The Small-World Social search—the Columbia experiment Phenomenon History An online experiment Previous theoretical work ◮ Motivation/Incentives/Perception matter. An improved model References Frame 11/47
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. Frame 11/47
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 Frame 11/47
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 Previous theoretical work An improved model References Frame 12/47
The Small-World Social search—the Columbia experiment Phenomenon History Successful chains disproportionately used An online experiment ◮ weak ties (Granovetter) Previous theoretical work An improved model References Frame 12/47
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 References Frame 12/47
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 Frame 12/47
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%) Frame 12/47
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 Frame 12/47
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) Frame 12/47
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%) Frame 12/47
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 Frame 13/47
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 Frame 13/47
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 Frame 13/47
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 Frame 13/47
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 Frame 13/47
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. Frame 14/47
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 Frame 15/47
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 Frame 15/47
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 References Frame 16/47
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) Frame 16/47
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 Frame 16/47
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 Frame 16/47
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 Frame 16/47
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 . Frame 17/47
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 An improved model References Frame 21/47
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 Frame 21/47
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, Frame 21/47
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, Frame 21/47
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, Frame 21/47
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,... Frame 21/47
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 Frame 21/47
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 Frame 24/47
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. Frame 24/47
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 theoretical work An improved model References Frame 25/47
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 An improved model References Frame 25/47
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 Frame 25/47
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 Frame 26/47
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 Frame 26/47
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
The Small-World Previous work—finding short paths Phenomenon History Kleinberg’s Network: An online experiment Previous theoretical work An improved model References Frame 27/47
The Small-World Previous work—finding short paths Phenomenon History Kleinberg’s Network: An online experiment 1. Start with regular d-dimensional cubic lattice. Previous theoretical work An improved model References Frame 27/47
The Small-World Previous work—finding short paths Phenomenon History Kleinberg’s Network: An online experiment 1. Start with regular d-dimensional cubic lattice. Previous theoretical work 2. Add local links so nodes know all nodes within a An improved distance q . model References Frame 27/47
The Small-World Previous work—finding short paths Phenomenon History Kleinberg’s Network: An online experiment 1. Start with regular d-dimensional cubic lattice. Previous theoretical work 2. Add local links so nodes know all nodes within a An improved distance q . model References 3. Add m short cuts per node. Frame 27/47
The Small-World Previous work—finding short paths Phenomenon History Kleinberg’s Network: An online experiment 1. Start with regular d-dimensional cubic lattice. Previous theoretical work 2. Add local links so nodes know all nodes within a An improved distance q . model References 3. Add m short cuts per node. 4. Connect i to j with probability p ij ∝ d ij − α . Frame 27/47
The Small-World Previous work—finding short paths Phenomenon History Kleinberg’s Network: An online experiment 1. Start with regular d-dimensional cubic lattice. Previous theoretical work 2. Add local links so nodes know all nodes within a An improved distance q . model References 3. Add m short cuts per node. 4. Connect i to j with probability p ij ∝ d ij − α . ◮ α = 0: random connections. ◮ α large: reinforce local connections. ◮ α = d : same number of connections at all scales. Frame 27/47
The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous theoretical work Theoretical optimal search: An improved model ◮ “Greedy” algorithm. References Frame 28/47
The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous theoretical work Theoretical optimal search: An improved model ◮ “Greedy” algorithm. References ◮ Same number of connections at all scales: α = d . Frame 28/47
The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous theoretical work Theoretical optimal search: An improved model ◮ “Greedy” algorithm. References ◮ Same number of connections at all scales: α = d . Search time grows slowly with system size (like log 2 N ). Frame 28/47
The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous theoretical work Theoretical optimal search: An improved model ◮ “Greedy” algorithm. References ◮ Same number of connections at all scales: α = d . Search time grows slowly with system size (like log 2 N ). But: social networks aren’t lattices plus links. Frame 28/47
The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous theoretical work ◮ If networks have hubs can also search well: Adamic An improved et al. (2001) [1] model References P ( k i ) ∝ k − γ i where k = degree of node i (number of friends). Frame 29/47
The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous theoretical work ◮ If networks have hubs can also search well: Adamic An improved et al. (2001) [1] model References P ( k i ) ∝ k − γ i where k = degree of node i (number of friends). ◮ Basic idea: get to hubs first (airline networks). Frame 29/47
The Small-World Previous work—finding short paths Phenomenon History An online experiment Previous theoretical work ◮ If networks have hubs can also search well: Adamic An improved et al. (2001) [1] model References P ( k i ) ∝ k − γ i where k = degree of node i (number of friends). ◮ Basic idea: get to hubs first (airline networks). ◮ But: hubs in social networks are limited. Frame 29/47
The problem The Small-World Phenomenon History An online experiment Previous If there are no hubs and no underlying lattice, how can theoretical work search be efficient? An improved model References Which friend of a is closest b to the target b? What does ‘closest’ mean? a What is ‘social distance’? Frame 30/47
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