Overview Class Admin Overview Overview of Complex Networks Class admin Class admin Basic definitions Basic definitions Complex Networks, Course 303A, Spring, 2009 Books Books Examples of Examples of Complex Networks Complex Networks ◮ Office hours: Prof. Peter Dodds Properties of Properties of Complex Networks Complex Networks ◮ Tuesday 2:30 pm–3:30 pm Modelling Complex Modelling Complex Department of Mathematics & Statistics ◮ Thursday 11:30 am–12:30 pm Networks Networks University of Vermont Nutshell ◮ Course outline Nutshell References References ◮ Projects Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/53 Frame 2/53 Basic definitions Overview Basic definitions Overview Class admin Class admin Basic definitions Basic definitions Complex System—Basic ingredients: Books Books Examples of Examples of ◮ Relationships are nonlinear Complex Networks Complex Networks Complex: (Latin = with + fold/weave (com + plex)) Properties of Properties of ◮ Relationships contain feedback loops Complex Networks Complex Networks Adjective ◮ Complex systems are open (out of equilibrium) Modelling Complex Modelling Complex ◮ Made up of multiple parts; intricate or detailed. Networks Networks ◮ Memory Nutshell Nutshell ◮ Not simple or straightforward. References ◮ Modular (nested)/multiscale structure References ◮ Opaque boundaries ◮ May result in emergent phenomena Frame 3/53 Frame 4/53
Basic definitions Overview Observation Overview Class admin Class admin Basic definitions Basic definitions Network: (net + work, 1500’s) Books Books Noun: Examples of Examples of Complex Networks Complex Networks 1. Any interconnected group or system ◮ Many complex systems can be regarded as complex Properties of Properties of Complex Networks Complex Networks 2. Multiple computers and other devices connected networks of physical or abstract interactions Modelling Complex Modelling Complex together to share information Networks Networks ◮ Opens door to mathematical and numerical analysis Nutshell Nutshell ◮ Dominant approach of last decade of a References References Verb: theoretical-physics/stat-mechish flavor. 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 5/53 Frame 6/53 Basic definitions Overview Basic definitions Overview Class admin Class admin Basic definitions Basic definitions Books Books Links = Connections between nodes Examples of Examples of Complex Networks Complex Networks ◮ links Nodes = A collection of entities which have Properties of Properties of Complex Networks Complex Networks ◮ may be real and fixed (rivers), properties that are somehow related to each other Modelling Complex ◮ real and dynamic (airline routes), Modelling Complex Networks Networks ◮ abstract with physical impact (hyperlinks), ◮ e.g., people, forks in rivers, proteins, webpages, Nutshell Nutshell ◮ or purely abstract (semantic connections between organisms,... References References concepts). ◮ Links may be directed or undirected. ◮ Links may be binary or weighted. Frame 7/53 Frame 8/53
Basic definitions Overview Basic definitions Overview Class admin Class admin Basic definitions Basic definitions Books Books Adjacency matrix: Examples of Examples of Complex Networks Complex Networks ◮ We represent a graph or network by a matrix A with Node degree = Number of links per node Properties of Properties of link weight a ij for nodes i and j in entry ( i , j ) . Complex Networks Complex Networks ◮ Notation: Node i ’s degree = k i . Modelling Complex Modelling Complex ◮ e.g., Networks Networks ◮ k i = 0,1,2,. . . . 0 1 1 1 0 Nutshell Nutshell ◮ Notation: the average degree of a network = � k � 0 0 1 0 1 References References A = 1 0 0 0 0 (and sometimes as z ) 0 1 0 0 1 0 1 0 1 0 Frame 9/53 Frame 10/53 Books Overview Books Overview Class admin Class admin Basic definitions Basic definitions Books Books Linked: How Everything Is Connected to Examples of Examples of Nexus: Small Worlds and the Groundbreaking Complex Networks Complex Networks Everything Else and What It Science of Networks—Mark Buchanan Properties of Properties of Means—Albert-Laszlo Barabási Complex Networks Complex Networks Modelling Complex Modelling Complex Networks Networks Nutshell Nutshell References References The Tipping Point: How Little Things can Six Degrees: The Science of a Connected make a Big Difference—Malcolm Gladwell [7] Age—Duncan Watts [17] Frame 11/53 Frame 12/53
Books Overview Books Overview Class admin Class admin Basic definitions Basic definitions Books Books Examples of Examples of Handbook of Graphs and Networks—editors: Complex Networks Complex Networks Stefan Bornholdt and H. G. Schuster [3] Properties of Properties of Complex Networks Complex Networks Social Network Analysis—Stanley Modelling Complex Modelling Complex Networks Wasserman and Kathleen Faust [16] Networks Nutshell Nutshell References References Evolution of Networks—S. N. Dorogovtsev . Mendes [6] and J. F . F Frame 13/53 Frame 14/53 Books Overview Examples Overview Class admin Class admin Basic definitions Basic definitions Numerous others: Books Books What passes for a complex network? ◮ Complex Social Networks—F. Vega-Redondo [15] Examples of Examples of Complex Networks Complex Networks ◮ Complex networks are large (in node number) ◮ Fractal River Basins: Chance and Properties of Properties of Complex Networks ◮ Complex networks are sparse (low edge to node Complex Networks Self-Organization—I. Rodríguez-Iturbe and A. Modelling Complex Modelling Complex Rinaldo [12] ratio) Networks Networks ◮ Random Graph Dynamics—R. Durette ◮ Complex networks are usually dynamic and evolving Nutshell Nutshell References References ◮ Scale-Free Networks—Guido Caldarelli ◮ Complex networks can be social, economic, natural, informational, abstract, ... ◮ Evolution and Structure of the Internet: A Statistical ◮ Isn’t this graph theory?: Yes, but emphasis is on data Physics Approach—Romu Pastor-Satorras and Alessandro Vespignani and mechanistic explanations... ◮ Complex Graphs and Networks—Fan Chung Frame 15/53 Frame 16/53
Examples Overview Examples Overview Interaction networks Class admin Class admin Physical networks Basic definitions Basic definitions ◮ The Blogosphere ◮ River networks Books Books ◮ The Internet ◮ Biochemical Examples of Examples of ◮ Neural networks Complex Networks Complex Networks ◮ Road networks networks ◮ Trees and leaves Properties of Properties of ◮ Power grids Complex Networks Complex Networks ◮ Gene-protein ◮ Blood networks Modelling Complex Modelling Complex networks Networks Networks ◮ Food webs: who Nutshell Nutshell References References eats whom ◮ The World Wide Web (?) ◮ Airline networks ◮ Call networks ◮ Distribution (branching) versus redistribution datamining.typepad.com ( ⊞ ) (AT&T) (cyclical) ◮ The Media Frame 17/53 Frame 18/53 Examples Overview Examples Overview Class admin Class admin Interaction networks: Basic definitions Basic definitions social networks Books Books ◮ Snogging Examples of Examples of Complex Networks Complex Networks ◮ Friendships Properties of Properties of Complex Networks Complex Networks ◮ Acquaintances Modelling Complex Modelling Complex Networks Networks ◮ Boards and Nutshell Nutshell directors References References ◮ Organizations ◮ myspace.com ( ⊞ ), facebook.com ( ⊞ ) (Bearman et al. , 2004) ◮ ‘Remotely sensed’ by: email activity, instant messaging, phone logs (*cough*). Frame 19/53 Frame 20/53
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