Exact Solution for Optimal Navigation with Total Cost Restriction

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by  Yong Li

epl draft
Exact Solution for Optimal Navigation with Total Cost Restriction
Y. Li, D. Zhou, Y. Hu(a), J. Zhang and Z. Di
Department of Systems Science, School of Management, Beijing Normal University - Beijing 100875, PRC
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1 PACS 89.75.Hc–Networks and genealogical trees
0 PACS 89.75.Fb–Structuresand organization in complex systems
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n Abstract.-Recently,Liet al. haveconcentratedonKleinberg’snavigationmodelwithacertain
a total length constraint Λ=cN, where N is the numberof total nodes and c is a constant. Their
J
simulation results for the 1- and 2-dimensional cases indicate that the optimal choice for adding

5 extralong-rangeconnectionsbetweenanytwositesseemstobeα=d+1,wheredisthedimension
of the lattice and α is the power-law exponent. In this paper, we prove analytically that for the
] 1-dimensional large networks, the optimal power-law exponent is α = 2 Further, we study the
h
p impactofthenetworksizeandprovideexactsolutionsfortimecostasafunctionofthepower-law
- exponentα. Wealsoshowthatouranalyticalresultsareinexcellentagreementwithsimulations.
c
o
s
.
s
c
i
s Introduction. – Since Milgram and his cooperators find that the network has the smallest average shortest
y
conducted the first small-world experiment in the 1960s, pathwhenα=2. Morerecently,Liet al. haveconsidered
h
muchattentionhasbeendedicatedtothe problemofnav- Kleinberg’s navigation model with a total cost constraint
p
[igationinrealsocialnetworks. Thefirstnavigationmodel [6]. Intheirmodel, the totallengthofthe long-rangecon-

