Asymptotic behavior of the Kleinberg model

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by  Shai Carmi

Asymptotic behavior of the Kleinberg model
Shai Carmi,1,2 Stephen Carter,3 Jie Sun,3 and Daniel ben-Avraham4,3,∗
1Minerva Center & Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel
2Center for Polymer Studies, Boston University, Boston, MA 02215, USA
3Department of Mathematics & Computer Science, Clarkson University, Potsdam, NY 13699-5815
4Physics Department, Clarkson University, Potsdam, New York 13699-5820, USA
WestudyKleinberg navigation (thesearch of a target in ad-dimensional lattice, whereeach site
9 is connected to one other random site at distance r, with probability ∼r−α) by means of an exact
0 master equation for the process. We show that the asymptotic scaling behavior for the delivery
0 time T to a target at distance L scales as T ∼ ln2L when α = d, and otherwise as T ∼ Lx, with
2 x=(d−α)/(d+1−α)forα<d,x=α−dford<α<d+1,andx=1forα>d+1. Thesevalues

n ofxexceedtherigorouslower-boundsestablishedbyKleinberg. Wealsoaddressthesituationwhere
a there is a finite probability for the message to get lost along its way and find short delivery times
J (conditioned upon arrival) for a wide range of α’s.

8
2 PACSnumbers: 05.40.Fb,02.50.-r,89.75.Hc,05.60.-k

]
h Inanowfamousstudy[1]thesocialpsychologistStan- Moreover, no local algorithm performs better, function-
c
ley Milgram asked randomly chosen people to send a ally, than the simple-minded greedy algorithm: Pass the
e
m postcard to a disclosed target in the USA. The partic- message forward to the neighbor node that is closest to
ipants were to send the card to a person they knew on the target (geographically).
-
t a first-name basis, who will then send it on to another In this letter we study the asymptotic long-time be-
a
acquaintance,andsoon,untilitreachedthedesireddes- havioroftheKleinbergsearchprocess. Wefindtheexact
t
s tination. 20%ofthecardssuccessfullyreachedthetarget formoftheexponentx(α),andweshowthatT ∼ln2Lis
.
t using, on average, chains of 6.5 acquaintances, confirm- the actualscaling (not just a bound) for the special case
a
m ingthenotionthatthenetworkofsocialcontactshasthe ofα=d. Ourapproachisbasedonamasterequationfor
small-world property[2]: averyshortpath,typicallylog- the full probability distribution for completing a search
-
d arithmic in the size of the system, connects between any within a given time. This formalism also enables one to
n two nodes. How does the message find its way, let alone consider the possibility of the message getting lost along
o so efficiently? Searching and navigation problems such its way, and we discuss briefly some surprising outcomes
c as this [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] are relevant to of that scenario.
[
severaldisciplines, fromsociology,to efficient algorithms Because the message gets closer to the target with

