The Small-World Phenomenon

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The Small-World The Small-World Phenomenon Outline Phenomenon The Small-World Phenomenon Complex Networks, CSYS/MATH 303, Spring, 2010 History History An online An online experiment experiment History Previous Previous Prof. Peter Dodds theoretical work theoretical work An improved An improved model model Department of Mathematics & Statistics An online experiment References References Center for Complex Systems Vermont Advanced Computing Center University of Vermont Previous theoretical work

An improved model

References

1/47 2/47 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

The Small-World The Small-World Some problems for people thinking about Phenomenon Social Search Phenomenon people?: History History

An online An online How are social networks structured? experiment experiment Previous A small slice of the pie: Previous theoretical work theoretical work I How do we deﬁne connections? An improved I Q. Can people pass messages between distant An improved model model I How do we measure connections? individuals using only their existing social References References I (remote sensing, self-reporting) connections? I A. Apparently yes... What about the dynamics of social networks? Handles: I How do social networks evolve? The Small World Phenomenon I How do social movements begin? I or “Six Degrees of Separation.” I How does collective problem solving work? I

I How is information transmitted through social networks? 3/47 4/47 The Small-World The Small-World The problem Phenomenon The problem Phenomenon

History History An online Lengths of successful chains: An online experiment experiment 18 Previous Previous theoretical work theoretical work Stanley Milgram et al., late 1960’s: An improved 15 An improved model model From Travers and References References I Target person worked in Boston as a stockbroker. 12 Milgram (1969) in [4]

I 296 senders from Boston and Omaha. ) Sociometry: L

( 9

I 20% of senders reached target. n “An Experimental Study of the Small I average chain length ' 6.5. 6 World Problem.”

0 1 2 3 4 5 6 7 8 9 10 11 12 L 5/47 6/47

The Small-World The Small-World The problem Phenomenon Social Search Phenomenon

History Milgram’s small world experiment with e-mail [2] History An online An online experiment experiment

Previous Previous theoretical work theoretical work

An improved An improved Two features characterize a social ‘Small World’: model model References References 1. Short paths exist and 2. People are good at ﬁnding them.

7/47 8/47 The Small-World The Small-World Social search—the Columbia experiment Phenomenon Social search—the Columbia experiment Phenomenon

History History

An online An online experiment experiment

Previous Previous I 60,000+ participants in 166 countries theoretical work theoretical work An improved I Milgram’s participation rate was roughly 75% An improved I 18 targets in 13 countries including model model Email version: Approximately 37% participation rate. I a professor at an Ivy League university, References I References I an archival inspector in Estonia, I Probability of a chain of length 10 getting through: I a technology consultant in India, 10 −5 I a policeman in Australia, .37 ' 5 × 10 and I a veterinarian in the Norwegian army. I ⇒ 384 completed chains (1.6% of all chains). I 24,000+ chains

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The Small-World The Small-World Social search—the Columbia experiment Phenomenon Social search—the Columbia experiment Phenomenon

History History

An online Successful chains disproportionately used An online experiment experiment Previous I weak ties (Granovetter) Previous theoretical work theoretical work Motivation/Incentives/Perception matter. I An improved I professional ties (34% vs. 13%) An improved model model I If target seems reachable I ties originating at work/college ⇒ participation more likely. References References I target’s work (65% vs. 40%) I Small changes in attrition rates ⇒ large changes in completion rates . . . and disproportionately avoided I e.g., & 15% in attrition rate ⇒ % 800% in completion rate I hubs (8% vs. 1%) (+ no evidence of funnels) I family/friendship ties (60% vs. 83%)

Geography → Work

11/47 12/47 The Small-World The Small-World Social search—the Columbia experiment Phenomenon Social search—the Columbia experiment Phenomenon

History History

An online An online experiment experiment Senders of successful messages showed Nevertheless, some weak discrepencies do exist... Previous Previous little absolute dependency on theoretical work An above average connector: theoretical work An improved An improved I age, gender model Norwegian, secular male, aged 30-39, earning over model References References I country of residence $100K, with graduate level education working in mass media or science, who uses relatively weak ties to people I income they met in college or at work. I religion

I relationship to recipient A below average connector: Italian, Islamic or Christian female earning less than $2K, Range of completion rates for subpopulations: with elementary school education and retired, who uses 30% to 40% strong ties to family members.

