Load distribution in weighted complex networks

0.17 MB·English

by  K. -I. Goh

Load distribution in weighted complex networks
K.-I. Goh1, J. D. Noh2, B. Kahng1, and D. Kim1
1School of Physics and Center for Theoretical Physics,
Seoul National University, Seoul 151-747, Korea
2Department of Physics, Chungnam National University, Daejon 305-764, Korea
6
(Dated: February 2, 2008)
0
0
Westudytheloaddistributioninweightednetworksbymeasuringtheeffectivenumberofoptimal
2
paths passing through a given vertex. The optimal path, along which the total cost is minimum,

n cruciallydependonthecost distributionfunctionpc(c). Inthestrongdisorderlimit,where pc(c)∼
a c−1, theload distribution follows apowerlaw bothintheErd˝os-R´enyi(ER) randomgraphsand in
J thescale-free(SF)networks,anditscharacteristicsaredeterminedbythestructureoftheminimum

1 spanning tree. The distribution of loads at vertices with a given vertex degree also follows the SF
1 nature similar to the whole load distribution, implying that the global transport property is not
correlated to the local structural information. Finally, we measure the effect of disorder by the

] correlationcoefficientbetweenvertexdegreeandload,findingthatitislargerforERnetworksthan
h
for SF networks.
c
e
PACSnumbers: 05.10.-a,89.70.+c,89.75.Da,89.75.Hc
m
-
t
a Studyofcomplexsystemsintheframeworkofthenet- router or optical cable. For example, the Abilene net-
t workrepresentationhasattractedconsiderableattention work consists of high-bandwidth backbone, while their
s
. as an interdisciplinary subject [1, 2, 3, 4]. Of particu- sub-connecting systems do of low-bandwidth [16]. In
t
a lar interest is the emerging pattern, a power-law behav- such weighted networks, the notion of the shortest path,
m ior in the degree distribution, p (k) ∼ k−γ, where the the path with minimal number of hops between two ver-
d
- degree k is the number of edges connecting a given ver- tices in the binary network, may not be as appropriate
d
tex. Such complex networks are called scale-free (SF) as the so-called optimal path, the path over which the
n
o networks [5]. Transport phenomena on SF networks sum of costs becomes minimal. Thus it is natural to
c suchasdatapackettransportonthecommunicationnet- generalize the load to that based on the optimal paths
[ work[6,7,8,9, 10,11], randomwalks[12]andthe infor- in weighted networks. In general, the optimal paths in

