Modeling Spatial Social Complex Networks for Dynamical ...downloads.hindawi.com/journals/complexity/2018/1428719.pdf · Modeling Spatial Social Complex Networks for Dynamical Processes

Research ArticleModeling Spatial Social Complex Networks forDynamical Processes

Shandeepa Wickramasinghe12 Onyekachukwu Onyerikwu3

Jie Sun 1234 and Daniel ben-Avraham124

1Clarkson Center for Complex Systems Science (C3S2) Potsdam NY 13699 USA2Department of Mathematics Clarkson University Potsdam NY 13699 USA3Department of Computer Science Clarkson University Potsdam NY 13699 USA4Department of Physics Clarkson University Potsdam NY 13699 USA

Correspondence should be addressed to Jie Sun sunjclarksonedu

Received 26 August 2017 Revised 7 December 2017 Accepted 25 December 2017 Published 13 February 2018

Academic Editor Albert Diaz-Guilera

Copyright copy 2018 Shandeepa Wickramasinghe et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The study of social networksmdashwhere people are located geographically and how they might be connected to one anothermdashis acurrent hot topic of interest because of its immediate relevance to important applications from devising efficient immunizationtechniques for the arrest of epidemics to the design of better transportation and city planning paradigms to the understandingof how rumors and opinions spread and take shape over time We develop a Spatial Social Complex Network (SSCN) model thatcaptures not only essential connectivity features of real-life social networks including a heavy-tailed degree distribution and highclustering but also the spatial location of individuals reproducing Zipf rsquos law for the distribution of city populations as well asother observed hallmarks We then simulate Milgramrsquos Small-World experiment on our SSCN model obtaining good qualitativeagreement with the known results and shedding light on the role played by various network attributes and the strategies used by theplayers in the game This demonstrates the potential of the SSCN model for the simulation and study of the many social processesmentioned above where both connectivity and geography play a role in the dynamics

1 Introduction

Much research has focused in recent years on a wide class ofdynamical processes that take place in large human popula-tions at the scale of cities whole countries and even world-wide Examples include epidemics spreading and strategies toarrest their spread [1ndash3] the evolution of the electoral mapduring elections [4] the spreading of rumors [5] memes[6 7] and opinions [8] the migration patterns of banknotes[9] and human populations [10] and the effects of citiesand infrastructure layouts on commerce and productivity[11 12] Many of these questions require specific knowledgeof individualsrsquo geographical location as well as their socialcontacts (many infections propagate by direct contact orphysical proximity we discuss and influence the opinions ofmostly those close to us etc)

In Milgramrsquos Small-World experiment [13] for exampleparticipants were asked to pass a message (a postcard) to aperson in a disclosed address but only through a chain ofsocial acquaintances each participant was allowed to pass themessage only to a person they know on a first-name basis Of160 messages started in Omaha Nebraska 44 or about 28reached the target in Boston Massachusetts with an averagepath length of about 54 links How does the message find itsway let alone in such a short numbers of steps

Kleinbergrsquos seminal work [14 15] for nodes in a squarelattice with random long-range connections provided a firstclue This was later extended to fractal [16] and anisotropic[17] latticesmdashstill a far cry however from the geographicalspread and network of connections typical of human societyRybski et al recently developed a spatial growth model forcities but such a model (and similar ones in other related

HindawiComplexityVolume 2018 Article ID 1428719 12 pageshttpsdoiorg10115520181428719

2 Complexity

studies) does not capture the social network of interactions[18] Dodds et al conducted a large-scale online experimentthat resembles Milgramrsquos original study highlighting the roleof information beyond just network structure [19] Liben-Nowell et al [20] proposed a spatial social network modelwith connections derived from an online bloggers com-munity and studied greedy routing on that model Similarstudies were conducted for online social networks [21] andcommunity structures from mobile phone records [22 23](see [24 25] for a more comprehensive review) Informationon peoplersquos location along with their social contacts isgenerally hard to come by and often relies on indirectproxies

In [26]we introduced a stochastic prototype Spatial SocialComplex Network (SSCN) relying on just two controllableparameters that simulates large populations including thelocations and the complex network of contacts betweenagents This ldquobaseline modelrdquo was designed with modestgoals in mind (a) The population density resembles the lightdensity observed in satellite pictures of earth at night (b) thepopulation of ldquocitiesrdquo (defined by percolation clusters [27])and their rankings follow Zipf rsquos law [28ndash30] (c) the socialnetwork of contacts exhibits a scale-free distribution and (d)highly connected nodes tend to be located in denser andlarger population areas In addition to meeting these basicgoals the SSCN baseline model also yielded good qualitativeagreement with census data for the population density as afunction of city size and for the weak super-linear depen-dence of the cumulative degree of nodes in a city on its totalpopulation as suggested from cell-phone data [12] Finally itallowed us to shed some light on the weak deviations [27]from Gibratrsquos law (that the rate of growth of a city and itsfluctuations are proportional to its population size)

Despite these initial successes the SSCN baseline modelfails to mimic real SSCNs in some crucial ways (i) The com-plex network of social contacts while displaying a realisticscale-free degree distribution is actually a tree in contrastwith the high degree of clustering observed in social netsand their proxies (ii)The network of contacts is built througha redirection mechanism [31ndash33] which is an adequatedescription of how individuals might join a social networkbut fails to account for the effect of relocations every so oftena person relocates to a far-away destination for study jobor other reasons This creates particular correlations in thenetwork of social contacts that are absent in the baselineSSCN

In this paper we resolve the baseline SSCN deficiencieswith some simple adjustments Connections to spatiallyclosest neighbors are added to mimic the clustering effect inreal social networks and relocations turn out to be crucial inreducing the average path length between nodes Simulationsof Milgramrsquos Small-World experiment on the revised SSCNmodel achieve a good qualitative fit with the empirical find-ings This demonstrates the suitability of the model as a sub-strate for simulations of other dynamic social processes thatdepend on both the contacts and the geographical locationsof the agents

2 Materials and Methods

21 The Baseline SSCN Model We now review the originalor ldquobaselinerdquo SSCN model established in our previous work[26] The model produces a spatially embedded network 119866 =(119881 119864119883) where 119881 = 1 2 119873 is a set of nodes and 119864 sub119881 times 119881 is a set of undirected edges The spatial embedding ofthe network is encoded in the set of coordinates 119883 =x(1) x(119873) where for 2D spatial networks (such as in ourcase) x(119894) isin R2 A unique feature of the model is that it notonly produces the requisite scale-free degree distribution forthe edges but also captures essential spatial features such as aZipf distribution of the populations emerging from the nodesclustering into ldquocitiesrdquo [26]

Consider first the creation of nodes and edges in thebaseline model defining 119881 and 119864 The starting point is aninitial ldquoseedrdquo network which in the baseline model consiststypically of a single node Nodes are added to the networkone at a time each contributing to a new edge according to avariant of the Krapivsky-Redner (KR) model [31ndash33] with asingle parameter 119903 isin [0 1]mdashthe redirection probability Eachtime a newnode 119894 joins the network one of the existing nodes119895 is chosen uniformly at random and 119894 is connected to 119895directly with probability 1 minus 119903 (creating a new edge 119894 harr 119895)otherwise with probability 119903 the connection is redirected to arandomly selected neighbor 1198951015840 of 119895 (edge 119894 harr 1198951015840) For large119873this leads to a scale-free degree distribution (in the originalKRmodel connections are redirected to the ancestor of 119895mdashthenode 119895 connected to upon joining the network our variantyields 120574 ≳ 120574KR = 1 + 1119903 we find that using the original KRrecipe results in a poorer resemblance with satellite picturesof earth at night furthermore the variant employed in ourmodel is somewhat simpler as there is no need to trackancestors)

119875 (119896) prop 119896minus120574 with 120574 asymp 1 + 1119903 (1)

