1 Dimension matching in Facebook and LinkedIn networks Anthony Bonato Ryerson University Seminar on...

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Dimension matching in Facebook and LinkedIn networks

Anthony BonatoRyerson University

Seminar on Social Networks, Big Data, Influence, and Decision-Making

University of Toronto

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Friendship networks

• network of on- and off-line friends form a large web of interconnected links

Dimension matching in OSNs

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6 degrees of separation

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


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6 Degrees in Facebook?• 1.31 billion users• (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


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Complex networks in the era of Big Data

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


What is a complex network?

• no precise definition• however, there is general consensus on the

following observed properties

1. large scale

2. evolving over time

3. power law degree distributions

4. small world properties

6Dimension matching in OSNs

Examples of complex networks

• technological/informational: web graph, router graph, AS graph, call graph, e-mail graph

• social: on-line social networks (Facebook, Twitter, LinkedIn,…), collaboration graphs, co-actor graph

• biological networks: protein interaction networks, gene regulatory networks, food networks


Properties of complex networks

1. Large scale: relative to order and size

• web graph: order > trillion– some sense infinite: number of strings entered into

Google• Facebook: > 1 billion nodes; Twitter: > 270 million

nodes– much denser (ie higher average degree) than the

web graph• protein interaction networks: order in thousands


Properties of complex networks

2. Evolving: networks change over time

• web graph: billions of nodes and links appear and disappear each day

• Facebook: grew to 1 billion users – denser than the web graph

• protein interaction networks:

order in the thousands– evolves much more slowly


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Properties of Complex Networks

3. Power law degree distribution

• for a graph G of order n and i a positive integer, let Ni,n denote the number of nodes of degree i in G

• we say that G follows a power law degree distribution if for some range of i and some b > 2,

• b is called the exponent of the power law

niN bni

,


Power laws in OSNs


Dimension matching in OSNs 12

Graph parameters

• average distance:

• clustering coefficient:

)(,

1

2),()(

GVvu

nvudGL

)(

1

-1

)()( ,2

)deg(|))((| )(

GVxxcnGC

xxNExc

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Properties of Complex Networks

4. Small world property

• introduced by Watts & Strogatz in 1998:– low distances

• diam(G) = O(log n)• L(G) = O(loglog n)

– higher clustering coefficient than random graph with same expected degree


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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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Other properties of complex networks

• many complex networks (including on-line social networks) obey two additional laws:

• Densification power law (Leskovec, Kleinberg, Faloutsos,05):

– networks are becoming more dense over time; i.e. average degree is increasing

|(E(Gt)| ≈ |V(Gt)|a

where 1 < a ≤ 2: densification exponent


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• Decreasing distances (Leskovec, Kleinberg, Faloutsos,05):

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

(Kumar et al,06):


Other properties

• Connected component structure: emergence of components; giant components

• Spectral properties: adjacency matrix and Laplacian matrices, spectral gap, eigenvalue distribution

• Small community phenomenon: most nodes belong to small communities (ie subgraphs with more internal than external links)

…


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Blau space

• OSNs live in social space or Blau space: – each user identified with a point in a

multi-dimensional space– coordinates correspond to socio-

demographic variables/attributes

• homophily principle: the flow of information between users is a declining function of distance in Blau space


Underlying geometry

Feature space thesis: every complex network is naturally associated with an underlying feature space.

For eg:– web graph: topic space– OSNs: Blau space– PPIs: biochemical space


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Dimensionality

• Question: What is the dimension of the Blau space of OSNs?

• what is a credible mathematical formula for the dimension of an OSN?


Six dimensions of separation 21

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Why model complex networks?

• uncover and explain the generative mechanisms underlying complex networks

• predict the future• nice mathematical challenges• models can uncover the hidden reality of

networks


Networks - Bonato 23

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


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Random geometric graphs• n nodes are randomly

placed in the unit square

• each node has a constant sphere of influence, radius r

• nodes are joined if their Euclidean distance is at most r

• G(n,r), r = r(n)


Some properties of G(n,r)

Theorem (Penrose,97) Let μ = nexp(-πr2n).

1. If μ = o(1), then asymptotically almost surely (a.a.s.) G is connected.

2. If μ = Θ(1), then a.a.s. G has a component of order Θ(n).

3. If μ →∞, then a.a.s. G is disconnected.

• many other properties studied of G(n,r): chromatic number, clique number, Hamiltonicity, random walks, …


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Spatially Preferred Attachment (SPA) model(Aiello, Bonato, Cooper, Janssen, Prałat,08),

(Cooper, Frieze, Prałat,12)

• volume of sphere of influence proportional to in-degree

• nodes are added and spheres of influence shrink over time

• a.a.s. leads to power laws graphs, low directed diameter, and small separators


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Ranking models(Fortunato, Flammini, Menczer,06),

(Łuczak, Prałat, 06), (Janssen, Prałat,09) • parameter: α in (0,1)• each node is ranked 1,2, …, n by some function r

– 1 is best, n is worst

• at each time-step, one new node is born, one randomly node chosen dies (and ranking is updated)

• link probability r-α

• many ranking schemes a.a.s. lead to power law graphs: random initial ranking, degree, age, etc.


