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Adversarial Deletion in Scale Free Random Graph
Process by A.D. Flaxman et al.

• Hammad Iqbal
• CS 3150
• 24 April 2006

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Talk Overview

• Background
• Large graphs
• Modeling large graphs
• Robustness and Vulnerability
• Problem and Mechanism
• Main Results
• Adversarial Deletions During Graph Generation
• Results
• Graph Coupling
• Construction of the proofs

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Large Graphs

• Modeling of large graphs has recently generated
  interest 1990s
• Driven by the computerization of data acquisition
  and greater computing power
• Theoretical models are still being developed
• Modeling difficulties include
• Heterogeneity of elements
• Non-local interactions

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Large Graphs Examples

• Hollywood graph 225,000 actors as vertices an
  edge connects two actors if they were cast in the
  same movie
• World Wide Web 800 million pages as vertices
  links from one page to another are the edges
• Citation pattern of scientific publications
• Electrical Power-grid of US
• Nervous system of the nematode worm
  Caenorhabditis elegans

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Small World of Large Graphs

• Large naturally occurring graphs tend to show
• Sparsity
• Hollywood graph has 13 million edges (25 billion
  for a clique of 225,000 vertices)
• Clustering
• In WWW, two pages that are linked to the same
  page have a higher prob of including link to one
  another
• Small Diameter
• log n
• D.J. Watts and S.H. Strogatz, Collective dynamics
  of 'small-world' networks, Nature (1998)

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Talk Overview

• Background
• Large graphs
• Modeling large graphs
• Robustness and Vulnerability
• Problem and Mechanism
• Main Results
• Adversarial Deletions During Graph Generation
• Results
• Graph Coupling
• Construction of the proofs

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Erdos-Renyi Random Graphs

• Developed around 1960 by Hungarian mathematicians
  Paul Erdos and Alfred Renyi.
• Traditional models of large scale graphs
• G(n,p) a graph on n where each pair is joined
  independently with prob p
• Weaknesses
• Fixed number of vertices
• No clustering

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Watts-Strogatz Model

• Starting from a ring lattice with n vertices and
  k edges per vertex, rewire each vertex with prob
  p to a randomly chosen destination figure
• A good model for Hollywood graph
• Web is also shown to fit small world model
• Weakness Constant n

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Barabasi model

• Incorporates growth and preferential attachment
• Evolves to a steady scale-free state the
  distribution of node degrees dont change over
  time
• Prob of finding a vertex with k edges k-3

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Degree Distribution

• Scale Free
• P X k ck-a
• Power Law distributed
• Heavy Tail
• Erdos- Renyi Graphs
• P X k e-? ?k / k!
• ? depends on the N
• Poisson distributed
• Decays rapidly for large k
• PXk ? 0 for large k

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Power Law distributions

• Also referred to as heavy-tail, Pareto, Zipfian
  distributions
• Pervasive in many naturally occurring phenomena
• Scale-free graph have power law distributions

P X k ck-a cgt0 and agt0
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Exponential (ER) vs Scale Free
130 vertices and 430 edges Red 5 highest
connected vertices Green Neighbors of red
Albert, Jeong, Barabasi 2000
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Degree Sequence of WWW

• In-degree for WWW pages is power-law distributed
  with x-2.1
• Out-degree x-2.45
• Av. path length between nodes 16

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Talk Overview

1. Background
2. Large graphs
3. Modeling large graphs
4. Robustness and Vulnerability
5. Problem and Mechanism
6. Main Results
7. Adversarial Deletions During Graph Generation
8. Results
9. Graph Coupling
10. Construction of the proofs

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Robustness and Vulnerability

• Many complex systems display inherent tolerance
  against random failures
• Examples genetic systems, communication systems
  (Internet)
• Redundant wiring is common but not the only
  factor
• This tolerance is only shown by scale-free graphs
  (Albert, Jeong, Barabasi 2000)

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Inverse Bond Percolation

• What happens when a fraction p of edges are
  removed from a graph?
• Threshold prob pc(N)
• Connected if edge removal probability pltpc(N)
• Infinite-dimensional percolation
• Worse for node removal

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General Mechanism

• Barabasi (2000) - Networks with the same number
  of nodes and edges, differing only in degree
  distribution
• Two types of node removals
• Randomly selected nodes
• Highly connected nodes (Worst case)
• Study parameters
• Size of the largest remaining cluster (giant
  component) S
• Average path length l

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Main Results(Deletion occurs after generation)
Why is this important?
? Random node removal ? Preferential node
removal
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Talk Overview

