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CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis

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The following slides include only the figures or videos that we use in class; they do not include detailed explanations, derivations or descriptions covered in class. Many of the following figures are copied from open sources at the Web. I do not claim any intellectual property for the following material. Disclaimers

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Outline As a reference point: – Poisson random graphs – Regular graphs Common properties of real-world networks – Size of largest connected component – Small-world property – Heavy-tailed degree distribution – Hierarchical organization – Network motifs Application paper: Small-world networks and functional connectivity in Azheimers disease Discuss course projects – project proposals due in a week Collect addresses Surprise visitor will talk about Sociology and NetSci

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Reference point-1: ER random graphs G(n,m) and G(n,p) models (see lecture notes for derivations)

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Emergence of giant connected component in G(n,p) as p increases

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Emergence of giant component See lecture notes for derivation of the following

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Emergence of giant connected component in G(n,p) as p increases https://www.youtube.com/watch?v=mpe 44sTSoF8 https://www.youtube.com/watch?v=mpe 44sTSoF8

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Reference point-2: Regular graphs Ring lattice with k connections to nearest neighbors (see lecture notes)

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Outline As a reference point: – Poisson random graphs – Regular graphs Common properties of real-world networks – Size of largest connected component – Small-world property – Heavy-tailed degree distribution – Hierarchical organization – Network motifs Application paper: Small-world networks and functional connectivity in Azheimers disease Discuss course projects – project proposals due in a week Collect addresses Surprise visitor will talk about Sociology and NetSci

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Outline As a reference point: – Poisson random graphs – Regular graphs Common properties of real-world networks – Size of largest connected component – Small-world property – Heavy-tailed degree distribution – Hierarchical organization – Network motifs Application paper: Small-world networks and functional connectivity in Azheimers disease Discuss course projects – project proposals due in a week Collect addresses Surprise visitor will talk about Sociology and NetSci

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More about power-laws (see derivations in class notes) Power-laws are everywhere (more normal than the Normal distribution) When is the mth moment of a power- law distribution finite? How to detect a power-law distribution? How to estimate the exponent of a power-law distribution?

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Outline As a reference point: – Poisson random graphs – Regular graphs Common properties of real-world networks – Size of largest connected component – Small-world property – Heavy-tailed degree distribution – Hierarchical organization – Network motifs Application paper: Small-world networks and functional connectivity in Azheimers disease Discuss course projects – project proposals due in a week Collect addresses Surprise visitor will talk about Sociology and NetSci

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Bow-tie structure of directed nets

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Outline As a reference point: – Poisson random graphs – Regular graphs Common properties of real-world networks – Size of largest connected component – Small-world property – Heavy-tailed degree distribution – Hierarchical organization – Network motifs Application paper: Small-world networks and functional connectivity in Azheimers disease Discuss course projects – project proposals due in a week Collect addresses Surprise visitor will talk about Sociology and NetSci

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How to control β and γ? The paper presents a stochastic model to do so But there are many other models that can do the same What is the main ingredient to get a power-law degree distribution? What is the main ingredient to get a hierarchical structure?

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Outline As a reference point: – Poisson random graphs – Regular graphs Common properties of real-world networks – Size of largest connected component – Small-world property – Heavy-tailed degree distribution – Hierarchical organization – Network motifs Application paper: Small-world networks and functional connectivity in Azheimers disease Discuss course projects – project proposals due in a week Collect addresses Surprise visitor will talk about Sociology and NetSci

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Outline As a reference point: – Poisson random graphs – Regular graphs Common properties of real-world networks – Size of largest connected component – Small-world property – Heavy-tailed degree distribution – Hierarchical organization – Network motifs Application paper: Small-world networks and functional connectivity in Azheimers disease Discuss course projects – project proposals due in a week Collect addresses Surprise visitor will talk about Sociology and NetSci

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Outline As a reference point: – Poisson random graphs – Regular graphs Common properties of real-world networks – Size of largest connected component – Small-world property – Heavy-tailed degree distribution – Hierarchical organization – Network motifs Application paper: Small-world networks and functional connectivity in Azheimers disease Discuss course projects – project proposals due in a week Collect addresses Surprise visitor will talk about Sociology and NetSci

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Course projects plz start with the following questions (and answer them in your project proposal) Do you want to do a research-oriented project? – Ok to work on something that relates to your research area – Not ok to submit something you have already done – Ok to do something that has no clear research potential (e.g., to reproduce the results of a published paper or to develop a tool that can be used in netsci research) What is the nature of the involved work? – Data collection, data analysis, simulation, math analysis, a combination of these? Do you want to do something domain-specific or general? – E.g., related only to computer networks? Social nets? Brain nets? – Or something general (e.g., an algorithm for community detection in general nets) Which topic of the course syllabus is your project most relevant to? – Have you read 1-2 papers about that topic? Solo or group project? – Which are the strengths or complementary backgrounds in your group? Some possible project types: – Reproduce the main results of a research paper with a different dataset(s) – Model a system that you understand well as a network and formulate some key questions about that system as network-related questions – Develop a simulator for a network model (ideally involving some sort of dynamics on the network) and investigate some concrete questions computationally – Develop an actual system (e.g., Web application) that will allow us to collect data about a network process in the background (e.g., a social game of some sort) – Prove analytically a property of a network model that has been shown only numerically in the published literature

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Duncan Watts (from the small world 98 paper) will talk to us about computational social science zWM0

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