2 mation exchange in social networks [13], are of vital im- weighted networks often take a detour with respect to
v portance in both theoretical and practical perspectives. the shortest path to reduce the total cost. Such a de-
7 As the first step, the transport property can be studied tour makes the optimal path longer than the shortest
1
through the quantity called the load introduced recently path [17, 18], which we expect leads to redistribution
3
[14], or the betweenness centrality in social network lit- of the load, the characteristics of which depends on the
0
1 erature [15]. To be specific, a packet leaves and arrives cost distribution function. Here we will concentrate on
4 between a pair of vertices, travelling along the shortest thedisorderinedgesonly,i.e.,thecasewhereedgescarry
0 pathway(s) between the pair. When the shortest path- theirownnon-uniformcostswithadistributionp (c). In
c
/
t waysbranch,the packetisassumedtobe dividedevenly. arecentpaper,Parketal.[19]studiedarelatedproblem
a
Then, the load ℓ of a vertex i is defined as the accumu- basedonthevertexcost,focusingontherelationbetween
m i
latedsumofthe amountofpacketspassingthroughthat the cost and the load of a vertex, finding that the load
-
d vertexwheneverypairofverticessendandreceiveaunit of a vertex decreases exponentially with the vertex cost
n packet. The load thus quantifies the level of burden of but scales as a power law with respect to the degree.
o vertices in the shortest path-based transport processes.
To study the effect of the weight in a systematic way,
c TheloaddistributionofSFnetworksalsofollowsapower
v: law, p (ℓ) ∼ ℓ−δ, with the load exponent δ [14]. When we consider a weighted network model by assigning ran-
L domcostoneachedgeofthestaticmodel[14]. Thestatic
i packets travel with constant velocity, the numerics indi-
X modelnetworkis composedofN verticesindexed by the
cate that the effect of time delay on the load does not
r integers i = 1,2,...,N and L edges that are added one
a have effect on the shape of the load distribution. There-
by one avoiding the multiple connections: Each step, an
fore the loadis usuallymeasuredasthe effective number
edgebetweenavertexpair(i,j)is chosenwiththe prob-
of pathways passing through a given vertex. So far, the ability p p where p =i−α/P j−α and added unless it
i j i j
power-lawbehavioroftheloaddistributionwasobserved
alreadyexists. αisacontrolparameterintherange[0,1)
onlyonbinarynetworks,wherethestrengthofeachedge
and we use L = 2N in this work. The network thereby
is either 1 (present) or 0 (absent) [14].
constructed is a SF network with the degree exponent
To describe transport phenomena in a more realistic γ = 1+1/α. Note that the α = 0 case corresponds to
way, one has to take into account of the heterogeneity the random graph of Erdo˝s and R´enyi [20]. Next, cost
of elements, e.g., buffer sizes and/or bandwidths of each of each edge is assigned randomly with a given distribu-
2
tion function, independent of the degrees of the vertices
located at its ends.
5 (a)
Partlymotivatedbytherecentobservationonthereal- P
S 4
worldweightednetworks[21,22],weconsiderafamilyof d
cth−eωc,owsthderisetrwibhuetnioωnf>un1cti(oωn<ina1)p,ocweisr-clahwosfeonrmaspcc(c>)∼1 d/opt 23
1
(0 < c < 1). The limit ω → ∞ corresponds to the
exponential cost distribution, and the limit ω → 0 does 0 0.5 1 1.5 2 2.5 3
to the uniform cost distribution. As ω is lowered from 1/ω
∞, the heterogeneity of the cost distribution increases 12
10
and we expect that the effect of disorder would increase, 1010 (b) 9
7
that is, longer optimal path length, dopt. The disorder pt 108 5
effect would be maximal at ω = 1, which corresponds c/Po 106 13
to the so-called strong disorder limit [18, 23] where the S 104
c 0 0.5 1 1.5 2 2.5 3
total cost is dominated by the maximum cost over the 102
path. Note that if the costs are assigned as {ci|c1 < 100
c < ... < c } such that Pn c < c , for instance, 0 0.5 1 1.5 2 2.5 3
2 L j=1 j n+1
c = 2n−1, then one would get the strong disorder limit 1/ω
n
and in this case we have ω = 1. As ω is lowered further
from ω = 1, the effect of disorder would decrease again. FIG.1: Therelativeoptimalpathlengthdopt/dSP(a)andthe
This expectation is confirmed by numerical simulations relativeadvantagecSP/copt (b)as afunction oftheexponent
ω of the cost distribution for SF network with γ = 3 and
asshowninFig.1(a)specificallyfortheSFnetworkwith N = 104. The effect of the disorder becomes maximal at
γ =3,wheretheratiooftheaverageoptimalpathlength ω = 1, which manifests itself as the peak in both quantities.
andthe averageshortestpathlengthdSP areshown. For The cross (triangle) symbol represents the the shortest path
ω 6=1, the network is in the weak disorder regime in the length of the underlying binary network (the uniform cost
terminologyofRef.[18]. Itisworthwhiletonotethatthe distribution case). The inset of (b) shows the same data in
