Core Models of Complex Networks Principles of Complex Systems - PowerPoint PPT Presentation

Core Models of Complex Networks Core Models of Complex Networks Principles of Complex Systems Generalized random networks CSYS/MATH 300, Spring, 2013 | #SpringPoCS2013 Small-world networks Main story Generalized affiliation networks Prof. Peter Dodds Nutshell Scale-free @peterdodds networks Main story A more plausible mechanism Department of Mathematics & Statistics | Center for Complex Systems | Robustness Vermont Advanced Computing Center | University of Vermont Redner & Krapivisky’s model Nutshell References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . 1 of 107

Core Models of These slides brought to you by: Complex Networks Generalized random networks Small-world networks Main story Generalized affiliation networks Nutshell Scale-free networks Main story A more plausible mechanism Robustness Redner & Krapivisky’s model Nutshell References 2 of 107

Core Models of Outline Complex Networks Generalized Generalized random networks random networks Small-world networks Small-world networks Main story Main story Generalized affiliation networks Nutshell Generalized affiliation networks Scale-free Nutshell networks Main story A more plausible mechanism Scale-free networks Robustness Redner & Krapivisky’s Main story model Nutshell A more plausible mechanism References Robustness Redner & Krapivisky’s model Nutshell References 3 of 107

Core Models of Models Complex Networks Generalized random networks Small-world networks Main story Generalized affiliation Some important models: networks Nutshell 1. Generalized random networks; Scale-free networks 2. Small-world networks; Main story A more plausible mechanism 3. Generalized affiliation networks; Robustness Redner & Krapivisky’s model 4. Scale-free networks; Nutshell 5. Statistical generative models ( p ∗ ). References 4 of 107

Core Models of Models Complex Networks Generalized random networks Small-world networks Main story Generalized random networks: Generalized affiliation networks Nutshell ◮ Arbitrary degree distribution P k . Scale-free networks ◮ Create (unconnected) nodes with degrees sampled Main story A more plausible from P k . mechanism Robustness ◮ Wire nodes together randomly. Redner & Krapivisky’s model Nutshell ◮ Create ensemble to test deviations from References randomness. 5 of 107

Core Models of Building random networks: Stubs Complex Networks Phase 1: Generalized random networks ◮ Idea: start with a soup of unconnected nodes with Small-world stubs (half-edges): networks Main story Generalized affiliation networks Nutshell Scale-free networks Main story A more plausible mechanism Robustness Redner & Krapivisky’s model ◮ Randomly select stubs Nutshell (not nodes!) and References connect them. ◮ Must have an even number of stubs. ◮ Initially allow self- and repeat connections. 6 of 107

Core Models of Building random networks: First rewiring Complex Networks Generalized random networks Small-world Phase 2: networks Main story ◮ Now find any (A) self-loops and (B) repeat edges and Generalized affiliation networks Nutshell randomly rewire them. Scale-free networks Main story A more plausible mechanism (A) (B) Robustness Redner & Krapivisky’s model ◮ Being careful: we can’t change the degree of any Nutshell node, so we can’t simply move links around. References ◮ Simplest solution: randomly rewire two edges at a time. 7 of 107

Core Models of General random rewiring algorithm Complex Networks i 2 e 1 i 1 Generalized random networks ◮ Randomly choose two edges. Small-world networks (Or choose problem edge and Main story a random edge) Generalized affiliation networks Nutshell ◮ Check to make sure edges Scale-free are disjoint. networks i 4 Main story e i 3 A more plausible 2 mechanism Robustness i 2 Redner & Krapivisky’s i model 1 Nutshell ◮ Rewire one end of each edge. References ◮ Node degrees do not change. e’ ◮ Works if e 1 is a self-loop or e’ 2 1 repeated edge. ◮ Same as finding on/off/on/off i 4 4-cycles. and rotating them. i 3 8 of 107

