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Navigationinnon-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. 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