1 incomputerscience,totheunderstandingoftheforaging each successive step, nodes are never revisited and one
v behavior of insects and animals. can view each long-range link as being created only as
5 the message arrives at its site of origin. The message
The problemhas been elegantly addressedin the sem-
3
closesinonthetargetinapeculiarkindofdirectedL´evy
5 inal work of Jon Kleinberg [3, 4]. Kleinberg considers
walk,consisting of a mix of “short”(one lattice spacing)
4 an (L×L)-square lattice, where in addition to the links
. between nearest neighbors each node i is connected to and “long” (power-law distributed) steps. In this search
1
a random node j with a probability p = r−α/ r−α thedimensionalityofthelatticeentersintoconsideration
0 ij ij k ik
9 (rij =|ri−rj|is the Euclideandistance betweennodes i mainlyasaJacobianinthevarioussums(orintegrals)of
P the equations describing the process. We therefore limit
0 and j, and the sum in the denominator excludes k =i).
: Suppose that a message is to be passed from a “source” the following derivations to one dimension and general-
v
ize the results for higher dimensions, having made the
i node s to a “target” node t, along the links of the net-
X necessary adjustments.
work, by a decentralized algorithm — an algorithm that
Forconvenience,torenderthep independentofi,we
r relies solely on local information. Kleinberg shows that ij
a adopt periodic boundary conditions. Specifically, con-
whentheexponentαequalsd,thelatticedimensionality,
analgorithmexiststhatrequires lessthanln2Lsteps to sider a ring of length 4L, with the source at 0 and the
target at L, and the range of the long-contact links lim-
complete the task. If α6= d, the delivery time, T, grows
as Lx, with rigorous lower bounds, x≥x [3, 4, 14]; ited to 2L−1 [15]. In that case,
K
2L−1
−1
p =Ak−α; A= 2 k−α , (2)
d−α 0≤α<d, i,i+k
T ∼Lx, x≥xK =(αd+α−−1d+d1 α>d. (1) (cid:16) Xk=1 (cid:17)
where A is a normalizing factor. Let P(n;l) denote the
probability that a message, at distance l from the tar-
get, takes n additional steps to reach the target. Once
∗Electronicaddress: [email protected] the message is at the target it takes no additional time
2
T , we obtain, after some rearranging,
k−1
l
D −D =A (l+1−k)−α
0.03 l+1 l
L=1000
T)L0.02 =1 nkX=1(cid:2) (5)
1000 P( 2l−1
0.01 −(l+k)−α D − k−αD ,
k l
TL 0 50 100 150 (cid:3) kX=1 o
TL for l = 1,2,...,L− 1, while for l = 0 we have D1 =
T −T =1.
1 0
100
L=1000 As a quick check, consider the limit of α → ∞, when
L=2000 allthelong-rangecontactsarerestrictedtolength1,and
L=4000
therefore one expects T =l. Indeed, in this case all the
l
k−αtermsintheequationtendtozero,unlessk =1,and
0 0.5 1.0 1.5 2.0 2.5 we get D −D = 0, which along with D = 1 yields
l+1 l 1
D =1, and T = l D =l, just as expected.
k l k=1 k
Next, consider the opposite limit, of α=0, where the
FIG. 1: (Color online) Mean delivery time, TL, as a function P
distributionoflong-ragecontactsishomogeneous. Inthis
of the long-contact exponent, α. Note the perfect agreement
between simulations (solid line) and the results from Eq. (4) case A=[2(2L−1)]−1 and we obtain from (5),
(symbols). Shown are results for three values of L. Inset:
2l−1
Distribution of the delivery time for the case of α = 1 and D −D =− D .
l+1 l l
2(2L−1)
L = 1000 as computed from (3) (solid line) is compared to
simulations (symbols).
Although this equation can be solved exactly, a continu-
ous approximation,
to reach it, so P(n;0) = δ . Likewise, P(0;l) = δ , d l
n,0 0,l D(l)=− D(l),
since the only way to reachthe target in 0 steps is if the dl 2L
message is already there to begin with.