Consider next the placement of the nodes in spacespecifying 119883 For a network of 119873 nodes the baseline modelplaces them within a square box of sides 119871 = radic119873 (with peri-odic boundary conditions) The initial seed node is placedat the origin x1 = 0 and the location of subsequent nodes119894 depends on whether it connects to node 119895 directly or to aneighbor 1198951015840 by redirection If 119894 joins directly it is placed at(119904 120579) from 119895 (using polar coordinates) where the angle 0 lt120579 le 2120587 is chosen randomly from the uniform distributionand 119904 is picked randomly from the distribution

119901 (119904) =

1ln (119904max)

119904minus1 1 lt 119904 lt 119904max0 otherwise

(2)

where 119904max = radic2119871 is themaximumpossible distance betweenany two points within the bounding square In the case ofredirection when node 119894 joins to 1198951015840 then we simply place 119894 atdistance 1 from 1198951015840 at a random angle 120579The growth algorithmis illustrated in Figure 1(a) Note that the choice of exponentminus1 in (2) is equivalent to 119901(s) sim |s|minus2 which is in line withKleinbergrsquos ldquomagicrdquo formula for optimal navigability

Complexity 3

r (redirection)Probability

j j

i (new node)i (new node)

1 minus r

j

(a) Baseline SSCN redirection mechanism and spatial location

i

q = 2 (for node i) q = 2 (for all nodes)

(b) Connecting to 119902 spatially closest neighbors

j

i

i

Step 1 Relocationmigration(carrying connections to newlocation)

i

Step 2 Connecting toq spatially closest neighbors

q = 2j

(c) Relocation of a node

Figure 1 Growth rules for the baseline (a) and the revised SSCNmodel (b amp c) (a) A new node 119894 joins the network and connects directly to arandomly selected node 119895 with probability 1minus119903 settling away from 119895 according to the rule of (2) (left panel)With probability 119903 the connectionis redirected to a random neighbor 1198951015840 of 119895 and 119894 settles at distance 1 from 1198951015840 (right panel) (b) Befriending 119902 closest neighbors (shown for 119902 = 2)Left panel Node 119894 needs to add a connection to the nearby node on its left in order to fulfill the requirement of connections to at least 119902 nearestneighbors The new link and 119894rsquos 2 nearest friends are highlighted in green Right The process is repeated for all nodes in the network until allfulfill the minimum-119902 requirement The new links added to the baseline model are highlighted in green (c) Relocation of node 119894 happens intwo stages Left In the first stage 119894 is translated to within distance 1 from a randomly selected node 119895 All of 119894rsquos old contacts (broken orangelines) are retained (orange lines) Right In the second stage links are added to ensure connection to at least 119902 new closest neighbors of 119894 (shownfor 119902 = 2) A new link and the 2 closest neighbors are highlighted in green Note that relocation does not alter the connectivity pattern in step1 but the relocating node 119894may acquire up to 119902 new links in step 2

While the above growth rules were ultimately selected tobest achieve the baseline modelrsquos goals they do make someintuitive sense as wellThe redirectionmechanism introducesa ldquorich-get-richerrdquo bias in that redirection favors the randomselection of nodes 1198951015840 of a higher degree This accounts for theemergence of the scale-free degree distribution In additionthe connection and placement rules capture some basicways of life a person 119894 joins an existing social net whenthey are born There is no choice in this matter and thesocial connection(s) established in this case is random (directconnection to node 119895) Eventually 119894 leaves home and settlesat some distant location The distribution of the distance to119894rsquos new home inversely proportional to the distance 119904 ismotivated by Kleinbergrsquos ldquomagicalrdquo condition for navigability[14 15]The other possibility is that 119894rsquos mostmeaningful socialconnection happens through redirection (119894 is referred to aworkplace or school etc) and in that case it makes sense tosettle nearby to the new contact (at distance 1mdashthe minimaldistance in our distance distribution)

The growth rules of the baseline SSCN model seem how-ever too simplistic in that they account for a bare minimumof social connections the connections to onersquos birth place arerepresented by a single link as are also the connections topeople in a referred (redirected) situation While the sparsityof connections can be justified on the grounds that the modelis a scaled-down version of real life (fewer nodes or people sofewer contacts per person) there is no getting around the factthat the baseline model network of connections is a tree incontrast with real-life social nets where clustering is large

(your friends have a higher than average probability to befriends among themselves) Another important effect is thatof relocations occasionally people move to a different placesometimes more than once over the course of their livesWhen people relocate they maintain friendship with someacquaintances in their place of origin and form friendshipswith their new neighbors Thus relocations have a profoundeffect on the network of social contacts In the next sectionwe describe a new version of the baseline SSCN that fixesthese shortcomings

22 A Revised SSCN Model For the present simulationswe use a redirection probability 119903 = 08 same as for thebaseline model This leads to a degree exponent 120574 asymp 23which is typical of large-scale social networks [34 35] Inaddition to the significant changes that we made to themodelrsquos connectivity we made some minor changes to theboundary conditions and to the initial seed and we describethese first

Free Boundary Condition In the baseline model we useda bounding box of side 119871 = radic119873 and periodic boundaryconditions For the present work we adopt a boundary-freeapproach Simply the first node is placed at the origin andeach subsequent node is placed in the same fashion as for thebaselinemodel but without regard to the bounding boxThatis the nodes are allowed to spread as far as the simulationtakes them Our simulations show that even with this free

4 Complexity

Zipf plot

104

103

102

101

City

pop

ulat

ion

100 101 102

Rank

104

102

100 101 102

r = rinfin (baseline)N0 = 10

N0 = 25

N0 = 50

(a)

Spatial layout of nodes

City 1

City 3

City 2

(b)

Figure 2 (a) Effect of1198730 on the distribution of city sizes by rank on a log-log scale The inset highlights the case of1198730 = 25 that we use forour simulations The fitted straight line has slope asymp minus132 (b) Spatial layout of a network of119873 = 51200 nodes generated with1198730 = 25 Forvisual clarity we divide the spatial domain into 200-by-200 equal-size square boxes (cells) and show only the nodes in populated cells whichare those with a population exceeding the average (per nonempty box) The three largest ldquocitiesrdquo are color-coded in red (pop 15072) blue(pop 5567) and green (pop 3743) As noted in the main text the cities were identified by the spatial City Clustering Algorithm as developedin [27] where the populated cells (boxes) are defined as above

boundary condition the radius of gyration scales quite accu-rately as radic119873 so that the average population density per unitarea remains constant even as the model is scaled up

Initial Seed Startingwith a single-node seed as in the baselinemodel tends to produce a few ldquomegacitiesrdquomdashcities that aredisproportionately larger than predicted by the Zipf distribu-tion [28 29] In our analysis the cities are identified usingthe spatial City Clustering Algorithm which was introducedin [25] and used in our baseline SSCN model [24] Themain idea and steps of the City Clustering Algorithm can besummarized as follows First the spatial domain is dividedinto a grid (typically equal-sized squares) where a cell isdetermined to be ldquopopulatedrdquo if the number of nodes in thatcell exceeds a given threshold Then a cell-to-cell graph isconstructed where the nodes are the populated cells and anedge exists between two nodes if the two corresponding cellsare spatially adjacent that is they share a border (diagonalneighbors do not count) Finally the cities are defined andcomputed as the connected components of the cell-to-cellgraph that is for any pair of (populated) cells of a city thereexists a path that connects them on the other hand no suchpath exists between two cells that belong to two differentcities Due to its objective formulation the algorithm enablesidentification of cities directly from spatial population data

In [26] we showed how the problem of megacities mightbe overcome by starting with seeds consisting of severalnodes Here we employ a single-node seed but let the

redirection probability varywith the number of nodes 119894 addedthereafter

119903119894 = (1 minus 119890minus(119894minus1)1198730) 119903infin (3)

