Limited Path Percolation in Complex Networks Eduardo Lpez Los - PowerPoint PPT Presentation

Limited Path Percolation in Complex Networks Eduardo López Los Alamos Nat. Lab. LA-UR 07-0432 Outline • Motivation. Percolation and its effects. • Presentation of new limited path length percolation model • Scaling theory of new model and results • Targeted percolation, theory and results. • Conclusions References “Limited path length percolation in complex networks”, López, Parshani, Cohen, Carmi and Havlin, Phys. Rev. Lett. (in press). cond-mat/0702691. Collaborators Roni Parshani Shlomo Havlin Shai Carmi Reuven Cohen

Motivation: How to go from Salem to Boston? But in Boston, Weatherman wrong: On a sunny day harsh winter! Bigger storm! many paths Question: How many roads need to be closed before most people cannot get to work? Answer: from Percolation theory

What is percolation theory? Theory to determine connectivity in systems p i , j distance S(p): # connected nodes i l’’ ij =1 l ij N <1 l’ ij P ∞ N l ij l’ ij >l ij due to removal j l’ ij =p c l’’ ij > l’ ij N df/d <p c most disconnec. log N p : occupied fraction of links 1 P ∞ : probability of random node to be in largest cluster p>p c p c : connectivity threshold Logarithmic Transition: Fractal Linear Slope 1 df : fractal dim., d dim. Log S P ∞ connected p=p c ↓ Slope d f /d c m i h i t a r o g L disconnected p<p c 0 p 0 1 Log N c

Motivation: What’s the problem with percolation? • Salem-Boston connected with any path! Storm hits • Long or short paths OK • Percolation finds critical percentage p c of roads needed to keep cities connected. • Percolation increases path lengths (and time), i.e., smaller p ⇒ longer path. • There is practical limit to connectivity ⇔ longer paths not useful. Commute time: 50min Commute time: 240min Commute time: 120min Commute time: 60-70min All day driving! Answer: sometimes percolation accepts useless paths.

Social contact network

New percolation model applied to complex networks •Definition of connection: i and j are connected if l’ ij ≤ al ij •Notation: S a (p) : Largest cluster size at occupation p, length condition a ~ = p p •Is there a critical occupation above which S a ~N? c Results: New limited path percolation transition •Scaling theory ~ p > p •Find new critical occupation c c 1 . c •Critical point is now a critical range: r e p LP perc. ( ) ~ δ δ = δ < < . S ~ N , ( a , p ) p p p g e R a c c Logarithmic Fractal Linear •Below and above range, behavior is P ∞ similar to regular percolation: ( ) < S ~ log N p p a c ( ) ~ 0 > S ~ N p p ~ p p 0 1 a c c c

Theory of model networks: Erd ő s-Rényi • Developed in the 1960’s by Erd ő s and Rényi. (Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 1960). • N nodes and each pair connected with probability φ. degree of i , k i =2 • Define k as the degree (number of links of a node), and ‹ k › node i is average number of links per node over the network. Construction degree of j , k j =3 a) Complete network c) Realization of network b) Annihilate links node j with probability − φ 1 ⎡ ⎤ k φ = ⎢ ⎥ − N 1 ⎣ ⎦ k k − k = • Distribution of degree is Poisson-like (exponential) P ( k ) e k !

Outline of scaling theory for Limited Path Percolation Example: Erd ő s-Rényi • Before percolation, typical path length l ~ log N /log <k> • After percolation, local structure is tree-like, with κ = + p k 1 branching factor • Tree approx. ⇒ S a ~ ( κ -1) l = ( p<k> ) a log N /log <k> = N δ • Scaling exponent 0 ≤ δ ≡ a (1+log p /log <k> ) ≤ 1 + p k 1 • δ ≤ 1 because S a cannot exceed N ~ − ( 1 a ) / a = p k • Solving δ = 1 ⇒ a log N / log k c ~ − 1 ⎯ ⎯→ ⎯ = p p k • Usual percolation recovered with a → ∞ : c → ∞ c a

Comparison of phase diagram of regular & Limited Path Percolation (Erd ő s-Rényi) Regular percolation Limited path percolation Regular percolation Communicating 1 1 1 ER Linear phase Concentration p Linear Phase ( δ =1 ) Concentration p Transition line p c Logarithmic Fractal P Linear ∞ ~ Fractal Phase ( δ <1 ) Fractal point p c =<k> -1 p c = <k> -1 0 Logarithmic Logarithmic Phase ( δ =0 ) p 0 1 0 c 0 phase ∞ 1 Length factor a Non-communicating Limited path percolation predicts a larger communication threshold.