assuming L ≥ l ≫ 1, works just as well. In view of
P(n;l) satisfies the equation
the boundary condition D(0) = 1, this has the solution
D(l)=exp(−l2/4L). Then, T(L)= LD(l)dl. The up-
2l−1 0
P(n;l)=A k−αP(n−1;|l−k|) perintegrationlimit maybe safelyreplacedwith∞,due
R
totherapiddecayofthegaussian,andasimplechangeof
k=1
X (3) variables yields T(L) ∼ L1/2, in perfect agreement with
2l−1
the Kleinberg bound for α=0.
+ 1−A k−α P(n−1;l−1).
For larger values of α we are not as fortunate as to
(cid:16) Xk=1 (cid:17) find a full analytic solution, but we can still obtain
The first term on the rhs represents the events that the the asymptotic behavior. For 0 ≤ α < 1 we take a
first of the additional n steps is a long step of length k, hint from the solution for α = 0 and make the ansatz
inwhichcasethemessagewouldcometowithindistance D(l) = f(lβ/L), where f(x) is a smoothly decreasing
|l−k|fromthetarget. Thesecondtermrepresentsashort function; f(x)=O(1) for x.1, and decaysvery rapidly
step, that advances the message a single lattice spacing. (e.g., exponentially) for x & 1. Consistent with this be-
For our main purpose here it is sufficient to consider havior, the derivative at the crossover point x∗ = 1 is
just the first moment hni ≡ Tl, that is, the mean deliv- f′(x∗) = −O(1). This ansatz is nicely confirmed by nu-
ery time from a site a distance l away from the target. merical integration of Eq. (5).
Multiplying Eq. (3) by n and summing over n, we get Apply now Eq. (5) to the crossover length l∗ = L1/β.
The lhs is
2l−1 d
Tl =A k−α 1+T|l−k| Dl∗+1−Dl∗ ≈ dlD(l)|l=l∗ ∼−L−1/β,
k=1
X (cid:0) (cid:1) (4) while the sums on the rhs can be estimated by replacing
2l−1
+ 1−A k−α (1+T ) , Dl withaconstantforl <l∗,andzeroforl>l∗,yielding
l−1 −Al1−α. But A ∼ 1/L1−α, leading to −1/β = (1 −
(cid:16) Xk=1 (cid:17) α)/β∗−(1−α), or β =(2−α)/(1−α). Finally,
for l =1,2,...,L. Numerical integrationof Eqs.(3) and
L ∞
(4) yields perfect agreement with the results from direct T(l)= D(l)dl≈ D(l)dl
simulation of the Kleinberg navigationprocess on a ring Z0 Z0
(Fig. 1). L1/β ∞
= f(x)x1/β−1dx∼L1/β,
Using the factthat T−k =Tk, anddefining Dk =Tk− β Z0
3
so
T(L)∼L(1−α)/(2−α), 0≤α<1. (6) nt
e1.00
n (a) - 1d
o
For α>1, we sum Eq. (5) over l, taking into account p
x
that D =1, and rearrange: e0.75
1 e
m
D −1=−A l k (l+m)−αD . (7) ry ti0.50
l+1 k e Simulation
v
k=1m=−k+2 e Theory (x)
X X Deli0.25 Kleinberg (xK)
This can be obtained more directly also by rearrang-
ing Eq (4). The inner sum over m can be approximated
0
by an integral, yielding
0 1 2 3 4 5
l
Dl+1−1≈ αA−1 (l+k)1−α−(l−k+2)1−α Dk. nent1.00 (b) - 2d
kX=1(cid:2) (cid:3) (8) xpo0.75
e
Sinceα>1,Aconverges,asL→∞,andwemaysimply e
followpowersofl. AssumefirstthatD ∼l−β. Thedom- m
inant−1 onthe lhs ofthe equationmulst be balancedby y ti0.50
r Simulation
the dominant term on the rhs, which scales as −l2−α−β, e
v Theory (x)
so β =2−α. This leads to eli0.25 Kleinberg (xK)
D
T ∼Lα−1, 1<α<2. (9)
0
0 1 2 3 4 5
TheupperlimitonαfollowsfromthefactthatT cannot
L
increase faster than linearly in L.
FIG. 2: (Color online) Delivery time exponent, x, as a func-
For the special case of α=1, we have
tionofαin(a)d=1and(b)d=2dimensions. Resultsfrom
Eq. (12) (solid lines) are compared to the Kleinberg bounds
l
l+k (broken lines) and to simulations (symbols).
D −1≈A ln D ,
l+1 k
l−k+2
k=1
X
andA∼(lnL)−1. Tocounterthe−1onthelhsonemust (or sums) through the Jacobian for d-dimensional inte-
then allow that D ∼(lnL)/l, for large l, which leads to gration. Making the necessary adjustments, we find
l
T ∼ln2L, α=1, (10) d−α 0≤α<d,
d+1−α
T ∼Lx, x= α−d d<α<d+1, (12)