The probability 119903119894 converges rapidly to 119903infin (we pick 119903infin = 08)and the parameter 1198730 controls the pace of the convergenceThus for1198730 ≪ 119873 the varying 119903119894 affects mainly the first simN0nodes but not the large-scale structure of the network Onthe other hand the fact that 119903119894 asymp 0 for the first few nodesreduces their capacity to attract further connections therebyalleviating the problem of megacities It is worth noting thatthe choice of the particular form of 119903119894 as in (3) is not crucialany (slowly) increasing function that saturates for large 119894 canin principle be used to achieve the effect of reducing theoccurrence and size of megacities The effect of 1198730 on thedistribution of city sizes is shown in Figure 2(a) In Figure 2(b)we show the spatial layout of a typical network producedwith 1198730 = 25 highlighting in color the first three largestcities This very same configuration is used for the studiesof connectivity and for the simulations of Milgramrsquos Small-World experiment reported below

Closest Neighbors and Clustering We now come to the moreserious revisions of the baseline SSCN model A big issue isthat the baseline modelrsquos network of social contacts is a treeThis means that the probability for two of your friends tobe friends among themselves is zero while in real life thatprobability is in fact much higher than the average density

Complexity 5

1 2 3 4 5 spatial nearest neighbors q

05

055

06

065

07

q = 5q = 10

1minus⟨C

⟩

100

10minus1

10minus2

C(k

)

101 102 103

Degree k

N = 800

N = 3200

N = 12800

N = 51200

Figure 3 Dependence of the average clustering coefficient ⟨119862⟩ on119902 for networks of size 119873 = 800 3200 12800 and 51200 (frombottom to top)The slope of the curves in this log-log plot is roughlyminus02 Inset Clustering coefficient 119862(119896) as a function of node degree119896 for networks of size119873 = 51200 with 119902 = 5 and 119902 = 10 The fittedstraight lines have slope asymp minus075 Each data point in the figures is theresult of an average over 20 independent network generations

of links possibly due to the nature of human social activitiesand interactions [36] Such an effect is best captured by theconcept of clustering [37 38] which for a given node 119894 in anetwork is defined as 119862119894 = 2119891119894[119896119894(119896119894 minus 1)] where 119896119894 is thedegree of node 119894 and 119891119894 is the number of links among theneighbors of 119894 (119862119894 = 0 if 119896119894 le 1) Then the clustering coef-ficient of the entire network is simply the average clusteringcoefficient of all nodes ⟨119862⟩ = (1119873)sum119873119894=1 119862119894

To fix the problem of (low) clustering in the baselinemodel we now require that each node be connected to atleast 119902 of its geographically closest neighbors mimicking thefact that one indeed tends to befriend ldquonext-doorrdquo neighborsNew edges are added in at the end of the growth process Theaddition of new edges is illustrated in Figure 1(b) Note thatthe baseline model corresponds to the special case of 119902 = 0

In Figure 3 we plot the clustering coefficient of thenetwork ⟨119862⟩ as a function of 119902 We see that ⟨119862⟩ is quitelarge and in line with real-life networks already for 119902 = 1⟨119862⟩ grows with 119902 (and decreases with the network size 119873)according to the empirical relation 1minus⟨119862⟩ prop log(119873)119902minus02Theinset of the figure shows the dependence of the clusteringcoefficient of individual nodes upon their degree 119896 Theemergent relation 119862(119896) sim 119896minus119909 (119909 asymp 075) is also typical ofmany real-life networks [38]

RelocationsThegrowth rules of the baselinemodel evenwiththe added rule for connecting 119902 closest neighbors still failto account for the very important effect of relocations Everyso often a person relocates to a new place changing jobs orpursuing education following marriage and so on When aperson relocates they retain many of their friendships at

their place of origin and form new friendships at their newlocationThis has a profound effect on the connectivity of thesocial network as we shall see below For now however wejust describe the way to incorporate relocations in the revisedSSCN model

To relocate a single node 119894 we first pick two nodes 119894 and 119895at random and move node 119894 to within distance 119904 = 1 fromnode 119895 and at a random angle 120579 from 119895 while retainingall of 119894rsquos connections In the second stage we examine thenew environs of node 119894 and add the necessary connections toenforce the minimum 119902 closest neighbors rule Note that thefirst stage entails merely changing x(119894) but not its contactsThe second stage ensures that agent 119894 not only keeps itsold social connections but also makes new acquaintancesin the new place The process of relocation is illustrated inFigure 1(c)

The random choice of the relocating node 119894 and the targetnode (or location) 119895 is motivated by the ldquogravity modelrdquo forhuman mobility [39] It basically assumes that any individual119894 is as likely to relocate as any other and that relocating toany particular place (near x(119895)) is more probable the morepopulated that place is

In the following section we study the effect of migratinga fraction 120576 of the119873 nodes in the system A single relocationaffects the degree of the relocating node 119894 in the same way asadding 119902 closest neighbors (But note that 119894 undergoes twosuch updates) Thus the combined effect of connecting 119902closest neighbors and migrating a fraction 120576 on the degreedistribution is similar to that of connecting 1199021015840 = 119902(1 + 120576)neighbors without migration On the other hand relocationshave a dramatic effect on the pattern of connections and onnavigation of the social network and they should not beneglected

3 Results Connectivity and MilgramrsquosSmall-World Experiment

We now turn to the main question of how well the social net-work is connected and what we can learn from simulations ofMilgramrsquos Small-World experiment For concreteness westudy the typical SSCN configuration shown in Figure 2(b)and focus on the connectivity between individuals in thelargest and second-largest cities in the figure (populations15072 and 5567 respectively) The two cities happen to beabout 190 units of length away from one another whichcompares nicely with 119904max = radic119873 asymp 226 and with the actualspan of the ldquocountryrdquo

31 Shortest Paths Consider first the shortest paths in thenetwork Shortest paths can be found very efficiently forexample by the Breadth-First Search (BFS) algorithm Theproblem is that efficient algorithms such as the BFS requireglobal knowledge of the whole network of contacts (or thefull adjacency matrix) This type of information is clearlynot available to any one person so the mere existence ofshortest paths cannot explain the results in Milgramrsquos Small-World experiment Nevertheless shortest paths constitute a

6 Complexity

0

2

4

6

8

10

Path length

00 5 10 15 20 25

01

02

0

02

04

0

02

04

0

02

04

0

02

04

P()

0 5 10 150 5 10 15

0 5 10 15

0 5 10 15

Aver

age s

hort

est p

ath

⟨⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)

Figure 4 Statistics of shortest path length between all node pairs (119894 119895)where 119894 and 119895 belong to cities 1 and 2 as shown in Figure 2(b) indicatinga decrease of the average path length as additional features are introduced into the model as well as narrowing of their distribution (insethistograms)

useful ldquobenchmarkrdquo to which one can compare variousdecentralized algorithms

Since the SSCN network of social contacts consists ofonly one connected component (even in the baseline model)there exists a shortest path of links between any two nodesWe explore first how shortest paths evolve as one addsconnectivity to the baseline model first by connecting 119902 = 5closest neighbors and then by migrating increasing fractions120576 = 005 01 and 02 of the nodes

Our results for the shortest paths between nodes 119894 in City1 and nodes 119895 in City 2 are summarized in Figure 4 For thebaseline model the shortest paths between nodes in the twocities follow a bell-shaped distribution and average to justunder 11 links Adding connections to 5-closest neighborsreduces the shortest paths average length to about 85 Thischange is actually less impressive than one would expectThe average degree of each node in the baseline model is⟨119896⟩ = 2 since the network is then a tree Adding links to 119902nearest neighbors of each node increases the average degreeto ⟨119896⟩ = 2 + 119902 We can now compare the results to a randomnetwork undergoing a similar change Since the shortestpath in a random network is simlog⟨119896⟩119873 the paths wouldhave shortened by a factor of log2+119902119873log2119873 asymp 281 afteradding 119902 = 5 neighbors Instead the average path length hasreduced only by a disappointing 1185 asymp 129 The reason

is of course that the added connections in our case are farfrom random andmdashwhile important in accounting for thecommon phenomenon of ldquonext-doorrdquo friendsmdashthey do notcreate efficient shortcuts The situation is quite opposite forrelocations Migrating a mere 005 fraction of the nodesresults in an additional shortening of the average path lengthsto about 7 a dramatic change for the tiny increase in ⟨119896⟩ from7 to 725 Increasing the migration rate results in furtherreduction of the average path lengths but the most dramaticchange is that seen between no relocations at all and a tinyfraction of relocations In that respect relocations seem toplay a similar role to that of random long-range connectionsin theWatts and Strogatz Small-World networks [37] Finallythe insets in the figure show the distribution of path lengthsfor each successive change The narrowing of these distri-butions can be traced to the homogenization of the degreedistribution as more links are added in