disorder effect does not vanish qualitatively even for the linear scale, excludingthose with cSP/copt ≥101.
uniform distribution, the ω → 0 limit. It is not obvious
if the crossover from ω =1+ to ω = 1− is continuous or
discontinuous under the present data, the test of which in the load distribution would be largest for the strong
would require much larger system sizes. disorder, moderate but substantial for the uniform cost
As the optimal path length increases, the total load distribution, and minimal for the exponential distribu-
of the system grows, as it satisfies the sum rule P ℓ = tion, as was the case for the change in the optimal path
i i
N(N−1)(d +1). Thatis,bytakingtheoptimalpaths, length. This picture is confirmed by numerical simula-
opt
we need more resource—router capacity, for example— tions, shown in Fig. 2, for the ER network, the SF net-
tomaintainthesysteminthefree-flowstate,wherepack- works with γ = 4, 3, and 2. The load of the weighted
etscantravelwithoutcongestion. Theadvantageintak- networkscanbeefficientlycomputedbythemodifiedDi-
ing the optimal paths then should compensate the in- jkstra algorithm [24, 25]. For the strong disorder limit
crease of resource requirement, for it to be optimal. We (ω =1),theequivalentcalculationcanbedonebynoting
show in Fig. 1(b) the advantage in taking the optimal that the optimal paths in this case lie on the minimum
paths, by the ratio of c , the average cost along the spanningtree(MST)[18],constructedbyremovinglinks
SP
shortest paths, to the c , that for the optimal paths. in the descending order of their costs one by one unless
opt
As anticipated, the advantage is always larger than 1, such a removal disconnects the graph. We exploit this
and it is more advantageous to take the optimal paths factandcomputetheloadinthestrongdisorderlimitby
as the disorder becomes strong, exhibiting a strong peak constructing MST via, e.g., the Kruskal algorithm [24].
near ω = 1. Furthermore, c /c > d /d in all There one can see that the load distribution becomes
SP opt opt SP
cases studied, that is, the increase in the optimal path broader as the strength of the disorder increases.
length(therequiredresource)iscompensatedbythecost Asurprisingresultisthat,inthestrongdisorderlimit,
advantage. the load distribution follows a power law even for the
Inthe following,wefocusonthe three specificcasesof ER network [Fig 2(a)]. Recall that the ER network has
thecostdistribution;(i)theexponentialcostdistribution theexponentiallydecayingloaddistributioninitsbinary
(ω →∞),(ii)theuniformcostdistribution(ω →0),and version. Theloadexponentismeasuredtobeδ ≈1.59(4)
(iii) the strong disorder limit (ω =1). forthe ERnetwork. Forthe SFnetworks,it is measured
Load distribution— First, in the most global level, we that δ = 1.64(4), 1,69(2), 1.73(3), 1.82(8), 1.95(5) and
study the load distribution of the networks in the pres- 1.96(5)forγ =5.0,4.0,3.5,3.0,2.5and2.0,respectively.
ence of the disorder. One may expect that the change Inthe strongdisorderlimit, the MSTcanbe regardedas
3
0 0 0 0
10 10 10 10
10-1 (a) 10-1 (b) 10-1 (a) 10-1 (b)
10-2 10-2 10-2 10-2
pl()wl1100---543 pl()wl111000---543 (|)plkw1100--43 (|)plkw111000---543
10 -6 -5
10 10 -6
10-6 10-7 10-6 1100-7
-7 -8
10 10 -7 -8
0 1 2 3 4 0 1 2 3 4 10 10
10 10 10 10 10 10 10 10 10 10 0 1 2 3 4 0 1 2 3 4
10 10 10 10 10 10 10 10 10 10
l l
100 w 100 w lw lw
-1 (c) -1 (d)
10 10
-2 -2 FIG. 3: The condition probability p(ℓ|k) with k = 2 for the
10 10
-3 -3 ERnetwork(a)andfortheSFnetworkwithγ =3(b). Data
) 10 ) 10
pl(wl1100--54 pl(wl1100--54 darisetfroibrutthieonex(p(cid:3)o)n,eanntidaltchoeststdroisntgribduistoiornde(r×l)im,tihte((cid:13)un)i.forPmlotctoesdt
are the guidelines with slopes −3 and −1.7 in (a); −2.5 and
-6 -6
10 10
−2 in (b).
-7 -7
10 10
-8 -8
10 10
0 1 2 3 4 0 1 2 3 4
10 10 10 10 10 10 10 10 10 10 Load-degree scaling—Next,toprobeloadsinthemore
lw lw microscopicsettings,wefocusonhowtheloadofanindi-
vidual vertex would change by the presence of disorder.
FIG.2: Theloaddistributionoftheweightednetworksinthe Inthebinarynetworks,thereisascalingrelationbetween
presence of disorder. Plotted are in each panel for the ER
the degree and the load of a vertex, as
networks (a), the SF networks with γ = 4 (b), γ = 3 (c),
and γ =2.0 (d). All networks are of size N =104. Data are ℓ ∼kη, (1)
b
logarithmically binned. The symbols are for the exponential
cost distribution (⋄), the uniform cost distribution ((cid:3)), and with η = (γ−1)/(δ−1) [14], where the subscript b de-
the strong disorder limit ((cid:13)). The guidelines are plotted notes the binary network. When the disorder becomes
together, with the slope −1.6 (a), −1.7 (b), −1.8 (solid) and
sufficiently strong, however, it is not clear if such a scal-
−2.2 (dotted) (c), and −2.0 (d).
ing relation would still hold. We find indeed there exists
