Moscow
June 11, 2014
The Erd˝os-R´enyi
Phase Transition
Joel Spencer
1
Any new possibility that existence ac-
quires, even the least likely, transforms
everything about existence.
– from Slowne...
Paul Erd˝os and Alfred R´enyi
On the Evolution of Random Graphs
Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl
volume 8, 17-61,...
The (Traditional) “Double Jump”
G(n, p), p = c
n (or ∼ c
2n edges)
(Average Degree c, “natural” model)
• c < 1
Biggest Com...
The Five Phases
Subcritical: p = c
n and c < 1
Barely subcritical: p ∼ 1
n and p = 1
n −λ(n)n−4/3
with λ(n) → ∞
The Critic...
• Barely Subcritical
p ∼ 1
n and p = 1
n − λ(n)n−4/3 with λ(n) → ∞
All components simple.
Top k components about same size...
Math Physics Bond Percolation
Zd. Bond “open” with probability p
There exists a critical probability pc
• Subcritical, p <...
Random 3-SAT
n Boolean x1, . . . , xn
L = {x1, x1, . . . , xn, xn} literals
Random Clauses Ci = yi1 ∨ yi2 ∨ yi3, yij ∈ L
f...
Evolution of n-Cube
Ajtai, Komlos, Szemeredi
Bollobas, Luczak, Kohayakawa
Borgs, Chayes, Slade, JS, van der Hofstad
p = c/...
The Bohman-Frieze Process
Begin empty. One edge each round.
Two random {x1, y1}, {x2, y2}
IF x1, y1 isolated ADD {x1, y1}
...
The Product Rule Process
Begin empty. One edge each round.
Two random {x1, y1}, {x2, y2}
IF |C(x1)|·|C(y1)| ≤ |C(x2)|·|C(y...
Poisson Birth Process
Root node “Eve”
Parameter c
Each node has Po(c) children
(Poisson: Pr[Po(c) = k] = e−cck/k!)
Zt ∼ Po...
Binomial Birth Process
Parameters m, p
Zt ∼ B[m, p], iid
T = Tbin
m,p total size.
For m large, p small, mp moderate:
Binom...
Graph Birth Process
Parameters n, p
Generate C(v) in G(n, p). BFS
Queue: Y0 = 1, Yt = Yt−1 + Zt − 1
Points Born: Zt ∼ B[Nt...
Poisson Birth Trichotomy
• c < 1
T finite
• c = 1
T finite
E[T] infinite (heavy tail)
• c > 1
Pr[T = ∞] = y = y(c) > 0
15
Poisson Birth Exact
Pr[Tc = u] =
e−uc(uc)u−1
u!
Pr[T1 = u] =
e−uuu−1
u!
= Θ(u−3/2
)
For c > 1, Pr[T = ∞] = y = y(c) > 0 wh...
Poisson Birth Near Criticality
c = 1 + ǫ, T = T
po
c
Pr[T = ∞] ∼ 2ǫ
Pr[T = u] ∼ (2π)−1/2u−3/2(ce1−c)u
ln[ce1−c] ∼ −ǫ2/2
• ...
Poisson Birth ∼ Graph Birth
Z1 ∼ B[n − 1, p] roughly Po(c), c = pn.
Ecological Limitation: Zt ∼ B[Nt−1, p].
Process succee...
Why n−4/3 for Critical Window
p = (1 + ǫ)/n, ǫ > 0, ǫ = o(1).
Pr[T
po
1+ǫ = ∞] ∼ 2ǫ.
The ∼ 2ǫn points “going to infinity” m...
The Barely Subcritical Region
p = (1 − ǫ)/n, ǫ = λn−1/3,
Pr[|C(v)| ≥ u] ≤ Pr[T1−ǫ ≥ u]
u = Kǫ−2 ln n ⇒ Pr = o(n−1)
No Such...
Barely Supercritical
p = (1 + ǫ)/n, ǫ = λn−1/3, λ → +∞
Trichotomy on Component Size
Small: |C| < Kǫ−2 ln n [can be impoved...
No Middle Ground
Yt = n − t − Nt = B[n − 1, 1 − (1 − p)t] − (t − 1)
At start E[Yt] ∼ ǫt [Negligible EcoLim]
When t ≫ ǫ−2 l...
Escape Probability
S := Kǫ−2 ln n, α := Pr[|C(v)| ≥ S]
Pr[|C(v)| ≥ S] ≤ Pr[Tbin
n−1,p ≥ S]
np = 1 + ǫ, S ≫ ǫ−2 so ∼ 2ǫ
Pr[...
