“Adversarial Deletion in Scale Free Random Graph Process” by A.D. Flaxman et al. Hammad Iqbal CS April 2006

Talk Overview 1. Background 1. Large graphs 2. Modeling large graphs 2. Robustness and Vulnerability 1. Problem and Mechanism 2. Main Results 3. Adversarial Deletions During Graph Generation 1. Results 2. Graph Coupling 3. 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)

Talk Overview 1. Background 1. Large graphs 2. Modeling large graphs 2. Robustness and Vulnerability 1. Problem and Mechanism 2. Main Results 3. Adversarial Deletions During Graph Generation 1. Results 2. Graph Coupling 3. Construction of the proofs

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

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 figurefigure  A good model for Hollywood graph  Web is also shown to fit small world model  Weakness: Constant n

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 λ k / k! λ depends on the N Poisson distributed Decays rapidly for large k P[X≥k]  0 for large k

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 -α c>0 and α>0

Exponential (ER) vs Scale Free Albert, Jeong, Barabasi vertices and 430 edges Red = 5 highest connected vertices Green = Neighbors of red

Degree Sequence of WWW  In-degree for WWW pages is power-law distributed with x -2.1  Out-degree x  Av. path length between nodes ~16

Talk Overview 1. Background 1. Large graphs 2. Modeling large graphs 2. Robustness and Vulnerability 1. Problem and Mechanism 2. Main Results 3. Adversarial Deletions During Graph Generation 1. Results 2. Graph Coupling 3. 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 p c (N): Connected if edge removal probability p<p c (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?

Talk Overview 1. Background 1. Large graphs 2. Modeling large graphs 2. Robustness and Vulnerability 1. Problem and Mechanism 2. Main Results 3. Adversarial Deletions During Graph Generation 1. Results 2. Graph Coupling 3. Construction of the proofs

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

Graph Coupling Random Graph G(n’,p) Red = Induced graph vertices Γ n

Informal 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

Formal Statements  Theorem 2 We can couple the construction of G n and random graph H n such that H n ~ G(Γ n,p) and whp e(H n \ G n ) ≤ Ae -Bm n  Difference in edge sets of G n and H n 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  Γ t = set of good vertices at time t  Good vertex can become bad  Bad vertex remains bad

Proof of Theorem 2 Construction  H[n/2] ~ G(Γ n/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 2 Type 0 failure  If not enough good vertices in Gk  Lemma 2: whp γ t ≥ 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 x k+1 in G[k]  r = number of good vertices selected  Let P[a given vertex is good] = ε 0  Failure if r ≤ (1-δ)ε 0 m  Upper bound: Proof of Theorem 2 Type 1 failure

 If the number of good vertices chosen by x k+1 in G[k] is less than the random vertices generated in H[k]  X~Bi(r, ε 0 ) and Y~Bi(γ k,p)  Failure if Y>X  Upper bound on type 2 failure prob: Ae - Bm Proof of Theorem 2 Type 2 failure

 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]  H n ~ G(Γ n,p)  Difference can only occur due to failure Proof of Theorem 2 Coupling and deletion

Proof of Theorem 2 Bound on failures  Prob of failure at each step Ae -Bm  Total number of misplaced edges added: E[M] ≤ Ae -Bm n

Lemma 1 Statement  Let G obtained by deleting fewer than n/100 edges from a realization of G n,c/n. if c≥10 then whp G has a component of size at least n/3

Proof of Lemma 1  G n,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-s s N/100

Proof of Lemma 1

Proof of Lemma 2 Statement and Notation  whp γ t ≥ t/10 for n/2 < t ≤ n  Let z t = number of deleted vertices ν’ t = number of vertices in G t  It is sufficient to show that

Proof of Lemma 2 Coupling  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 z t (P*)

Theorem 1 Statement 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 G n has a “giant” connected component with size at least θn

Proof of Theorem 1  Let G 1 =G n and G 2 = G(Γ n,p)  Let G = G 1 ∩ G 2  e(G 2 \ G) ≤ Ae -Bm n by theorem 2  whp |G|= γ n ≥ n/10 by lemma 2  Let m be large so that p>10/ γ n  Proof by lemma 1