Complex Networks Basic definitions Principles of Complex Systems - PowerPoint PPT Presentation

Overview of Complex Networks Complex Networks Basic definitions Principles of Complex Systems Books Course 300, Fall, 2008 Examples of Complex Networks Properties of Complex Networks Prof. Peter Dodds Basic models of complex networks Generalized random networks Department of Mathematics & Statistics Scale-free networks University of Vermont Small-world networks Generalized affiliation networks References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/95

Overview of Outline Complex Networks Basic definitions Basic definitions Books Examples of Books Complex Networks Properties of Complex Networks Examples of Complex Networks Basic models of complex networks Generalized random Properties of Complex Networks networks Scale-free networks Small-world networks Generalized affiliation Basic models of complex networks networks Generalized random networks References Scale-free networks Small-world networks Generalized affiliation networks References Frame 2/95

Overview of Basic definitions Complex Networks Basic definitions Network: (net + work, 1500’s) Books Examples of Noun: Complex Networks Properties of 1. Any interconnected group or system Complex Networks Basic models of 2. Multiple computers and other devices connected complex networks Generalized random together to share information networks Scale-free networks Small-world networks Generalized affiliation networks Verb: References 1. To interact socially for the purpose of getting connections or personal advancement 2. To connect two or more computers or other computerized devices Frame 3/95

Overview of Observation Complex Networks Basic definitions Books Examples of Complex Networks Properties of Complex Networks ◮ Many complex systems can be regarded as complex Basic models of networks of physical or abstract interactions complex networks Generalized random ◮ Opens door to mathematical and numerical analysis networks Scale-free networks Small-world networks ◮ Dominant approach of last decade of a Generalized affiliation networks theoretical-physics/stat-mechish flavor. References Frame 4/95

Overview of Basic definitions Complex Networks Basic definitions Books Examples of Complex Networks Properties of Complex Networks Nodes = A collection of entities which have Basic models of properties that are somehow related to each other complex networks Generalized random networks ◮ e.g., people, forks in rivers, proteins, webpages, Scale-free networks Small-world networks organisms,... Generalized affiliation networks References Frame 5/95

Overview of Basic definitions Complex Networks Basic definitions Books Examples of Links = Connections between nodes Complex Networks Properties of ◮ links Complex Networks ◮ may be real and fixed (rivers), Basic models of complex networks ◮ real and dynamic (airline routes), Generalized random networks ◮ abstract with physical impact (hyperlinks), Scale-free networks ◮ or purely abstact (semantic connections between Small-world networks Generalized affiliation networks concepts). References ◮ Links may be directed or undirected. ◮ Links may be binary or weighted. Frame 6/95

Overview of Basic definitions Complex Networks Basic definitions Books Examples of Complex Networks Properties of Node degree = Number of links per node Complex Networks Basic models of ◮ Notation: Node i ’s degree = k i . complex networks Generalized random networks ◮ k i = 0,1,2,. . . . Scale-free networks Small-world networks ◮ Notation: the average degree of a network = � k � Generalized affiliation networks (and sometimes as z ) References Frame 7/95

Overview of Basic definitions Complex Networks Basic definitions Books Adjacency matrix: Examples of Complex Networks Properties of ◮ We represent a graph or network by a matrix A with Complex Networks link weight a ij for nodes i and j in entry ( i , j ) . Basic models of complex networks ◮ e.g., Generalized random networks   0 1 1 1 0 Scale-free networks Small-world networks 0 0 1 0 1 Generalized affiliation   networks   A = 1 0 0 0 0 References     0 1 0 0 1   0 1 0 1 0 Frame 8/95

Overview of Books Complex Networks Basic definitions Books Examples of Complex Networks Nexus: Small Worlds and the Groundbreaking Properties of Complex Networks Science of Networks—Mark Buchanan Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References The Tipping Point: How Little Things can make a Big Difference—Malcolm Gladwell Frame 9/95

Overview of Books Complex Networks Basic definitions Books Examples of Complex Networks Linked: How Everything Is Connected to Properties of Everything Else and What It Complex Networks Means—Albert-Laszlo Barabási Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References Six Degrees: The Science of a Connected Age—Duncan Watts Frame 10/95

Overview of Books Complex Networks Basic definitions Books Examples of Complex Networks Handbook of Graphs and Networks—editors: Properties of Complex Networks Stefan Bornholdt and H. G. Schuster Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References Evolution of Networks—S. N. Dorogovtsev and J. F . F . Mendes. Frame 11/95

Overview of Books Complex Networks Basic definitions Books Examples of Complex Networks Social Network Analysis—Stanley Properties of Complex Networks Wasserman and Kathleen Faust Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References In the Beat of a Heart: Life, Energy, and the Unity of Nature—John Whitfield Frame 12/95

Overview of Books Complex Networks Basic definitions Books Numerous others: Examples of Complex Networks ◮ Complex Social Networks—F. Vega-Redondo Properties of ◮ Fractal River Basins: Chance and Complex Networks Self-Organization—I. Rodríguez-Iturbe and A. Basic models of complex networks Rinaldo Generalized random networks Scale-free networks ◮ Random Graph Dynamics—R. Durette Small-world networks Generalized affiliation ◮ Scale-Free Networks—Guido Caldarelli networks References ◮ Evolution and Structure of the Internet: A Statistical Physics Approach—Romu Pastor-Satorras and Alessandro Vespignani ◮ Complex Graphs and Networks—Fan Chung Frame 13/95

Overview of Examples Complex Networks Basic definitions Books Examples of Complex Networks What passes for a complex network? Properties of Complex Networks ◮ Complex networks are large (in node number) Basic models of complex networks ◮ Complex networks are sparse (low edge to node Generalized random networks ratio) Scale-free networks Small-world networks ◮ Complex networks are usually dynamic and evolving Generalized affiliation networks References ◮ Complex networks can be social, economic, natural, informational, abstract, ... Frame 14/95

Overview of Examples Complex Networks Basic definitions Physical networks Books ◮ River networks Examples of ◮ The Internet Complex Networks ◮ Neural networks Properties of ◮ Road networks Complex Networks ◮ Trees and leaves ◮ Power grids Basic models of ◮ Blood networks complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References ◮ Distribution (branching) versus redistribution (cyclical) Frame 15/95

Overview of Examples Complex Networks Interaction networks Basic definitions Books ◮ The Blogosphere Examples of Complex Networks ◮ Biochemical Properties of networks Complex Networks ◮ Gene-protein Basic models of complex networks networks Generalized random networks Scale-free networks ◮ Food webs: who Small-world networks Generalized affiliation eats whom networks References ◮ The World Wide Web (?) ◮ Airline networks ◮ Call networks datamining.typepad.com ( ⊞ ) (AT&T) ◮ The Media Frame 16/95

Overview of Examples Complex Networks Basic definitions Interaction networks: Books social networks Examples of Complex Networks ◮ Snogging Properties of Complex Networks ◮ Friendships Basic models of ◮ Acquaintances complex networks Generalized random networks ◮ Boards and Scale-free networks Small-world networks directors Generalized affiliation networks ◮ Organizations References ◮ myspace.com ( ⊞ ), facebook.com ( ⊞ ) (Bearman et al. , 2004) ◮ ‘Remotely sensed’ by: email activity, instant messaging, phone logs (*cough*). Frame 17/95

Overview of Examples Complex Networks Basic definitions Books Examples of Complex Networks Properties of Complex Networks Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References Frame 18/95