The Small-World Phenomenon History An online Complex Networks, - PowerPoint PPT Presentation

The Small-World Phenomenon The Small-World Phenomenon History An online Complex Networks, Course 295A, Spring, 2008 experiment Previous theoretical work An improved Prof. Peter Dodds model References Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/47

The Small-World Outline Phenomenon History An online experiment History Previous theoretical work An improved model An online experiment References Previous theoretical work An improved model References Frame 2/47

The Small-World Some problems for sociologists Phenomenon History An online How are social networks structured? experiment Previous ◮ How do we define connections? theoretical work ◮ How do we measure connections? An improved model ◮ (remote sensing, self-reporting) References What about the dynamics of social networks? ◮ How do social networks evolve? ◮ How do social movements begin? ◮ How does collective problem solving work? ◮ How is information transmitted through social networks? Frame 3/47

The Small-World Social Search Phenomenon History An online experiment A small slice of the pie: Previous theoretical work ◮ Q. Can people pass messages between distant An improved model individuals using only their existing social References connections? ◮ A. Apparently yes... Handles: ◮ The Small World Phenomenon ◮ or “Six Degrees of Separation.” Frame 4/47

The problem The Small-World Phenomenon History An online experiment Previous theoretical work Stanley Milgram et al., late 1960’s: An improved model References ◮ Target person worked in Boston as a stockbroker. ◮ 296 senders from Boston and Omaha. ◮ 20% of senders reached target. ◮ average chain length ≃ 6.5. Frame 5/47

The problem The Small-World Phenomenon History Lengths of successful chains: An online experiment 18 Previous theoretical work 15 An improved model From Travers and References 12 Milgram (1969) in Sociometry: [4] n ( L ) 9 “An Experimental Study of the Small 6 World Problem.” 3 0 1 2 3 4 5 6 7 8 9 10 11 12 L Frame 6/47

The problem The Small-World Phenomenon History An online experiment Previous theoretical work An improved Two features characterize a social ‘Small World’: model References 1. Short paths exist and 2. People are good at finding them. Frame 7/47

The Small-World Social Search Phenomenon History Milgram’s small world experiment with e-mail [2] An online experiment Previous theoretical work An improved model References Frame 8/47

The Small-World Social search—the Columbia experiment Phenomenon History An online experiment Previous ◮ 60,000+ participants in 166 countries theoretical work An improved ◮ 18 targets in 13 countries including model ◮ a professor at an Ivy League university, References ◮ an archival inspector in Estonia, ◮ a technology consultant in India, ◮ a policeman in Australia, and ◮ a veterinarian in the Norwegian army. ◮ 24,000+ chains Frame 9/47

The Small-World Social search—the Columbia experiment Phenomenon History An online experiment Previous theoretical work ◮ Milgram’s participation rate was roughly 75% An improved model ◮ Email version: Approximately 37% participation rate. References ◮ Probability of a chain of length 10 getting through: . 37 10 ≃ 5 × 10 − 5 ◮ ⇒ 384 completed chains (1.6% of all chains). Frame 10/47

The Small-World Social search—the Columbia experiment Phenomenon History An online experiment Previous theoretical work ◮ Motivation/Incentives/Perception matter. An improved model ◮ If target seems reachable References ⇒ participation more likely. ◮ Small changes in attrition rates ⇒ large changes in completion rates ◮ e.g., ց 15% in attrition rate ⇒ ր 800% in completion rate Frame 11/47

The Small-World Social search—the Columbia experiment Phenomenon History Successful chains disproportionately used An online experiment ◮ weak ties (Granovetter) Previous theoretical work ◮ professional ties (34% vs. 13%) An improved model ◮ ties originating at work/college References ◮ target’s work (65% vs. 40%) . . . and disproportionately avoided ◮ hubs (8% vs. 1%) (+ no evidence of funnels) ◮ family/friendship ties (60% vs. 83%) Geography → Work Frame 12/47