was proposed by Kleinberg [1]. He employed an L L nections isrestrictedto Λ=c N,whereN representsthe
4 × ·
squarelattice,whereinadditiontothelinksbetweennear- total number of nodes in the lattice based network and c
v
1estneighborseachnodeiwasconnectedtoarandomnode is a positive constant. Their results show that the best
8j withaprobabilityPij ∝ri−jα (rij denotesthelatticedis- transportcondition(minimalnumber ofsteps to reachthe
2tance between nodes i and j). Kleinberg has proved that target) is obtained with a power-law exponent α = d+1
1when α = d, where d is the dimension of the lattice, the forconstrainednavigationinad-dimensionallatticeinthe
.
7optimal time cost of navigation with a decentralized al- 1 and 2-dimensional cases. In this paper, we give a rig-
0gorithm is at most O(log2L). The optimal case indicates orous theoretical analysis of the optimal condition α = 2
0
that individuals are able to find short paths effectively for navigation and provide the exact time cost for vari-
1
withonlylocalinformation,whichcanexplainsixdegrees ouspower-lawexponentsαonthe1-dimensionalcostcon-
:
vof separation quite well. In recent years, further stud- strained network.
i
Xies on Kleinberg’s navigation model have been developed
[2–8]. Roberson et al. study the navigation problem in Dynamical Equations for One-Dimensional Nav-
r
afractal small-world networks [2], where they prove that igation with Cost Restriction. – We consider the
α=d is also the optimal power-law exponent in the frac- one-dimensional navigation problem on a cycle with N =
tal case. Cartozo et al. use dynamical equations to study 4n nodes. For the sake of simplicity, we assume that each
the process of Kleinberg navigation [3,4]. They provide nodehasonlytwoshort-rangeconnectionstoitstwonear-
an exact solution for the asymptotic behavior of such a est neighbors, and the probability of having a long-range
greedy algorithm as a function of the dimension d of the connection satisfies to a certain power-law distribution.
lattice and the power-law exponent α. Yang et al. con- Obviously,the largestpossible lengthofa long-rangecon-
struct a network with limited cost Λ = C. The limited nectionis2ninthiscycle. Wenumberallnodesinclusively
cost Λ represents the total length of the long-range con- from 0 to 4n 1 and assume that the navigation process
−
nections which are added with power-law distance distri- starts from the node 0 and ends at the node n for further
bution P(r) = ar α [5] in one-dimensional space. They simplification. The network is illustrated in FIG.1.
−
According to the discussion above, the probability of
(a)E-mail: [email protected] a long-range connection between any given pair of nodes
p-1
Y. Li et al.
E(J ). Then we have
s
n s 2n 2s 1
− − −
E(J )=λe λ E(Lα)+1 λe λ [ p(r,α)+ p(r,α)]
s − · s − − ·
r=1 r=n s+1
X X−
(3)
For simplicity, we consider the continuous form of all
equations provided above. Then λ can be written as
c 2 , α=0
2n+1
cln2n, α=1
λ=cc1ααl−n2−n22211nn·,((22nn))21−−αα−11, eαls=e.2 (4)
Fstiagr.ti1n:gTnohdee,naavnidgantoiodne nmoisdtehleintatrhgiest.paper. Node 0 is the Whenthenetworksize−islargeen−ough,Eq.(4)canbesim-
plified to
with a distance r is 0, 0 α 2
λ= ≤ ≤ (5)
r α (cαα−12, else.
p(r,α)= − ,α 0, (1) −
2n ≥ The method of dynamical equations is used to deduce
2 r α
− the searching time with limited total cost. Suppose that
r=1
P at time t, the corresponding position is s(t). Obviously,
where α is the power exponent of the power-law distribu- s(0)=0holds. Thedynamicalequationcanbewrittenas
tion. The expected length of the long-range connection
2n ds =E(J ),
from any node satisfies E(Lα) = 2 r p(r,α). To be dt s (6)
· s(0)=0.
r=1 (cid:26)
consistentwithLi’swork[6],inthispPaperwesetthetotal
costlimit to Λ=c 4n,where c is a positive constantand BeforesolvingEq.(6),wefirststudythe optimalpower-
·
4n is the number of nodes on the cycle. Subject to this law exponent α by comparing E(J ) (Eq.(3)) under dif-
s
limit, the expected number of long-range connections on ferent values of α. We rewrite the distance to the target
the whole cycle can be written as E(N )= Λ . n s as εn, where 0 < ε 1 is a constant. For any
α E(Lα) − ≤
Since all nodes are homogeneous, we know that the given 0<ε 1, we assume εn is large enough, such that
≤
numberofdirectedlong-rangeconnectionsfromeachnode long-rangeconnections will be used in the search process.
should obey the Poisson distribution with a parameter Finally, Eq.(3) can be simplified to the following forms,
λ = E(Nα). Thus, the probability of a one long-range
connect4ionn from each node can be given by λe λ. When 1ε2c+1, α=0
− 4
λ is small enough, the probability of the existence two or ln2εc+1, α=1
2
mcra,ontrhebeetrheiagnanrotewreoodn.lloyMngto-wrreaonongvoeedrc,eosfnonwrehacitcgiohivnessnaftonirsofaydeatrahbneitdrcaodrniysdtniatoniodcnee E(Js)∼21e2−c(αc+αα−−1112),c+1, αα>=22 (7)
toShnoa,wtifhtλhicehisdttihsoteoalnleacnreggstehb,ewotefwlceoaennngn-torhatenmcgoncasontndrnutechctetaiogsnivpseaintsianploondweeetrwbeloarrwk. It is not difficultto22α(sα−h−−1o−1w)1εt2h−eαcri+gh1t,sideleseof the Eq.(7)
distribution. According to the above two reasons, in this monotonically increases with α for 0 α 2 and de-
≤ ≤
paper we only consider the case where navigation process creaseswith α for α>2. We canalso provethat E(Js) is
is carried out by at most one long-range connection (c is continuous at the point α=2. Overall, E(Js) is continu-
small) for each node. ous and reaches its maximal value at α=2 for any given
If we use E(Lα) to denote the expected distance by a ε. Ithasbeenrevealedthattheoptimalconditionfornav-
s
long-range connection from a node s toward the target igation with limited cost is a tradeoff between the length
node n, then we have and the number of long-range connections added to the
cycle. Prior to solving the dynamical equations theoreti-
n s 2n 2s 1 cally, we have already shown that the optimal power-law
− − −
E(Lαs)= r·p(r,α)+ (2n−2s−r)·p(r,α). exponent is α=2 with some proper simplifications.
r=1 r=n s+1
X X− (2) Results.– Asdiscussedabove,theoptimalpower-law
We further denote the expecteddistance by anedge (long exponent for navigation in a 1-dimensional large network
orshortrange)fromanodes towardthe targetnodenas is α=2. FIG.2 representsthe size effect onE(J ). It can
s
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Exact Solution for Optimal Navigation with Total Cost Restriction
1.5 1
1.45
0.95
1.4
1.35 0.9
)s c=0.5
J 1.3
( n
E /0.85
T
1.25
1.2 0.8
c=1
1.15
0.75
1.1
0 0.5 1 1.5 2 2.5 3
α

