On the intersection of random graphs with an application to random - PowerPoint PPT Presentation

NIST, ACMD Seminar Series, February 2014 1 On the intersection of random graphs with an application to random key pre-distribution ab Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu a Supported by NSF Grants CCF-0830702 and CCF-1217997. b Joint work with N. Prasanth Anthapadmanabhan and O. Ya˘ gan

NIST, ACMD Seminar Series, February 2014 2 The big picture

NIST, ACMD Seminar Series, February 2014 3 Intersecting graphs Assume given two graphs with vertex set V , say G 1 ≡ ( V, E 1 ) G 2 ≡ ( V, E 2 ) and The intersection of the two graphs G 1 ≡ ( V, E 1 ) and G 2 ≡ ( V, E 2 ) is the graph ( V, E ) with E := E 1 ∩ E 2 We write G 1 ∩ G 2 := ( V, E 1 ∩ E 2 )

NIST, ACMD Seminar Series, February 2014 4 Capturing multiples constraints Adjacency expresses constraints/relationships which can be physical , logical , sociological , etc. E.g., for two constraints: • Communication constraint and link quality (e.g., fading) • Communication constraint and secure link (e.g., via shared key) • Membership in two different social networks

NIST, ACMD Seminar Series, February 2014 5 Random graphs For vertex set V , let E ( V ) denote the collection of all sets of (undirected) edges on V . A random graph with vertex set V is simply an E ( V )-valued rv defined on some probability triple (Ω , F , P ), say E : Ω → E ( V ). We write G ≡ ( V, E ) Erd˝ os-R´ enyi graphs , generalized random graphs, geometric random graphs, random key graphs, small worlds, random threshold graphs, multiplicative attribute graphs, growth models (e.g., preferential attachment models, fitness-based models)

NIST, ACMD Seminar Series, February 2014 6 Constructing (undirected) random graphs Convenient to write V ≡ { 1 , . . . , n } . Random link assignments encoded through { 0 , 1 } -valued rvs { L ij , 1 ≤ i < j ≤ n } with  1 if ( i, j ) up    L ij =   0 if ( i, j ) down 

NIST, ACMD Seminar Series, February 2014 7 Distinct nodes i, j = 1 , . . . , n are adjacent if L ij = 1, and an undirected link is assigned between nodes i and j . Examples: • Erd˝ os-Renyi (Bernoulli) graphs • Geometric random graphs – Disk models • Random key graphs

NIST, ACMD Seminar Series, February 2014 8 Intersecting random graphs Assume given two random graphs with same vertex set V , say G 1 ≡ ( V, E 1 ) G 2 ≡ ( V, E 2 ) and The intersection of the two random graphs G 1 ≡ ( V, E 1 ) and G 2 ≡ ( V, E 2 ) is the random graph ( V, E ) where E := E 1 ∩ E 2 We write G 1 ∩ G 2 = ( V, E 1 ∩ E 2 )

NIST, ACMD Seminar Series, February 2014 9 Equivalently, L ij = L 1 ,ij · L 2 ,ij , 1 ≤ i < j ≤ n Throughout the component random graphs G 1 and G 2 are assumed to be independent : The collections { L 1 ,ij , 1 ≤ i < j ≤ n } and { L 2 ,ij , 1 ≤ i < j ≤ n } are independent.

NIST, ACMD Seminar Series, February 2014 10 A basic objective Inheritance – Understand how the structural properties of the random graph G 1 ∩ G 2 are shaped by those of the component random graphs G 1 and G 2 Focus on graph connectivity and on the absence of isolated nodes – Easier and hopefully asymptotically equivalent After all n ( n − 1) 2 possible graphs on V 2 and typical behavior explored asymptotically via Zero-one Laws

NIST, ACMD Seminar Series, February 2014 11 A basic source of difficulty G 1 ∩ G 2 connected implies G 1 and G 2 both connected But the converse is false! E 1 : 1 ∼ 2 ∼ 3 V = { 1 , 2 , 3 } : E 2 : 1 ∼ 3 ∼ 2 E 1 ∩ E 2 : 2 ∼ 3 Similar comment when considering the absence of isolated nodes

NIST, ACMD Seminar Series, February 2014 12 Examples of random graphs and their zero-one laws

NIST, ACMD Seminar Series, February 2014 13 Erd˝ os-Renyi (ER) graphs G ( n ; p ) Random link assignment encoded through i.i.d. { 0 , 1 } -valued rvs { L ij , 1 ≤ i < j ≤ n } with P [ L ij = 1] = p for some 0 < p < 1. Also known as Bernoulli graphs

NIST, ACMD Seminar Series, February 2014 14 Strong zero-one law for graph connectivity in ER graphs G ( n ; p ) (0 < p < 1) [ Erd˝ os and Renyi ]: Whenever p n ∼ c log n n for some c > 0, we have  0 if 0 < c < 1    n →∞ P [ G ( n ; p n ) is connected] = lim   1 if 1 < c  Same zero-one law for absence of isolated nodes Critical scaling for graph connectivity: n := log n p ⋆ n = 1 , 2 , . . . , n

NIST, ACMD Seminar Series, February 2014 15 We also have the weak zero-one law:  p n 0 if lim n →∞ n = 0  p ⋆   n →∞ P [ G ( n ; p n ) is connected] = lim  p n  1 if lim n →∞ n = ∞  p ⋆ Simple consequence of strong zero-one law by the monotonicity of the mapping p → P [ G ( n ; p ) is connected]