Almost Done
Not Small implies Large ∼ 2ǫn
Expected 2ǫn in Large components
BUT
Can we have two
of size 2ǫn
half the time?
...
Sprinkling
Add sprinkle of n−4/3, p ← p+
If G(n, p) had two Large they would merge
That would give ≥ 4ǫn in G(n, p+)
But p...
Computer Experiment (Try It!)
n = 500000 vertices. Start: Empty
Add random edges
Parametrize e/ n
2 = (1 + λn−1/3)/n
Merge...
If I have seen further it is by standing
on the shoulders of giants.
Isaac Newton, 1676
27
If I have seen further it is by standing
on the shoulders of Hungarians.
Peter Winkler
28
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The Phase Transition

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More than fifty years ago, Paul Erd˝os and Alfred R´enyi discovered that the random graph G(n, p) underwent a phase transition (in modern language) near p = 1/n. For p a bit smaller (e.g., 0.99/n) all of the components were very small and had simple structures. But for p a bit bigger (e.g., 1.01/n) a “giant component” had emerged with a complex behavior. We now understand how to slow down this process so as to see the incipient giant component (the dominant component) at an early stage and to define a Critical Window through which the process moves from subcriticality to supercriticality.

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The Phase Transition

  1. 1. Moscow June 11, 2014 The Erd˝os-R´enyi Phase Transition Joel Spencer 1
  2. 2. Any new possibility that existence ac- quires, even the least likely, transforms everything about existence. – from Slowness by Milan Kundera 2
  3. 3. Paul Erd˝os and Alfred R´enyi On the Evolution of Random Graphs Magyar Tud. Akad. Mat. Kutat´o Int. K¨ozl volume 8, 17-61, 1960 Γn,N(n): n vertices, random N(n) edges [. . .] the largest component of Γn,N(n) is of or- der log n for N(n) n ∼ c < 1 2, of order n2/3 for N(n) n ∼ 1 2 and of order n for N(n) n ∼ c > 1 2. This double “jump” when c passes the value 1 2 is one of the most striking facts concerning random graphs. 3
  4. 4. The (Traditional) “Double Jump” G(n, p), p = c n (or ∼ c 2n edges) (Average Degree c, “natural” model) • c < 1 Biggest Component O(ln n) |C1| ∼ |C2| ∼ . . . All Components simple (= tree/unicyclic) • c = 1 Biggest Component Θ(n2/3) |C1|n−2/3 nontrivial distribution |C2|/|C1| nontrivial distribution Complexity of C1 nontrivial distribution • c > 1 Giant Component |C1| ∼ yn, y = y(c) > 0 All other |Ci| = O(ln n) and simple 4
  5. 5. The Five Phases Subcritical: p = c n and c < 1 Barely subcritical: p ∼ 1 n and p = 1 n −λ(n)n−4/3 with λ(n) → ∞ The Critical Window p = 1 n + λn−4/3 λ arbitrary real, but constant. Barely supercritical: p ∼ 1 n and p = 1 n+λ(n)n−4/3 with λ(n) → ∞ Supercritical: p = c n and c > 1 5
  6. 6. • Barely Subcritical p ∼ 1 n and p = 1 n − λ(n)n−4/3 with λ(n) → ∞ All components simple. Top k components about same size |C1| = o(n2/3) • Barely Supercritical p ∼ 1 n and p = 1 n + λ(n)n−4/3 with λ(n) → ∞ Dominant Component |C1| ≫ n2/3, High Complexity All other |C| ≪ n2/3, Simple Duality: Remove Dominant Component and get Subcritical Picture. 6
