“adversarial deletion in scale free random graph process” by a.d. flaxman et al. hammad iqbal cs...

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

Hammad Iqbal

CS 3150

24 April 2006

Talk Overview1. Background

1. Large graphs2. Modeling large graphs

2. Robustness and Vulnerability1. Problem and Mechanism2. Main Results

3. Adversarial Deletions During Graph Generation

1. Results2. Graph Coupling3. Construction of the proofs

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

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

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)

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

Barabasi model

Incorporates growth and preferential attachment

Evolves to a steady ‘scale-free’ state: the distribution of node degrees don’t change over time

Prob of finding a vertex with k edges ~k-3

Degree Distribution

Scale Free P [X ≥ k] ~ ck-α

Power Law distributed Heavy Tail

Erdos- Renyi Graphs P [X = k] = e-λ λk / k! λ depends on the N Poisson distributed Decays rapidly for large k P[X≥k] 0 for large k

Exponential (ER) vs Scale Free

Albert, Jeong, Barabasi 2000

130 vertices and 430 edgesRed = 5 highest connected verticesGreen = Neighbors of red

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




3. Adversarial Deletions During Graph Generation1. Results2. Graph Coupling3. Construction of the proofs

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)

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 p<pc(N)

Infinite-dimensional percolation

Worse for node removal

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

Main Results(Deletion occurs after generation)

□ Random node removal ○ Preferential node removal

Why is this important?Why is this important?

Main Result

Time steps {1,…,n} New vertex with m edges using preferential att. Total deleted vertices ≤ δn (Adversarially) m >> δ w.h.p a component of size ≥ n/30

Formal Statements

Theorem 1 For any sufficiently small constant δ there

exists a sufficiently large constant m=m(δ) and a constant θ=θ(δ,m) such that whp Gn

has a “giant” connected component with size at least θn

Graph Coupling

Random Graph G(n’,p)

Red = Induced graph vertices Γn

InformalInformal 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

Formal Statements

Theorem 2 We can couple the construction of Gn and

random graph Hn such that Hn ~ G(Γn,p) and whp

e(Hn \ Gn) ≤ Ae-Bmn

Difference in edge sets of Gn and Hn

decreases exponentially with the number of edges

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

Γt = set of good vertices at time t Good vertex can become bad Bad vertex remains bad

Proof of Theorem 2Construction

H[n/2] ~ G(Γn/2,p)

For k > n/2, both G[k] and H[k] are constructed inductively: Gk is generated by preferential attachment

model. H[k] is constructed by connecting a new

vertex with the vertices that are good in G[k] A difference will only happen in case of

‘failure’

Proof of Theorem 2Type 0 failure

If not enough good vertices in Gk Lemma 2: whp γt ≥ t/10

Prob of occurrence is therefore o(1) Generate G[n] and H[n] independently if

this occurs

If not enough good vertices are chosen by xk+1 in G[k]

r = number of good vertices selected Let P[a given vertex is good] = ε0

Failure if r ≤ (1-δ)ε0m

Upper bound:


If the number of good vertices chosen by xk+1 in G[k] is less than the random vertices generated in H[k]

X~Bi(r, ε0) and Y~Bi(γk,p)

Failure if Y>X Upper bound on type 2 failure prob: Ae-

Bm


Take a random subset of size Y of the good chosen vertices in G[k] and connect them with the new vertex in H[k]

Delete vertices in H[k] that are deleted by the adversary in G[k]

Hn ~ G(Γn,p)

Difference can only occur due to failure

Proof of Theorem 2Coupling and deletion

Proof of Theorem 2Bound on failures

Prob of failure at each step Ae-Bm

Total number of misplaced edges added:

E[M] ≤ Ae-Bmn

Lemma 1Statement

Let G obtained by deleting fewer than n/100 edges from a realization of Gn,c/n. if c≥10 then whp G has a component of size at least n/3

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 ε so that n/100 ≤(1-ε)s(n-s)c/n

n-ss

N/100

Proof of Lemma 1

Proof of Lemma 2Statement and Notation

whp γt ≥ t/10 for n/2 < t ≤ n

Let zt = number of deleted vertices

ν’t = number of vertices in Gt

It is sufficient to show that

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*)

Theorem 1Statement

For any sufficiently small constant δ there exists a sufficiently large constant m=m(δ) and a constant θ=θ(δ,m) such that whp Gn

has a “giant” connected component with size at least θn

Proof of Theorem 1

Let G1=Gn and G2= G(Γn,p)

Let G = G1 ∩ G2

e(G2 \ G) ≤ Ae-Bmn by theorem 2

whp |G|= γn ≥ n/10 by lemma 2

Let m be large so that p>10/ γn

Proof by lemma 1

“adversarial deletion in scale free random graph process” by a.d. flaxman et al. hammad iqbal cs...

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