1. Miniconference on the Mathematics of Computation
1st Symposium on Spatial Networks
Oxford University
The Geometry of
Social Networks
Anthony Bonato
Ryerson University

2. Geometry of Social Networks
Friendship networks
network of on- and off-line friends form a large web of interconnected links
Geometry of Social Networks

3. Geometry of Social Networks
6 degrees of separation
(Stanley Milgram,67): famous chain letter experiment
Geometry of Social Networks

4. Geometry of Social Networks
4 Degrees in Facebook
1.71 billion users
(Backstrom,Boldi,Rosa, Ugander,Vigna,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
Geometry of Social Networks

5. Are we really that similar?
Geometry of Social Networks

6. Geometry of Social Networks
Social distance
4 or 6 degrees of separation does not reflect our true social distance
D. Liben-Nowell, J. Kleinberg,
Tracing information flow on a global scale using Internet chain-letter data
PNAS 105 (2008)
Geometry of Social Networks

7. Geometry of Social Networks
Hidden geometry vs
Geometry of Social Networks

8. Complex networks in the era of Big Data
web graph, social networks, biological networks, internet networks, …
Geometry of Social Networks

9. What is a complex network?
no precise definition
however, there is general consensus on the following observed properties
large scale
evolving over time
power law degree distributions
small world properties
Geometry of Social Networks

10. Examples of complex networks
technological/informational: web graph, router graph, AS graph, call graph, graph
social: on-line social networks (Facebook, Twitter, LinkedIn,…), collaboration graphs, co-actor graph
biological networks: protein interaction networks, gene regulatory networks, food networks, connectomes
Geometry of Social Networks

11. Geometry of Social Networks
Other properties
densification power law (Leskovec, Kleinberg, Faloutsos,05):
|(E(Gt)| ≈ |V(Gt)|a where 1 < a ≤ 2: densification exponent
community structure
spectral expansion
Geometry of Social Networks

12. Geometry of Social Networks
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
Geometry of Social Networks

13. Geometry of Social Networks
Dimensionality
Question: What is the dimension of the Blau space of OSNs?
what is a credible mathematical formula for the dimension of an OSN?
Geometry of Social Networks

14. Geometry of Social Networks

15. 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)
Geometry of Social Networks

16. Some 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, …
Geometry of Social Networks

17. Geometry of Social Networks
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
Geometry of Social Networks

18. Geometry of Social Networks
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.
Geometry of Social Networks

19. 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
Geometry of Social Networks

20. 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
Geometry of Social Networks

21. Geometry of Social Networks
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
Geometry of Social Networks

22. Geometry of Social Networks
Simulation with 5000 nodes
Geometry of Social Networks

23. Geometry of Social Networks
Simulation with 5000 nodes
random geometric
GEO-P
Geometry of Social Networks

24. Properties 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/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
Geometry of Social Networks

25. Geometry of Social Networks
Dimension of OSNs
given the order of the network n and diameter D, we can calculate m
gives formula for dimension of OSN:
Geometry of Social Networks

26. 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)
Geometry of Social Networks

27. Few dimensions implies greater difference
low dimensional separation
high dimensional separation
Geometry of Social Networks

28. 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 Blau space
Geometry of Social Networks

29. 6 Dimensions of Separation
OSN
Dimension
Facebook 7
YouTube 6
Twitter 4
Flickr
Cyworld
Geometry of Social Networks

30. Geometry of Social Networks

31. 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
Geometry of Social Networks

32. 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
Geometry of Social Networks

33. 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)
Geometry of Social Networks

34. 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
Geometry of Social Networks

35. Geometry of Social Networks
Back to question…
How would we measure the dimensionality of Blau space?
Geometry of Social Networks

36. Aside: machine learning
machine learning is a branch of AI that infers structure from data
examples:
spam filters
Netflix recommender systems
text and image categorization
especially useful when the data or number of decisions are too large for humans to process
Geometry of Social Networks

37. Model selection in complex networks
(Middendorf,Ziv,Wiggins,05)
used ADTs and motifs for model selection in protein networks
predicted 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

38. Support Vector Machine (SVM)
Sec. 15.1
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
Geometry of Social Networks

39. Geometry of Social Networks
Facebook100
Geometry of Social Networks

40. Geometry of Social Networks
Validating the LDH
we tested the dimensionality of large-scale samples from real OSN data
FB100 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
Geometry of Social Networks

41. Miniconference on the Mathematics of Computation
Motifs/Graphlets
Geometry of Social Networks

42. Miniconference on the Mathematics of Computation
Experimental design
Geometry of Social Networks

43. Miniconference on the Mathematics of Computation
Sample: Michigan
Geometry of Social Networks

44. Miniconference on the Mathematics of Computation
Stanford3: n: edges: avgdeg: plexp: GeoP parameters alphabeta: alpha: beta: python geop_dim_experiment.py --logcount -s 50 -t 0 --mmax 12 --prob Stanford M-GeoP dimensions: LADTree: J48: 3 Logistic: SVM:
Geometry of Social Networks

45. Miniconference on the Mathematics of Computation
FB and LinkedIn - SVM
Geometry of Social Networks

46. FB and LinkedIn - Eigenvalues
Miniconference on the Mathematics of Computation
FB and LinkedIn - Eigenvalues
Geometry of Social Networks

47. Miniconference on the Mathematics of Computation
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): e doi: /journal.pone
Geometry of Social Networks

48. Miniconference on the Mathematics of Computation
Underlying 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 space
For e.g.:
web graph: topic space
OSNs: Blau space
PPIs: biochemical space
Geometry of Social Networks

49. Miniconference on the Mathematics of Computation
Implications of FST
new way of viewing complex networks
not just graph structure, but underlying, hidden geometry that matters
graph structure can help uncover this hidden geometry
Geometry of Social Networks

50. Miniconference on the Mathematics of Computation
Future directions
other data sets
fractal or other dimension
underlying metric?
what are the attributes?
what implications does LDH have for OSNs or social networks in general?
Geometry of Social Networks

51. Miniconference on the Mathematics of Computation
Character 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-occurrence
edges may be weighted
Geometry of Social Networks

52. Miniconference on the Mathematics of Computation
E.g.: Marvel universe
10K nodes
diameter 9
10 communities
average degree 41
Character networks

53. Miniconference on the Mathematics of Computation
moviegalaxies.com
moviegalaxies.com,catalogues the social networks in 800+ movies
Geometry of Social Networks

54. Dimensionality of character networks?
Miniconference on the Mathematics of Computation
Dimensionality of character networks?
Geometry of Social Networks