN any site is connected to any
other site with equal probability and therefore the av-
erage link length hℓ i is large and of the order of the
system size L. In aijtoppling the grains therefore jump for D = 2 and β = 3.2, giving τun−opt = β/D = 1.6.
ThestochasticMannasandpileisalsostudiedontheun-
largedistancesontheaverage. Wefirststudyadetermin-
optimized BASFN onthe square lattice for system sizes
istic sandpile model on such a network. In this sandpile
upto L=1024 again. We estimated D = 2 and β = 3.0
model a grain jumps a distance around L in a toppling.
Consequently, the spatial extent of all avalanches, small giving τunopt = 1.5. We believe that τun−opt for both
the BTW and Manna sandpiles on the un-optimized BA
or big, are around L.
SFNs are indeed mean-field like and both the exponents
The avalanche size is measured in two ways: (i) the
should be actually 1.5. Similar slight deviation from 1.5
total number of topplings s and the number of distinct
was also observed in [19] for the deterministic case. It is
sitestoppleda. Apowerlawdistributionoftheavalanche
also observed that the area a has a similar distribution.
sizeswithasimplefinitesizescalingimpliesthatthedis-
RecentlyacostoptimizedSFNonEuclideanspacehas
tribution function Prob(s,L) obeys the following scaling
been constructed [20] where the total sum of the link
form:
lengths is optimized keeping the nodal degree distribu-
s
Prob(s,L)∼L−βf( ), (2) tion exactly same as that of the original SFN. For such
LD
a construction one defines a cost function C(N) as the
where the scaling function f(x) ∼ x−τ in the limit of total wiring length in terms of the symmetric adjacency
x → 0 giving τ = β/D, τ and D are the exponents matrix A of size N ×N (which has elements a = 1 if
ij
of the avalanche size distribution. One immediate way there is a link between the pair of nodes i and j and 0
to check validity of Eqn. (1) is to attempt a data col- otherwise) and the distance ℓ between nodes i and j
ij
lapse by plotting LβProb vs. s/LD with trial values of as C(N) = Σ a ℓ . The optimization process is es-
i>j ij ij
the exponents. It is now well known in the literature sentiallyarewiringprocessmaintainingthenodaldegree
that for the Manna stochastic sandpile model the distri- distributionintact. Itstartswith aBASFN constructed
bution obeys FSS with τ ≈ 1.28 where as for the onasquarelatticeasmentionedabove. Apairofdistinct
Manna
BTW sandpile the probability distribution Prob(s,L) of linksoftheSFNischosenwhosenodesarenotlinkedoth-
this measure has been found recently to obey a multi- erwise. One end of each link is then opened and rewired
scaling ansatz [22, 23]. In Fig. 1 we plot the scaling of suitablytoanothernodeofthequartetsothattotalsum
the avalanche size distribution data for the BTW sand- of the rewired length is smaller. More precisely, the first
pile on the un-optimized BA SFN on the square lattice node n is randomly selected from the set of N nodes
1
for L = 256, 512 and 1024. The best collapse works andthe secondnode n israndomlyselectedfromthe k
2 1
4
neighbours of n . In the same way n (6= n 6= n ) is se- q values. In stead, the curve gradually increases with
1 3 1 2
lectedrandomlyfromN nodesandn (6=n 6=n )ischo- q, very similar to the multiscaling behaviour of BTW
4 1 2
senfromk neighboursofn . Clearlythismoveconserves model. Tocompareweplotbothσ(q)anddσ(q)/dqofthe
3 3
thelinknumbersaswellasdegreedistribution. Rewiring ordinaryBTWonsquarelatticestudiedforsamesystem
isdonefollowingthisdecision: Ifbothn n andn n are sizes with different line styles. We see that in both plots
1 3 2 4
not linked and also ℓ +ℓ is greater than ℓ +ℓ we the behaviour is very similar and the difference between
12 34 13 24
link n n and n n . Another possibility is if n n and the two curves is very small, within 2-3 %.
1 3 2 4 1 4
n n arenotlinkedbutℓ +ℓ is greaterthanℓ +ℓ
2 3 12 34 14 23 The stochastic Manna sandpile is also studied on the
then we link n n and n n . If both cases are possible
1 4 2 3 optimized SFNs for small system sizes L 32, 64 and 128.
we accept one of them with probability 1/2. If only one
The scaling exponents are estimated as: D = 2.62 and
is satisfiedwe acceptthat. After rewiringwe removethe
β = 3.4 giving τ ≈ 1.3. This value of τ is compared
opt
linksn n andn n . Ifnone ofthe twoissatisfiedwego
1 2 2 4 with the corresponding τ ≈ 1.28 value of the ordinary
for a fresh trial. On repeated trials of these moves the
Manna sandpile.
cost function gradually decreases. Initially it decreases
To summarize, we studied the BTW sandpile model
very fast but eventually the success rate becomes very
onaBaraba´si-Albertscale-freenetworkofN =L2 nodes
slow. To monitor the optimizationprocesswe kept track
wherethenodesarethesitesofasquarelatticeofsizeL.
ofthe averagelinklength. Ourbestpossibleeffortyields
The SFN is then optimized minimizing the total wiring
the average link length hℓ i ≈ 1.75 lattice constant. A
ij
length but keeping the degree distribution intact. On
picture of the optimized network is given in Fig. 2. In
such an optimized SFN on the Euclidean space we ob-
this best possible optimized network the link lenghs ℓ
serve that the sandpile model has the same scaling be-
have an exponential distribution as: D(ℓ) ∼ exp(−gℓ)
haviour as the BTW model where as the deterministic
with g ≈ 1.16. Also the diameter of the network D(N)
sandpile on the un-optimized SFN has a mean-field like
is measured and is observed to grow as Nµ where µ is
behaviour.
estimated to be 0.40±0.02. Therefore this network is
scale-free but not a small-world network.
ThedeterministicBTWsandpilemodelisthenstudied
on such a network. The avalanche size distribution is
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