Large-deviation properties of random graphs Alexander K. Hartmann - - PowerPoint PPT Presentation

Large-deviation properties of random graphs

Alexander K. Hartmann

Institut für Physik Universität Oldenburg

MAPCON 12, 15. May 2012

1 / 15

Rare events

Storms Stock-market crashs Floodings − → Earth quakes ↓ ...

Oldenburg August 2010

San Fracisco 1906

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 0.5 1 1.5 2 2.5 3 frequency Richter scale magnitude M Gutenberg-Richter Gaussian

2 / 15

Outline

Algorithm Large-deviation graph properties (ER/2d lattice random graphs) largest component number of components Sampling of graphs

3 / 15

Graphs

Graph G = (V, E) connected components: transitive closure of “connectivity relation”

4 / 15

Graphs

Graph G = (V, E) connected components: transitive closure of “connectivity relation” diameter: longest among all pairwise shortest paths (within components)

4 / 15

Graphs

Graph G = (V, E) connected components: transitive closure of “connectivity relation” diameter: longest among all pairwise shortest paths (within components) Random graphs: N vertices, each edge tentative (ij) with prob. p. Erdös-Rényi: (ij) ∈ N(2), p = c/N → finite connect. c two-dim. percolation: (ij) ∈ square lattice, p = const N vertices, given (sampled) degree sequence k1. . . . , ki, e.g., scale-free P(k) ∼ k−γ each graph with same probability (“configuration model”)

4 / 15

Physics Approach

Idea: model ↔ physical system quenched realisation ↔ degrees of freedom x (state) quantity “score” S ↔ energy E( x) (ground state: often known) simulate at finite T Monte Carlo moves: change realisat. a bit Simulation at different T (using (MC)3/PT) Example (sequence alignment) equilibration: start with ground state/ with random state Wang-Landau approach

5000 10000 15000 20000 t 5 10 15 20 S(t) T=0.5 T=0.57 T=0.69 T=0.98

5 / 15

Distribution of Scores

Raw result − → (simple ↔ T = ∞) at low T: high scores prefered MC moves: x → x′ change on “element” probability = fa

5 10 15 20

S

10

−5

10

−4

10

−3

10

−2

10

−1

10

p

*(S)

simple T=0.69 T=0.57

Pr(acceptance) = min{1, exp(S(

x′)/T) exp(S( x)/T) } = min{1, e∆S/T}

⇒ equilibrium distribution QT( x) = P( x)eS(

x)/T/Z(T)

with P( x) =

i fxi, Z(T) =

• x P(

x)eS(

x)/T

⇒ pT(S) =

• x,S(

x)=S QT(

x) = exp(S/T)

Z(T)

• x,S(

x)=S P(

x) ⇒ p(S) = pT(S)Z(T)e−S/T

[AKH, PRE 2001]

6 / 15

Match Distriutions

• p(S) = pT(S)Z(T) exp(−S/T)
• rescaling with exp(−S/T)

10 20

S

10

• 8

10

• 6

10

• 4

10

• 2

10 10

2

pT(S)exp(-S/T)

simple T=0.69 T=0.57

Z(T) by “matching”

10 20

S

10

• 12

10

• 10

10

• 8

10

• 6

10

• 4

10

• 2

10 10

2

p(S)

simple, N=10

9

simple, N=10

4

T=0.69 T=0.57

agrees with large statistics simple sampling agrees with (for this example) known exact result

7 / 15

Results: Erd˝

• s-Rényi

Size S of largest component (connectivity c)

[AKH, Eur. Phys. J. B (2011)]

8 / 15

Rate function Φ(s) ≡ − 1

N log P(s), s = S/N

Comparison with exact asymptotic result

[M. Biskup, L. Chayes, S.A. Smith, Rand. Struct. Alg. 2007]

→ evaluate algorithm → works very well → finite-size corrections visible

9 / 15

Phase transition

Cluster size as function of (artificial) temperature 1st order transition in percolating phase → large system sizes not fully accessible (→ use Wang-Landau algorithm here)

10 / 15

Bias in Configuration model

Configuration model: k “stubs” for each node of degree k. Randomly draw pairs of stubs. If multiple/self edge: refusal: start graph from scratch repetition: redraw pair Repetition is biased: relevant for measurements (N → ∞)?

100 200 300 400 realization 9000 10000 11000

• No. graphs

refusal repetition n=8 10 50 100 200 n 5 10 15 20 diameter

refusal repetition refusal with cutoff repetition with cutoff

γ = 2.5

[H. Klein-Hennig, AKH, Phys. Rev. E 2012]

→ Markov chains/ hidden variables/ throw-away edges/ ...

11 / 15

Two-dimensional percolation

N = L × L, edge density p No exact result known (to me) Results comparable to Erd˝

• s-Rényi random graphs

but stronger finite-size effects

12 / 15

Graph Diameter

Diameter d⋆ := Longest of all shortest i → j paths Random graphs: (c < 1): Gumbel distribution PrG(d⋆ = d) = λe−λ(d−d0)e−e−λ(d−d0) (sloppy) explanation: graph = forest d = maxtrees T d(T) → Gumbel distribution Fit to P(d) = PG(d)e−a(d−d0)2 “gaussianized” Gumbel

[AKH, M. Mézard, in preparation]

50 100

d

10

• 30

10

• 25

10

• 20

10

• 15

10

• 10

10

• 5

10

P(d) N=100 N=200 N=1000 N=5000 fits

10

2

10

3

10

4

N 0.1 0.2 0.3 0.4 λ(N)

c=0.9 13 / 15

Close to c = 1, asymptotically λ(c) = − log c

[T. Luczak, Rand.Struct.Alg., 1998]

0.2 0.4 0.6 0.8 1

c

0.0 0.5 1.0 1.5 2.0 2.5 3.0

λ(c) N → ∞

• log(c)

Percolating region: more complex distributions

0.2 0.4 0.6 0.8 1

d/N

• 1.0
• 0.5

0.0

φ(d/N)

N=100 N=200 N=500 N=1000 c=3.0

14 / 15

Summary

Large-deviation properties Physics approach: study system at artificial finite temperature

(or, in principle, Wang-Landau algorithm + modifications)

Full distribution of size of largest component Erd˝

• s-Rényi random graphs: matches well analytics

1st order transition in percolating phase (also: number of components, 2d percolation, diameter) Simple sampling of configuration model is biased

Work more efficiently: read/write/edit scientific paper summaries

www.papercore.org

(open access)

Summer school: Efficient Algorithms in Computational Physics Bad Honnef (Germany), 10-14. September 2012

15 / 15