Table Of ContentNavigationinnon-uniform density socialnetworks
Yanqing Hu, Yong Li, Zengru Di, Ying Fan∗
Department of Systems Science, School of Management and Center for
Complexity Research, Beijing Normal University, Beijing 100875, China
(Dated:January6,2011)
Recentempiricalinvestigationssuggestauniversalscalinglawforthespatialstructureofsocialnetworks.Itis
foundthattheprobabilitydensitydistributionofanindividualtohaveafriendatdistancedscalesasP(d)∝d−1.
Since population density is non-uniform in real social networks, a scale invariant friendship network(SIFN)
basedontheaboveempiricallawisintroducedtocapturethisphenomenon. Weprovethetimecomplexityof
navigationin2-dimensionalSIFNisatmostO(log4n).Intherealsearchingexperiment,individualsoftenresort
1 toextrainformationbesidesgeographylocation.Thus,real-worldsearchingprocessmaybeseenasaprojection
1 ofnavigationinak-dimensionalSIFN(k>2).Therefore,wealsodiscusstherelationshipbetweenhighandlow
0 dimensionalSIFN.Particularly,weprovea2-dimensionalSIFNistheprojectionofa3-dimensionalSIFN.As
2
amatteroffact,thisresultcanalsobegeneratedtoanyk-dimensionalSIFN.
n
a PACSnumbers:
J
5
I. INTRODUCTION operators have provided exact solutions respectively for the
] asymptoticbehaviorofKleinberg’snavigationmodel[20,21].
h More recently, the navigation probolem with a total cost re-
p Tounderstandthestructureofthesocialnetworksinwhich
striction has also been discussed, where the cost denotesthe
- we live is a very interesting problem. As part of the re-
c lengthofthelong-rangeconnections[22,23].
cent surge of interest in networks, there have been active
o
research about social networks[1–6]. Besides some well Meanwhile, recent empirical investigations suggest a uni-
s
. knowncommonpropertiessuchas small-worldand commu- versal spatial scaling law on social networks. Liben-Nowell
s
c nitystructure[7–9],muchattentionhasbeendedicatedtonav- et al explored the role of geography alone in routing mes-
i igationinrealsocialnetworks. sageswithintheLiveJournalsocialnetwork[24]. Theyfound
s
y In the 1960s, Milgram and his co-workers conducted the thattheprobabilitydensityfunction(PDF)ofgeographicdis-
h firstsmall-worldexperiment[10]. Randomlychosenindivid- tance d between friendship was about P(d) ∝ d−1. Adamic
p and Ada also observed the P(d) ∝ d−1 law when investigat-
ualsintheUnitedStateswereaskedtosendalettertoapartic-
[ ingtheHewlett-PackardLabsemailnetwork[25]. Lambiotte
ularrecipientusingonlyfriendsoracquaintances.Theresults
2 oftheexperimentrevealthattheaveragenumberofinterme- et al analyzed the statistical properties of a communication
v diatestepsinasuccessfulchainisaboutsix. Sincethen,“six networkconstructedfromtherecordsofamobilephonecom-
5 pany [26]. Their empiricalresults showed that the probabil-
degreesofseparation”hasbecamethesubjectofbothexper-
4 ity that two people u and v living at a geographic distance
imentalandtheoreticalresearch[11, 12]. Recently, Doddset
8 d(u,v)wereconnectedbyalinkwasproportionaltod(u,v)−2.
alcarriedoutanexperimentstudyinaglobalsocialnetwork
1
Becausethenumberofnodeshavingdistanced toanygiven
. consistingabout60,000emailusers[13]. Theyestimatedthat
0 nodeisproportionaltodin2-dimensionalworld,sotheprob-
social navigation can reach their targets in a median of five
1 abilityforanindividualtohavea friendatdistanced should
to seven steps, which is similar to the results of Milgram’s
0 be P(d) ∝ d · d−2 = d−1. More recently, Goldenberg et al
1 experiment.
studied the effect of IT revolutionon social interactions[27].
: The first theoretical navigation model was proposed by
v ThroughanalyzinganextensivedatasetoftheFacebookon-
Kleinberg[14,15]. Heintroducedann×nlatticetomodelso-
Xi cialnetworks. Inadditiontothelinksbetweennearestneigh- linesocialnetwork,theypointedoutthatsocialcommunica-
tiondecreaseinverselywiththedistancedfollowingthescal-
r bors, each node u is connected to a random node v with a
a probabilityproportionaltod(u,v)−r,whered(u,v)denotesthe inglawP(d)∝d−1aswell.
