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CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis

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1 CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis

2 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

3 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

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

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

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

7 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

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

9 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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14 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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18 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?

19 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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23 Bow-tie structure of directed nets

24 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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31 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?

32 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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39 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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50 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

51 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

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


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