Outline Overview of Complex Networks Complex Networks Complex - PowerPoint PPT Presentation

Overview of Outline Overview of Complex Networks Complex Networks Complex Networks Basic definitions Basic definitions Basic definitions Principles of Complex Systems Books Books Course 300, Fall, 2008 Examples of Examples of Books Complex Networks Complex Networks Properties of Properties of Complex Networks Complex Networks Examples of Complex Networks Prof. Peter Dodds Basic models of Basic models of complex networks complex networks Generalized random Generalized random Properties of Complex Networks networks networks Department of Mathematics & Statistics Scale-free networks Scale-free networks University of Vermont Small-world networks Small-world networks Generalized affiliation Generalized affiliation Basic models of complex networks networks networks References Generalized random networks References Scale-free networks Small-world networks Generalized affiliation networks References Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/95 Frame 2/95 Basic definitions Overview of Observation Overview of Complex Networks Complex Networks Basic definitions Basic definitions Network: (net + work, 1500’s) Books Books Examples of Examples of Noun: Complex Networks Complex Networks Properties of Properties of 1. Any interconnected group or system Complex Networks Complex Networks ◮ Many complex systems can be regarded as complex Basic models of Basic models of 2. Multiple computers and other devices connected networks of physical or abstract interactions complex networks complex networks Generalized random Generalized random together to share information ◮ Opens door to mathematical and numerical analysis networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks ◮ Dominant approach of last decade of a Generalized affiliation Generalized affiliation networks networks Verb: theoretical-physics/stat-mechish flavor. References 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 Frame 4/95

Basic definitions Overview of Basic definitions Overview of Complex Networks Complex Networks Basic definitions Basic definitions Books Books Examples of Examples of Links = Connections between nodes Complex Networks Complex Networks Properties of Properties of ◮ links Complex Networks Complex Networks Nodes = A collection of entities which have ◮ may be real and fixed (rivers), Basic models of Basic models of properties that are somehow related to each other complex networks complex networks ◮ real and dynamic (airline routes), Generalized random Generalized random networks networks ◮ abstract with physical impact (hyperlinks), ◮ e.g., people, forks in rivers, proteins, webpages, Scale-free networks Scale-free networks Small-world networks ◮ or purely abstact (semantic connections between Small-world networks organisms,... Generalized affiliation Generalized affiliation networks networks concepts). References References ◮ Links may be directed or undirected. ◮ Links may be binary or weighted. Frame 5/95 Frame 6/95 Basic definitions Overview of Basic definitions Overview of Complex Networks Complex Networks Basic definitions Basic definitions Books Books Examples of Adjacency matrix: Examples of Complex Networks Complex Networks Properties of Properties of ◮ We represent a graph or network by a matrix A with Node degree = Number of links per node Complex Networks Complex Networks link weight a ij for nodes i and j in entry ( i , j ) . Basic models of Basic models of ◮ Notation: Node i ’s degree = k i . complex networks complex networks ◮ e.g., Generalized random Generalized random networks networks ◮ k i = 0,1,2,. . . .   0 1 1 1 0 Scale-free networks Scale-free networks Small-world networks Small-world networks ◮ Notation: the average degree of a network = � k � 0 0 1 0 1 Generalized affiliation Generalized affiliation   networks networks   A = 1 0 0 0 0 (and sometimes as z ) References References     0 1 0 0 1   0 1 0 1 0 Frame 7/95 Frame 8/95

Books Overview of Books Overview of Complex Networks Complex Networks Basic definitions Basic definitions Books Books Examples of Examples of Complex Networks Linked: How Everything Is Connected to Complex Networks Nexus: Small Worlds and the Groundbreaking Properties of Properties of Everything Else and What It Complex Networks Complex Networks Science of Networks—Mark Buchanan Means—Albert-Laszlo Barabási Basic models of Basic models of complex networks complex networks Generalized random Generalized random networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation networks networks References References The Tipping Point: How Little Things can Six Degrees: The Science of a Connected make a Big Difference—Malcolm Gladwell Age—Duncan Watts Frame 9/95 Frame 10/95 Books Overview of Books Overview of Complex Networks Complex Networks Basic definitions Basic definitions Books Books Examples of Examples of Complex Networks Complex Networks Handbook of Graphs and Networks—editors: Social Network Analysis—Stanley Properties of Properties of Complex Networks Complex Networks Stefan Bornholdt and H. G. Schuster Wasserman and Kathleen Faust Basic models of Basic models of complex networks complex networks Generalized random Generalized random networks networks Scale-free networks Scale-free networks Small-world networks Small-world networks Generalized affiliation Generalized affiliation networks networks References References In the Beat of a Heart: Life, Energy, and the Evolution of Networks—S. N. Dorogovtsev Unity of Nature—John Whitfield and J. F . F . Mendes. Frame 11/95 Frame 12/95

Books Overview of Examples Overview of Complex Networks Complex Networks Basic definitions Basic definitions Books Books Numerous others: Examples of Examples of ◮ Complex Social Networks—F. Vega-Redondo Complex Networks Complex Networks What passes for a complex network? Properties of Properties of ◮ Fractal River Basins: Chance and Complex Networks Complex Networks ◮ Complex networks are large (in node number) Self-Organization—I. Rodríguez-Iturbe and A. Basic models of Basic models of complex networks complex networks ◮ Complex networks are sparse (low edge to node Rinaldo Generalized random Generalized random networks networks ratio) Scale-free networks Scale-free networks ◮ Random Graph Dynamics—R. Durette Small-world networks Small-world networks Generalized affiliation ◮ Complex networks are usually dynamic and evolving Generalized affiliation ◮ Scale-Free Networks—Guido Caldarelli networks networks References References ◮ Complex networks can be social, economic, natural, ◮ Evolution and Structure of the Internet: A Statistical informational, abstract, ... Physics Approach—Romu Pastor-Satorras and Alessandro Vespignani ◮ Complex Graphs and Networks—Fan Chung Frame 13/95 Frame 14/95 Examples Overview of Examples Overview of Complex Networks Complex Networks Interaction networks Basic definitions Basic definitions Physical networks Books Books ◮ The Blogosphere ◮ River networks Examples of Examples of ◮ The Internet Complex Networks ◮ Biochemical Complex Networks ◮ Neural networks Properties of Properties of ◮ Road networks networks Complex Networks Complex Networks ◮ Trees and leaves ◮ Power grids ◮ Gene-protein Basic models of Basic models of ◮ Blood networks complex networks complex networks networks Generalized random Generalized random networks networks Scale-free networks Scale-free networks ◮ Food webs: who Small-world networks Small-world networks Generalized affiliation Generalized affiliation eats whom networks networks References References ◮ The World Wide Web (?) ◮ Airline networks ◮ Call networks ◮ Distribution (branching) versus redistribution datamining.typepad.com ( ⊞ ) (AT&T) (cyclical) ◮ The Media Frame 15/95 Frame 16/95