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The Small-World The Small-World Social search—the Columbia experiment Phenomenon Social search—the Columbia experiment Phenomenon

History History

An online An online experiment experiment

Previous Previous Mildly bad for continuing chain: theoretical work Basic results: theoretical work

choosing recipients because “they have lots of friends” or An improved An improved I hLi = 4.05 for all completed chains because they will “likely continue the chain.” model model References I L∗ = Estimated ‘true’ median chain length (zero References Why: attrition) I Intra-country chains: L∗ = 5 I Speciﬁcity important I Inter-country chains: L∗ = 7 I Successful links used relevant information. (e.g. connecting to someone who shares same I All chains: L∗ = 7 profession as target.) I Milgram: L∗ ' 9

15/47 16/47 The Small-World The Small-World Previous work—short paths Phenomenon Previous work—short paths Phenomenon

History History

An online An online experiment experiment

Previous Previous theoretical work theoretical work I Connected random networks have short average An improved An improved model model path lengths: References References hdABi ∼ log(N) Need “clustering” (your N = population size, friends are likely to know each other): dAB = distance between nodes A and B.

I But: social networks aren’t random...

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The Small-World The Small-World Non-randomness gives clustering Phenomenon Randomness + regularity Phenomenon

History History

B An online B An online experiment experiment

Previous Previous theoretical work theoretical work

An improved An improved model model

References References

A A

= h i dAB = 10 → too many long paths. 19/47 Now have dAB 3 d decreases overall 20/47 letters to nature