NIST, ACMD Seminar Series, February 2014 16 Geometric random graphs G ( n ; ρ ) Population of n nodes located at X 1 , . . . , X n in a bounded convex region A ⊂ R 2 . With ρ > 0, nodes i and j are adjacent if ∥ X i − X j ∥ ≤ ρ so that L ij = 1 [ ∥ X i − X j ∥ ≤ ρ ] Usually, i.i.d. node locations X 1 , . . . , X n which are uniformly distributed on unit square or unit disk – Disk model

NIST, ACMD Seminar Series, February 2014 17 Strong zero-one law for graph connectivity in geometric random graphs G ( n ; ρ ) ( ρ > 0) [ Penrose, Gupta and Kumar ]: Whenever n ∼ c log n πρ 2 n for some c > 0, we have  0 if 0 < c < 1    n →∞ P [ G ( n ; ρ n ) is connected] = lim   1 if 1 < c  Same zero-one law for absence of isolated nodes Critical scaling for graph connectivity: n ) 2 = log n π ( ρ ⋆ n = 1 , 2 , . . . , n

NIST, ACMD Seminar Series, February 2014 18 A random key pre-distribution scheme (Eschenauer and Gligor 2002) For integers P and K with 1 ≤ K < P , let P K denote the collection of all subsets of { 1 , . . . , P } with exactly K elements For each node i = 1 , . . . , n , with θ = ( P, K ), let K i ( θ ) denote the random set of K distinct keys assigned to node i Under the EG scheme, the rvs K 1 ( θ ) , . . . , K n ( θ ) are assumed to be i.i.d. rvs, each of which is uniformly distributed over P K with ) − 1 ( P P [ K i ( θ ) = S ] = S ∈ P K , i = 1 , . . . , n , K

NIST, ACMD Seminar Series, February 2014 19 The random key graph K ( n ; θ ) Distinct nodes i, j = 1 , . . . , n are said to be adjacent if they share at least one key in their key rings, namely K i ( θ ) ∩ K j ( θ ) ̸ = ∅ . In other words, L ij ( θ ) := 1 [ K i ( θ ) ∩ K j ( θ ) ̸ = ∅ ] For distinct i, j = 1 , . . . , n , ( P − K ) K q ( θ ) = P [ K i ( θ ) ∩ K j ( θ ) = ∅ ] = ) . ( P K

NIST, ACMD Seminar Series, February 2014 20 Strong zero-one law for graph connectivity in random key graphs K ( n ; θ ) ( K < P ) [ Di Pietro et al., Burbank and Gerke, Rybarczyk, YM ]: Whenever K 2 ∼ c log n n P n n for some c > 0, we have  0 if 0 < c < 1    n →∞ P [ K ( n ; θ n ) is connected] = lim   1 if 1 < c  Same zero-one law for absence of isolated nodes Observation: With lim n →∞ q ( θ n ) = 1, K 2 n ∼ 1 − q ( θ n ) P n

NIST, ACMD Seminar Series, February 2014 21 Observation All cases discussed so far are “ homogeneous ” with a well-defined link probability p ( G ): p ( G ) = Probability that two nodes are adjacent in G Zero-one laws for connectivity and absence of isolated nodes are determined by conditions on p ( G ), or proxy thereof: p (?( n, ? n ) ∼ c log n n for some c > 0

NIST, ACMD Seminar Series, February 2014 22 ER graphs G ( n ; p ): p . . . but πρ 2 Random geometric graphs G ( n ; ρ ): 1 − q ( θ ) but K 2 Random key graphs K ( n ; θ ): P

NIST, ACMD Seminar Series, February 2014 23 Intersecting random graphs and their zero-one laws

NIST, ACMD Seminar Series, February 2014 24 Three examples Secure links via key sharing under partial visibility with an on-off communication model: G ( n ; p ) ∩ K ( n ; θ ) Disk model with possibility of defective links due to fading: G ( n ; ρ ) ∩ G ( n ; p ) Disk model with possibility of secure links via key sharing: G ( n ; ρ ) ∩ K ( n ; θ )

NIST, ACMD Seminar Series, February 2014 25 With n → ∞ , In all cases mentioned earlier, elements of a limiting theory are available for the component random graphs: Zero-one laws hold for graph connectivity and absence of isolated nodes when the parameters are properly scaled with n Inheritance – For a given random intersection graph, • Zero-one laws for graph connectivity and for the absence of isolated nodes? • Critical thresholds? • Width of phase transitions?

NIST, ACMD Seminar Series, February 2014 26 A silly detour: Intersecting ER graphs With G 1 ≡ G ( n, p 1 ) and G 2 ≡ G ( n, p 2 ) , then G 1 ∩ G 2 = st G ( n, p ) with p := p 1 · p 2 under the independence of the components. Whenever p n = p 1 ,n · p 2 ,n ∼ c log n n for some c > 0, we have  0 if 0 < c < 1    n →∞ P [ G ( n ; p n ) is connected] = lim   1 if 1 < c 

NIST, ACMD Seminar Series, February 2014 27 Zero-law holds for G ( n, p 1 ) ∩ G ( n, p 2 ) whenever p n = p 1 ,n · p 2 ,n = 1 log n n = 1 , 2 , . . . , 2 n Yet one-law holds for G ( n, p 1 ) and G ( n, p 2 ) with √ 1 log n p 1 ,n = p 2 ,n = n = 1 , 2 , . . . , 2 n since √ log n 1 √ 1 √ n 2 n 2 · log n = ∞ lim = lim log n n →∞ n →∞ n