lecture 6 - models of complex networks ii
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AM8002 Fall 2014. Lecture 6 - Models of Complex Networks II. Dr. Anthony Bonato Ryerson University. Key properties of complex networks. Large scale. Evolving over time. Power law degree distributions. Small world properties. - PowerPoint PPT PresentationTRANSCRIPT
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AM8204 Topics in Discrete Mathematics
Week 6 Models of Complex Networks II
Dr. Anthony BonatoRyerson University
Ryerson Mathematics Winter 2016
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Key properties of complex networks
1. Large scale.
2. Evolving over time.
3. Power law degree distributions.
4. Small world properties.
• Other properties are also important: densification power law, shrinking distances,…
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Geometry of the web?
• idea: web pages exist in a topic-space
– a page is more likely to link to pages close to it in topic-space
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Random geometric graphs• nodes are randomly
placed in space
• nodes are joined if their distance is less than a threshold value d; nodes each have a region of influence which is a ball of radius d
(Penrose, 03)
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Simulation with 5000 nodes
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Geometric Preferential Attachment (GPA) model(Flaxman, Frieze, Vera, 04/07)
• nodes chosen on-line u.a.r. from sphere with surface area 1
• each node has a region of influence with constant radius
• new nodes have m out-neighbours, chosena) by preferential attachment; andb) only in the region of influence
• a.a.s. model generates power law,
low diameter graphs with small separators/sparse cuts
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Spatially Preferred Attachment graphs
• regions of influence shrink over time (motivation: topic space growing with time), and are functions of in-degree
• non-constant out-degree
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Spatially Preferred Attachment (SPA) model
(Aiello,Bonato,Cooper,Janssen,Prałat, 08)• parameter: p a real number in (0,1]• nodes on a 3-dimensional sphere with surface area 1• at time 0, add a single node chosen u.a.r.• at time t, each node v has a region of influence Bv
with radius
• at time t+1, node z is chosen u.a.r. on sphere• if z is in Bv, then add vz independently with probability
p
t
vtG
1deg
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Simulation: p=1, t=5,000
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• as nodes are born, they are more likely to enter some Bv with larger radius (degree)
• over time, a power law degree distribution results
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power law exponent 1+1/p
Theorem 6.1 (ACBJP, 08)Define
Then a.a.s. for t ≤ n and i ≤ if,
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Rough sketch of proof• derive an asymptotic expression for E(Ni,t)
1-t0,t0,1t0, N1)|N(Nt
pGt E
1-ti,1-t1,-iti,1ti, N)1(
N)|N(Nt
ip
t
piGt
E :0i
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• solve the recurrence asymptotically:
)(N ti,E
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• prove that Ni,t is concentrated on E(Ni,t) via martingales
• standard approach is to use c-Lipshitz condition: change in Ni,t is bounded above by constant c
• c-Lipschitz property may fail: new nodes may appear in an unbounded number of overlapping regions of influence
• prove this happens with exponentially small probabilities using the differential equation method
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Models of OSNs
• few models for on-line social networks
• goal: find a model which simulates many of the observed properties of OSNs,– densification and shrinking distance– must evolve in a natural way…
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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
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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?
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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
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Geometric Protean (GEO-P) Model(Bonato, Janssen, Prałat, 10)
• parameters: α, β in (0,1), α+β < 1; positive integer m• nodes live in 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
Theorem (Bonato, Janssen, Prałat, 2010)
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.
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Density
• average number of edges added at each time-step
• parameter β controls density• if β < 1 – α, then density grows with n (as
in real OSNs)
n
i
nni1
1
1
1
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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
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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
YouTube 6Twitter 4Flickr 4
Cyworld 7
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Experiential component
1. Speculate as to what the feature space would be for protein interaction networks.
2. Verify that for fixed constants α, β in (0,1):
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n
i
nni1
1
1
1
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Transitivity
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Iterated Local Transitivity (ILT) model(Bonato, Hadi, Horn, Prałat, Wang, 08)
• key paradigm is transitivity: friends of friends are more likely friends
• nodes often only have local influence
• evolves over time, but retains memory of initial graph
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ILT model
• start with a graph of order n• to form the graph Gt+1 for each node x from
time t, add a node x’, the clone of x, so that xx’ is an edge, and x’ is joined to each node joined to x
• order of Gt is n2t
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G0 = C4
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Properties of ILT model
• average degree increasing to with time
• average distance bounded by constant and converging, and in many cases decreasing with time; diameter does not change
• clustering coefficient higher than in a random generated graph with same average degree
• bad expansion: small gaps between 1st and 2nd eigenvalues in adjacency and normalized Laplacian matrices of Gt
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Densification
• nt = order of Gt, et = size of Gt
Lemma 6.2: For t > 0,
nt = 2tn0, et = 3t(e0+n0) - nt.
→ densification power law:et ≈ nt
a, where a = log(3)/log(2).
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Average distance
Theorem 6.3: If t > 0, then
• average distance bounded by a constant, and converges; for many initial graphs (large cycles) it decreases
• diameter does not change from time 0
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Clustering Coefficient
Theorem 6.4: If t > 0, then
c(Gt) = ntlog(7/8)+o(1).
• higher clustering than in a random graph G(nt,p) with same order and average degree as Gt, which satisfies
c(G(nt,p)) = ntlog(3/4)+o(1)
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…Degree distribution–generate power
law graphs from ILT?• ILT model
gives a binomial-type distribution
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Experiential component
1. Let degt(z) be the degree of z at time t.
a) Show that if x is in Gt, then degt +1(x) = 2degt (x)+1.
b) Show that degt +1(x’) = degt(x) +1.
2. Let nt = order of Gt and et = size of Gt
Prove the following Lemma.
Lemma 5.2: For t > 0,
nt = 2tn0, et = 3t(e0+n0) - nt.
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