removed from a clustered neighbourhood to make a short cut has, at model—it is probably generic for many large, sparse networks most, a linear effect on C; hence C(p) remains practically unchanged found in nature. for small p even though L(p) drops rapidly. The important implica- We now investigate the functional significance of small-world tion here is that at the local level (as reflected by C(p)), the transition connectivity for dynamical systems. Our test case is a deliberately to a small world is almost undetectable. To check the robustness of simplified model for the spread of an infectious disease. The these results, we have tested many different types of initial regular population structure is modelled by the family of graphs described graphs, as well as different algorithms for random rewiring, and all in Fig. 1. At time t ¼ 0, a single infective individual is introduced give qualitatively similar results. The only requirement is that the into an otherwise healthy population. Infective individuals are rewired edges must typically connect vertices that would otherwise removed permanently (by immunity or death) after a period of8 be much farther apart than Lrandom. sickness that lasts one unit of dimensionless time. During this time, The idealized construction above reveals the key role of short each infective individual can infect each of its healthy neighbours cuts. It suggests that the small-world phenomenon might be with probability r. On subsequent time steps, the disease spreads common in sparse networks with many vertices, as even a tiny along the edges of the graph until it either infects the entire fraction of short cuts would suffice. To test this idea, we have population, or it dies out, having infected some fraction of the computed L and C for the collaboration graph of actors in feature population in the process. films (generated from data available at http://us.imdb.com), the letters to nature electrical power grid of the western United States, and the neural network of the nematode worm C. elegans17. All three graphs are of Table 1 Empirical examples of small-world networks scientific interest. The graph of film actors is a surrogate for a social 18 Lactual Lrandom Cactual Crandom removed from a clustered neighbourhood to make a short cut has, at model—it is probably generic for many large, sparse networks network , with the advantage of being much more easily specified...... most, a linear effect on C; hence C(p) remains practically unchanged found in nature. It is also akin to the graph of mathematical collaborations centred, Film actors 3.65 2.99 0.79 0.00027 traditionally, on P. Erdo¨s (partial data available at http:// Power grid 18.7 12.4 0.080 0.005 for small p even though L(p) drops rapidly. The important implica- We now investigate the functional significance of small-world C. elegans 2.65 2.25 0.28 0.05 www.acs.oakland.edu/ϳgrossman/erdoshp.html). The graph of ...... tion here is that at the local level (as reflected by C(p)), the transition connectivity for dynamical systems. Our test case is a deliberately Characteristic path length L and clustering coefficient C for three real networks, compared the power grid is relevant to the efficiency and robustness of to random graphs with the same number of vertices (n) and average number of edges per 19 to a small world is almost undetectable. To check the robustness of simplified model for the spread of an infectious disease. The power networks . And C. elegans is the sole example of a completely vertex (k). (Actors: n ¼ 225;226, k ¼ 61. Power grid: n ¼ 4;941, k ¼ 2:67. C. elegans: n ¼ 282, these results, we have tested many different types of initial regular population structure is modelled by the family of graphs described k ¼ 14.) The graphs are defined as follows. Two actors are joined by an edge if they have mapped neural network. acted in a film together. We restrict attention to the giant connected component16 of this graphs, as well as different algorithms for random rewiring, and all in Fig. 1. At time t ¼ 0, a single infective individual is introduced Table 1 shows that all three graphs are small-world networks. graph, which includes ϳ90% of all actors listed in the Internet Movie Database (available at These examples were not hand-picked; they were chosen because of http://us.imdb.com), as of April 1997. For the power grid, vertices represent generators, give qualitatively similar results. The only requirement is that the into an otherwise healthy population. Infective individuals arThee Small-World transformerThe Small-Worlds and substations, and edges represent high-voltage transmission lines rewired edges must typically connect vertices that would otherwise remSmall-worldoved permanently networks(by immunity or death) after a period ofPhenomenon Toytheir modelinherent interest and because complete wiring diagrams were betwePhenomenonen them. For C. elegans, an edge joins two neurons if they are connected by either available. Thus the small-world phenomenon is not merely a a synapse or a gap junction. We treat all edges as undirected and unweighted, and all 8 vertices as identical, recognizing that these are crude approximations. All three networks be much farther apart than Lrandom. sickness that lasts one unit of dimensionless time. During this time, 13,14 curiosity of social networks nor an artefact of an idealized show the small-world phenomenon: L Ռ Lrandom but C q Crandom. The idealized construction above reveals the key role of short eachIntroducedinfective indiv by idual can infect each of its healthy neighboursHistory History [6] An online An online cuts. It suggests that the small-world phenomenon might be withWattsprobabilit andy Strogatzr. On subsequent (Nature, 1998)time steps, the disease spreadsexperiment experiment1 “Collective dynamics of ‘small-world’ networks.” Previous Previous common in sparse networks with many vertices, as even a tiny along the edges of the graph until it either infects the entiretheoretical work theoretical work Regular Small-world Random 0.8 C(p) / C(0) fraction of short cuts would suffice. To test this idea, we have population,Small-worldor it networksdies out, having were foundinfected everywhere:some fraction of theAn improved An improved computed L and C for the collaboration graph of actors in feature population in the process. model model films (generated from data available at http://us.imdb.com), the I neural network of C. elegans, References References0.6 I semantic networks of languages, electrical power grid of the western United States, and the neural 0.4 17 network of the nematode worm C. elegans . All three graphs are of I actor collaboration graph, Table 1 Empirical examples of small-world networks L(p) / L(0) scientific interest. The graph of film actors is a surrogate for a social 0.2 I food webs, 18 Lactual Lrandom Cactual Crandom network , with the advantage of being much more easily specified...... I social networks of comic book characters,... It is also akin to the graph of mathematical collaborations centred, Film actors 3.65 2.99 0.79 0.00027 0 Power grid 18.7 12.4 0.080 0.005 p = 0 p = 1 0.0001 0.001 0.01 0.1 1 traditionally, on P. Erdo¨s (partial data available at http:// Increasing randomness C. elegVeryans weak requirements:2.65 2.25 0.28 0.05 p www.acs.oakland.edu/ϳgrossman/erdoshp.html). The graph of ...... Characteristic path length L and clustering coefficient C for three real networks, compared Figure 1 Random rewiring procedure for interpolating between a regular ring Figure 2 Characteristic path length L(p) and clustering coefficient C(p) for the the power grid is relevant to the efficiency and robustness of to randomIgrlocalaphs with regularitythe same+number randomof verticesshort(n cuts) and average number of edges per 19 lattice and a random network, without altering the number of vertices or edges in family of randomly rewired graphs described in Fig. 1. Here L is defined as the power networks . And C. elegans is the sole example of a completely vertex (k). (Actors: n ¼ 225;226, k ¼ 61. Power grid: n ¼ 4;941, k ¼ 2:67. C. elegans: n ¼ 282, k ¼ 14.) The graphs are defined as follows. Two actors are joined by an edge if they hav21/47e the graph. We start with a ring of n vertices, each connected to its k nearest number22/47 of edges in the shortest path between two vertices, averaged over all mapped neural network. acted in a film together. We restrict attention to the giant connected component16 of this neighbours by undirected edges. (For clarity, n ¼ 20 and k ¼ 4 in the schematic pairs of vertices. The clustering coefficient C(p) is defined as follows. Suppose

