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ICMCM’09 December, 2009. Dynamic Models of On-line Social Networks. Anthony Bonato Ryerson University. Toronto in December…. Complex Networks. web graph, social networks, biological networks, internet networks , …. nodes : web pages edges : links

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Dynamic models of on line social networks

ICMCM’09

December, 2009

Dynamic Models of On-line Social Networks

Anthony Bonato

Ryerson University

On-line Social Networks - Anthony Bonato


Dynamic models of on line social networks

Toronto in December…

On-line Social Networks - Anthony Bonato


Complex networks

Complex Networks

  • web graph, social networks, biological networks, internet networks, …

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

The web graph

On-line Social Networks - Anthony Bonato


Social networks

Social Networks

nodes: people

edges:

social interaction

(eg friendship)

On-line Social Networks - Anthony Bonato


On line social networks osns facebook twitter orkut linkedin gupshup

On-line Social Networks (OSNs)Facebook, Twitter, Orkut, LinkedIn, GupShup…

On-line Social Networks - Anthony Bonato


A new paradigm

A new paradigm

  • half of all users of internet on some OSN

    • 250 million users on Facebook, 45 million on Twitter

  • unprecedented, massive record of social interaction

  • unprecedented access to information/news/gossip

On-line Social Networks - Anthony Bonato


Properties of complex networks

Properties of Complex Networks

  • observed properties:

    • massive, power law, small world, decentralized

(Broder et al, 01)

On-line Social Networks - Anthony Bonato


Small world property

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


Paths in twitter

Paths in Twitter

Dalai Lama

Arnold

Schwarzenegger

Queen Rania

of Jordan

Christianne Amanpour

Ashton Kutcher

On-line Social Networks - Anthony Bonato


Why model complex networks

Why model complex networks?

  • uncover the generative mechanisms underlying complex networks

  • models are a predictive tool

  • nice mathematical challenges

  • models can uncover the hidden reality of networks

    • in OSNs:

      • community detection

      • advertising

      • security

On-line Social Networks - Anthony Bonato


Many different models

Many different models

On-line Social Networks - Anthony Bonato


Social network analysis

Social network analysis

On-line

  • Milgram (67): average distance between two Americans is 6

  • Watts and Strogatz (98): introduced small world property

  • Adamic et al. (03): early study of on-line social networks

  • Liben-Nowell et al. (05): small world property in LiveJournal

  • Kumar et al. (06): Flickr, Yahoo!360;average distances decrease with time

  • Golder et al. (06): studied 4 million users of Facebook

  • Ahn et al. (07): studiedCyworld in South Korea, along with MySpace and Orkut

  • Mislove et al. (07): studiedFlickr, YouTube, LiveJournal, Orkut

  • Java et al. (07): studied Twitter: power laws, small world

On-line Social Networks - Anthony Bonato


Key parameters

Key parameters

  • power law degree distributions:

  • average distance:

  • clustering coefficient:

Wiener index, W(G)

On-line Social Networks - Anthony Bonato


Power laws in osns

Power laws in OSNs

On-line Social Networks - Anthony Bonato


Flickr and yahoo 360

Flickr and Yahoo!360

  • (Kumar et al,06):shrinking diameters

On-line Social Networks - Anthony Bonato


Sample data flickr youtube livejournal orkut

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

(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

    • e(t) ≈ n(t)a

  • where 1 < a ≤ 2:densification exponent

    • a=1: linear growth – constant average degree, such as in web graph models

    • a=2: quadratic growth – cliques

  • On-line Social Networks - Anthony Bonato


    Densification physics citations

    Densification – Physics Citations

    e(t)

    1.69

    n(t)

    On-line Social Networks - Anthony Bonato


    Densification autonomous systems

    Densification – Autonomous Systems

    e(t)

    1.18

    n(t)

    On-line Social Networks - Anthony Bonato


    Dynamic models of on line social networks

    • Decreasing distances

    • distances (diameter and/or average distances) decrease with time

      • noted by Kumar et al. in Flickr and Yahoo!360

    • Preferential attachment model (Barabási, Albert, 99), (Bollobás et al, 01)

      • diameter O(log t)

    • Random power law graph model (Chung, Lu, 02)

      • average distance O(log log t)

    On-line Social Networks - Anthony Bonato


    Diameter arxiv citation graph

    Diameter – ArXiv citation graph

    diameter

    time [years]

    On-line Social Networks - Anthony Bonato


    Diameter autonomous systems

    Diameter – Autonomous Systems

    diameter

    number of nodes

    On-line Social Networks - Anthony Bonato


    Models for the laws

    Models for the laws

    • (Leskovec, Kleinberg, Faloutsos, 05, 07):

      • Forest Fire model

        • stochastic

        • densification power law, decreasing diameter, power law degree distribution

    • (Leskovec, Chakrabarti, Kleinberg,Faloutsos, 05, 07):

      • Kronecker Multiplication

        • deterministic

        • densification power law, decreasing diameter, power law degree distribution

    On-line Social Networks - Anthony Bonato


    Models of osns

    Models of OSNs

    • many models exist for general complex networks

    • few models for on-line social networks

    • goal: find a model which simulates many of the observed properties of OSNs

      • must be simple and evolve in a natural way

      • must be different than previous complex network models: densification and constant diameter!

