Frontiers of Network Science Fall 2019 Class 17: Robustness II (Chapter 8 in Textbook) Boleslaw Szymanski based on slides by Albert-László Barabási and Roberta Sinatr a www.BarabasiLab.com
Self-organized Criticality (BTW Sandpile Model) Initial Setup Homogenous case • Random graph with N nodes • Each node i has height h i = 0 . Cascade • At each time step, a grain is added at a randomly chosen node i : h i ← h i +1 • If the height at the node i reaches a prescribed threshold z i = k i , then it becomes unstable and all the grains at the node topple to its adjacent nodes: h i = 0 and h j ← h j +1 Scale-free network • if i and j are connected. Homogenous network: < k 2 > converges P ( S ) ~ S −3/2 Scale-free network : p k ~ k - γ (2< γ <3) P ( S ) ~ S − γ /( γ −1) Network Science: Robustness Cascades K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)
Branching Process Model Branching Process Starting from a initial node, each node in generation t produces k number of offspring nodes in the next t + 1 generation, where k is selected randomly from a fixed probability distribution q k =p k-1 . Fix <k>=1 to be Hypothesis critical power • No loops (tree structure) law P(S) • No correlation between branches Narrow distribution: < k 2 > converged P ( S ) ~ S −3/2 Fat tailed distribution: q k ~ k - γ (2< γ <3) P ( S ) ~ S − γ /( γ −1) Network Science: Robustness Cascades K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Phys. Rev. Lett. 91, 148701 (2003)
Short Summary of Models: Universality Models Networks Exponents Failure Propagation Model ER 1.5 Overload Model Complete Graph 1.5 1.5 (ER) BTW Sandpile Model ER/SF γ /( γ - 1) (SF) 1.5 (ER) Branching Process Model ER/SF γ /( γ - 1) (SF) Universal for homogenous networks P ( S ) ~ S −3/2 Same exponent for percolation too (random failure, attacking, etc.) Network Science: Robustness Cascades
Explanation of the 3/2 Universality Simplest Case: q 0 = q 2 = 1/2 , < k > = 1 S : number of nodes X : number of open branches S = S+ 1 ½ chance S = 2, X = 0 X = X -1 k = 0 S = 2, X = 2 S = 1 S = S+ 1 ½ chance k =2 X = 1 X = X +1 X >0, Branching process stops X when X = 0 Dead S
Explanation of 3/2 Universality X Dead S Equivalent to 1D random walk model , where X and S are the position and time , respectively. Question : what is the probability that X = 0 after S steps? First return probability ~ S -3/2 M. Ding, W. Yang, Phys. Rev. E. 52, 207-213 (1995)
Size Distribution of Branching Process (Cavity Method) S 1 S 1 S 2 k = 0 k = 1 k = 2 S = 1 S = 1 + S 1 S = 1 + S 1 + S 2 ∑ ∑ = δ + δ + − + δ + + − + ( S ) P ( 1 ) ( ) ( 1 ) ( ) ( ) ( 1 ) q q P S S S q P S P S S S S 0 1 1 1 2 1 2 1 2 , S S S 1 1 2 K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Physica A 346, 93-103 (2005) Network Science: Robustness Cascades
Solving the Equation by Generating Function Definition: Property: G S ( x ) = Σ S =0 P ( S ) x S G S (1) = G k (1) = 1 G k ( x ) = Σ k =0 q k x k G S ’ (1) = < S > , G k ’(1) = < k > k ∑ ∑ ∑ = δ + − ( ) ( ) ( ) ( ) ( 1 ) P S q P S P S P S S S 1 2 k k j = , 1 k S S j 1 k ∑ + ∑ ∑ ∑ 1 S = = j k ( ) ( ) ( ) ( ) j G x q P S P S x q xG x 1 S k k k S 1 , k S S k k = ( ( )) xG G x k S Phase Transition < S > = G S ’ (1) = 1+ G k ’ (1) G S ’ (1) = 1 + < k > < S >, then < S > = 1/(1- < k > ) The average size < S > diverges at < k > c = 1
Finding the Critical Exponent from Expansion Definition: G S ( x ) = Σ S =0 P ( S ) x S G k ( x ) = Σ k =0 q k x k Theorem: If P ( k ) ~ k - γ (2< γ <3) , then for δ x < 0, | δ x | << 1 G (1+ δ x ) = 1 + < k > δ x + < k ( k -1)/2> ( δ x ) 2 + … + O(| δ x | γ - 1 ) P ( S ) ~ S − α ,1< α < 2 G S (1+ δ x ) ≈ 1 + A | δ x | α -1 Homogenous case: < k 2 > converged Inhomogeneous case: < k 2 > diverged < k > = 1, < k 2 > < ∞ < k > = 1, q k ~ k - γ (2< γ <3) G k (1+ δ x ) ≈ 1 + δ x + B | δ x | γ - 1 G k (1+ δ x ) ≈ 1 + δ x + B δ x 2 Network Science: Robustness Cascades
Critical Exponent for Homogenous Case Homogenous case G k (1+ δ x ) ≈ 1 + δ x + B δ x 2 G S (1+ δ x ) ≈ 1 + A | δ x | α -1 G S ( x ) = xG k ( G S ( x )) G S ( x ) ≈ 1 + A | δ x | α -1 xG k ( G S ( x )) ≈ (1+δ x )[1+ ( G S (1+ δ x )-1) + B ( G S (1+ δ x )-1) 2 ] ≈ (1+δ x )[1+ A | δ x | α -1 + AB | δ x | 2 α -2 ] = 1 + A | δ x | α -1 + AB | δ x | 2 α -2 + δ x + O (| δ x | α ) The lowest order reads AB | δ x | 2 α -2 + δ x = 0 , which requires 2 α -2 = 1 and A = 1/ B . Or, α = 3/2 Network Science: Robustness Cascades
Critical Exponent for Inhomogeneous Case Inhomogeneous case G k (1+ δ x ) ≈ 1 + δ x + B | δ x | γ - 1 G S (1+ δ x ) ≈ 1 + A | δ x | α -1 G S ( x ) = xG k ( G S ( x )) G S ( x ) ≈ 1 + A | δ x | α -1 xG k ( G S ( x )) ≈ (1+δ x )[1+ ( G S (1+ δ x )-1) + B | G S (1+ δ x )-1| γ -1 ] ≈ (1+δ x )[1+ A | δ x | α -1 + AB | δ x | ( α -1)( γ -1) ] = 1 + A | δ x | α -1 + AB | δ x | ( α -1)( γ -1) + δ x + O (| δ x | α ) The lowest order reads AB | δ x | ( α -1)( γ -1) + δ x = 0 , which requires ( α -1)( γ -1) = 1 and A = 1/ B . Or, α = γ /( γ −1) Network Science: Robustness Cascades
Compare the Prediction with the Real Data γ > 3 / 2 , 3 − α α = ( ) ~ , P S S γ γ − < γ < /( 1 ), 2 3 Blackout Blackout Source Exponent Quantity North America 2.0 Power Sweden 1.6 Energy Norway 1.7 Power New Zealand 1.6 Energy China 1.8 Energy Earthquake α ≈ 1.67 I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, CHAOS 17, 026103 (2007) Y. Y. Kagan, Phys. Earth Planet. Inter. 135 (2–3), 173–209 (2003) Network Science: Robustness Cascades
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