1. Background
2. Large graphs
3. Modeling large graphs
4. Robustness and Vulnerability
5. Problem and Mechanism
6. Main Results
7. Adversarial Deletions During Graph Generation
8. Results
9. Graph Coupling
10. Construction of the proofs

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Main Result

• Time steps 1,,n
• New vertex with m edges using preferential att.
• Total deleted vertices dn (Adversarially)
• m gtgt d
• w.h.p a component of size n/30

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Formal Statements

• Theorem 1
• For any sufficiently small constant d there
  exists a sufficiently large constant mm(d) and a
  constant ??(d,m) such that whp Gn has a giant
  connected component with size at least ?n

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Graph Coupling
Random Graph G(n,p)
Red Induced graph vertices Gn
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Informal Proof Construction

• A random graph can be tightly coupled with the
  scale free graph on the induced subset (Theorem
  2)
• Deleting few edges from a random graph with
  relatively many edges will leave a giant
  connected component (Lemma 1)
• There will be a sufficient number of vertices
  for the construction of induced subset (Lemma 2)

w.h.p
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Formal Statements

• Theorem 2
• We can couple the construction of Gn and random
  graph Hn such that Hn G(Gn,p) and whp
• e(Hn \ Gn) Ae-Bmn
• Difference in edge sets of Gn and Hn decreases
  exponentially with the number of edges

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Induced Sub-graph Properties

• Vertex classification at each time step t
• Good if
• Created after t/2
• Number of original edges that remain undeleted
  m/6
• Bad otherwise
• Gt set of good vertices at time t
• Good vertex can become bad
• Bad vertex remains bad

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Proof of Theorem 2Construction

• Hn/2 G(Gn/2,p)
• For k gt n/2, both Gk and Hk are constructed
  inductively
• Gk is generated by preferential attachment model.
• Hk is constructed by connecting a new vertex
  with the vertices that are good in Gk
• A difference will only happen in case of failure

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Proof of Theorem 2Type 0 failure

• If not enough good vertices in Gk
• Lemma 2 whp ?t t/10
• Prob of occurrence is therefore o(1)
• Generate Gn and Hn independently if this
  occurs

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Proof of Theorem 2Type 1 failure

• If not enough good vertices are chosen by xk1 in
  Gk
• r number of good vertices selected
• Let Pa given vertex is good e0
• Failure if r (1-d)e0m
• Upper bound

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Proof of Theorem 2Type 2 failure

• If the number of good vertices chosen by xk1 in
  Gk is less than the random vertices generated
  in Hk
• XBi(r, e0) and YBi(?k,p)
• Failure if YgtX
• Upper bound on type 2 failure prob Ae-Bm

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Proof of Theorem 2Coupling and deletion

• Take a random subset of size Y of the good chosen
  vertices in Gk and connect them with the new
  vertex in Hk
• Delete vertices in Hk that are deleted by the
  adversary in Gk
• Hn G(Gn,p)
• Difference can only occur due to failure

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Proof of Theorem 2Bound on failures

• Prob of failure at each step Ae-Bm
• Total number of misplaced edges added
• EM Ae-Bmn

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Lemma 1Statement

• Let G obtained by deleting fewer than n/100 edges
  from a realization of Gn,c/n. if c10 then whp G
  has a component of size at least n/3

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Proof of Lemma 1

• Gn,c/n contains a set S of size n/3 s n/2
• P at most n/100 edges joining s to n-s is small
• E number of edges across this cut s(n-s)c/n
• Pick some e so that n/100 (1-e)s(n-s)c/n

s
n-s
N/100
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Proof of Lemma 1
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Proof of Lemma 2Statement and Notation

• whp ?t t/10 for n/2 lt t n
• Let
• zt number of deleted vertices
• ?t number of vertices in Gt
• It is sufficient to show that

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Proof of Lemma 2Coupling

• Couple two generative processes
• P adversary deletes vertices at each time step
• P no vertices are deleted until t and then
  same vertices are deleted as P
• Difference can only occur because of failure
• Upper bound on zt(P)

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Theorem 1Statement

• For any sufficiently small constant d there
  exists a sufficiently large constant mm(d) and a
  constant ??(d,m) such that whp Gn has a giant
  connected component with size at least ?n

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Proof of Theorem 1

• Let G1Gn and G2 G(Gn,p)
• Let G G1 n G2
• e(G2 \ G) Ae-Bmn by theorem 2
• whp G ?n n/10 by lemma 2
• Let m be large so that pgt10/ ?n
• Proof by lemma 1

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