Table 1 shows that all three graphs are small-world networks. graph, which includes ϳ90% of all actors listed in the Internet Movie Database (available at examples shown here, but much larger n and k are used in the rest of this Letter.) that a vertex v has kv neighbours; then at most kvðkv Ϫ 1Þ=2 edges can exist http://us.imdb.com), as of April 1997. For the power grid, vertices represent generators, We choose a vertex and the edge that connects it to its nearest neighbour in a between them (this occurs when every neighbour of v is connected to every other These examples were not hand-picked; they were chosen because of transformers and substations, and edges represent high-voltage transmission lines their inherent interest and because complete wiring diagrams were between them. For C. elegans, an edge joins two neurons if they are connected by either clockwise sense. With probability p, we reconnect this edge to a vertex chosen neighbour of v). Let Cv denote the fraction of these allowable edges that actually a synapse or a gap junction. We treat all edges as undirected and unweighted, and all uniformly at random over the entire ring, with duplicate edges forbidden; other- exist. Define C as the average of Cv over all v. For friendship networks, these The Small-World The Small-World available. Thus the small-world phenomenon is not merely a vertices as identical, recognizing that these are crude approximations. All three networks wise we leave the edge in place. We repeat this process by moving clockwise statistics have intuitive meanings: L is the average number of friendships in the 13,14 The structural small-world property Phenomenon Previous work—finding short paths Phenomenon show the small-world phenomenon: L Ռ Lrandom but C q Crandom. curiosity of social networks nor an artefact of an idealized around the ring, considering each vertex in turn until one lap is completed. Next, shortest chain connecting two people; Cv reflects the extent to which friends of v we consider the edges that connect vertices to their second-nearest neighbours are also friends of each other; and thus C measures the cliquishness of a typical History History 1 clockwise. As before, we randomly rewire each of these edges with probability p, friendship circle. The data shown in the figure are averages over 20 random An online An online experiment and continue this process, circulating around the ring and proceeding outward to realizationsexperiment of the rewiring process described in Fig.1, and have been normalized more distant neighbours after each lap, until each edge in the original lattice has by the values L(0), C(0) for a regular lattice. All the graphs have n ¼ 1;000 vertices Regular Small-world Random 0.8 C(p) / C(0) Previous Previous theoretical work been considered once. (As there are nk/2 edges in the entire graph, the rewiring andtheoreticalan av workerage degree of k ¼ 10 edges per vertex. We note that a logarithmic An improved process stops after k/2 laps.) Three realizations of this process are shown, for horizontalAn improvedscale has been used to resolve the rapid drop in L(p), corresponding to model But are these short cuts findable? model 0.6 different values of p. For p ¼ 0, the original ring is unchanged; as p increases, the the onset of the small-world phenomenon. During this drop, C(p) remains almost References References No.graph becomes increasingly disordered until for p ¼ 1, all edges are rewired constant at its value for the regular lattice, indicating that the transition to a small randomly. One of our main results is that for intermediate values of p, the graph is world is almost undetectable at the local level. 0.4 a small-world network: highly clustered like a regular graph, yet with small Nodescharacteriscannottic path length,find eachlike a rando otherm gr quicklyaph. (See Fig. 2.) L(p) / L(0) with any local search method. 0.2 Nature © Macmillan Publishers Ltd 1998 NATURE | VOL 393 | 4 JUNE 1998 441