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Geometric model for OSNs

• we consider a geometric model of OSNs, where– nodes are in m-

dimensional Euclidean space

– volume of spheres of influence variable: a function of ranking of nodes


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Geometric Protean (GEO-P) Model(Bonato, Janssen, Prałat, 12)

• parameters: α, β in (0,1), α+β < 1; positive integer m• nodes live in an m-dimensional hypercube• 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


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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


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Simulation with 5000 nodes


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Simulation with 5000 nodes

random geometric GEO-P


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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 = nΘ(1/m)

• small world: constant order if m = Clog n– bad spectral expansion and high clustering coefficient


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Dimension of OSNs

• given the order of the network n and diameter D, we can calculate m

• gives formula for dimension of OSN:

D

nm

log

log


Logarithmic Dimension Hypothesis

In an OSN of order n and diameter D, the dimension of its Blau space is

• posed independently by (Leskovec,Kim,11), (Frieze, Tsourakakis,11)

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D

n

log

log


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Uncovering the hidden reality• reverse engineering approach

– given network data (n, 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


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6 Dimensions of Separation

OSN Dimension

Facebook 7

YouTube 6

Twitter 4

Flickr 4

Cyworld 7


MITACS team, UBC 2012

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L to R: Amanda Tian, David Gleich, Myughwan Kim, Me, Stephen Young, Dieter Mitsche



MGEO-P(Bonato, Gleich, Mitsche, Prałat, Tian, Young,14)

• time-steps in GEO-P form a computational bottleneck

• consider a GEO-P where we forget the history of ranks– memoryless GEO-P (MGEO-P)

• place n points u.a.r. in the hypercube • assign ranks from via a random permutation σ• for each pair i > j, ij is an edge if j is in the ball of

volume

σ(i)–αn-β


Contrasting the models

• by considering the evolution of ranks in GEO-P, the probability that an edge is present in GEO-P and not in MGEO-P is:

• intuitively, the models generate similar graphs

• many a.a.s properties hold in MGEO-P with similar parameters

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)1()log( 2

41

2

2

onnO


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Properties of the MGEO-P model (BGMPTY,14)

• a.a.s. the MGEO-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 = nΘ(1/m)


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Proof sketch: diameter• eminent node:

– highly ranked: ranking greater than some fixed R

• partition hypercube into small hypercubes• choose size of hypercubes and R so that

– each hypercube contains at least log2n eminent nodes

– sphere of influence of each eminent node covers each hypercube and all neighbouring hypercubes

• choose eminent node in each hypercube: backbone

• show all nodes in hypercube distance at most 2 from backbone


Back to question…

• How would we measure the dimensionality of Blau space?


Aside: machine learning

• machine learning is a branch of AI where computers make decisions and answer questions based on data sets

• examples: – spam filters– Netflix recommender systems

• especially useful when the data or number of decisions are too large for humans to process


Facebook100


Validating the LDH

• we tested the dimensionality of large-scale samples from real OSN data– Facebook100 and LinkedIn (sampled over

time)• IDEA: use machine learning (SVM) to predict

dimensions– features: small subgraph counts (3- and 4-

vertex subgraphs)– compared sampled data vs simulations of

MGEO-P with dimensions 1 through 1248Dimension matching in OSNs

Graphlets


Experimental design


Sample: Michigan


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Stanford3: n: 11621 edges: 568330 avgdeg: 97.81086 plexp: 3.730000 GeoP parameters alphabeta: 0.510389 alpha: 0.366300 beta: 0.144089

python geop_dim_experiment.py --logcount -s 50 -t 0 --mmax 12 --prob 0.001 Stanford3 11621 568330 0.366300 0.144089 M-GeoP dimensions: LADTree: 2 J48: 3 Logistic: 5 SVM: 5


FB and LinkedIn - SVM


FB and LinkedIn - Eigenvalues


Figure 6. For three of the Facebook networks, we show the eigenvalue histogram in red, the eigenvalue histogram from the best fit MGEO-P network in blue, and the eigenvalue histograms

for samples from the other dimensions in grey.

Bonato A, Gleich DF, Kim M, Mitsche D, et al. (2014) Dimensionality of Social Networks Using Motifs and Eigenvalues. PLoS ONE 9(9): e106052. doi:10.1371/journal.pone.0106052http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106052

http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106052

Future directions

• Other data sets

• Fractal dimension

• What are the attributes?

• What implications does LDH have for OSNs or social networks in general?


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1 Dimension matching in Facebook and LinkedIn networks Anthony Bonato Ryerson University Seminar on...

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