ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Random regular graphs
Nick Wormald University of Waterloo
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Random regular graphs
• What are they (models) • Typical and powerful results • Handles for analysis • One or two general techniques • Some open problems
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Regular graphs
A vertex has degree d if it is incident with d edges.
A d-regular graph has all vertices of degree d.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
sparrow Australia pigeon Christmas Island lyrebird lark Ireland spotted Noumea treecreeper PEI dodo Tasmania pegasus Traffic Island Kentucky fried chicken Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Random regular graphs What are they?
Faultily faultless, icily regular, splendidly null, dead perfection; no more.
- Lord Alfred Tennyson
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Random permutation
Choose a permutation of {1, . . . , n} uniformly at random. 1 2 e.g. (1 2 3 5) (4 6 7) 6 7 3
5 4 Then forget orientations 1 2
6 7 3
5 4
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Uniform model
n,d d G : Probability space, elements are the - n regular graphs on vertices. 1 Each has the same probability: |G n,d |
But is not known exactly. Hard to analyse.|G n,d |
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Algorithmic models
e.g. Degree-restricted process: add edges to random places, keeping all vertex degrees at most d.
1 2 1 2
6 3 6 3
5 4 5 4
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Superposition models
Random ∈ H6 ∈ G6,1 Hamilton cycle
Take one Throw it away member from if a multiple each of two ∈ H6 G6,1 edge is models, and created. superimpose.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
a.a.s.
A property Q holds asymptotically almost surely (a.a.s.) in a random graph model if
P n (G has Q) → 1 as → ∞
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Questions - uniform model
n,d Do graphs in G a.a.s. satisfy the following? d connected, and moreover -connected n contain a perfect matching (for even)
hamiltonian (have cycle through all vertices)
trivial automorphism group ... and how are the following distributed?
subgraph counts chromatic number
eigenvalues
independent & dominating set sizes
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
n,d Some answers for G Many things are known. Some examples: 3 d n 4 d For : a.a.s. -connected and hamiltonian (Bollobas, Wormald, Frieze, Robinson, Cooper, Reed, Krivelevich, Sudakov, Vu), trivial automorphism group (B, McKay, W, K, S, V),
For fixed d: distribution of eigenvalues (McKay), < 2√d 1 + second eigenvalue a.a.s. (Friedman) Chromatic number bounds known (Achlioptas & Moore, Shi & Wormald)
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Contiguity
n n Two sequences of models G and F are contiguous if for any sequence of events An
An n is a.a.s. true in G if and only if An n is a.a.s. true in F
n n Notation: G ≈ F
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Contiguity of Superposition models
d 3 Thm 1 (~Robinson & W; Janson) For n, d n, n, d n G G ≈ G ( even).
Thm 2 (Janson; Molloy, Reed, Robinson & Wormald)
n, n, n, n, n G G G ≈ G ( even). 3 n, n, i.e. G ≈ G d 3 Thm 3 (Robalewska) For . n, d n, n, d G G ≈ G
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Arithmetic of contiguity
Example with n even:
n, n, n, G ≈ G G (Thm 3) n, 2 n, ≈ G G (Thm 3) n, n, 2 n, ≈ G G G (Thm 1) n, 2 n, 2 n, ≈ G G G (Thm 1) 3 n, 2 n, 2 n, ≈ G G G (Thm 2) 5 n, 2 n, = G G
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
1 + 1 is not 2
n, d In general all such equations with G and respecting degree sums are true (n even):
n, d n, d . . . n, d n, d G G G k ≈ G d + d + . . . + d = d 3. provided k
There is one failure:
n, n, n, G G ≈ G
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
n Contiguity with H n H = random Hamilton cycle on n vertices Thm 4 (Frieze, Jerrum, Molloy, Robinson & Wormald)
n, d n n, d d 3 G H ≈ G (fixed ). Thm 5 (Kim & Wormald) n n n, H H ≈ G n, d n Thus all equations involving various G and H respecting degree sums are true, provided the total degree is at least 3.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
n Contiguity with F
n F = graph formed from uniformly chosen random permutation of n vertices (with no loops or multiple edges).
Thm 6 (Greenhill, Janson, Kim & Wormald)
n n n, n, d n n, d d 3 F F ≈ G , G F ≈ G ( ).
n, d n n Thus equations involving various G , F and H respecting degree sums are true, provided the total degree is at least 3.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Corollaries n n n, n, d n F F ≈ G ≈ G H n n implies that F F is a.a.s. hamiltonian. (Also proved algorithmically by Frieze.)
n, d n n, d G H ≈ G n, d implies that G is a.a.s. hamiltonian. d n, n, d n G ≈ G ( even) n, d implies that G is a.a.s. decomposable into d perfect matchings (so is d-edge-colourable).
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Other corollaries Each model is a.a.s. decomposable into d/ 2 d 4 edge-disjoint Hamilton cycles (even ). Bipartite version of this is also true (Greenhill, Kim & Wormald). So there exist 4-regular bipartite graphs with a hamiltonian decomposition and arbitrarily large girth (=length of shortest cycle). This gives examples of complexes with in- coherent fundamental group (McCammond & Wise).
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Other corollaries
Anything a.a.s. true in one of the models is also a.a.s. true in the others.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Handles for analysis
1. Pairing model (esp. for small d)
2. Switchings (esp. for moderate d)
3. Enumeration results (esp. for large d)
For this talk: only 1 in detail.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
n, d Pairing model - P To stand for a vertex, take d points in a “cell” or bucket.
vertex 1 vertex 2 ...