0 p = 0 p = 1 0.0001 0.001 0.01 0.1 1 Increasing randomness p

Figure 1 Random rewiring procedure for interpolating between a regular ring Figure 2 Characteristic path length L(p) and clustering coefficient C(p) for the23/47 24/47 lattice and a random network, without altering the number of vertices or edges in family of randomly rewired graphs described in Fig. 1. Here L is defined as the the graph. We start with a ring of n vertices, each connected to its k nearest number of edges in the shortest path between two vertices, averaged over all neighbours by undirected edges. (For clarity, n ¼ 20 and k ¼ 4 in the schematic pairs of vertices. The clustering coefficient C(p) is defined as follows. Suppose examples shown here, but much larger n and k are used in the rest of this Letter.) that a vertex v has kv neighbours; then at most kvðkv Ϫ 1Þ=2 edges can exist We choose a vertex and the edge that connects it to its nearest neighbour in a between them (this occurs when every neighbour of v is connected to every other clockwise sense. With probability p, we reconnect this edge to a vertex chosen neighbour of v). Let Cv denote the fraction of these allowable edges that actually uniformly at random over the entire ring, with duplicate edges forbidden; other- exist. Define C as the average of Cv over all v. For friendship networks, these wise we leave the edge in place. We repeat this process by moving clockwise statistics have intuitive meanings: L is the average number of friendships in the around the ring, considering each vertex in turn until one lap is completed. Next, shortest chain connecting two people; Cv reflects the extent to which friends of v we consider the edges that connect vertices to their second-nearest neighbours are also friends of each other; and thus C measures the cliquishness of a typical clockwise. As before, we randomly rewire each of these edges with probability p, friendship circle. The data shown in the figure are averages over 20 random and continue this process, circulating around the ring and proceeding outward to realizations of the rewiring process described in Fig.1, and have been normalized more distant neighbours after each lap, until each edge in the original lattice has by the values L(0), C(0) for a regular lattice. All the graphs have n ¼ 1;000 vertices been considered once. (As there are nk/2 edges in the entire graph, the rewiring and an average degree of k ¼ 10 edges per vertex. We note that a logarithmic process stops after k/2 laps.) Three realizations of this process are shown, for horizontal scale has been used to resolve the rapid drop in L(p), corresponding to different values of p. For p ¼ 0, the original ring is unchanged; as p increases, the the onset of the small-world phenomenon. During this drop, C(p) remains almost graph becomes increasingly disordered until for p ¼ 1, all edges are rewired constant at its value for the regular lattice, indicating that the transition to a small randomly. One of our main results is that for intermediate values of p, the graph is world is almost undetectable at the local level. a small-world network: highly clustered like a regular graph, yet with small characteristic path length, like a random graph. (See Fig. 2.)

Nature © Macmillan Publishers Ltd 1998 NATURE | VOL 393 | 4 JUNE 1998 441 The Small-World The Small-World Previous work—ﬁnding short paths Phenomenon Previous work—ﬁnding short paths Phenomenon

History History

An online An online I What can a local search method reasonably use? experiment experiment Previous [3] Previous I How to ﬁnd things without a map? theoretical work Jon Kleinberg (Nature, 2000) theoretical work I Need some measure of distance between friends An improved “Navigation in a small world.” An improved model model and the target. References References Allowed to vary: Some possible knowledge: 1. local search algorithm and I Target’s identity 2. network structure. I Friends’ popularity

I Friends’ identities

I Where message has been

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The Small-World The Small-World Previous work—ﬁnding short paths Phenomenon Previous work—ﬁnding short paths Phenomenon

Kleinberg’s Network: History History An online An online experiment experiment 1. Start with regular d-dimensional cubic lattice. Previous Previous 2. Add local links so nodes know all nodes within a theoretical work Theoretical optimal search: theoretical work An improved An improved distance q. model model “Greedy” algorithm. 3. Add m short cuts per node. References I References Same number of connections at all scales: α = d. 4. Connect i to j with probability I

−α 2 pij ∝ dij . Search time grows slowly with system size (like log N).

But: social networks aren’t lattices plus links.

I α = 0: random connections.

I α large: reinforce local connections. α = I d: same number of connections at all scales. 27/47 28/47 The Small-World The Small-World Previous work—ﬁnding short paths Phenomenon The problem Phenomenon

History History

An online An online experiment experiment

Previous Previous theoretical work If there are no hubs and no underlying lattice, how can theoretical work I If networks have hubs can also search well: Adamic [1] An improved search be efﬁcient? An improved et al. (2001) model model −γ References References P(ki ) ∝ ki Which friend of a is closest where k = degree of node i (number of friends). b to the target b? Basic idea: get to hubs ﬁrst I What does ‘closest’ mean? (airline networks). a I But: hubs in social networks are limited. What is ‘social distance’?

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The Small-World The Small-World The model Phenomenon Social distance—Bipartite afﬁliation networks Phenomenon

One approach: incorporate identity. History History contexts (See “Identity and Search in Social Networks.” Science, 2002, An online 1 2 3 4 An online Watts, Dodds, and Newman [5]) experiment experiment Previous Previous theoretical work theoretical work

An improved An improved Identity is formed from attributes such as: model model

References References Geographic location I individuals I Type of employment a b c d e

I Religious beliefs

I Recreational activities. b d

Groups are formed by people with at least one similar attribute. a unipartite network e c Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. 31/47 32/47 The Small-World The Small-World Social distance—Context distance Phenomenon The model Phenomenon

History History

An online An online occupation experiment experiment Previous Distance between two individuals xij is the height of Previous theoretical work theoretical work lowest common ancestor. education health care An improved An improved model model

References l=4 References b=2 high school kindergarten teacher teacher nurse doctor g=6

j v i k

xij = 3, xik = 1, xiv = 4.

a b c d e

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The Small-World The Small-World The model Phenomenon Social distance—Generalized context space Phenomenon

History History

An online An online experiment experiment Previous geography occupation age Previous I Individuals are more likely to know each other the theoretical work theoretical work

closer they are within a hierarchy. An improved 0 100 An improved model model

I Construct z connections for each node using References References

pij = c exp{−αxij }.

a b c d e I α = 0: random connections. (Blau & Schwartz, Simmel, Breiger) I α large: local connections.

