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SOCIAL NETWORKS ANALYSIS SEMINAR INTRODUCTORY LECTURE Danny Hendler and Yehonatan Cohen Advanced Topics in on-line Social Networks Analysis

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Seminar requirements Select a paper and notify me by March, 17'th Study the paper well and prepare a good presentation Give an excellent seminar talk Participate in at least 80% of seminar talks

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Seminar schedule Introductory lecture #1 5/3/14 No seminar (Purim!) Semester ends 12/3/14 Introductory lecture #2 Papers list published, students send their 3 preferences 14/3/14 11 weeks of Student talks 19/3/14 Student talks start All students preferences must be received 10/3/14 26/3/14

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Talk outline Social networks Properties of on-line social networks Small-world phenomenon Power-law distribution Community structure Community detection

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Social networks What is a social network? A network, were nodes represent actors and edges represent interactions/relationships

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Social networks: an example

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Giant component

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Social networks: an example Some nodes are very active

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Types of online social media

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Top 20 USA websites 1Google.com11Blogger.com 2Facebook.com12msn.com 3Yahoo.com13Myspace.com 4YouTube.com14Go.com 5Amazon.com15Bing.com 6Wikipedia.org16AOL.com 7Craigslist.org17LinkedIn.com 8Twitter.com18CNN.com 9Ebay.com19Espn.go.com 10Live.com20Wordpress.com Source: Alexa report, February, 2014

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Top 20 USA websites 1Google.com11Blogger.com 2Facebook.com12msn.com 3Yahoo.com13Myspace.com 4YouTube.com14Go.com 5Amazon.com15Bing.com 6Wikipedia.org16AOL.com 7Craigslist.org17LinkedIn.com 8Twitter.com18CNN.com 9Ebay.com19Espn.go.com 10Live.com20Wordpress.com 30% social network sites Source: Alexa report, February, 2014

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Top 20 USA websites 1Google.com11Blogger.com 2Facebook.com12msn.com 3Yahoo.com13Myspace.com 4YouTube.com14Go.com 5Amazon.com15Bing.com 6Wikipedia.org16AOL.com 7Craigslist.org17LinkedIn.com 8Twitter.com18CNN.com 9Ebay.com19Espn.go.com 10Live.com20Wordpress.com 30% social network sites 30% additional sites with social network aspects Source: Alexa report, February, 2014

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Social networks Properties of on-line social networks Small-world phenomenon Power-law distribution Community structure Community detection Properties of social networks

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Milgram's small world phenomenon experiment (1967) Six degrees of separation:I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation between us and everyone else on this planet. (*) (*) John Guare. Six Degrees of Separation: A Play. Vintage Books, 1990. So, Milgram decided to check…

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Milgram's experiment (cont'd) Budget: $680!!! A set of starters, all try to forward a letter to a single target person Starters notified of targets name/address/occupation Must forward letter to someone known on first-name basis Image taken from Wiki.

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Milgram's experiment: results 64 chains arrived Median length: 6 Taken from: Networks, crowds and Markets, D. Easley & J. Kleinberg. (Book is online)

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A slightly more modern example (2008)

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Social networks Properties of on-line social networks Small-world phenomenon Power-law distribution Community structure Community detection Properties of social networks

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A matter of popularity… As a function of k: what fraction of Web pages have k in-links? ~1/k 2.1 (*) (*) Broder et al. Graph structure in the Web. WWW 2000, pp. 309-320.

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The power low distribution Degrees Nodes A.k.a. long tail distribution, scale-free distribution Most nodes have low degrees Few nodes have extremely high degrees

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Web pages in-degree: log-log scale

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Some more examples Friendship Network in FlickrFriendship Network in YouTube

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Why is popularity power-law?

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A simple game… Procedure for creating Web page j {1,2…N} Choose page i<j uniformly: a.With probability p, create a link to page i b.With probability 1-p, create a link to the page pointed to by page i As a function of k: what fraction of Web pages have k in-links? ~k c, lim c =-2 p 0

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Rich get richer… Procedure for creating Web page j {1,2…N} Choose page i<j uniformly: a.With probability p, create a link to page i b.With probability 1-p, create a link to the page pointed to by page i = a.With probability p, choose page i<j uniformly and create a link to page i b.With probability 1-p, choose a page i<j with probability proportional to ith number of links and create a link to i

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The situation in random graphs Nodes connected at random Node degrees follow a binomial distribution

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1 6 54 63 67 2 94 number of nodes found Power-law graph: BFS Animation taken from a presentation by Ofrit Lesser.

