18th Ontario Combinatorics Workshop On-line Social Networks

Slides:



Advertisements
Similar presentations
1 Dynamics of Real-world Networks Jure Leskovec Machine Learning Department Carnegie Mellon University
Advertisements

Analysis and Modeling of Social Networks Foudalis Ilias.
Week 5 - Models of Complex Networks I Dr. Anthony Bonato Ryerson University AM8002 Fall 2014.
Lecture 21 Network evolution Slides are modified from Jurij Leskovec, Jon Kleinberg and Christos Faloutsos.
SILVIO LATTANZI, D. SIVAKUMAR Affiliation Networks Presented By: Aditi Bhatnagar Under the guidance of: Augustin Chaintreau.
Information Networks Small World Networks Lecture 5.
1 Dimension matching in Facebook and LinkedIn networks Anthony Bonato Ryerson University Seminar on Social Networks, Big Data, Influence, and Decision-Making.
CS 599: Social Media Analysis University of Southern California1 The Basics of Network Analysis Kristina Lerman University of Southern California.
Universal Random Semi-Directed Graphs
The influence of search engines on preferential attachment Dan Li CS3150 Spring 2006.
CSE 522 – Algorithmic and Economic Aspects of the Internet Instructors: Nicole Immorlica Mohammad Mahdian.
Hierarchy in networks Peter Náther, Mária Markošová, Boris Rudolf Vyjde : Physica A, dec
Topology Generation Suat Mercan. 2 Outline Motivation Topology Characterization Levels of Topology Modeling Techniques Types of Topology Generators.
CS728 Lecture 5 Generative Graph Models and the Web.
TDTS21: Advanced Networking Lecture 8: Online Social Networks Based on slides from P. Gill Revised 2015 by N. Carlsson.
Small Worlds Presented by Geetha Akula For the Faculty of Department of Computer Science, CALSTATE LA. On 8 th June 07.
CS Lecture 6 Generative Graph Models Part II.
1 Algorithms for Large Data Sets Ziv Bar-Yossef Lecture 7 May 14, 2006
Measurement and Evolution of Online Social Networks Review of paper by Ophir Gaathon Analysis of Social Information Networks COMS , Spring 2011,
Networks - Bonato1 Modelling, Mining, and Searching Networks Anthony Bonato Ryerson University Master’s Seminar November 2012.
Lecture 6 - Models of Complex Networks II Dr. Anthony Bonato Ryerson University AM8002 Fall 2014.
Topic 13 Network Models Credits: C. Faloutsos and J. Leskovec Tutorial
Log Dimension Hypothesis1 The Logarithmic Dimension Hypothesis Anthony Bonato Ryerson University MITACS International Problem Solving Workshop July 2012.
Complex networks - Bonato1 Complex networks and their models Anthony Bonato Ryerson University Graduate Seminar October 2011.
1 Vertex-pursuit in heirarchical social networks Anthony Bonato Ryerson University TAMC’12 Complex Networks.
Week 3 - Complex Networks and their Properties
1 Burning a graph as a model of social contagion Anthony Bonato Ryerson University Institute of Software Chinese Academy of Sciences.
Jure Leskovec Computer Science Department Cornell University / Stanford University Joint work with: Jon Kleinberg (Cornell), Christos.
On-line Social Networks - Anthony Bonato 1 Dynamic Models of On-Line Social Networks Anthony Bonato Ryerson University WAW’2009 February 13, 2009 nt.
Lecture 10: Network models CS 765: Complex Networks Slides are modified from Networks: Theory and Application by Lada Adamic.
Week 1 – Introduction to Graph Theory I Dr. Anthony Bonato Ryerson University AM8002 Fall 2014.
1 Modelling, Mining, and Searching Networks Anthony Bonato Ryerson University Graduate Seminar October 2015.
Most of contents are provided by the website Network Models TJTSD66: Advanced Topics in Social Media (Social.
RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs Leman Akoglu, Mary McGlohon, Christos Faloutsos Carnegie Mellon University School.
Miniconference on the Mathematics of Computation
1 How to burn a graph Anthony Bonato Ryerson University GRASCan 2015.
On-line Social Networks - Anthony Bonato 1 Dynamic Models of On-line Social Networks Anthony Bonato Ryerson University ICMCM’09 December, 2009.
A short course on complex networks
1 Dimension matching in Facebook and LinkedIn networks Anthony Bonato Ryerson University Toronto, Canada ICMCE 2015.
Cmpe 588- Modeling of Internet Emergence of Scale-Free Network with Chaotic Units Pulin Gong, Cees van Leeuwen by Oya Ünlü Instructor: Haluk Bingöl.
Graph clustering to detect network modules
Network (graph) Models
Cohesive Subgraph Computation over Large Graphs
Modelling, Mining, and Searching Networks
Lecture 1: Complex Networks
Topics In Social Computing (67810)
Miniconference on the Mathematics of Computation
Peer-to-Peer and Social Networks
Community detection in graphs
The Watts-Strogatz model
Lecture 13 Network evolution
Peer-to-Peer and Social Networks Fall 2017
Discrete Mathematics and its Applications Lecture 1 – Graph Theory
Graph and Tensor Mining for fun and profit
Clustering Coefficients
Modelling and Searching Networks Lecture 3 – ILT model
Discrete Mathematics and its Applications Lecture 3 – ILT model
Miniconference on the Mathematics of Computation
Modelling and Searching Networks Lecture 4 – ILAT model
Lecture 21 Network evolution
Miniconference on the Mathematics of Computation
Modelling and Searching Networks Lecture 2 – Complex Networks
Modelling and Searching Networks Lecture 5 – Random graphs
Modelling and Searching Networks Lecture 6 – PA models
Affiliation Network Models of Clusters in Networks
Discrete Mathematics and its Applications Lecture 5 – Random graphs
Network Models Michael Goodrich Some slides adapted from:
Discrete Mathematics and its Applications Lecture 6 – PA models
Advanced Topics in Data Mining Special focus: Social Networks
Cops and Robbers from a Distance
Presentation transcript:

