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lecture2 notes - 6.207/14.15 Networks Lecture 2 Graph...

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6.207/14.15: Networks Lecture 2: Graph Theory and Social Networks Daron Acemoglu and Asu Ozdaglar MIT September 14, 2009 1
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Networks: Lecture 2 Introduction Outline Types of networks Graphs: notation and terminology Properties of networks: Diameter, average path length, clustering, degree distributions, centrality Reading: Jackson, Chapters 2 and 3 EK, Chapters 2 and 13 2
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Networks: Lecture 2 Introduction Networks in the Real World A network is a set of items ( nodes or vertices ) connected by edges or links . Systems taking the form of networks abound in the world. Types of Networks: Social and economic networks : A set of people or groups of people with some pattern of contacts or interactions between them. Facebook, friendship networks, business relations between companies, intermarriages between families, labor markets Questions: Degree of connectedness, homophily, small-world effects Information networks: Connections of “information” objects. Network of citations between academic papers, World Wide Web (network of Web pages containing information with links from one page to other), semantic (how words or concepts link to each other) Questions: Ranking, navigation 3
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Networks: Lecture 2 Introduction Networks in the Real World (Continued) Types of Networks: Technological networks: Designed typically for distribution of a commodity or service. Infrastructure networks: e.g., Internet (connections of routers or administrative domains), power grid, transportation networks (road, rail, airline, mail) Temporary networks: e.g., ad hoc communication networks, sensor networks, autonomous vehicles Questions: Does network structure support performance? Fragility? Cascading failures? Biological networks: A number of biological systems can also be represented as networks. Food web, protein interaction network, network of metabolic pathways 4
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Networks: Lecture 2 Introduction Network Study Historical study of networks: Mathematical graph theory: One of the pillars of discrete mathematics Started with Euler’s celebrated 1735 solution of the K o ¨ nigsberg bridge problem. Networks also studied extensively in sociology. Typical studies involve circulation of questionnaires, leading to small networks of interactions. Recent years witnessed a substantial change in network research. From analysis of single small graphs (10-100 nodes) to statistical properties of large scale networks (million-billion nodes). Motivated by availability of computers and computer networks that allow us to gather and analyze large scale data. New Analytical Approach: Find statistical properties that characterize the structure of these networks and ways to measure them Create models of networks Predict behavior of networks on the basis of measured structural properties and models 5
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Networks: Lecture 2 Graphs Graphs—1 We represent a network by a graph ( N , g ) , which consists of a set of nodes N = { 1, . . . , n } and an n × n matrix g = [ g ij ] i , j N (referred to as an adjacency matrix ), where g ij ∈ { 0, 1 } represents the availability of an edge from node i to node j .
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