Core Models of Sampling random networks Complex Networks Generalized random networks Small-world networks Phase 2: Main story Generalized affiliation networks ◮ Use rewiring algorithm to remove all self and repeat Nutshell Scale-free loops. networks Main story A more plausible mechanism Phase 3: Robustness Redner & Krapivisky’s ◮ Randomize network wiring by applying rewiring model Nutshell algorithm liberally. References ◮ Rule of thumb: # Rewirings ≃ 10 × # edges [10] . 9 of 107

Core Models of People thinking about people: Complex Networks How are social networks structured? Generalized random networks ◮ How do we define and measure connections? Small-world networks ◮ Methods/issues of self-report and remote sensing. Main story Generalized affiliation networks Nutshell What about the dynamics of social networks? Scale-free networks ◮ How do social networks/movements begin & evolve? Main story A more plausible mechanism ◮ How does collective problem solving work? Robustness Redner & Krapivisky’s model ◮ How does information move through social networks? Nutshell References ◮ Which rules give the best ‘game of society?’ Sociotechnical phenomena and algorithms: ◮ What can people and computers do together? (google) ◮ Use Play + Crunch to solve problems. Which problems? 11 of 107

Core Models of Social Search Complex Networks Generalized random networks Small-world networks Main story Generalized affiliation networks A small slice of the pie: Nutshell Scale-free ◮ Q. Can people pass messages between distant networks Main story individuals using only their existing social A more plausible mechanism connections? Robustness Redner & Krapivisky’s model ◮ A. Apparently yes... Nutshell References 12 of 107

Core Models of Milgram’s social search experiment (1960s) Complex Networks Generalized random networks ◮ Target person = Small-world networks Boston stockbroker. Main story Generalized affiliation ◮ 296 senders from Boston and networks Nutshell Omaha. Scale-free networks ◮ 20% of senders reached Main story A more plausible target. mechanism Robustness ◮ chain length ≃ 6.5. Redner & Krapivisky’s model Nutshell References Popular terms: ◮ The Small World Phenomenon; ◮ “Six Degrees of Separation.” http://www.stanleymilgram.com 13 of 107

Core Models of The problem Complex Networks Generalized Lengths of successful chains: random networks Small-world 18 networks Main story Generalized affiliation networks 15 Nutshell From Travers and Scale-free networks 12 Milgram (1969) in Main story Sociometry: [13] A more plausible mechanism n ( L ) 9 Robustness “An Experimental Redner & Krapivisky’s model Study of the Small Nutshell 6 References World Problem.” 3 0 1 2 3 4 5 6 7 8 9 10 11 12 L 14 of 107

Core Models of The problem Complex Networks Generalized random networks Small-world networks Main story Generalized affiliation networks Two features characterize a social ‘Small World’: Nutshell Scale-free 1. Short paths exist, (= Geometric piece) networks Main story and A more plausible mechanism Robustness 2. People are good at finding them. (= Algorithmic Redner & Krapivisky’s model piece) Nutshell References 15 of 107

Core Models of Social Search Complex Networks Milgram’s small world experiment with email: Generalized random networks Small-world networks Main story Generalized affiliation networks Nutshell Scale-free networks Main story A more plausible mechanism Robustness Redner & Krapivisky’s model Nutshell References “An Experimental study of Search in Global Social Networks” P . S. Dodds, R. Muhamad, and D. J. Watts, Science , Vol. 301, pp. 827–829, 2003. [6] 16 of 107

Core Models of Social search—the Columbia experiment Complex Networks Generalized random networks ◮ 60,000+ participants in 166 countries Small-world networks ◮ 18 targets in 13 countries including Main story ◮ a professor at an Ivy League university, Generalized affiliation networks ◮ an archival inspector in Estonia, Nutshell Scale-free ◮ a technology consultant in India, networks ◮ a policeman in Australia, Main story A more plausible and mechanism Robustness ◮ a veterinarian in the Norwegian army. Redner & Krapivisky’s model ◮ 24,000+ chains Nutshell References We were lucky and contagious (more later): “Using E-Mail to Count Connections” ( ⊞ ), Sarah Milstein, New York Times, Circuits Section (December, 2001) 17 of 107