18th Ontario Combinatorics Workshop On-line Social Networks May 8, 2010 The Geometry of On-line Social Networks Anthony Bonato Ryerson University On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Friendship networks network of friends (some real, some virtual) form a large web of interconnected links On-line Social Networks - Anthony Bonato

Ashton Kutcher is the centre of Twitterverse Dalai Lama Arnold Schwarzenegger Queen Rania of Jordan Christianne Amanpour Ashton Kutcher On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato 6 degrees of separation Stanley Milgram: famous chain letter experiment in 1967 On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato 6 Degrees of Kevin Bacon On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato 6 Degrees in Twitter Java et al. (2009) 6 degrees of separation in Twitter other researchers found similar results in Facebook, Myspace, … On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Complex Networks web graph, social networks, biological networks, internet networks, … On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato The web graph nodes: web pages edges: links over 1 trillion nodes, with billions of nodes added each day On-line Social Networks - Anthony Bonato

Math Behind Web Search - Anthony Bonato Chris Godsil Ontario Travel UW City of Toronto Four Seasons Hotel Frommer’s Greenland Tourism 5/15/2019 Math Behind Web Search - Anthony Bonato

On-line Social Networks - Anthony Bonato nodes: people edges: social interaction (eg friendship) On-line Social Networks - Anthony Bonato

On-line Social Networks (OSNs) Facebook, Twitter, LinkedIn, MySpace… On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato A new paradigm half of all users of internet on some OSN 400 million users on Facebook, 100 million on Twitter unprecedented, massive record of social interaction unprecedented access to information/news/gossip On-line Social Networks - Anthony Bonato

“Putting people at the centre of the web” Mark Zuckerberg, co-founder + CEO of FaceBook, April 21, 2010, F8 Conference, discussing Open Graph On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Key parameters power law degree distributions: average distance: clustering coefficient: Wiener index, W(G) On-line Social Networks - Anthony Bonato

Properties of Complex Networks observed properties: massive, power law, small world, decentralized (Broder et al, 01) On-line Social Networks - Anthony Bonato

Interpreting a power law Many low-degree nodes Few high-degree nodes Introducing the Web Graph - Anthony Bonato

Introducing the Web Graph - Anthony Bonato Binomial Power law Highway network Air traffic network Introducing the Web Graph - Anthony Bonato

On-line Social Networks - Anthony Bonato Small World Property small world networks introduced by social scientists Watts & Strogatz in 1998 low diameter/average distance (“6 degrees of separation”) globally sparse, locally dense (high clustering coefficient) On-line Social Networks - Anthony Bonato

Example of community structure W. Zachary’s Ph.D. thesis (1972): observed social ties and rivalries in a university karate club (34 nodes,78 edges) during his observation, conflicts intensified and group split On-line Social Networks - Anthony Bonato