35/47 36/47 The Small-World The Small-World The model Phenomenon The model Phenomenon

h=1 h=2 History History An online An online experiment experiment

Previous Previous theoretical work Triangle inequality doesn’t hold: theoretical work

An improved An improved i j i j model h=1 h=2 model References References h=3

i j,k i, j k

i, j yik = 4 > yij + yjk = 1 + 1 = 2.

T T ~vi = [1 1 1] , ~vj = [8 4 1] Social distance: 1 2 3 h xij = 4, xij = 3, x = 1. yij = min xij . ij h

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The Small-World The Small-World The model Phenomenon The model-results—searchable networks Phenomenon

History α = 0 versus α = 2 for N ' 105: History An online −0.5 An online experiment experiment q ≥ r Individuals know the identity vectors of Previous −1 Previous I theoretical work q < r theoretical work An improved q An improved 1. themselves, 10 −1.5 model r = 0.05 model 2. their friends, log References References and −2 3. the target. −2.5 I Individuals can estimate the social distance between 1 3 5 7 9 11 13 15 H their friends and the target. q = probability an arbitrary message chain reaches a I Use a greedy algorithm + allow searches to fail target. randomly. I A few dimensions help.

I Searchability decreases as population increases.

I Precise form of hierarchy largely doesn’t matter. 39/47 40/47 The Small-World The Small-World The model-results Phenomenon Social search—Data Phenomenon

History History

An online An online Milgram’s Nebraska-Boston data: experiment experiment Previous Previous Model parameters: theoretical work theoretical work 12 An improved Adamic and Adar (2003) An improved 8 model model 10 I N = 10 , References I For HP Labs, found probability of connection as References 8 ) I z = 300, g = 100, function of organization distance well ﬁt by L ( 6 n I b = 10, exponential distribution. 4 I α = 1, H = 2; I Probability of connection as function of real distance 2 ∝ 1/r. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 L I hLmodeli ' 6.7 I Ldata ' 6.5

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The Small-World The Small-World Social Search—Real world uses Phenomenon Social Search—Real world uses Phenomenon

History History

An online An online experiment experiment

Previous Previous theoretical work theoretical work I Tags create identities for objects An improved Recommender systems: An improved I Website tagging: http://www.del.icio.us model model References References I (e.g., Wikipedia) I Amazon uses people’s actions to build effective connections between books. I Photo tagging: http://www.flickr.com Conﬂict between ‘expert judgments’ and I Dynamic creation of metadata plus links between I information objects. tagging of the hoi polloi.

I Folksonomy: collaborative creation of metadata

43/47 44/47 The Small-World The Small-World Conclusions Phenomenon References I Phenomenon

History [1] L. Adamic, R. Lukose, A. Puniyani, and History An online An online experiment B. Huberman. experiment

Previous Search in power-law networks. Previous theoretical work theoretical work I Bare networks are typically unsearchable. Phys. Rev. E, 64:046135, 2001. pdf () An improved An improved model model I Paths are ﬁndable if nodes understand how network [2] P. S. Dodds, R. Muhamad, and D. J. Watts. References References is formed. An experimental study of search in global social I Importance of identity (interaction contexts). networks.

I Improved social network models. Science, 301:827–829, 2003. pdf () I Construction of peer-to-peer networks. [3] J. Kleinberg. I Construction of searchable information databases. Navigation in a small world. Nature, 406:845, 2000. pdf () [4] J. Travers and S. Milgram. An experimental study of the small world problem. 45/47 Sociometry, 32:425–443, 1969. pdf () 46/47

The Small-World References II Phenomenon

History

An online experiment

Previous theoretical work [5] D. J. Watts, P. S. Dodds, and M. E. J. Newman. An improved model Identity and search in social networks. References Science, 296:1302–1305, 2002. pdf () [6] D. J. Watts and S. J. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393:440–442, 1998. pdf ()

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