Dynamic Models of On-Line Social Networks

On-line Social Networks - Anthony Bonato

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Dynamic Models of On-Line Social Networks

Anthony BonatoRyerson University

WAW’2009February 13, 2009

nt


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Complex Networks

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


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Social Networks

nodes: people

edges: social interaction


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On-line networks: Flickr graph


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Facebook graph


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On-line gaming networks


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Properties of Complex Networks• observed properties:

– massive, power law, small world, decentralized

– many bipartite subgraphs, high clustering, sparse cuts, strong conductance, eigenvalue power law, …

(Broder et al, 01)


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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)


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Social network analysis• 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): studied Cyworld in South Korea, along with MySpace and Orkut

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

On-line


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Key parameters

• power law degree distributions:

• average distance:

• clustering coefficient:

tiNb bti

, ,1

)(,

1

2),()(

GVvu

tvudGL

)(

1

-1

)()( ,2

)deg(|))((|)(

GVxxctGC

xxNExc

Wiener index, W(G)


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Facebook• Golder et al (06):

• current number of users (nodes): > 120 million


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Flickr and Yahoo!360

• Kumar et al (06): shrinking diameters


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Sample data: Flickr, YouTube, LiveJournal, Orkut

• Mislove et al (07): short average distances and high clustering coefficients


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Leskovec, Kleinberg, Faloutsos (05):

– many complex networks obey two laws:

1. 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


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Densification – Physics Citations

n(t)

e(t)

1.69


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Densification – Autonomous Systems

n(t)

e(t)

1.18


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2. Decreasing distances

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

– noted by Kumar et al in Flickr and Yahoo!360

• in contrast with Preferential attachment model

– a.a.s. diameter O(log t)


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Diameter – ArXiv citation graph

time [years]

diameter


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Diameter – Autonomous Systems

number of nodes

diameter


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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


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Models of On-line Social Networks

• many models exist for general complex networks (preferential attachment, random power law, copying, duplication, geometric, rank-based, Forest fire, …)

• few models for on-line social networks• goal: design and analyze a model which

simulates many of the observed properties of on-line social networks– model should be simple and evolve in a natural way


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“All models are wrong, but some are more useful.” – G.P.E. Box


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

• key paradigm is transitivity: friends of friends are more likely friends; eg (Frank,80),

(White, Harrison, Breiger, 76), (Scott, 00)– 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


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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 of x, so that xx’ is an edge, and x’ is joined to each neighbour of x


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G0 = C4


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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

• cop and domination numbers do not change with time

• bad expansion: small gaps between 1st and 2nd eigenvalues in adjacency and normalized Laplacian matrices of Gt


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Densification

• nt = order of Gt, et = size of Gt

Lemma: For t > 0,

nt = 2tn0, et = 3t(e0+n0) - 2tn0.

→ densification power law:et ≈ nt

a, where a = log(3)/log(2).


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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


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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)


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Sketch of proof

• 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 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 1’s in the binary sequence of x,

then e(x,t) ≥ 3k-2e(x,2) = Ω(3k)


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Sketch of proof, continued

• there are many nodes with k many

0’s in their binary sequence• hence,

k

tn0

2

2

0

2

0 0

8

7

243

1

2

43

)( tt

n

tk

tn

GCt

t

t

t

kt

k

t


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• example: (Zachary, 72) observed social ties and rivalries in a university karate club (34 nodes,78 edges)• during his observation, conflicts intensified and group split• see also (Girvan, Newman, 02)

Community structure in social networks


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Spectral results• the spectral gap λ 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, λ tends to 1 as order grows• in the ILT model, λ < ½• similar results for adjacency matrix A• bad expansion/small spectral gaps in the ILT model found in

social networks but not in the web graph or biological networks (Estrada, 06) – in social networks, there are a higher number of intra-

rather than inter-community links


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Random model

• randomize the ILT model – add random edges independently to new

nodes, with probability a function of t– makes densification tuneable

• 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


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Missing ingredient: Power laws

– generate power law graphs from ILT?

• deterministic ILT model gives a binomial-type distribution


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• preprints, reprints, contact:

Google: “Anthony Bonato”

http://www.cse-cst.gc.ca/

Dynamic Models of On-Line Social Networks

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