Social network analysis On-line Milgram (67): average distance between Americans is 6 Watts and Strogatz (98): introduced small world property Adamic et al. (03): OSN at Stanford Liben-Nowell et al. (05): studied LiveJournal Kumar et al. (06): Flickr, Yahoo!360 Golder et al. (06): Facebook Ahn et al. (07): Cyworld (South Korea), MySpace and Orkut Mislove et al. (07): Flickr, YouTube, LiveJournal, Orkut Java et al. (07): Twitter On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Power laws in OSNs On-line Social Networks - Anthony Bonato

Sample data: Flickr, YouTube, LiveJournal, Orkut (Mislove et al,07): short average distances and high clustering coefficients On-line Social Networks - Anthony Bonato

(Leskovec, Kleinberg, Faloutsos,05): many complex networks (including on-line social networks) obey two additional laws: Densification Power Law networks are becoming more dense over time; i.e. average degree is increasing et ≈ nta where 1 < a ≤ 2: densification exponent On-line Social Networks - Anthony Bonato

Densification – Physics Citations 1.69 On-line Social Networks - Anthony Bonato

Densification – Autonomous Systems e(t) 1.18 n(t) On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Decreasing distances distances (diameter and/or average distances) decrease with time Preferential attachment model (Barabási, Albert, 99), (Bollobás et al, 01) diameter O(log t) Diameter first, DPL second Check diameter formulas As the network grows the distances between nodes slowly grow On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Flickr and Yahoo!360 (Kumar et al,06): shrinking distances On-line Social Networks - Anthony Bonato

Diameter – ArXiv citation graph time [years] On-line Social Networks - Anthony Bonato

Why model complex networks? uncover the generative mechanisms underlying complex networks nice mathematical challenges models can uncover the hidden reality of networks in OSNs: community detection advertising security and counterterrorism On-line Social Networks - Anthony Bonato

Preferential attachment model Albert-László Barabási Réka Albert

Preferential Attachment Model (Barabási, Albert, 99), (Bollobás,Riordan,Spencer,Tusnady,01) Wilensky, U. (2005). NetLogo Preferential Attachment model. http://ccl.northwestern.edu/netlogo/models/PreferentialAttachment.

Properties of the PA model (BRST,01) A.a.s. (that is, with probability tending to 1 as t→∞) for all k satisfying 0 ≤ k ≤ t1/15 (Bollobás, Riordan, 04) A.a.s. the diameter of the graph at time t is Anthony Bonato - The web graph

On-line Social Networks - Anthony Bonato Many different models On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Models of OSNs few models for on-line social networks goal: find a model which simulates many of the observed properties of OSNs must evolve in a natural way… On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato “All models are wrong, but some are more useful.” – G.P.E. Box On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Transitivity On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Iterated Local Transitivity (ILT) model (Bonato, Hadi, Horn, Prałat, Wang, 08) key paradigm is transitivity: friends of friends are more likely friends start with a graph of order n to form the graph Gt+1 for each node x from time t, add a node x’, the clone of x, so that xx’ is an edge, and x’ is joined to each node joined to x On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato G0 = C4 On-line Social Networks - Anthony Bonato

Properties of ILT model average degree increasing to with time average distance bounded by constant and converging, and in many cases decreasing with time; diameter does not change clustering higher than in a random generated graph with same average degree bad expansion: small gaps between 1st and 2nd eigenvalues in adjacency and normalized Laplacian matrices of Gt On-line Social Networks - Anthony Bonato

Densification nt = order of Gt, et = size of Gt Lemma: For t > 0, nt = 2tn0, et = 3t(e0+n0) - 2tn0. → densification power law: et ≈ nta, where a = log(3)/log(2). On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Average distance Theorem 2: If t > 0, then average distance bounded by a constant, and converges; for many initial graphs (large cycles) it decreases diameter does not change from time 0 On-line Social Networks - Anthony Bonato

Clustering Coefficient Theorem 3: If t > 0, then c(Gt) = ntlog(7/8)+o(1). higher clustering than in a random graph G(nt,p) with same order and average degree as Gt, which satisfies c(G(nt,p)) = ntlog(3/4)+o(1) On-line Social Networks - Anthony Bonato

Sketch of proof of lower bound each node x at time t has a binary sequence corresponding to descendants from time 0, with a clone indicated by 1 let e(x,t) be the number of edges in N(x) at time t we may show that e(x,t+1) = 3e(x,t) + 2degt(x) e(x’,t+1) = e(x,t) + degt(x) if there are k many 0’s in the binary sequence of x, then e(x,t) ≥ 3k-2e(x,2) = Ω(3k) On-line Social Networks - Anthony Bonato

