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The Logarithmic Dimension Hypothesis

Anthony BonatoRyerson University

MITACS International Problem Solving Workshop

July 2012


Workshop team• David Gleich (Purdue)

• Dieter Mitsche (Ryerson)

• Stephen Young (UCSD)

• Myunghwan Kim (Stanford)

• Amanda Tian (York)


Friendship networks• network of friends (some real, some virtual) form

a large web of interconnected links


6 degrees of separation

• (Stanley Milgram, 67): famous chain letter experiment


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


Complex Networks• web graph, social networks, biological networks, internet

networks, …


The web graph

• nodes: web pages

• edges: links

• over 1 trillion nodes, with billions of nodes added each day


On-line Social Networks (OSNs)Facebook, Twitter, LinkedIn, Google+…

http://www.tweetwheel.com/AJCann


Key parameters• degree distribution:

• average distance:

• clustering coefficient:

|})deg(:)({| , iuGVuN ni

)(,

1

2),()(

GVvu

nvudGL

)(

1-1

)()( ,2

)deg(|))((| )(

GVxxcnGC

xxNExc


Properties of Complex Networks• power law degree distribution

(Broder et al, 01)

2 some ,, bniN bni

Power laws in OSNs (Mislove et al,07):



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


Sample data: Flickr, YouTube, LiveJournal, Orkut

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


• (Zachary, 72)

• (Mason et al, 09)

• (Fortunato, 10)

• (Li, Peng, 11): small community property

Community structure


(Leskovec, Kleinberg, Faloutsos,05):• densification power law: average degree is

increasing with time• decreasing distances

• (Kumar et al, 06): observed in Flickr, Yahoo! 360


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


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?


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


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


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


Simulation with 5000 nodes


Simulation with 5000 nodes

random geometric GEO-P


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


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


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


6 Dimensions of Separation

OSN DimensionFacebook 7YouTube 6Twitter 4Flickr 4

Cyworld 7


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


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)


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


Example


• preprints, reprints, contact:search: “Anthony Bonato”

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