exactly as the Kleinberg upper limit for this case.
1 α>d+1.
The dominant−1 term onthe lhs of (8) mightalso be
cancelledifD(l)=1−g(l),whereg(l)vanishesasl→∞. Once again, these results agree with the Kleinberg
Substituting this ansatz in (8) we find g(l) ∼ l2−α, and
bounds x ≥ x . Eq. (12) fits simulations results for
integration of D(l) yields K
d = 2,3,4 nicely (we did not test higher dimensions).
T ∼L+O(L3−α), α>2. (11) Results for d=2 are shown in Fig. 2b.
Returningnowto the probabilitydistribution, P(n;l),
In this case, the condition α>2 prevents the correction astandardwaytotackleEq.(3)isthroughthegenerating
term from growing faster than linearly. function
In Fig. 2a we compare our various results for x(α) in
∞
one dimension to computer simulations [16] and to the Pˆ(z;l)≡ P(n;l)zn.
Kleinbergbounds,x (α). Forα→2weexpectlogarith-
K n=0
mic terms, as the main contribution and the correction X
term in (11) approach then the same power-law. The The“fugacity”zmaybeinterpretedastheprobabilityto
logarithmicbehaviormakesitverydifficulttoextractre- completeasinglestepsuccessfully. InthatcaseP(z;l)is
liable estimates from simulations for the exponent x(α) thetotalprobabilitytocompletethedeliverysuccessfully
near α=1 and 2. (toatargetatdistancel). Wedeferamoredetailedstudy
The foregoing results, argued for one-dimensional lat- of P(n;l) for future work and focus for the moment on
tices, are easily generalized to higher dimensions: the the conditional average of the delivery time, T , that
z,l
space dimensionality, d, enters in the various integrals is, the averagedelivery time conditionedupon successful
4
We are now ready to address the conditional average,
at least numerically. Fixing the value of z, we solve
Eqs. (14) and (15) recursively, up to l = L, and use
Eq. (13) to compute the average. Typical results are
z,L shownin Fig.3. For a fixed distance L the delivery time
T is remarkably small throughout a wide range of long-
contact exponents, α . α (z;L), and saturates rapidly,
∗
100 T ∼ L, as soon as α exceeds α . Thus, it seems that
z,L ∗
in the presence of losses small-world behavior does not
L=200
imply a particular value of the long-contact exponent.
L=400
If anything, there appears to be a small local maximum
L=600
(barelyperceptibleinthefigure)aroundavalueofαthat
creepstowardsd asL increases. These results seemrele-
10
vant to the Milgram experiment, where it was estimated
0 3 6 9 12
that z ≈0.75 [17].
FIG. 3: (Color online) Conditional delivery time, T , as a In summary, we have found the actual delivery time
z,l
function of α, for L = 200 (squares), 400 (circles), and 600 exponent in the Kleinberg navigationmodel, for the non
(triangles). Resultsforz =0.9(emptysymbols)arecompared small-world cases of α 6= d, and we have confirmed that
to perfect transmission (solid symbols). T ∼ln2Lis the actualscaling(notjust abound) for the
special case of α = d. We have also introduced master
equations that allow one to study the Kleinberg model
arrival at the target: analyticallyingreatergenerality,andhavelookedbriefly
into the delivery time in the case of imperfect transmis-
T = n=0nP(n;l)zn =z∂Pˆ(z;l)/∂z . (13) sion,whenthereisafinite probabilityforthe messageto
z,l P(n;l)zn Pˆ(z;l) get lost along its way. The distribution of delivery times
P n=0
and the analytical treatment of imperfect transmission
MultyplingP(3) by zn and summing over n, minding are important problems left open to further inquiry.
the boundary conditions, we obtain
2l−1
Pˆ(z;l)=A k−αzPˆ(z;|l−k|)
k=1
X (14)
2l−1
+ 1−A k−α zPˆ(z;l−1), Acknowledgments
(cid:16) kX=1 (cid:17)
Taking the derivative of this equation with respect to z,
We thank L. M. Glasser and O. Gromenko for many
and writing ∂Pˆ(z;l)/∂z ≡Fˆ(z;l), we have
useful discussions. We are grateful to the Israel Science
Foundation, to the Adams Fellowship Program of the
2l−1
Fˆ(z;l)=A k−α[zFˆ(z;|l−k|)+Pˆ(z;|l−k|)] IsraelAcademyofSciences andHumanities (SC), andto
theNSF,awardPHY-0555312(DbA),forpartialsupport
k=1
X (15) of this work.
2l−1
+ 1−A k−α [zFˆ(z;l−1)+Pˆ(z;l−1)].
(cid:16) kX=1 (cid:17)
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