Sketch of proof, continued there are many nodes with k many 0’s in their binary sequence hence, On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Spectral results the spectral gap λ of G is defined by max{|λ1-1|, |λn-1-1|} where 0 = λ0 ≤ λ1 ≤ … ≤ λn-1 ≤ 2 are the eigenvalues of the normalized Laplacian of G: I-D-1/2AD1/2 (Chung, 97) for random graphs, λ = o(1) in the ILT model, λ > ½ bad spectral expansion found in the ILT model characteristic of social networks but not the web graph (Estrada, 06) in social networks, there are a higher number of intra- rather than inter-community links On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato …Degree distribution generate power law graphs from ILT? deterministic ILT model gives a binomial-type distribution On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Geometry of OSNs? OSNs live in social space: proximity of nodes depends on common attributes (such as geography, gender, age, etc.) IDEA: embed OSN in 2-, 3- or higher dimensional space On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Dimension of an OSN dimension of OSN: minimum number of attributes needed to classify nodes like game of “20 Questions”: each question narrows range of possibilities what is a credible mathematical formula for the dimension of an OSN? On-line Social Networks - Anthony Bonato

Random geometric graphs nodes are randomly placed in space nodes are joined if their distance is less than a threshold value (Penrose, 03) On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Simulation with 5000 nodes On-line Social Networks - Anthony Bonato

Geometric model for OSNs we consider a geometric model of OSNs, where nodes are in m-dimensional Euclidean space threshold value variable: a function of ranking of nodes On-line Social Networks - Anthony Bonato

Geometric Protean (GEO-P) Model (Bonato, Janssen, Prałat, 10) parameters: α, β in (0,1), α+β < 1; positive integer m nodes live in m-dimensional hypercube each node is ranked 1,2, …, n by some function r 1 is best, n is worst we use random initial ranking at each time-step, one new node v is born, one randomly node chosen dies (and ranking is updated) each existing node u has a region of influence with volume add edge uv if v is in the region of influence of u On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Notes on GEO-P model models uses both geometry and ranking dynamical system: gives rise to ergodic (therefore, convergent) Markov chain users join and leave OSNs number of nodes is static: fixed at n order of OSNs has ceiling top ranked nodes have larger regions of influence On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Simulation with 5000 nodes On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Simulation with 5000 nodes random geometric GEO-P On-line Social Networks - Anthony Bonato

Properties of the GEO-P model (Bonato, Janssen, Prałat, 09) with high probability, the GEO-P model a.a.s. generates graphs with the following properties: power law degree distribution with exponent b = 1+1/α average degree d = (1+o(1))n(1-α-β)/21-α dense graph diameter D = (1+o(1))nβ/(1-α)m m = clog n, then diameter is a constant bad spectral expansion if m = clog n, then On-line Social Networks - Anthony Bonato

Introducing the Web Graph - Anthony Bonato rich get richer: as nodes are born, they are more likely to enter some larger region of influence over time, a power law degree distribution results rigorous proof follows by calculating expected degrees and applying the Chernoff bounds Introducing the Web Graph - Anthony Bonato

On-line Social Networks - Anthony Bonato Dimension of OSNs given the order of the network n, power law exponent b, average degree d, and diameter D, we can calculate m gives formula for dimension of OSN: On-line Social Networks - Anthony Bonato

Uncovering the hidden reality reverse engineering approach given network data (n, b, d, D), dimension of an OSN gives smallest number of attributes needed to identify users that is, given the graph structure, we can (theoretically) recover the social space On-line Social Networks - Anthony Bonato

“6 Dimensions of Separation” OSN Dimension Facebook 6 MySpace 8 Twitter 4 Flickr Cyworld 7 On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Future directions what is a community in an OSN? (Porter, Onnela, Mucha,09): a set of graph partitions obtained by some “reasonable” iterative hierarchical partitioning algorithm motifs Pott’s method from statistical mechanics betweeness centrality lack of a formal definition, and few theorems On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato GEO-P model validating the GEO-P model fit model to data is theoretical estimate of the dimension of an OSN accurate? On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato Who is popular? how to find popular users? not just degree If you have popular friends, then you should be more popular “SocialRank” ? OSN version of Google’s PageRank algorithm On-line Social Networks - Anthony Bonato

preprints, reprints, contact: Google: “Anthony Bonato” On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato WOSN’2010 On-line Social Networks - Anthony Bonato

Graphs at Ryerson (G@R) On-line Social Networks - Anthony Bonato

On-line Social Networks - Anthony Bonato