on-line social networks - anthony bonato 1 dynamic models of on-line social networks anthony bonato...
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On-line Social Networks - Anthony Bonato
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Dynamic Models of On-Line Social Networks
Anthony BonatoRyerson University
WAW’2009February 13, 2009
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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”