  7. 7. Math Physics Bond Percolation Zd. Bond “open” with probability p There exists a critical probability pc • Subcritical, p < pc. All C finite, E[|C(0)|] finite Pr[|C(0)| ≥ u] exponential tail • Supercritical, p > pc. Unique Infinite Component E[|C(0)|] infinite Pr[|C(0)| ≥ u|finite] exponential tail • Critical, p = pc. All C finite, E[|C(0)|] infinite, heavy tail Key topic: p = pc ± ǫ as ǫ → 0. 7
  8. 8. Random 3-SAT n Boolean x1, . . . , xn L = {x1, x1, . . . , xn, xn} literals Random Clauses Ci = yi1 ∨ yi2 ∨ yi3, yij ∈ L f(m) := Pr[C1 ∧ · · · ∧ Cmsatisfiable] Conjecture: There exists critical c0 • Subcritical, c < c0, f(cn) ∼ 1 • Supercritical, c > c0, f(cn) ∼ 0 Friedgut: Criticality, but possibly nonuniform But what is the Critical Window I.e.: Parametrize m = m(n) to “see” f(m) go ǫ to 1 − ǫ Open Question! 8
  9. 9. Evolution of n-Cube Ajtai, Komlos, Szemeredi Bollobas, Luczak, Kohayakawa Borgs, Chayes, Slade, JS, van der Hofstad p = c/n c < 1 subcritical c > 1 giant Ω(2n) component Critical p0 ∼ n−1 At p0(1 − ǫ) all “small” At p0(1 + ǫ). For ǫ = Ω(n−100) and more: Giant 2ǫn. Second open But what is the Critical Window Open Question! 9
  10. 10. The Bohman-Frieze Process Begin empty. One edge each round. Two random {x1, y1}, {x2, y2} IF x1, y1 isolated ADD {x1, y1} ELSE ADD {x2, y2} Criticality at p = tcr/n proprotion tcr > 1 Bohman Frieze Exact tcr: Wormald, JS Critical Window: Kang, Perkins, Bhamidi Budhiraja, Wang, Sen · · · But what is the Critical Window? Same as Erd˝os-R´enyi 10
  11. 11. The Product Rule Process Begin empty. One edge each round. Two random {x1, y1}, {x2, y2} IF |C(x1)|·|C(y1)| ≤ |C(x2)|·|C(y2)| ADD {x1, y1} ELSE ADD {x2, y2} JS, D’Souza, Achlioptas: Simulation gives “Explosive” percolation DeCosta, Dorogovtsev, Goltsev, Mendes: Physics says NO Riordan, Warnke: Math also says NO But what is the Critical Window? Open Question! 11
  12. 12. Poisson Birth Process Root node “Eve” Parameter c Each node has Po(c) children (Poisson: Pr[Po(c) = k] = e−cck/k!) Zt ∼ Po(c), iid t-th node has Zt children Queue Size Yt. Y0 = 1 (Eve) Yt = Yt−1 + Zt − 1 (Has children and dies) Fictional Continuation: Yt defined though pro- cess stops when some Ys = 0. Size T = T po c is minimal t with Yt = 0. T = ∞: All Yt > 0. T = Tc is total size 12
  13. 13. Binomial Birth Process Parameters m, p Zt ∼ B[m, p], iid T = Tbin m,p total size. For m large, p small, mp moderate: Binomial is very close to Poisson c = mp. Binomial Birth Process very close to Poisson Birth Process 13
  14. 14. Graph Birth Process Parameters n, p Generate C(v) in G(n, p). BFS Queue: Y0 = 1, Yt = Yt−1 + Zt − 1 Points Born: Zt ∼ B[Nt−1, p] Dead Points (popped): t Live Points (in Queue): Yt Neutral Points(in Reservoir): Nt t + Yt + Nt = n N0 = n−1, Nt = Nt−1−Zt, Nt ∼ B[n−1, (1−p)t] T = T gr n,p: minimal t with Yt = 0 T = t implies Nt = n − t 14
  15. 15. Poisson Birth Trichotomy • c < 1 T finite • c = 1 T finite E[T] infinite (heavy tail) • c > 1 Pr[T = ∞] = y = y(c) > 0 15
  16. 16. Poisson Birth Exact Pr[Tc = u] = e−uc(uc)u−1 u! Pr[T1 = u] = e−uuu−1 u! = Θ(u−3/2 ) For c > 1, Pr[T = ∞] = y = y(c) > 0 where 1 − y = e−cy For c < 1, α := ce1−c < 1 Pr[Tc > u] = O(αu) Exponential Tail 16