Results for S a ~N δ (Erd ő s-Rényi) Regular Percolation Limited path percolation (a) ER <k>=3.0 (random) 5 10 a=1.0 1.1 p>p c 1.2 1.4 4 10 1.7 Slope 1 Log S S a 4 10 3 p=p c 10 3 Slope d f /d 10 S a Logarithmic 2 10 p<p c 2 10 N 4 5 10 10 Log N 3 4 5 6 10 10 10 10 > ⎧ N ( p p ) N ~ > ⎧ N ( p p ) c ⎪ c ⎪ = 2 / 3 ~ S ~ N ( p p ) ⎨ δ ≤ ≤ S ~ N ( p p p ) ⎨ c c c ⎪ ⎪ < log N ( p p ) ⎩ < log N ( p p ) ⎩ c c ~ − 1 − ( 1 a ) / a = = p c k p k c δ ≡ a log pk log k

Complex Networks Poisson distribution Scale-free distribution - λ K m Erd ő s-Rényi Network Scale-free Network

Some basic network properties Erd ő s-Rényi networks Scale-free networks •Narrow range of typical degree •Wide range of typical degree ( ) λ − ≤ ≤ 1 / 1 − ≤ ≤ + k k k N k k k k k min min ( k min is minimum degree) •Small diameter •Small or ultra-small diameter < λ < D ~ ln(ln N ) [ 2 3 ] D ~ ln N λ > D ~ ln N [ 3 ]

Scaling theory for limited path percolation on scale-free networks • For λ >3: [ ] ( ) + κ − a 1 log p / log 1 S ~ N o a ( ) ( ~ ) a − 1 a = κ − p 1 c o • For 2< λ <3: Tree approximation invalid. Networks are ultra-small: λ − l ~ log log N log( 2 ) λ − l ' ~ log log P N log( 2 ) ∞ Therefore: l ' log log P N = → ∞ a ~ 1 l log log N → ∞ N

Phase Diagram of Limited Path Percolation on scale-free networks Communicating 1 1 SF 2< λ <3 SF λ >3 Linear Phase ( δ =1 ) Concentration p ) Concentration p 0 = ( δ Transition line p c e s a h p ~ r a e n i L Fractal Phase ( δ <1 ) p c = ( κ o -1 ) -1 Logarithmic Phase ( δ =0 ) ~ = p 0 0 c ∞ ∞ 1 1 Length factor a Length factor a Non-communicating

Results for S a ~N δ Scale-free (b) SF ( λ =3.5) (random) (c) SF (2< λ <3) (random) λ =2.2 a=1.01 4 10 4 10 2.3 1.1 2.4 1.2 1 1.5 3 10 S a S a 5 10 4 10 S a 3 3 10 10 2 10 2 10 N 4 5 10 10 3 4 4 5 10 10 10 10 N N

Targeted attacks on scale-free networks • Scale-free networks have sensitive nodes (hubs) with large k. • Examples: Airline hubs, central communication nodes, disease super-spreaders. Model for targeted percolation • p : fraction of lowest degree nodes present. • In targeted percolation (no length restriction) p c is large: p c =1 ( λ → 2) hub p c close to 1 ( λ >2) Network falls apart with few node removals. Question: What happens for limited path percolation?

Scaling theory for limited path targeted percolation on scale-free networks • For λ >3: ( ) ( ) κ − κ − a log 1 log 1 S ~ N o a ~ ~ = κ κ p p ( a , , ) c c o • For 2< λ <3: Tree approximation valid again after percolation: ( ) ( ) ( ) κ − λ − 2 a log 1 log 2 S ~ log N a Any finite a fails to produce transition to linear phase: ~ = p 1 c

Phase Diagram of Limited Path Percolation Scale-free targeted removal Communicating ~ = 1 p 1 c SF 2< λ <3 SF λ >3 Linear Phase ( δ =1 ) Concentration p e Concentration p Random T r s a n a s i t h i o n p l i n ~ e c p c i m h t i r a g o Fractal Phase ( δ <1 ) L p c = ( κ o -1 ) -1 Logarithmic Phase ( δ =0 ) 0 ∞ ∞ 1 1 Length factor a Length factor a Non-communicating

Results for S a ~N δ Scale-free targeted removal 5 4.25 10 (d) SF ( λ =2.3) (targeted) (d) SF ( λ =3.5) (targeted) 1.2 1.5 2.0 4 10 3.75 Slope=6.0+/ − 0.1 Log S a S a 4 3 10 10 3.25 S a 3 10 2 10 N 4 5 10 10 2.75 4 5 10 10 0.50 0.55 0.60 0.65 0.70 0.75 N Log (Log N)

Differences in Limited Path Percolation due to ~ network structure and removal method at p c ≤ p ≤ p c Random removal Erd ő s-Rényi Scale-free ( λ >3) Scale-free (2 ≤λ≤ 3) Quantity ( ) ~ ( − 1 a ) / a − ( 1 a ) / a κ − 1 p k 0 Transition c o ⎛ + ⎞ ⎛ ⎞ log p log p ⎜ ⎟ ⎜ ⎟ + δ a 1 a 1 1 ⎜ ⎟ ⎜ ⎟ κ − log( 1 ) log k ⎝ ⎠ ⎝ ⎠ No Transition Transition o N δ N δ N δ S a Targeted removal ~ ~ κ κ p c a ( , , ) p 1 - o c κ − log( 1 ) κ − log( 1 ) - 2 a δ a λ − κ − log( 2 ) log( 1 ) o N δ - (log N) δ S a