strong dispersion in the scaling relation for the uniform
costdistribution. Tocharacterizethedispersion,wecon-
the composition of the percolation clusters at the per- sider the conditional probability p(ℓ |k) that the load
w
colation threshold and inter-links, so called hot bonds, of a vertex with degree k is ℓ . If this distribution is
w
connecting disconnected clusters [18]. This picture can sufficiently broad, it is meaningless to speak of a scaling
be applied for the case of γ > 3, where the percolation relation as in the binary network version. We show the
thresholdisfinite. ThedegreeofeachvertexintheMST result of numerical simulation specifically for k = 2 in
is proportional to that in the original network [26]. For Fig. 3. In the strong disorder limit, p(ℓ |k) even follows
w
example, the degree distribution of the MST of the ER the similar power-law decay as p (ℓ ), meaning that the
ℓ w
network is not scale-free. In this case, it is measured degree of a vertex has essentially no correlations with
that the load exponent depends on the degree exponent. the load. This picture also applies to the SF network,as
When 2 < γ < 3, however, due to the vanishing perco- showninFig.3(b)forthecaseofγ =3. Wealsofindthat
lation threshold [27] and the presence of non-negligible it is also the case for intermediate degree nodes, k = 10
fraction of short cycles [28], the percolation cluster net- for example, in the SF networks with γ = 3. Thus in
workpicturenolongerholds. The formationofthe MST the presenceofdisorder,onemaynotpredictthe levelof
is not random and the proportionality of the degrees of trafficatarouterorthecentralityofanindividualbased
thevertexintheMSTandintheoriginalnetworkbreaks solely on the connectivity information.
down. Inthis case we find numerically that the loaddis- Disordervs.Networkheterogeneity—Wenowturnour
tribution follows a power law with the exponent about 2 attention to the interplay between the disorder in edge
with an additional fatter tail. costs and the network heterogeneity. To characterize
In the weak disorder limit, the sum of all the costs the extent of the effect due to the disorder by a simple
along the path determines the optimal path. The load scalarmeasure, we introduce the Pearsoncorrelationco-
distribution depends on the strength of disorder. Note efficientsbetweenthedegree(k)andthe load(ℓ )inthe
w
thatinthiscase,theloaddistributionfortheERnetwork weighted network, and between ℓ and the load (ℓ ) in
w b
does not follow a power law. Numerical data for other the binary network, of the same vertex. The correlation
values of γ are also shown in Fig. 2. coefficientisdefinedbyr ≡(xy−x·y)/σ σ ,wherex
xy x y
4
tail. In the weak disorder regime, it is found that for
1
0.9 sufficientlynarrowcostdistributiontheloaddistribution
0.8 changes little. As the heterogeneity of weights increases,
0.7 however,theeffectbecomessignificantandbecomesmax-
b 0.6
rllw 00..45 ibmeianlgaseqiutivaaplpernotatcohetshtehsetrcoonstgddiisstorridbeurtiolinmiptc.(cT)h∼is1s/itc-,
0.3 uation also holds on the individual vertex level, in that
0.2 the fluctuation of the load of the vertices of a given de-
0.1
greegrowsunboundedlyasthecostdistributionbecomes
0 0.5 1 1.5 2 2.5 3
broader. Theeffectofdisorderismanifestlylargerforthe
1/ω
ERnetworksthaninthe SF networks,sincethe disorder
mustcompete withthe networkheterogeneityinSFnet-
FIG. 4: The correlation coefficient rℓwℓb as a function of ω
for the ER network (◦), the SF network with γ =3 ((cid:3)) and works. Finally, we note that the time delay effect by
γ =2 (⋄). The dotted lines interpolating the data are drawn packets travelling with constant speed does not change
fortheeye. Theplotofrkℓw,whichisnotshownhere,follows the load exponent even in weighted networks, which is
very similar feature, owing to the high correlation between k checked through the weighted ER network.
and ℓb. InthisLetter,wehaveconsideredtheuncorrelatedcost
distributions on uncorrelated networks only. In the real-
world as well as the model evolving networks, however,
and σ denote the average and the standard deviation,
x theweightofalinkandthedegreesoftheverticesateach
respectively,ofavariablexoverallvertices. (x,y)stands
end are often correlated [21]. The effect of such corre-
for (k,ℓ ) or (ℓ ,ℓ ).
w w b lated disorder on the transport property in the weighted
As the scaling relation Eq. (1) implies, the correlation
networks is an open question.
r for binary network is high, typically larger than 0.9.
kℓ ThisworkissupportedbytheKOSEFGrantNo. R14-
This correlationdecreases as the strength of disorder in-
2002-059-01000-0in the ABRL program.
creases and the effect of disorder is larger for the ER
network than for the SF network.
The result that the effect of disorder is larger for the
ERnetworkthanforthe SF networkmaybe understood
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