32 Greedy Paths Consider now Milgramrsquos Small-Worldexperiment [13] Participants in the experiment have accessonly to local information You knowwho your friends are andwhere they live and so on but have little information abouttheir friends down the line The puzzle is how the messagefinds its way under these circumstances let alone in a shortnumber of steps Local or decentralized algorithms for passing

Complexity 7

0

10

20

30

40

4 6 8 102Path length

0

02

04

06 Success 012

0 20 40 600

002

004

006Success 25

0

002

004

006Success 33

0

002

004

006Success 38

0

002

004

006Success 39

P()

0 20 40 60

0 20 40 60

0 20 40 60Aver

age g

reed

y pa

th ⟨

⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)

City 1 2

City 2 1

Figure 5 Statistics of greedy path length obtained by randomly sampling 5 times 105 node pairs (119894 119895) where 119894 and 119895 belong to the cities 1 and2 see Figure 2(b) In the baseline model there are very few short greedy paths Connecting to closest neighbors increases the success ratesignificantly but the paths found are quite longer Even a tiny percentage of relocations not only further increases the success rate but alsoreduces the greedy path length significantly

the message may be quite involved and we shall test a fewscenarios For now however we stick to the simplest greedyalgorithm

Pass the message to the contact that is geographi-cally closest to the target (provided that it is closerthan yourself)

Kleinberg [14 15] had shown that for his Small-World latticeno other decentralized algorithm can obtain paths that scalemore favorably with the population 119873 than the greedyalgorithm In other words greedy paths give us a good ideaof how well any other decentralized method might perform(at least functionally in119873)

The proviso that each subsequent node is closer to the tar-get is important On the one hand it guarantees convergenceon the other hand it means that the message might get stuckwhen there is not a single contact that is closer to the targetthan oneself In such a case there is no greedy path betweenthe source and the target When a greedy path exists we saythat the source and target are greedily connected Greedy con-nectivity was explored for some benchmark networks (butnot for SSCN models) in [40] Some of the more importantproperties of greedy connectivity are as follows

(i) Nodes that are connected in the usual sensemight notbe greedily connected (but not the other way around)

(ii) Greedy paths are never shorter than shortest paths(iii) Greedy connectivity is not transitive If 119906 is greedily

connected to V and V is greedily connected to 119908 it isnot necessarily the case that 119906 is greedily connected to119908

(iv) Greedy connectivity is not symmetric there might bea greedy path from 119906 to V but no greedy path from Vto 119906

We have selected 500000 random pairs of nodes (119894 119895)with 119894 isin City 1 and 119895 isin City 2 and then searched for greedypaths from 119894 to 119895 and from 119895 to 119894 The results are summarizedin Figure 5

The average greedy path length for the baseline modelof about 7 links is pleasingly short however only 012of the pairs are greedily connected Adding connections to119902 = 5 closest neighbors dramatically increases the greedyconnectivity to about 25 of the pairs but the averagegreedy path lengthens to about 39 links These results can beunderstood as follows In the baseline model the network of

8 Complexity

contacts is a tree and there is a unique path between any pairof nodes (This path is also the shortest path) Since the spatialconnections are lain at a random angle 120579 the probability thatan ℓ-links path from 119894 to 119895 is also a greedy path is (12)ℓThus the typical shortest paths of average length ⟨ℓ⟩ = 11are greedy paths with probability (12)11 asymp 005 in generalagreement with the observed result Connecting 119902 closestneighbors makes for multiple paths between pairs of nodesThe probability that a greedy search might have to be aban-doned at any particular step is roughly (12)119902 (assuming thatthe closest neighbors are randomly distributed and neglect-ing the underlying baseline tree) For 119902 = 5 the probabilityof the typical greedy paths (of length 39) making it through istherefore (1minus(12)5)39 asymp 29 quite in line with the observedresults Despite the dramatic increase in the success rate forgreedy searches the typical path length is too large to explainthe observations in Milgramrsquos Small-World experiment

Migrating even a small fraction 120576 = 005 of the nodesfurther increases the success rate to about 33 but moreimportantly it slashes the typical greedy path length by afactor of 2 (Note that the total number of links increasesafter migration from (1 + 119902)119873 to (1 + (1 + 120576)119902)119873 but the42 increase resulting from 120576 = 005 cannot explain thesedramatic results)Migrating larger fractions of the populationachieves only modest improvements Once again the role ofrelocations seems analogous to that of random long-rangeconnections in Watts and Strogatz Small-World networks[37] Nevertheless the typical greedy path lengths of about15 even for 120576 = 02 migrations still seem too long toaccount for Milgramrsquos results Our SSCNmodel suggests thatthe difference is due largely to clever strategies adopted byparticipants in the experimentmdashpeople act more cleverlythan the simple-minded greedy algorithmmdashand partly due tothe effect of attrition the nonzero probability to abandonthe task at any particular step before the search is completedeffectively shortens the length of successfully completedpaths We turn to these issues next

33 Complex Strategies and Attrition The greedy path algo-rithm cannot by itself explain the results from MilgramrsquosSmall-World experiment and we are led to consider morecomplex strategies A possible strategy is to prefer friends thatlive closer to the target to some extent but give also someweight to friends that are exceptionally well-connected (sincethey might be more likely to make a better choice thanourselves) The following algorithm captures the gist of thisidea

Suppose that node 119894 currently holds the message that isdestined for the (disclosed) target 119905 Node 119894 assigns a score 119878119895to each of his 119896119894 acquaintances (119895 = 1 2 119896119894)

119878119895 = 120582 119904119894119904119895+ (1 minus 120582)

119896119895119896119894 (4)

Here 119904119894 and 119904119895 are the geographical distances between 119894 and119905 and 119895 and 119905 and 119896119894 and 119896119895 are the degrees of node 119894 andof its 119895th contact respectively In other words agent 119894 scoreshis acquaintances relative to himself (his own score is 119878119894 = 1)assigning higher value to friends that are closer to 119905 than

himself and that are better connected than himself Theparameter 120582 isin [0 1] controls the relative importance ofeach attribute With the scores at hand the strategy proceedsexactly as in the greedy algorithm but with the aim ofmaximizing 119878119895 (rather than minimizing the distance)

Pass the message to the contact that has the largestscore (provided that its score is larger than 1)

Kleinbergrsquos greedy algorithm corresponds to the specialcase of 120582 = 1 For any other 0 lt 120582 lt 1 the strategy stillguarantees convergence to the target (if a path is available)since the distance from 119905 to itself is zero so that the scoreof 119905 is infinite and overwhelms all other considerations (Thecase of 120582 = 0 is problematic for the message may then fail toreach the target evenwhen 119905 is a contact of 119894 andwe thereforerequire 120582 gt 0) The search for a path to 119905 is aborted when theproviso that 119878119895 gt 1 is not fulfilled In addition for 120582 lt 1the path may revisit a previously touched node creating aclosed loopThe search is of course abandoned in such casesas well We note that the search strategy considered here is bynomeans exclusive Several other heuristic search algorithmsbeyond Kleinbergrsquos greedy algorithm have been investigatedin previous work such as [41] on both synthetic and real-world spatial networks