  17. 17. Poisson Birth Near Criticality c = 1 + ǫ, T = T po c Pr[T = ∞] ∼ 2ǫ Pr[T = u] ∼ (2π)−1/2u−3/2(ce1−c)u ln[ce1−c] ∼ −ǫ2/2 • u small: u = o(ǫ−2) Pr[Tc = u] ∼ Pr[T1 = u] = Θ(u−3/2) Scaling: u = Aǫ−2 Pr[∞ > T1+ǫ > Aǫ−2] = ǫe−(1+o(1))A/2 Pr[T1−ǫ > Aǫ−2] = ǫe−(1+o(1))A/2 17
  18. 18. Poisson Birth ∼ Graph Birth Z1 ∼ B[n − 1, p] roughly Po(c), c = pn. Ecological Limitation: Zt ∼ B[Nt−1, p]. Process succeeds, Nt−1 gets smaller Fewer new vertices Death is inevitable Upper: Pr[T gr n,p ≥ u] ≤ Pr[Tbin n−1,p ≥ u] Proof: Replenish reservoir Lower: Pr[T gr n,p ≥ u] ≥ Pr[Tbin n−u,p ≥ u] Proof: Hold reservoir to n − u. 18
  19. 19. Why n−4/3 for Critical Window p = (1 + ǫ)/n, ǫ > 0, ǫ = o(1). Pr[T po 1+ǫ = ∞] ∼ 2ǫ. The ∼ 2ǫn points “going to infinity” merge to form dominant component. Tpo finite is O(ǫ−2), corresponds to component sizes O(ǫ−2). Finite/Infinite Poisson Dichotomy becomes Small/Dominant Graph Dichotomy if ǫ−2 ≪ 2nǫ, or ǫ ≫ n−1/3. 19
  20. 20. The Barely Subcritical Region p = (1 − ǫ)/n, ǫ = λn−1/3, Pr[|C(v)| ≥ u] ≤ Pr[T1−ǫ ≥ u] u = Kǫ−2 ln n ⇒ Pr = o(n−1) No Such component. More delicately: Parametrize u = Kǫ−2 ln λ = Kn2/3λ−2 ln λ K big: Pr[|C(v)| ≥ u] = O(ǫλ−10) Expected nǫλ−10 = n2/3λ−9 vertices in com- ponents of size ≥ Kn2/3λ−2 ln λ No such component! 20
  21. 21. Barely Supercritical p = (1 + ǫ)/n, ǫ = λn−1/3, λ → +∞ Trichotomy on Component Size Small: |C| < Kǫ−2 ln n [can be impoved!] Large: (1 − δ)2ǫn < |C| < (1 + δ)2ǫn Awkward: All else No Middle Ground No Awkward Components Suffices: Pr[C(v) awkward] = o(n−1) 21
  22. 22. No Middle Ground Yt = n − t − Nt = B[n − 1, 1 − (1 − p)t] − (t − 1) At start E[Yt] ∼ ǫt [Negligible EcoLim] When t ≫ ǫ−2 ln n, E[Yt] ≫ Var[Yt]1/2 ∼ t1/2, Pr[Yt = 0] = o(n−10) Later E[Yt] = (n−1)[1−(1−p)t]−(t−1) ∼ ǫt− t2 2n For t ∼ 2ǫn, E[Yt] ∼ 0, dominant component. |C(v)| = t implies Yt = 0. For t ∼ yǫn, y = 2: Pr[|C(v)| = t] ≤ Pr[Yt = 0] = o(n−10). 22
  23. 23. Escape Probability S := Kǫ−2 ln n, α := Pr[|C(v)| ≥ S] Pr[|C(v)| ≥ S] ≤ Pr[Tbin n−1,p ≥ S] np = 1 + ǫ, S ≫ ǫ−2 so ∼ 2ǫ Pr[Tbin n−S,p ≥ S] ≤ Pr[|C(v)| ≥ S] (Here ǫ ≫ n−1/3 ln1/3 n but with care . . . ) • As Sp = o(ǫ) EcoLim negligible! p(n − S) = 1 + ǫ + o(ǫ) so Pr ∼ 2ǫ Sandwich: Escape Prob ∼ 2ǫ 23
  24. 24. Almost Done Not Small implies Large ∼ 2ǫn Expected 2ǫn in Large components BUT Can we have two of size 2ǫn half the time? 24
  25. 25. Sprinkling Add sprinkle of n−4/3, p ← p+ If G(n, p) had two Large they would merge That would give ≥ 4ǫn in G(n, p+) But p+ = (1 + ǫ + o(ǫ))/n has nothing ≥ 4ǫn Conclusion: • G(n, p) has precisely one Large component • It has size ∼ 2ǫn • As no middle ground: All other component sizes ≤ Kǫ−2 ln n. So Large Component is Dominant Component 25
  26. 26. Computer Experiment (Try It!) n = 500000 vertices. Start: Empty Add random edges Parametrize e/ n 2 = (1 + λn−1/3)/n Merge-Find for Component Size/Complexity −4 ≤ λ ≤ +4, |Ci| = cin2/3 See biggest merge into dominant 26
  27. 27. If I have seen further it is by standing on the shoulders of giants. Isaac Newton, 1676 27
  28. 28. If I have seen further it is by standing on the shoulders of Hungarians. Peter Winkler 28

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