    On-line Social Networks - Anthony Bonato


    Dynamic models of on line social networks

    “All models are wrong, but some are more useful.”

    – G.P.E. Box

    On-line Social Networks - Anthony Bonato


    Iterated local transitivity ilt model bonato hadi horn pra at wang 08

    Iterated Local Transitivity (ILT) model(Bonato, Hadi, Horn, Prałat, Wang, 08)

    • key paradigm is transitivity: friends of friends are more likely friends; eg (Girvan and Newman, 03)

      • iterative cloning of closed neighbour sets

    • deterministic: amenable to analysis

    • local: nodes often only have local influence

    • evolves over time, but retains memory of initial graph

    On-line Social Networks - Anthony Bonato


    Ilt model

    ILT model

    • parameter: finite simple undirected graph G = G0

    • to form the graph Gt+1 for each vertex x from time t, add a vertex x’, the clone ofx, so that xx’ is an edge, and x’ is joined to each neighbour of x

    • order of Gt is 2tn0

    On-line Social Networks - Anthony Bonato


    G 0 c 4

    G0 = C4

    On-line Social Networks - Anthony Bonato


    Properties of ilt model

    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

    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


    Average distance

    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

    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

    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

    Sketch of proof, continued

    • there are many nodes with k many

      0’s in their binary sequence

    • hence,

    On-line Social Networks - Anthony Bonato


    Example of community structure

    Example of community structure

    • Wayne Zachary’s Ph.D. thesis (1970-72): 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


    Adjacency matrix a

    Adjacency matrix, A

    eigenvalue spectrum: (-2)41531

    On-line Social Networks - Anthony Bonato


    Spectral results

    Spectral results

    • the spectral gapλ of G is defined by

      min{λ1, 2 - λn-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, λtends to 1 as order grows

    • in the ILT model, λ < ½

    • bad expansion/small spectral gaps in the ILT model found in social networks but not in 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


    Random ilt model

    Random ILT model

    • randomize the ILT model

      • add random edges independently to new nodes, with probability a function of t

      • makes densification tunable

        • densification exponent becomes

          log(3 + ε) / log(2),

          where ε is any fixed real number in (0,1)

          • gives any exponent in (log(3)/log(2), 2)

        • similar (or better) distance, clustering and spectral results as in deterministic case

    On-line Social Networks - Anthony Bonato


    Degree distribution

    Degree distribution

    • generate power law graphs from ILT?

      • deterministic ILT model gives a binomial-type distribution

    On-line Social Networks - Anthony Bonato


    Geometric model for social networks

    Geometric model for social networks

    • OSNs live in social space: proximity of nodes depends on common attributes (such as geography, gender, age, etc.)

    • IDEA: embed OSN in m-dimensional Euclidean space

    On-line Social Networks - Anthony Bonato


    Dimension of an osn

    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

    Random geometric graphs

    • nodes are randomly distributed in Euclidean space according to a given distribution

    • nodes are joined by an edge if and only if their distance is less than a threshold value

      (Penrose, 03)

    On-line Social Networks - Anthony Bonato


    Spatial model for osns

    Spatial model for OSNs

    • we consider a spatial model of OSNs, where

      • nodes are embedded in m-dimensional Euclidean space

      • number of nodes is static

      • threshold value variable: a function of ranking of nodes

    On-line Social Networks - Anthony Bonato


    Prestige based spatial pbs model bonato janssen pra at 09

    Prestige-Based Spatial (PBS) Model(Bonato, Janssen, Prałat, 09)

    • parameters: α, β in (0,1), α+β < 1; positive integer m

    • nodes live in hypercube of dimension m, measure 1

    • 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 node chosen u.a.r. 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


    Notes on pbs model

    Notes on PBS 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


    Properties of the pbs model bonato janssen pra at 09

    Properties of the PBS model (Bonato, Janssen, Prałat, 09)

    • with high probability, the PBS model 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

        • tends to infinity with n

      • diameter D = (1+o(1))nβ/(1-α)m

        • depends on dimension m

        • m = clog n, then diameter is a constant

    On-line Social Networks - Anthony Bonato


    Dimension of an osn continued

    Dimension of an OSN, continued

    • given the order of the network n, power law exponentb, average degree d, and diameterD, we can calculate m

    • gives formula for dimension of OSN:

    On-line Social Networks - Anthony Bonato


    Uncovering the hidden reality

    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


    Examples

    Examples

    On-line Social Networks - Anthony Bonato


    Future directions

    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


    Spatial ranking models

    Spatial ranking models

    • rigorously analyze spatial model with ranking by

      • age

      • degree

    • simulate PBS model

      • fit model to data

      • is theoretical estimate of the dimension of an OSN accurate?

    On-line Social Networks - Anthony Bonato


    Who is popular

    Who is popular?

    • how to find popular users?

    • PageRank in OSNs

    • domination number

      • constant in ILT model

      • in OSN data, domination number is large (end-vertices)

      • which is the correct graph parameter to consider?

    On-line Social Networks - Anthony Bonato


    Dynamic models of on line social networks

    preprints, reprints, contact:

    Google: “Anthony Bonato”

    On-line Social Networks - Anthony Bonato


    Wosn 2010

    WOSN’2010

    On-line Social Networks - Anthony Bonato


    Dynamic models of on line social networks

    On-line Social Networks - Anthony Bonato


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