Figure 6 summarizes the results of this mixed strategy asapplied to the case of 119902 = 5 closest neighbors and 120576 = 005fraction of relocations For clarity we include only the resultsfor searches from City 2 to City 1 (the small differences foundfor the reverse direction are discussed in the next subsection)Panel (a) shows the fraction of pairs119877(120582) that are successfullyconnected The overall trend shown in the inset is of a rapiddecay to zero as 120582 decreases For 120582 close to 1 however thereis first an increase from 119877(1) asymp 037 to a maximum of 045success rate for 120582 asymp 0998 At the same time the average pathlength (Figure 6(b)) decreases from ⟨ℓ⟩ = 197 at 120582 = 1 to⟨ℓ⟩ = 160 at 120582 = 0998 There is in fact a whole range of1205821 lt 120582 lt 1 for which the mixed strategy performs better(higher success rate and shorter paths) than the pure greedyalgorithm of 120582 = 1 At 1205821 asymp 0986 for example the successrate is as good as for 120582 = 1 but the average path length isslashed by nearly 5 links

As 120582 decreases beyond 1205821 it becomes harder to judge thesuccess of the mixed strategy On the one hand there is theattractive effect of decreasing ⟨ℓ⟩ on the other hand fewerand fewer pairs remain connected One way out of thisconundrum is to select the point for which 119877 matches thereported success rate of Milgramrsquos Small-World experimentof roughly 28 This occurs for 1205822 asymp 0982 where ⟨ℓ⟩ isreduced to nearly 134 links

An important conclusion is that geographical proximityis the largest factor in finding decentralized paths as evidentfrom the large values of 120582 that are optimal in our mixedstrategyThis understanding is also in linewith the findings ofLiben-Nowell et al [20] Our mixed strategy shows that onecan do better than geography alone (the case of 120582 = 1) yet notas well as reported by Milgram The reason is that our mixedstrategy fails to incorporate much of the intuition and socialcleverness that are second-nature to people In Milgramrsquos

Complexity 9

09 092 094 096 098 1Mixing parameter

005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)

Mixing parameter 09 092 094 096 098 1

3

7

11

15

19

To target nodeTo target city

0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)

0 10 20 30 40Path length

0

003

006

009

No attritionWith attrition

P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)

Figure 6 Decentralized paths found with the mixed greedy strategy (a) Fraction of completed searches 119877(120582) in the range 09 le 120582 le 1 Themixed strategy beats the pure greedy algorithm in the pink-shaded region 1205821 lt 120582 lt 1 At 120582 = 1205822 the success rate of the mixed strategymatches the 28 rate reported in Milgramrsquos work [13] Inset 119877(120582) for the whole range of 0 le 120582 le 1 (b) Average path length from points inCity 2 to City 1 (top purple curve) and average number of links to reach City 1 (bottom orange curve) in the range 09 le 120582 le 1 Inset Samefor the full range of 0 le 120582 le 1 (c) The effect of incidental attrition Shown is the distribution of path lengths 119875(ℓ) for 120582 = 0998 (solid line)along with (09)ℓ119875(ℓ) accounting for 01 probability of incidental dropout (broken line) The overall success rate reduces from 45 to 11and the conditional average path length ⟨ℓ⟩ decreases from 16 to 117 (Both curves are normalized in the figure to highlight the change inshape that results from incidental attrition)

experiment for example the targetrsquos occupation (stockbro-ker) was disclosed in addition to name and address Thename holds clues to the targetrsquos gender and ethnicity and theaddress might hint at social status None of this informationis accounted for in our naive approach

Amore realistic approachwould probably still relymostlyon geography at least until the message reaches the targetrsquoscity Once inside the city the additional clues of occupationgender ethnicity social status and so on provide effectivemeans for finding shorter paths (eg the stockbrokers inBoston tend to know one another) Indeed subject reports inMilgram-like experiments strongly support this idea [19]The

average path to the targetrsquos city in our simulations is signif-icantly shorter than the total path (Figure 6(b)) At 1205822 =0982 (wherewe reproduceMilgramrsquos success rate of 28) forexample the average path length is ⟨ℓ⟩ = 134 but only 4 ofthose links are needed to reach City 1 At this stage Milgramrsquosresults seem quite within reach

So far we have considered attrition only due to the strat-egy or strategical attrition the search is dropped when thealgorithm fails to find a next valid step In real life howeverthere are other reasons for defecting besides the unavailabilityof an attractive option Participants may drop out from theexperiment because of busyness laziness lack of motivation

10 Complexity

and so onWe refer to this effect as incidental attritionWe canlump both types of attrition into a single probability 119901 that anindividual drops out of the experimentmdashthis means a pathof length ℓ has (1 minus 119901)ℓ chance of being completed FromMilgramrsquos second study [42] for example it can be estimatedthat 119901 asymp 038 To illustrate the effect of incidental attritionin Figure 6(c) we plot the probability distribution for paths oflength ℓ119875(ℓ) for the case of 120582 = 0998 (solid line) alongwiththe distribution (09)ℓ119875(ℓ) that results from an incidentaldropout probability of 01 (broken line) As onewould expectthe overall success rate drops from 45 to 11 but the(conditional) average path length is reduced by 43 links Thetwo types of attrition are a significant factor in the selectionof shorter paths

34 Asymmetry Consider finally the asymmetry of greedyor decentralized paths paths from 119894 in City 1 to 119895 in City2 are not necessarily the same as paths from 119895 to 119894 We seethis effect quite clearly in Figure 5 where the average pathlength for City 1 rarr 2 is systematically shorter than forCity 2 rarr 1 through all stages of the modelrsquos buildup Thesuccess rates too are systematically smaller for paths fromCity 1 to 2 than the reverse (the differences are small and inthe figure we put for simplicity only the average of the tworates)

A simple explanation to this asymmetry is that purelygreedy paths from City 1 to City 2 can go through City 3but those from City 2 to City 1 cannot (City 3 is fartheraway from the target) see Figure 2(b) The situation isstatistically symmetric for a ldquodirectrdquo commute City 1 harr2 without City 3 in the picture same expected number ofsuccessful paths and average path lengths in either directionThe extra 2 rarr 3 rarr 1 routes tend to be longer than thedirect commute and account both for the higher success rateand the longer average path lengths in the City 2 rarr 1direction

We observe small similar asymmetries also with ourmixed strategy for all values of 120582The regionwhere themixedstrategy beats the pure greedy algorithm for example issomewhat narrower for the City 1 rarr 2 direction with1205821 = 0988 (instead of 1205821 = 0986 for City 2 rarr 1) butwe do not have a simple explanation to account for thesefindings

4 Discussion and Conclusion

In summary we have proposed improvements to the baselineSSCN model of [26] that render it suitable for simulationsof dynamic social processes such as Milgramrsquos Small-Worldexperiment [13 42] The most important revisions call forconnecting each node to a number of spatially closest nearestneighbors to account for ldquonext-doorrdquo friends and relocatinga fraction 120576 of the nodes to account for relocations (due tojob change study marriage etc) These two revisions have aminor effect on the degree distribution of the baseline modelbut a dramatic effect on the connectivity properties of thenetwork of social contacts The connections to closest neigh-borsmake for a robust clustering effect (absent in the baseline

model) and even a tiny fraction 120576 of relocations intro-duces long-range connections that decrease the average pathlength between pairs of nodes substantially similarly to therandom long-range links inWatts and Strogatzrsquos Small-Worldnetworks [37]

Our simulations of theMilgram Small-World experimentshow that Kleinbergrsquos greedy algorithmmdashbased only onthe geographical distance between nodesmdashis successful infinding decentralized paths between pairs of nodes but thepaths are too long to explain Milgramrsquos results We haveshown that more complex strategies such as occasionallypassing the message to acquaintances that are especiallywell-connected can result in a significant reduction ofthe path length We have also confirmed the notion thatgeography is the most important consideration in findingshort paths [19 20] at least in the initial stages untilthe message reaches the targetrsquos city The remaining pathto the target within the city could be shortened con-siderably using the additional explicit information (egoccupation) and implicit information (ethnicity social sta-tus) known about the target We have also discussed theeffect of attrition (the fact that participants drop out ofthe experiment for various reasons) and showed how ithelps select for shorter paths Note that alternative mod-els of navigable spatial networks have been recently stud-ied for example based on mapping to a hypergeomet-ric space [43] or some iterative optimization techniques[44]

