Anthony Bonato Ryerson University 1 st Symposium on Spatial Networks Oxford University 1 Friendship networks network of on and offline friends form a large web of interconnected links 2 Geometry of Social Networks ID: 647347
Download Presentation The PPT/PDF document "The Geometry of Social Networks" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
The Geometry of Social Networks
Anthony BonatoRyerson University
1st Symposium on Spatial NetworksOxford University
1Slide2
Friendship networks
network of on- and off-line friends form a large web of interconnected links2
Geometry of Social NetworksSlide36 degrees of separation
3(Stanley Milgram,67)
: famous chain letter experiment
Geometry of Social NetworksSlide44 Degrees in Facebook
1.71 billion users(Backstrom,Boldi,Rosa, Ugander,Vigna,
2012)4 degrees of separation in Facebookwhen considering another person in the world, a friend of your friend knows a friend of their friend, on averagesimilar results for Twitter and other OSNs
4Geometry of Social NetworksSlide5Are we really that similar?
Geometry of Social Networks5Slide6Social distance
4 or 6 degrees of separation does not reflect our true
social distanceGeometry of Social Networks
6
D. Liben-Nowell, J.
Kleinberg,
Tracing information flow on a global scale using Internet chain-letter data
PNAS
105
(2008) 4633-4638
.Slide7Hidden geometry
Geometry of Social Networks7
vsSlide8
8Complex networks in the era of Big Data
web graph, social networks, biological networks, internet networks, …
Geometry of Social NetworksSlide9What is a complex network?
no precise definitionhowever, there is general consensus on the following observed propertieslarge scaleevolving over time
power law degree distributionssmall world properties9
Geometry of Social NetworksSlide10Examples of complex networks
technological/informational: web graph, router graph, AS graph, call graph, e-mail graphsocial: on-line social networks (Facebook, Twitter, LinkedIn,…), collaboration graphs, co-actor graph
biological networks: protein interaction networks, gene regulatory networks, food networks, connectomes10
Geometry of Social NetworksSlide11Other properties
densification power law (Leskovec, Kleinberg, Faloutsos,05):
|(E(Gt)| ≈ |V(Gt)|a
where 1 < a ≤ 2: densification exponentcommunity structurespectral expansion
Geometry of Social Networks
11Slide12
Blau spaceOSNs live in social space or Blau space: each user identified with a point in a multi-dimensional space
coordinates correspond to socio-demographic variables/attributeshomophily principle: the flow of information between users is a declining function of distance in Blau space12
Geometry of Social NetworksSlide13Dimensionality
Question: What is the dimension of the Blau space of OSNs?
what is a credible mathematical formula for the dimension of an OSN?13Geometry of Social NetworksSlide14
Geometry of Social Networks14Slide15
15Random geometric graphs
n nodes are randomly placed in the unit squareeach node has a constant sphere of influence, radius
rnodes are joined if their Euclidean distance is at most rG(n,r), r = r(n)
Geometry of Social NetworksSlide16Some properties of G(n,r)
Theorem (Penrose,97) Let μ = nexp(-
πr2n).If μ = o(1), then
asymptotically almost surely (a.a.s.) G is connected.If μ = Θ(1), then a.a.s. G has a component of order
Θ
(n).
If
μ
→∞
, then a.a.s.
G
is disconnected.
many other properties studied of
G(n,r)
: chromatic number, clique number, Hamiltonicity, random walks, …
16
Geometry of Social NetworksSlide17
Spatially Preferred Attachment (SPA) model(Aiello, Bonato, Cooper, Janssen, Prałat,08), (Cooper, Frieze,
Prałat,12)17
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
Geometry of Social NetworksSlide18Ranking 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 r1 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.