lattice distance between u and v. Kleinberg has proved that SuchasintheLiveJournalsocialnetwork,populationden-
the optimal navigation can be obtained when the power-law sityisnon-uniforminrealsocialnetworks[24]. Todealwith
exponentr equals to d, where d is the dimensionalityof the the navigation problem with non-uniform population den-
lattice, and the time complexityof navigation in that case is sity, a scale invariant friendship network (SIFN for short)
at most O(log2n). Since then, muchattention has been ded- model based on the above spatial scaling law P(d) ∝ d−1
icated to Kleinberg’snavigationmodel[16–18]. Robersonet of social networks is proposed in this paper. We prove the
al. studiedthenavigationprobleminfractalnetworks,where time complexityof navigationin a 2-dimensionalSIFN is at
they proved that r = d was also the optimal power-law ex- most O(log4n), which indicates social networks is naviga-
ponentin the fractal case[19]. Carmi, Cartozo and their co- ble. Dodds et al have pointed out that individuals often re-
sort to extra information such as education and professional
informationbesides geographylocationin the realsearching
experiment[13]. Considering this phenomenon, navigation
∗[email protected] process in real world may be seen as the projection of nav-
2
igation in a higher dimensionalSIFN. Therefore, we further
discuss the relationship between high and low dimensional
SIFN. Particularly, we prove that a 2-dimensionalSIFN can
be seen as the projection of any k-dimensionalSIFN(k > 2)
throughtheoreticalanalysis.
II. NAVIGATIONINNON-UNIFORMDENSITYSOCIAL
NETWORKS
Todealwiththenon-uniformpopulationdensityinrealso-
cialnetworks,wedividethewholepopulationintosmallareas
andgivethefollowingtwoassumptions. First,thepopulation
densityisuniformin eachsmallarea. Second,theminimum
populationdensityamongtheareasism,whilethemaximum
is M. We set m > 0 to guaranteethat a searchingalgorithm FIG.1:Twostrategiesofsendingmessageina2-dimensionalSIFN.
Strategy A, send the message directly to target t from the current
can always make some progress toward any target at every
messageholder usingKleinberg’sgreedyroutingstrategy. Ateach
stepofthechain.
step,themessageissenttooneofitsneighborswhoismostcloseto
Like Kleinberg’s network (KN for short) and Liben-
thetargetinthesenseoflatticedistance. StrategyB,themessageis
Nowell’srank-basedfriendshipnetwork(RFN forshort), we
firstsenttoagivennode jusingKleinberg’sgreedystrategyandthen
employann×nlatticetoconstructSIFN.Withoutlossofgen-
tothetargetnodetusingthesamestrategy.Supposewestartfroma
erality,weassumeeachnodeuhasqdirectedlong-rangecon- sourcenode s,afteronestep,themessagereachesnodes A and B
1 1
nections,whereqisaconstant[15]. Togeneratealong-range respectivelywithstrategyAandB. Consider B asthenewsource
1
connectionof nodeu, we first randomlychoosea distance d node, thenweshould get A and B respectively withstrategiesA
2 2
according to the observed scaling law P(d) ∝ d−1 in social andBinthenextstep.
networks. Thenrandomlychooseanodevfromthenodeset,
whose elements have the same lattice distance d to node u,
andcreateadirectedlong-rangeconnectionfromutov. The in social networks compared with KN and RFN. Further,
latticeisassumedtobelargeenoughthatthelong-rangecon- Pr (u,v,k),Pr (u,v) and Pr (u,v) can be quite differ-
KN SIFN RFN
nectionswillnotoverlap. entwhenthepopulationdensityisnon-uniform.
For simplicity, we set q = 1. Let S denote the set of all
Since our 2-dimensional SIFN captures the non-uniform
nodes, then the probability that u chooses v as its long-rang
population density property in the real social networks, we
connectioninSIFNcanbegivenbyeq.(1).
purposefullydividethenavigationprocessintotwostagesfor
simplicity. First send messages inside a small area and then
1 d(u,v)−1
Pr (u,v)= (1) amongthe areas. To analyze the time complexityof naviga-
SIFN c(u,v) n d−1
d=1 tion in a 2-dimensional SIFN, we first compare the follow-
where c(u,v) = |{x|d(u,x) = d(u,v),Px ∈ S}| and d(u,v) de- ing two searchingstrategies as shownin FIG.1. StrategyA,
send the message directly to target t from the current mes-
notes the lattice distance between nodes u and v. Likewise,
sageholderusingKleinberg’sgreedyroutingstrategy.Ateach
theprobabilitythatuchoosesvasitslong-rangconnectionin
step, themessageissentto oneofitsneighborswhoismost
KNandRFNaregivenrespectivelybyeq.(2)andeq.(3).