Simulations of Milgramrsquos experiment pose a particularlystrict test to the SSCN model in that finding decentralizedpaths relies quite sensitively both on the location of thenodes and on their network of connections The modelrsquossuccess makes it a promising substrate for the simulation ofother dynamical processes on social networks where suchconsiderations are important (epidemics opinion modelsetc)

Appendix

Algorithmic Description of the Spatial SocialComplex Network (SSCN) Model

In Algorithm 1 we provide pseudocode on using the (revised)SSCN model to generate a spatial social network Typicalchoices of the redirection parameters as discussed in themain text are 119903infin = 08 and1198730 = 25

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was funded in part by the Simons FoundationGrant no 318812 and the Army Research Office Grant noW911NF-16-1-0081

Complexity 11

Input 119873 (number of nodes) 119903infin isin [0 1](asymptotic redirection probability)1198730 gt 0(additional parameter for redirection) 119902 isin N cup 0(min number of spatial nearest neighbors per node)120576 isin [0 1] (relocation probability)Output 119860 = [119860 119894119895]119873times119873 (network adjacency matrix)and119883 = [x(1) x(119873)]2times119873 (nodes spatial coordinates)(1) x(1) larr [0 0]⊤ andN1 larr (2) for 119894 = 2 3 119873 do(3) 119903 larr (1 minus 119890minus(119894minus2)1198730 )119903infin(4) Choose 119895 at random from 1 119894 minus 1(5) Choose 119911 at random from the interval (0 1)(6) Choose 120579 at random from the interval [0 2120587)(7) if 119911 lt 1 minus 119903 then(8) N119894 larr 119895 andN119895 larr N119895 cup 119894(9) Choose 119904 sim 119901(119904) = (1 log(119904max))119904minus1 (1 lt 119904 lt 119904max)(10) x(119894) larr [x(119895)1 + 119904 cos(120579) x(119895)2 + 119904 sin(120579)]⊤(11) else(12) Choose 1198951015840 at random from the setN119895(13) N119894 larr 1198951015840 andN1198951015840 larr N119895 cup 119894(14) x(119894) larr [x(119895

1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤

(15) end if(16) end for(17) if 119902 ge 1 then(18) for 119894 = 1 2 119873 do(19) Q larr 119902 spatially nearest neighbors of node 119894(20) N119894 larr N119894 cup Q

(21) end for(22) end if(23) Choose a random permutation 120587 on the set 1 119873(24) for 119894 = 1 2 119873 do(25) Choose 119911 at random from the interval (0 1)(26) if 119911 lt 120576 then(27) Choose 119895 at random from 1 119873120587119894(28) Choose 120579 at random from the interval [0 2120587)(29) x(120587119894) larr [x(119895)1 + cos(120579) x(119895)2 + sin(120579)]⊤(30) if 119902 ge 1 then(31) Q larr 119902 spatially nearest neighbors

of node120587119894(32) N120587119894 larr N120587119894 cup Q

(33) end if(34) end if(35) end for(36) for 119894 = 1 2 119873 do(37) for every 119895 isin N119894 do(38) 119860 119894119895 larr 1(39) end for(40) end for

Algorithm 1 Network generation using the SSCN model

References

[1] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14 pp3200ndash3203 2001

[2] RCohen SHavlin andD Ben-Avraham ldquoEfficient immuniza-tion strategies for computer networks andpopulationsrdquoPhysicalReview Letters vol 91 no 24 Article ID 247901 2003

[3] V Belik T Geisel and D Brockmann ldquoNatural HumanMobil-ity Patterns and Spatial Spread of Infectious Diseasesrdquo PhysicalReview X vol 1 no 1 Article ID 011001 pp 1ndash5 2011

[4] J Kim E Elliott and D M Wang ldquoA spatial analysis ofcounty-level outcomes in US Presidential elections 1988-2000rdquoElectoral Studies vol 22 no 4 pp 741ndash761 2003

[5] S Kwon M Cha K Jung W Chen and Y Wang ldquoProminentfeatures of rumor propagation in online social mediardquo inProceedings of the 13th IEEE International Conference on DataMining ICDM 2013 pp 1103ndash1108 USA December 2013

[6] Y Hu S Havlin and H A Makse ldquoConditions for viral influ-ence spreading through multiplex correlated social networksrdquoPhysical Review X vol 4 no 2 Article ID 021031 2014

[7] J P Gleeson K P OrsquoSullivan R A Banos and Y MorenoldquoEffects of network structure competition andmemory time onsocial spreading phenomenardquo Physical Review X vol 6 no 2Article ID 021019 2016

[8] LWeng A Flammini A Vespignani and FMenczer ldquoCompe-tition amongmemes in aworldwith limited attentionrdquo ScientificReports vol 2 article 335 8 pages 2012

[9] D Brockmann L Hufnagel and T Geisel ldquoThe scaling laws ofhuman travelrdquo Nature vol 439 no 7075 pp 462ndash465 2006

[10] S H Lee R Ffrancon D M Abrams B J Kim and M APorter ldquoMatchmaker matchmaker make me a match Migra-tion of populations via marriages in the pastrdquo Physical ReviewX vol 4 no 4 Article ID 041009 2014

[11] LM A Bettencourt ldquoThe origins of scaling in citiesrdquoAmericanAssociation for the Advancement of Science Science vol 340 no6139 pp 1438ndash1441 2013

[12] M Schlapfer L M Bettencourt S Grauwin et al ldquoThe scalingof human interactionswith city sizerdquo Journal of the Royal SocietyInterface vol 11 no 98 pp 20130789-20130789 2014

[13] S Milgram ldquoThe small-world problemrdquo Psychology Today vol1 pp 60ndash67 1967

[14] J M Kleinberg ldquoNavigation in a small worldrdquo Nature vol 406no 6798 p 845 2000

[15] J Kleinberg ldquoThe small-world phenomenon An algorithmicperspectiverdquo in Proceedings of the 32nd Annual ACM Sympo-sium onTheory of Computing STOC2000 pp 163ndash170 usaMay2000

[16] M R Roberson and D Ben-Avraham ldquoKleinberg navigationin fractal small-world networksrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 74 no 1 Article ID017101 2006

[17] JM Campuzano J P Bagrow andD ben-Avraham ldquoKleinbergNavigation on Anisotropic Latticesrdquo Research Letters in Physicsvol 2008 pp 1ndash4 2008

[18] D Rybski A Garcıa Cantu Ros and J P Kropp ldquoDistance-weighted city growthrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 87 no 4 Article ID 042114 2013

[19] P S Dodds R Muhamad and D J Watts ldquoAn experimentalstudy of search in global social networksrdquo Science vol 301 no5634 pp 827ndash829 2003

[20] D Liben-Nowell J Novak R Kumar P Raghavan and ATomkins ldquoGeographic routing in social networksrdquo Proceedingsof the National Acadamy of Sciences of the United States ofAmerica vol 102 no 33 pp 11623ndash11628 2005

[21] S Scellato A Noulas R Lambiotte and C Mascolo ldquoSocio-spatial properties of online location-based social networksrdquo inProceedings of Fifth International AAAI Conference on Weblogsand Social Media (ICWSM 2011 p 5 Barcelona Spain 2011

12 Complexity

[22] P Expert T S Evans V D Blondel and R Lambiotte ldquoUncov-ering space-independent communities in spatial networksrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 108 no 19 pp 7663ndash7668 2011

[23] J P Onnela S Arbesman M C Gonzalez A L Barabasi andN A Christakis ldquoGeographic constraints on social networkgroupsrdquo PLoS ONE vol 6 no 4 Article ID e16939 2011

[24] M Barthelemy ldquoSpatial networksrdquo Physics Reports vol 499 no1-3 pp 1ndash101 2011

[25] M Barthelemy The Structure and Dynamics of Cities Cam-bridge University Press Cambridge 2016

[26] G F Frasco J Sun H D Rozenfeld and D Ben-AvrahamldquoSpatially distributed social complex networksrdquo Physical ReviewX vol 4 no 1 Article ID 011008 2014