0 0.5 1 1.5 2 2.5 3
α
Fig. 2: Effects of the network size n on the numerical results.
The network size is 10β, and β is chosen from 3 to 10 from
Fig. 3: Comparison between numerical results and simulation
bottomtotop. Itcanbeseenthattheoptimalchoiceofαgets
resultsforc=0.5andc=1respectively. Thecurvesrepresent
closer to α=2 as n increases.
the numerical results while the squares denote the simulation
results with n = 10000. It is shown that our numerical so-
be found that α = 2 is the optimal power-law exponent lutions are consistent with the simulation results and both of
them havetheoptimal valueat α=2 approximately.
when the network size goes to infinity.
Toobtainexacttimecostofnavigationwithcostrestric-
tion,Eq.(6)shouldbesolved. Inthefollowing,wewillfirst
The above results suggest that the relationship between
give its numerical results and then derive its exact solu-
the exact time cost and the distance is linear for most
tions for various values of α. The Ronge-Kutta method
values of α. Meanwhile,we havestudied the size effect on
has been introduced to solve the dynamical equation nu-
navigationwithcostconstraint. BasedonEq.(4),weknow
mericallyandthe resultsarepresentedinFIG.3. Itshows
thatλapproachesitslimitmuchmoreslowerwhenαgets
that the optimal power-law exponent is α = 2. To check
closerto2asnincreases. Thetimecostofnavigationwith
up our method, we also perform search experiments on
different network sizes are provided in FIG.4. It can be
the one-dimensional cycle. The comparison between our
verifiedthatitwillapproachitslimit asngoestoinfinity,
analytical results and the simulation results with c = 0.5
which is given by Eq.(10).
and c=1 are given by FIG.3. As can be seen, they agree
In summary, we constructed a the dynamical equation
quitewellandbothofthemobtaintheoptimalnavigation
for the 1-dimensionalnavigationwith limited cost. Based
at α=2.
on the equation, we proved that for large networks and
It is able to get the exact solutions of navigation with
comparativelysmallcost the optimalpower-lawexponent
limited cost for various values of α. For instance, the
is α=2. Our analytical results confirm the previous sim-
dynamical equation in the case α=0 is,
ulations [6] well.
Acknowledgement.We wish to thank Prof. Shlomo
ds c(n s 1)2
= − − +1. (8) Havlinforsomeusefuldiscussionsandtwoanonymousref-
dt 4n2
erees for their helpful suggestions. This work is partially
Based on the initial condition s(0) = 0, we can get the supported by the Fundamental Research Funds for the
exact solution of Eq.(8). CentralUniversitiesandNSFCunderGrantNo. 70771011
Here, we use T to denote the time cost for getting the and 60974084. Y. Hu is supported by Scientific Research
destination node n. We should have T = 2n arctan√c FoundationandExcellentPh.DProjectofBeijingNormal
√c 2
University.
for large enough n. Thus, the average required time to
navigate from the source node to the target satisfies
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T 2 √c
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p-3
Y. Li et al.
0.95
0.9
0.85
n
/ 0.8
T
0.75
0.7
0.65
0 0.5 1 1.5 2 2.5 3
α
Fig. 4: Size effect on navigation. The exact solutions are ob-
tained on the 1-dimensional cycle with parameter c= 1. The
network size n is 10β, and parameter β values of the upper
three solid curves are chosen from 3 to 5 from top to bottom.
Itisshown thatα=2isalways theoptimalchoice forvarious
values of n. The bottom line, obtained when n goes infinity,
represents the exact solutions of the navigation process with
limited cost. Notice that we only provide the exact solutions
atα=0andα=1whenα<2,thusweonlyconnect thetwo
exact points with dashed lines.
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leyH.E.andAndrade,Jr.J.S.,Phys.Rev.Lett.,104
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