close to the target in the sense of lattice distance. Strategy
d(u,v)−r B, the message is first sent to a given node j using Klein-
PrKN(u,v,r)= d(u,w)−r (2) berg’sgreedystrategyandthentothetargetnodet usingthe
w,u samestrategy.ItcanbeprovedthatstrategyAperformsbetter
P
thanstrategyBonaverage.Supposewestartsendingmessage
PrRFN(u,v)= ranrkaun(kv)(−w1)−1 (3) rferospmecthtievesloyurwceithnosdtreaste,gthyeAmeasnsdagBeraefatcehreosnneosdteespA. 1ItanisdaBl1-
w,u u ways correctthat lattice distance d(A ,t) ≤ d(B ,t), because
P 1 1
where rank (v) = |{w|d(u,w) < d(u,v),x ∈ S}| denotes greedyroutingstrategyalwayschoosethenodemostcloseto
u
the number of nodes within distance d(u,v) to node u in targettfromitsneighbors.Accordingtotheresultsof[18,21],
RFN[15, 24]. Notice that, the number of nodes with a dis- the longer the distance between a source and a given tar-
tanced(u,v)inak-dimensional(k > 1)latticeisproportional get, the more is the expected steps. Thus we should have
to d(u,v)k−1. Thus, a node u connectsto nodev with proba- T(A1 → t) ≤ T(B1 → t), where T(A1 → t) and T(B1 → t)
bilityproportionaltod(u,v)−a doesnotmean P(d) ∝ d−a but denotetheexpecteddeliverytime to targett from A1 and B1
P(d) ∝ d−a+k−1 instead. Therefore, Pr (u,v,k), Pr (u,v) respectively.
KN SIFN
andPr (u,v)areexactlythesameforanyk-dimensionallat- LetT(s → j → t)denotetheexpecteddeliverytimefrom
RFN
ticebasednetworkwhenpopulationdensityisuniform.How- stotviaatransportnode j,thenwehaveT(s→ t) ≤ T(s→
ever, SIFN always satisfies the empirical results P(d) ∝ d−1 B →t).ConsiderB asanewsourcenode,thenmessagewill
1 1
3
reach A and B with strategies A and B respectivelyin the complexityof navigationin SIFN with strategy B is at most
2 2
next step. Following the same deduction, we have T(B → O(M log4n).However,actualnavigationprocessinrealworld
1 m
A → t) ≤ T(B → B → t). Repeat this process until shouldbecarriedoutregardlessoftheabovetwoassumptions,
2 1 2
the message reaches the given node j with strategy B, then whichindicatesindividualsshouldusestrategyA. Basedon
weshouldhaveamonotoneincreasingsequenceofexpected the above analysis, strategy A performs better than strategy
deliverytime{T(s → B → t),T(s → B → t),···,T(s → B on average. Therefore, the time complexityof navigation
1 2
j → t) }. Therefore,we can obtainT(s → t) ≤ T(s → j → in2-dimensionalSIFNisatmostO(log4n)withnon-uniform
t), which means strategy A is better than strategy B. This populationdensity.
analysiscanbeextendedtoanyk-dimensionalSIFN.
BasedonthefirstassumptionandthefactthatSIFNisiden-
ticaltoKNwhenpopulationdensityisuniform,theexpected III. RELATIONSHIPBETWEENHIGHANDLOW
steps spent in each small area using Kleinberg greedy algo- DIMENSIONALSIFN
rithmisatmostO(log2n).Considereachsmallareaasanode,
wewillgetanew2-dimensionalweightedlattice. Theweight Theempiricalresultsshowindividualsalwaysresorttoex-
(population)of the nodes is between m and M based on the tra information such as profession and education informa-
secondassumption.Thuswehave tion besides the target’s geography location when routing
messages[13]. Then, real navigation process in social net-
m M
c d−1 ≤Pr′ (u,v)≤c d−1 (4) worksmaybemodeledwithahigherdimensionalSIFN.Inthe
M SIFN m following, we will discuss the relationship between the high
and low dimensional SIFN and prove that a 2-dimensional
where c is a constant and Pr′ (u,v) represents the proba-
SIFN SIFN can be obtained by any k-dimensional SIFN (k > 2).
bility that area u is connected to area v in the new weighted
Particularly,wewillprovidethetheoreticanalysisforthecase
lattice. wherek = 3. Theanalysiscanbegeneratedtoanyk dimen-
We say that the executionof greedyalgorithm is in phase
sionalcases.
j (j > 0) when the lattice distance from the currentnode to
WeemployearandomvariableD todenotethefriendship
target t is greater than 2j and at most 2j+1. Obviously, we 3
distanceina3-dimensionalSIFN.Forsimplicity,acontinuous
have
expressionsis used. Since, the long-rangeconnectionsin 3-
n dimensionalSIFNsatisfiestheaboveempiricallaw,thePDF
n
d−1 ≤1+ x−1dx=1+logn<2logn. (5) ofD3canbeexpressedby
Z
Xd=1 1 P(D =d)= 1 1,d ≤d≤d (9)
3 lnd −lnd d m M
Further, we define B as the node set whose elements are M m
j
within lattice distance 2j +2j+1 < 2j+2 to u. Let|B | denote whered andd denotetheminimumandmaximumdistance
j m M
thenumberofnodesinsetB ,weshouldhave respectivelyinthe3-dimensionalSIFN.