[27] H D Rozenfeld D Rybski J S Andrade Jr M Batty HE Stanley and H A Makse ldquoLaws of population growthrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 105 no 48 pp 18702ndash18707 2008

[28] G Zipf Human Behavior And The Principle of Least EffortAddison-Wesley Cambridge Mass USA 1949

[29] M Cristelli M Batty and L Pietronero ldquoThere is more than apower law in Zipfrdquo Scientific Reports vol 2 article no 812 2012

[30] T Fluschnik S Kriewald A G C Ros et al ldquoThe size dis-tribution scaling properties and spatial organization of urbanclusters A global and regional percolation perspectiverdquo ISPRSInternational Journal of Geo-Information vol 5 no 7 Article ID638868205 2016

[31] P L Krapivsky and S Redner ldquoOrganization of growing randomnetworksrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 63 no 6 Article ID 066123 2001

[32] P L Krapivsky and S Redner ldquoFiniteness and fluctuationsin growing networksrdquo Journal of Physics A Mathematical andGeneral vol 35 no 45 pp 9517ndash9534 2002

[33] J Kim P L Krapivsky B Kahng and S Redner ldquoInfinite-order percolation and giant fluctuations in a protein interactionnetworkrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 66 no 5 Article ID 055101 p 05510142002

[34] R Albert and A L Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[35] M E Newman ldquoThe structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[36] L K Gallos D Rybski F Liljeros S Havlin and H A MakseldquoHow people interact in evolving online affiliation networksrdquoPhysical Review X vol 2 no 3 Article ID 031014 2012

[37] D J Watts and S H Strogatz ldquoCollective dynamics of rsquosmall-worldrsquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[38] S Boccaletti V Latora Y Moreno M Chavez and D-UHwang ldquoComplex networks Structure and dynamicsrdquo PhysicsReports vol 424 no 4-5 pp 175ndash308 2006

[39] N Bharti Y Xia O N Bjornstad and B T Grenfell ldquoMeasleson the edge Coastal heterogeneities and infection dynamicsrdquoPLoS ONE vol 3 no 4 Article ID e1941 2008

[40] J Sun and D Ben-Avraham ldquoGreedy connectivity of geo-graphically embedded graphsrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 82 no 1 Article ID016109 2010

[41] H P Thadakamalla R Albert and S R T Kumara ldquoSearchin spatial scale-free networksrdquo New Journal of Physics vol 9article no 190 2007

[42] J Travers and S Milgram ldquoAn experimental study of the smallworld problemrdquo Sociometry vol 32 no 4 pp 425ndash443 1969

[43] D Krioukov F Papadopoulos M Kitsak A Vahdat and MBoguna ldquoHyperbolic geometry of complex networksrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 82no 3 Article ID 036106 2010

[44] Y A Malkov and A Ponomarenko ldquoGrowing homophilicnetworks are natural navigable small worldsrdquo PLoS ONE vol11 no 6 Article ID e0158162 2016

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2 Complexity

studies) does not capture the social network of interactions[18] Dodds et al conducted a large-scale online experimentthat resembles Milgramrsquos original study highlighting the roleof information beyond just network structure [19] Liben-Nowell et al [20] proposed a spatial social network modelwith connections derived from an online bloggers com-munity and studied greedy routing on that model Similarstudies were conducted for online social networks [21] andcommunity structures from mobile phone records [22 23](see [24 25] for a more comprehensive review) Informationon peoplersquos location along with their social contacts isgenerally hard to come by and often relies on indirectproxies

In [26]we introduced a stochastic prototype Spatial SocialComplex Network (SSCN) relying on just two controllableparameters that simulates large populations including thelocations and the complex network of contacts betweenagents This ldquobaseline modelrdquo was designed with modestgoals in mind (a) The population density resembles the lightdensity observed in satellite pictures of earth at night (b) thepopulation of ldquocitiesrdquo (defined by percolation clusters [27])and their rankings follow Zipf rsquos law [28ndash30] (c) the socialnetwork of contacts exhibits a scale-free distribution and (d)highly connected nodes tend to be located in denser andlarger population areas In addition to meeting these basicgoals the SSCN baseline model also yielded good qualitativeagreement with census data for the population density as afunction of city size and for the weak super-linear depen-dence of the cumulative degree of nodes in a city on its totalpopulation as suggested from cell-phone data [12] Finally itallowed us to shed some light on the weak deviations [27]from Gibratrsquos law (that the rate of growth of a city and itsfluctuations are proportional to its population size)

Despite these initial successes the SSCN baseline modelfails to mimic real SSCNs in some crucial ways (i) The com-plex network of social contacts while displaying a realisticscale-free degree distribution is actually a tree in contrastwith the high degree of clustering observed in social netsand their proxies (ii)The network of contacts is built througha redirection mechanism [31ndash33] which is an adequatedescription of how individuals might join a social networkbut fails to account for the effect of relocations every so oftena person relocates to a far-away destination for study jobor other reasons This creates particular correlations in thenetwork of social contacts that are absent in the baselineSSCN

In this paper we resolve the baseline SSCN deficiencieswith some simple adjustments Connections to spatiallyclosest neighbors are added to mimic the clustering effect inreal social networks and relocations turn out to be crucial inreducing the average path length between nodes Simulationsof Milgramrsquos Small-World experiment on the revised SSCNmodel achieve a good qualitative fit with the empirical find-ings This demonstrates the suitability of the model as a sub-strate for simulations of other dynamic social processes thatdepend on both the contacts and the geographical locationsof the agents

2 Materials and Methods

21 The Baseline SSCN Model We now review the originalor ldquobaselinerdquo SSCN model established in our previous work[26] The model produces a spatially embedded network 119866 =(119881 119864119883) where 119881 = 1 2 119873 is a set of nodes and 119864 sub119881 times 119881 is a set of undirected edges The spatial embedding ofthe network is encoded in the set of coordinates 119883 =x(1) x(119873) where for 2D spatial networks (such as in ourcase) x(119894) isin R2 A unique feature of the model is that it notonly produces the requisite scale-free degree distribution forthe edges but also captures essential spatial features such as aZipf distribution of the populations emerging from the nodesclustering into ldquocitiesrdquo [26]

Consider first the creation of nodes and edges in thebaseline model defining 119881 and 119864 The starting point is aninitial ldquoseedrdquo network which in the baseline model consiststypically of a single node Nodes are added to the networkone at a time each contributing to a new edge according to avariant of the Krapivsky-Redner (KR) model [31ndash33] with asingle parameter 119903 isin [0 1]mdashthe redirection probability Eachtime a newnode 119894 joins the network one of the existing nodes119895 is chosen uniformly at random and 119894 is connected to 119895directly with probability 1 minus 119903 (creating a new edge 119894 harr 119895)otherwise with probability 119903 the connection is redirected to arandomly selected neighbor 1198951015840 of 119895 (edge 119894 harr 1198951015840) For large119873this leads to a scale-free degree distribution (in the originalKRmodel connections are redirected to the ancestor of 119895mdashthenode 119895 connected to upon joining the network our variantyields 120574 ≳ 120574KR = 1 + 1119903 we find that using the original KRrecipe results in a poorer resemblance with satellite picturesof earth at night furthermore the variant employed in ourmodel is somewhat simpler as there is no need to trackancestors)

119875 (119896) prop 119896minus120574 with 120574 asymp 1 + 1119903 (1)

Consider next the placement of the nodes in spacespecifying 119883 For a network of 119873 nodes the baseline modelplaces them within a square box of sides 119871 = radic119873 (with peri-odic boundary conditions) The initial seed node is placedat the origin x1 = 0 and the location of subsequent nodes119894 depends on whether it connects to node 119895 directly or to aneighbor 1198951015840 by redirection If 119894 joins directly it is placed at(119904 120579) from 119895 (using polar coordinates) where the angle 0 lt120579 le 2120587 is chosen randomly from the uniform distributionand 119904 is picked randomly from the distribution

119901 (119904) =

1ln (119904max)