18
Geometry of Social NetworksSlide19Geometric model for OSNs
we consider a geometric model of OSNs, wherenodes are in m
-dimensional Euclidean spacevolume of spheres of influence variable: a function of ranking of nodes19
Geometry of Social NetworksSlide20
Geometric Protean (GEO-P) Model(Bonato,Janssen,Prałat,12)parameters:
α, β in (0,1), α+β
< 1; positive integer mnodes live in an m-dimensional hypercubeeach node is ranked 1,2, …, n by some function r1 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
20
Geometry of Social NetworksSlide21Notes on GEO-P model
models uses both geometry and rankingnumber of nodes is static: fixed at norder of OSNs at most number of people (roughly…)
top ranked nodes have larger regions of influence 21Geometry of Social NetworksSlide22Simulation with 5000 nodes
22
Geometry of Social NetworksSlide23Simulation with 5000 nodes
23
random geometric
GEO-P
Geometry of Social NetworksSlide24Properties of the GEO-P model
(BJP,2012)a.a.s. the GEO-P model generates graphs with the following properties:power law degree distribution with exponent
b = 1+1/αaverage degree d =
(1+o(1))n(1-α-β)/21-α
densification
diameter
D =
n
Θ
(1/m)
small world:
constant order if
m
= Clog n
bad spectral expansion
and
high clustering coefficient
24
Geometry of Social NetworksSlide25Dimension of OSNs
given the order of the network n and diameter
D, we can calculate m gives formula for dimension of OSN:
25
Geometry of Social NetworksSlide26
Logarithmic Dimension HypothesisIn an OSN of order n and diameter D, the dimension of its Blau space is
posed independently by (Leskovec,Kim,11), (Frieze, Tsourakakis,11)26
Geometry of Social NetworksSlide27Few dimensions implies
greater differenceGeometry of Social Networks27
low dimensional separation
high dimensional separationSlide28Uncovering the hidden reality
reverse engineering approachgiven 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 Blau space28
Geometry of Social NetworksSlide296 Dimensions of Separation
OSN
DimensionFacebook7
YouTube6Twitter
4
Flickr
4
Cyworld
7
29
Geometry of Social NetworksSlide30
Geometry of Social Networks30Slide31MGEO-P
(Bonato,Gleich,Mitsche,Prałat,Tian,Young,14)time-steps in GEO-P form a computational bottleneckconsider 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
31
Geometry of Social NetworksSlide32Contrasting 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 graphsmany a.a.s properties hold in MGEO-P with similar parameters
32
Geometry of Social NetworksSlide33Properties 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/αaverage degree
d = (1+o(1))n(1-α-β)/2
1-
α
densification
diameter
D = n
Θ
(1/m)
33
Geometry of Social NetworksSlide34
Proof sketch: diametereminent node: highly ranked
: ranking greater than some fixed Rpartition hypercube into small hypercubeschoose size of hypercubes and R so thateach hypercube contains at least log2
n eminent nodessphere of influence of each eminent node covers each hypercube and all neighbouring hypercubeschoose eminent node in each hypercube: backboneshow all nodes in hypercube distance at most 2 from backbone
34
Geometry of Social NetworksSlide35Back to question…
How would we measure the dimensionality of Blau space?
35Geometry of Social NetworksSlide36Aside: machine learning
machine learning is a branch of AI that infers structure from dataexamples: spam filtersNetflix recommender systems
text and image categorizationespecially useful when the data or number of decisions are too large for humans to process36
Geometry of Social NetworksSlide37Model selection in
complex networks (Middendorf,Ziv,Wiggins,05) used ADTs and motifs for model selection in protein networkspredicted
duplication/mutation model (Memišević,Milenković,Pržulj,10) model selection predicting
random geometric graphs as best fit for protein networks (Janssen,Hurshman,Kalyaniwalla,12)ADT with motif classifiers predict PA and SPA models best fit Facebook 100 graphs
Geometry of Social Networks
37Slide38
38
Support Vector Machine (SVM)
support
vectors
maximizes
margin
SVM maximizes the
margin
around the separating hyperplane
solving SVMs is a
quadratic programming
problem
successful text and image classification method
Sec. 15.1
Geometry of Social NetworksSlide39Facebook100
Geometry of Social Networks39Slide40Validating the LDH
we tested the dimensionality of large-scale samples from real OSN dataFB100 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 12
40Geometry of Social NetworksSlide41Motifs/Graphlets
Geometry of Social Networks
41Slide42Experimental design
Geometry of Social Networks
42Slide43Sample: Michigan
Geometry of Social Networks
43Slide44
44Stanford3:
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
Geometry of Social NetworksSlide45FB and LinkedIn - SVM
Geometry of Social Networks45Slide46FB and LinkedIn - Eigenvalues
46Geometry of Social NetworksSlide47
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.0106052
http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106052
Geometry of Social Networks
47Slide48Underlying geometry
Feature space thesis (B,16+)every complex network has an underlying metric (or
feature) space, where nodes are identified with points in the feature space, and edges are influenced by node similarity and proximity in the spaceFor e.g.:
web graph: topic spaceOSNs: Blau spacePPIs: biochemical space48
Geometry of Social NetworksSlide49Implications of FST
new way of viewing complex networksnot just graph structure, but underlying, hidden geometry that mattersgraph structure can help uncover this hidden geometry
Geometry of Social Networks49Slide50Future directions
other data setsfractal or other dimensionunderlying metric?what are the attributes?
what implications does LDH have for OSNs or social networks in general?50Geometry of Social NetworksSlide51Character networks
cultural work:fictional works such novels or short stories, movies, biographies, historical works, religious texts
character networks: nodes: characters or persona in a cultural work
edges: co-occurrenceedges may be weightedGeometry of Social Networks51Slide52E.g.:
Marvel universeCharacter networks
52
10K nodesdiameter 9
10
communities
average degree
41Slide53moviegalaxies.com
Geometry of Social Networks53
moviegalaxies.com,catalogues
the social networks in 800+ moviesSlide54Dimensionality
of character networks?
Geometry of Social Networks54