the logarithmic dimension hypothesis
DESCRIPTION
MITACS International Problem Solving Workshop July 2012. The Logarithmic Dimension Hypothesis. Anthony Bonato Ryerson University. Workshop team. David Gleich (Purdue) Dieter Mitsche (Ryerson) Stephen Young (UCSD) Myunghwan Kim (Stanford) Amanda Tian (York). Friendship networks. - PowerPoint PPT PresentationTRANSCRIPT
Log Dimension Hypothesis 1
The Logarithmic Dimension Hypothesis
Anthony BonatoRyerson University
MITACS International Problem Solving Workshop
July 2012
Log Dimension Hypothesis 2
Workshop team• David Gleich (Purdue)
• Dieter Mitsche (Ryerson)
• Stephen Young (UCSD)
• Myunghwan Kim (Stanford)
• Amanda Tian (York)
Log Dimension Hypothesis 3
Friendship networks• network of friends (some real, some virtual) form
a large web of interconnected links
Log Dimension Hypothesis 4
6 degrees of separation
• (Stanley Milgram, 67): famous chain letter experiment
Log Dimension Hypothesis 5
6 Degrees in Facebook?• 900 million users, > 70
billion friendship links• (Backstrom et al., 2012)
– 4 degrees of separation in Facebook
– when considering another person in the world, a friend of your friend knows a friend of their friend, on average
• similar results for Twitter and other OSNs
Log Dimension Hypothesis 6
Complex Networks• web graph, social networks, biological networks, internet
networks, …
Log Dimension Hypothesis 7
The web graph
• nodes: web pages
• edges: links
• over 1 trillion nodes, with billions of nodes added each day
Log Dimension Hypothesis 8
On-line Social Networks (OSNs)Facebook, Twitter, LinkedIn, Google+…
Log Dimension Hypothesis 9
Key parameters• degree distribution:
• average distance:
• clustering coefficient:
|})deg(:)({| , iuGVuN ni
)(,
1
2),()(
GVvu
nvudGL
)(
1-1
)()( ,2
)deg(|))((| )(
GVxxcnGC
xxNExc
Log Dimension Hypothesis 10
Properties of Complex Networks• power law degree distribution
(Broder et al, 01)
2 some ,, bniN bni
Power laws in OSNs (Mislove et al,07):
Log Dimension Hypothesis 11
Log Dimension Hypothesis 12
Small World Property• small world networks
(Watts & Strogatz,98)– low distances
• diam(G) = O(log n)• L(G) = O(loglog n)
– higher clustering coefficient than random graph with same expected degree
Log Dimension Hypothesis 13
Sample data: Flickr, YouTube, LiveJournal, Orkut
• (Mislove et al,07): short average distances and high clustering coefficients
Log Dimension Hypothesis 14
• (Zachary, 72)
• (Mason et al, 09)
• (Fortunato, 10)
• (Li, Peng, 11): small community property
Community structure
Log Dimension Hypothesis 15
(Leskovec, Kleinberg, Faloutsos,05):• densification power law: average degree is
increasing with time• decreasing distances
• (Kumar et al, 06): observed in Flickr, Yahoo! 360
Log Dimension Hypothesis 16
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
Log Dimension Hypothesis 17
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?
Log Dimension Hypothesis 18
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
Log Dimension Hypothesis 19
Geometric Protean (GEO-P) Model(Bonato, Janssen, Prałat, 12)
• parameters: α, β in (0,1), α+β < 1; positive integer m• nodes live in m-dimensional hypercube (torus metric)• 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 nr
Log Dimension Hypothesis 20
Notes on GEO-P model
• models uses both geometry and ranking• number of nodes is static: fixed at n
– order of OSNs at most number of people (roughly…)
• top ranked nodes have larger regions of influence
Log Dimension Hypothesis 21
Simulation with 5000 nodes
Log Dimension Hypothesis 22
Simulation with 5000 nodes
random geometric GEO-P
Log Dimension Hypothesis 23
Properties of the GEO-P model (Bonato, Janssen, Prałat, 2012)
• a.a.s. the GEO-P 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
• densification– diameter D = O(nβ/(1-α)m log2α/(1-α)m n)
• small world: constant order if m = Clog n– clustering coefficient larger than in comparable
random graph
Log Dimension Hypothesis 24
Spectral properties• the spectral gap λ of G is defined by the
difference between the two largest eigenvalues of the adjacency matrix of G
• for G(n,p) random graphs, λ tends to 0 as order grows
• in the GEO-P model, λ is close to 1• (Estrada, 06): bad spectral expansion in real
OSN data
Log Dimension Hypothesis 25
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:
Dn
nd
bb
Dnm
loglog
loglog
211
loglog
Log Dimension Hypothesis 26
6 Dimensions of Separation
OSN DimensionFacebook 7YouTube 6Twitter 4Flickr 4
Cyworld 7
Log Dimension Hypothesis 27
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
Log Dimension Hypothesis 28
Logarithmic Dimension Hypothesis
• Logarithmic Dimension Hypothesis (LDH): the dimension of an OSN is best fit by about log n, where n is the number of users OSN– theoretical evidence GEO-P and MAG
(Leskovec, Kim,12) models–empirical evidence? – (Sweeney, 2001)
Log Dimension Hypothesis 29
Experimental design• supervised machine learning
– Alternating Decision Trees (ADT)– approach of (Janssen et al, 12+) based on earlier
work on PIN by (Middendorf et al, 05)• classify OSN data vs simulated graphs from GEO-P
model in various dimensions• develop a feature vector (graphlets, degree distribution
percentiles, average distance, etc) to classify the correct dimension
• ADT will classify which dimension best fits the data – cross-validation and robustness testing
Log Dimension Hypothesis 30
Example
Log Dimension Hypothesis 31
• preprints, reprints, contact:search: “Anthony Bonato”
Log Dimension Hypothesis 32