119904minus1 1 lt 119904 lt 119904max0 otherwise

(2)

where 119904max = radic2119871 is themaximumpossible distance betweenany two points within the bounding square In the case ofredirection when node 119894 joins to 1198951015840 then we simply place 119894 atdistance 1 from 1198951015840 at a random angle 120579The growth algorithmis illustrated in Figure 1(a) Note that the choice of exponentminus1 in (2) is equivalent to 119901(s) sim |s|minus2 which is in line withKleinbergrsquos ldquomagicrdquo formula for optimal navigability

Complexity 3


j j


1 minus r

j


i



j

i

i


i


q = 2j








4 Complexity

Zipf plot

104

103

102

101

City

pop

ulat

ion

100 101 102

Rank

104

102

100 101 102


N0 = 25

N0 = 50

(a)


City 1

City 3

City 2

(b)









Complexity 5


05

055

06

065

07

q = 5q = 10

1minus⟨C

⟩

100

10minus1

10minus2

C(k

)

101 102 103

Degree k

N = 800

N = 3200

N = 12800

N = 51200













6 Complexity

0

2

4

6

8

10

Path length

00 5 10 15 20 25

01

02

0

02

04

0

02

04

0

02

04

0

02

04

P()

0 5 10 150 5 10 15

0 5 10 15

0 5 10 15

Aver

age s

hort

est p

ath

⟨⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)







Complexity 7

0

10

20

30

40


0

02

04

06 Success 012

0 20 40 600

002

004

006Success 25

0

002

004

006Success 33

0

002

004

006Success 38

0

002

004

006Success 39

P()

0 20 40 60

0 20 40 60

0 20 40 60Aver

age g

reed

y pa

th ⟨

⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)

City 1 2

City 2 1












8 Complexity





119878119895 = 120582 119904119894119904119895+ (1 minus 120582)

119896119895119896119894 (4)








Complexity 9


005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)


3

7

11

15

19


0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)


0

003

006

009


P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)






10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3


j j


1 minus r

j


i



j

i

i


i


q = 2j








4 Complexity

Zipf plot

104

103

102

101

City

pop

ulat

ion

100 101 102

Rank

104

102

100 101 102


N0 = 25

N0 = 50

(a)


City 1

City 3

City 2

(b)









Complexity 5


05

055

06

065

07

q = 5q = 10

1minus⟨C

⟩

100

10minus1

10minus2

C(k

)

101 102 103

Degree k

N = 800

N = 3200

N = 12800

N = 51200













6 Complexity

0

2

4

6

8

10

Path length

00 5 10 15 20 25

01

02

0

02

04

0

02

04

0

02

04

0

02

04

P()

0 5 10 150 5 10 15

0 5 10 15

0 5 10 15

Aver

age s

hort

est p

ath

⟨⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)







Complexity 7

0

10

20

30

40


0

02

04

06 Success 012

0 20 40 600

002

004

006Success 25

0

002

004

006Success 33

0

002

004

006Success 38

0

002

004

006Success 39

P()

0 20 40 60

0 20 40 60

0 20 40 60Aver

age g

reed

y pa

th ⟨

⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)

City 1 2

City 2 1












8 Complexity





119878119895 = 120582 119904119894119904119895+ (1 minus 120582)

119896119895119896119894 (4)








Complexity 9


005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)


3

7

11

15

19


0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)


0

003

006

009


P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)






10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity

Zipf plot

104

103

102

101

City

pop

ulat

ion

100 101 102

Rank

104

102

100 101 102


N0 = 25

N0 = 50

(a)


City 1

City 3

City 2

(b)









Complexity 5


05

055

06

065

07

q = 5q = 10

1minus⟨C

⟩

100

10minus1

10minus2

C(k

)

101 102 103

Degree k

N = 800

N = 3200

N = 12800

N = 51200













6 Complexity

0

2

4

6

8

10

Path length

00 5 10 15 20 25

01

02

0

02

04

0

02

04

0

02

04

0

02

04

P()

0 5 10 150 5 10 15

0 5 10 15

0 5 10 15

Aver

age s

hort

est p

ath

⟨⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)







Complexity 7

0

10

20

30

40


0

02

04

06 Success 012

0 20 40 600

002

004

006Success 25

0

002

004

006Success 33

0

002

004

006Success 38

0

002

004

006Success 39

P()

0 20 40 60

0 20 40 60

0 20 40 60Aver

age g

reed

y pa

th ⟨

⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)

City 1 2

City 2 1












8 Complexity





119878119895 = 120582 119904119894119904119895+ (1 minus 120582)

119896119895119896119894 (4)








Complexity 9


005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)


3

7

11

15

19


0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)


0

003

006

009


P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)






10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5


05

055

06

065

07

q = 5q = 10

1minus⟨C

⟩

100

10minus1

10minus2

C(k

)

101 102 103

Degree k

N = 800

N = 3200

N = 12800

N = 51200













6 Complexity

0

2

4

6

8

10

Path length

00 5 10 15 20 25

01

02

0

02

04

0

02

04

0

02

04

0

02

04

P()

0 5 10 150 5 10 15

0 5 10 15

0 5 10 15

Aver

age s

hort

est p

ath

⟨⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)







Complexity 7

0

10

20

30

40


0

02

04

06 Success 012

0 20 40 600

002

004

006Success 25

0

002

004

006Success 33

0

002

004

006Success 38

0

002

004

006Success 39

P()

0 20 40 60

0 20 40 60

0 20 40 60Aver

age g

reed

y pa

th ⟨

⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)

City 1 2

City 2 1












8 Complexity





119878119895 = 120582 119904119894119904119895+ (1 minus 120582)

119896119895119896119894 (4)








Complexity 9


005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)


3

7

11

15

19


0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)


0

003

006

009


P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)






10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

0

2

4

6

8

10

Path length

00 5 10 15 20 25

01

02

0

02

04

0

02

04

0

02

04

0

02

04

P()

0 5 10 150 5 10 15

0 5 10 15

0 5 10 15

Aver

age s

hort

est p

ath

⟨⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)







Complexity 7

0

10

20

30

40


0

02

04

06 Success 012

0 20 40 600

002

004

006Success 25

0

002

004

006Success 33

0

002

004

006Success 38

0

002

004

006Success 39

P()

0 20 40 60

0 20 40 60

0 20 40 60Aver

age g

reed

y pa

th ⟨

⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)

City 1 2

City 2 1












8 Complexity





119878119895 = 120582 119904119894119904119895+ (1 minus 120582)

119896119895119896119894 (4)








Complexity 9


005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)


3

7

11

15

19


0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)


0

003

006

009


P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)






10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7

0

10

20

30

40


0

02

04

06 Success 012

0 20 40 600

002

004

006Success 25

0

002

004

006Success 33

0

002

004

006Success 38

0

002

004

006Success 39

P()

0 20 40 60

0 20 40 60

0 20 40 60Aver

age g

reed

y pa

th ⟨

⟩

Baseline(N0 = 25)

Spatial NN(q = 5)

Migration 1( = 005)

Migration 2( = 01)

Migration 3( = 02)

City 1 2

City 2 1












8 Complexity





119878119895 = 120582 119904119894119904119895+ (1 minus 120582)

119896119895119896119894 (4)








Complexity 9


005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)


3

7

11

15

19


0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)


0

003

006

009


P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)






10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity





119878119895 = 120582 119904119894119904119895+ (1 minus 120582)

119896119895119896119894 (4)








Complexity 9


005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)


3

7

11

15

19


0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)


0

003

006

009


P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)






10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9


005

015

025

035

045

0 05 10

02

04

Succ

ess r

ate

R(

)2 1

(a)


3

7

11

15

19


0 05 10

10

20

Path

leng

th ⟨

⟩

2 1

(b)


0

003

006

009


P()

asymp 117

asymp 160⟨⟩

⟨⟩

(c)






10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity










Appendix





Acknowledgments


Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11


1015840)1 + cos(120579) x(119895

1015840)2 + sin(120579)]⊤






References






















12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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