The Small-World Social search—the Columbia experiment Phenomenon History An online experiment Senders of successful messages showed Previous theoretical work little absolute dependency on An improved ◮ age, gender model References ◮ country of residence ◮ income ◮ religion ◮ relationship to recipient Range of completion rates for subpopulations: 30% to 40% Frame 13/47

The Small-World Social search—the Columbia experiment Phenomenon History An online experiment Nevertheless, some weak discrepencies do exist... Previous theoretical work An above average connector: An improved model Norwegian, secular male, aged 30-39, earning over References $100K, with graduate level education working in mass media or science, who uses relatively weak ties to people they met in college or at work. A below average connector: Italian, Islamic or Christian female earning less than $2K, with elementary school education and retired, who uses strong ties to family members. Frame 14/47

The Small-World Social search—the Columbia experiment Phenomenon History An online experiment Previous Mildly bad for continuing chain: theoretical work choosing recipients because “they have lots of friends” or An improved model because they will “likely continue the chain.” References Why: ◮ Specificity important ◮ Successful links used relevant information. (e.g. connecting to someone who shares same profession as target.) Frame 15/47

The Small-World Social search—the Columbia experiment Phenomenon History An online experiment Previous Basic results: theoretical work An improved ◮ � L � = 4 . 05 for all completed chains model ◮ L ∗ = Estimated ‘true’ median chain length (zero References attrition) ◮ Intra-country chains: L ∗ = 5 ◮ Inter-country chains: L ∗ = 7 ◮ All chains: L ∗ = 7 ◮ Milgram: L ∗ ≃ 9 Frame 16/47

Previous work—short paths The Small-World Phenomenon History An online experiment Previous theoretical work ◮ Connected random networks have short average An improved model path lengths: References � d AB � ∼ log ( N ) N = population size, d AB = distance between nodes A and B . ◮ But: social networks aren’t random... Frame 17/47

Previous work—short paths The Small-World Phenomenon History An online experiment Previous theoretical work An improved model References Need “clustering” (your friends are likely to know each other): Frame 18/47

Non-randomness gives clustering The Small-World Phenomenon History B An online experiment Previous theoretical work An improved model References A d AB = 10 → too many long paths. Frame 19/47

Randomness + regularity The Small-World Phenomenon History B An online experiment Previous theoretical work An improved model References A Now have d AB = 3 � d � decreases overall Frame 20/47

The Small-World Small-world networks Phenomenon History Introduced by Watts and Strogatz (Nature, 1998) [5] An online experiment “Collective dynamics of ‘small-world’ networks.” Previous theoretical work Small-world networks were found everywhere: An improved model ◮ neural network of C. elegans, References ◮ semantic networks of languages, ◮ actor collaboration graph, ◮ food webs, ◮ social networks of comic book characters,... Very weak requirements: ◮ local regularity + random short cuts Frame 21/47

Toy model The Small-World Phenomenon History An online experiment Previous theoretical work Regular Small-world Random An improved model References p = 0 p = 1 Increasing randomness Frame 22/47

The structural small-world property The Small-World Phenomenon History 1 An online experiment Previous C ( p ) / C (0) 0.8 theoretical work An improved model 0.6 References 0.4 L ( p ) / L (0) 0.2 0 0.0001 0.001 0.01 0.1 1 p Frame 23/47

The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous theoretical work An improved But are these short cuts findable? model References No. Nodes cannot find each other quickly with any local search method. Frame 24/47

The Small-World Previous work—finding short paths Phenomenon History An online ◮ What can a local search method reasonably use? experiment Previous ◮ How to find things without a map? theoretical work ◮ Need some measure of distance between friends An improved model and the target. References Some possible knowledge: ◮ Target’s identity ◮ Friends’ popularity ◮ Friends’ identities ◮ Where message has been Frame 25/47

The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous Jon Kleinberg (Nature, 2000) [3] theoretical work An improved “Navigation in a small world.” model References Allowed to vary: 1. local search algorithm and 2. network structure. Frame 26/47