j
Wecanobtaina2-dimensionalnetworkmodelifweproject
2j a3-dimensionalSIFNtoa2-dimensionalworld. Similarly,a
|Bj|>1+ i>22j−1. (6) randomvariable D2 isusedto denotethefriendshipdistance
Xi=1 inthenew2-dimensionalnetworkmodel. Itisnotdifficultto
understandthattheconditionfora2-dimensionalSIFNshould
Supposethatthemessageholderiscurrentlyinphase j,then
be the PDF of D satisfies P(d) ∝ d−1. Since D is the pro-
theprobabilitythatthenodeisconnectedbyalong-rangelink 2 2
toanodeinphase j−1isatleast(Mm−12logn·4·22j+4)−1. jection of D3, then D2 can be seen as the productof D3 and
X. HererandomvariableXisindependentonD anditsPDF
Theprobabilityψ(x)toreachthenextphase j−1inmorethan 3
canbegivenbyeq.(10).
xstepscanbegivenby
ψ(x)=(1−(Mm−12logn·4·22j+4)−1)x (7) 1
P(X = x)= ,0≤ x≤λ (10)
λ
andtheaveragenumberofstepsrequiredtoreachphase j−1
where0≤λ≤1. Finally,thePDFofD canbewrittenas
is 2
∞
m i−1 256Mlogn
< x>= (1− ) = . (8) 0, d≤0
tOSoi(tnaMmclelnotuhgme2bnine)i.rtioaflXi=vs1taelpuseroef2q5ju6iirsMeadtlotmogonrsetalcohgtnh,ethtaernmgethteisexaptemctoesdt P(D2 =d)=0dlnM,dd1dM−m−λddMl(1nlMλnd−dmdM,m−lndm), 0ddm<>λdd<M≤dλd≤mλdMλ (11)
As a matter of fact, it means that we are using strategy B When taking account of real social networks, d is large
M
tosendmessagein2-dimensionalSIFNwhenthenavigation enoughthattheterm 1 willapproachitslimitof0. Mean-
process is divided into the above 2 stages. Thus, the time while, the term d λ cdaMnλ be neglected when compared with
m
4
d λ,becauseλ ≤ 1andd isrelativelysmall. ThusthePDF tion density. It has been proved that the time complexity of
M m
ofD canbesimplifiedintoP(d) ∝ d−1,whichisidenticalto navigationin2-dimensionalSIFNisatmostO(log4n),which
2
thatofD ina3-dimensionalSIFN. correspondstotheupperbondofnavigationinrealsocialnet-
3
Through theoretical analysis, we have proved a 2- works.Giventhefactthatindividualsarealwaysrestrictedon
dimensional SIFN can be seen as the projection of a 3- the 2-dimensionalgeographyworld even they possess infor-
dimensional SIFN. Likewise, we can get a 2-dimensional mationofthehigherdimensions,actualsearchingprocesscan
SIFN from any k-dimensional(k > 2) SIFN. Notice that in- beseenasaprojectionofnavigationinahigherk-dimensional
dividualsare always restricted on the 2-dimensionalgeogra- SIFN.Throughtheoreticalanalysis,weprovethattheprojec-
phyworldeventheypossessextrainformationfromotherdi- tionofahigherk-dimensionalSIFNresultsina2-dimensional
mensions.Thus,real-worldsearchingprocessmaybeseenas SIFN.Therefore,SIFNmodelmayexplainthenavigabilityof
theprojectionofnavigationinahighdimensionalSIFN.Our realsocialnetworkseventakeaccountoftheinformationfrom
analysisindicatethatSIFNmodelmayexplainthenavigabil- higherdimensionsotherthangeographydimensions.
ity of real social networkseven take accountof the fact that
individualsalways resortto extra informationin real search-
ingexperiments.
V. ACKNOWLEDGEMENT
IV. CONCLUSION WethankProf.ShlomoHavlinforsomeusefuldiscussions.
ThisworkispartiallysupportedbytheFundamentalResearch
Recentinvestigationssuggestthatthe probabilitydistribu- Funds for the Central Universities and NSFC under Grants
tion of having a friend at distance d scales as P(d) ∝ d−1. No.70771011and No. 60974084and NCET-09-0228. Yan-
We propose an SIFN model based on this spatial property qing Hu is supportedby Scientific Research Foundationand
to deal with navigation problem with non-uniform popula- ExcellentPh.DProjectofBeijingNormalUniversity.
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