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93 number of nodes found 1 3 7 1115 19 Random graph: BFS Animation taken from a presentation by Ofrit Lesser.

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Communities (a.k.a. clusters/modules) Community structure: the organization of vertices in clusters, with many edges joining vertices of the same community and relatively few edges joining different communities Often represent sets of actors sharing similar properties/roles.

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Social networks Properties of on-line social networks Small-world phenomenon Power-law distribution Community structure Community detection

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Community detection applications Clustering web clients with geographical proximity and similar access patterns cache servers positioning [Krishnamurty & Wang, SIGCOMM 2000] Clustering customers with similar interests Recommendation systems [Reddy et al., DNIS 2002] Analysing structural positions Identifying central actors and inter-community mediators …

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Edge-betweenness based detection A divisive method (as opposed to agglomerative methods) Look for an edge that is most between pairs of nodes o Responsible for connecting many pairs Remove edge and recalculate Newman and Girvan. Finding and evaluating community structure in networks, 2003

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Shortest-path betweenness Compute all-pairs shortest paths For each edge, compute the number of such paths it belongs to Remove a maximum-weight edge Repeat until no edges (more on this later)

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Shortest-path betweeness: an example 0 1 2 3 5 4 6 7 9 8

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0 1 2 3 5 4 6 7 9 8 24

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Shortest-path betweeness: an example 0 1 2 3 5 4 6 7 9 8

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0 1 2 3 5 4 6 7 9 8 9

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0 1 2 3 5 4 6 7 9 8 3

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0 1 2 3 5 4 6 7 9 8 1

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0 1 2 3 5 4 6 7 9 8 1

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0 1 2 3 5 4 6 7 9 8 1

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0 1 2 3 5 4 6 7 9 8 1

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0 1 2 3 5 4 6 7 9 8 1

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0 1 2 3 5 4 6 7 9 8 1

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6 7 9 8 1 0 1 2 3 5 4

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6 7 9 8 1 0 1 2 3 5 4

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6 7 9 8 1 0 1 2 3 5 4

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6 7 9 8 0 1 2 3 5 4

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Dendrograms (hierarchical trees) A dendrogram (hierarchical tree) illustrates the output of hierarchical clustering algorithms Leaves represent graph nodes, top represents original graph As we move down the tree, larger communities are partitioned to smaller ones 1234567890

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Shortest-path betweeness: an example 0 1 2 3 5 4 6 7 9 8 24 1234567890

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Shortest-path betweeness: an example 0 1 2 3 5 4 6 7 9 8 9 1234567890

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0 1 2 3 5 4 6 7 9 8 3 1234567890

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0 1 2 3 5 4 6 7 9 8 1 1234567890

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0 1 2 3 5 4 6 7 9 8 1 1234567890

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0 1 2 3 5 4 6 7 9 8 1 1234567890

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0 1 2 3 5 4 6 7 9 8 1 1234567890

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0 1 2 3 5 4 6 7 9 8 1 1234567890

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0 1 2 3 5 4 6 7 9 8 1 1234567890

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6 7 9 8 1 1234567890 0 1 2 3 5 4

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6 7 9 8 1 1234567890 0 1 2 3 5 4

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6 7 9 8 1 1234567890 0 1 2 3 5 4

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6 7 9 8 123456789 0 1 2 3 5 4 0

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Evaluation: computer-generated networks Large number of graphs with 128 nodes and 4 communities of 32-nodes each Probability p in for intra-community edges Probablilty p ext for inter-community edges Chosen such that expected vertex degree is 16

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Results (for 64-nodes networks) z in =6, z out =2

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Evaluation: the Zachary karate club

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Results on Zachary club network Shortest-pathShortest-path no recalculation Shortest path 2- communities partition missed just a single person! Re-calculation of betweenness essential

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Quality functions Hierarchical clustering algorithms create numerous partitions In general, we do not know how many communities we should seek. How may we know that our clustering is good We need a quality function

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The modularity quality function Newman and Girvan. Finding and evaluating community structure in networks, 2003 No communities in random graphs Equal probabilities for all edges Check how far intra-community and inter-community densities are from those you would expect in a random graph with identical nodes and same degree-distribution

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The modularity quality function (con'd) Clauset, Newman and Moore. Finding community structure in very large networks, 2004 Modularity value # edges Graph adjacency matrix Degrees of nodes-pair Probability of an edge if only a function of node-degrees In-same-cluster indicator variable

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Computer-generated networks: modularity Modularity maximized at correct partition

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Zachary club network: modularity One of two local maxima at correct partition

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