Then take a random pairing (perfect matching) of all the points. The pairs determine the edges of the graph.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
n, d Pairing model - P
2 n = 6 d = 3
1 3
4 6 Reject if loops or 5 multiple edges
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
n, d Pairing model - P
2 n = 6 d = 3
1 3
4 6 Uniformly uniformly random random pairing gives 5 graph
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Analysis of random pairings For analysis, permit loops and multiple edges. Example: distribution of number of triangles.
For this we use the method of moments as in Lecture 4. But now pairs are dependent. (c.f. G (n , p ) , where edges are independent.)
Create an indicator variable I j for each triple of pairs that induce a triangle in the graph. Call such a triple a triangle of the pairing.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Expected number of triangles
If X is the total number of triangles in the random pairing then since I j is an indicator EX = E I j = P(I j = 1 ) j j X X
P(I j = 1 ) = M (dn 6) / M (dn ) where M ( k ) is the number of perfect matchings of k points.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
triangles (cont.)
M ( k ) = ( k 1)(k 3) . . . 1 We easily get 3 P(I j = 1 ) (dn ) and then . The number of ways to choose a triangle in the pairing is
(d(d 1 )) 3 n 3
3 EX (d 1 ) / 6 Thus .
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
ASIDE: Easy exercise
Show that if F has more edges than vertices then it a.a.s. does not occur as a subgraph of n, d G .
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Distribution of cycle counts
Higher moments easily computed in a similar way. Conclusion:
X has asymptotically Poisson distribution with 3 = (d 1 ) / 6 expectation .
X r - cycles of length r - can be done similarly and again the distribution is asymptotically = (d 1 ) r / 2 r Poisson with expectation r .
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Joint distribution Joint moments also computed in the same fashion. For instance i j E(X ) (X ) i j from which we may conclude X and X are asymptotically jointly independent Poisson. One implication of this is P(X = X = 0 ) exp( ) (1 d 2 ) / 4 = e .
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Simple graphs
Let simple denote the event that the random pairing produces no loops or multiple edges. Then we have found P(simple) e (1 d 2)/ 4 .
Joint moments of X , X and other Xr ‘s give the asymptotic distribution of Xr Xs n,d , , . . . , in G .
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
a.a.s. properties of simple graphs Since P(simple) is bounded below, if for A P(A) = 1 o(1) any event we show that n,d P(A)= 1 o(1) n,d in P then also in G . This is the basis of attack for many problems. Example: “1+1” is not “2” because the n, probability of G having no odd cycle of of length less than 2g is asymptotically exp( g ) , which tends to 0 as g goes to infinity.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Variance and Hamilton cycles Now let Y be the total number of Hamilton cycles in (the graph of) the random pairing. Use an indicator variable I j for each possible set of pairs inducing a Hamilton cycle to find E(Y ) var(Y ) d 3 n,d and . For we find in P var(Y ) / E ( Y ) 2 d /(d 2 ) 1 , a positive constant. By second moment method this is an upper bound on P(Y = 0 ).
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Small subgraph conditioning - for proving contiguity Implicit in work of Robinson & Wormald, dis- tilled by Janson, also Molloy, Robalewska, R&W. The technique may apply when var(Y ) is of 2 the order of E(Y ) and the variability signified by the large variance is “induced” by some variables describing local properties.
Often these are X , X , . . . (short cycle counts).
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
The hypotheses
Let Y count decompositions of a graph of a specific type. For example, Hamilton cycle + (d 2 ) -regular graph.
1. X , X , . . . , Xk are asymptotically independent Poisson with expectations i.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
The hypotheses (PART 2) Y X X X 2. E ( ) j ( ) j ( k ) j k k → ji EY ( i (1 + i )) i= Y for every finite sequence j , j , . . . , jk of > 1 non-negative integers, where all i . ∞ 2 2 2 3. E Y (E Y) exp i i i = X and the sum converges.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Small subgraph conditioning - conclusion
n n, d R ≈ G n where R is the space of random regular graphs each with probability proportional to the num- ber of decompositions of the specified type. Superposition models relate to decompositions! Calculations in the Hamilton cycle example d 3 verify the conditions for , so n, d n n, d G H ≈ G
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
and so on
All the contiguity results stated before are proved by that method.
If we have time, let’s look at the differential equation method.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Greedy algorithms
Problem: what is the size of the largest independent set in a random regular graph? Largest dominating set?
Greedy algorithms often achieve good a.a.s. bounds.
How do we analyse them?
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Differential equation method A randomised algorithm is applied to a graph.
When the algorithm is applied to a random regular graph, its steps depend on some variables that a.a.s. follow close to the solutions of some system of differential equations. (Justification by martingale techniques.)
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
DE method for random pairings
We will need to compute the expected changes in these variables, in each step.
Consider the algorithm applied to a random pairing. In each step of the algorithm, one may generate at random just those pairs involving whichever points are relevant for the next step of the algorithm.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
example: max independent set An independent set is a set of vertices, no two of which are adjacent. (G) denotes the largest size of an independent set in G. From expectation arguments, ( G ) < ( d ) n a.a.s., where e.g. (McKay) (3) = 0.4554, (4) = 0.4163. Lower bounds come from greedy algorithms.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Greedy alg for max independent set Simple algorithm: select vertices consecutively at random to build an independent set. Upon selecting a vertex, delete it and its neighbours. Yi(t): number of vertices of degree i after t steps of the algorithm. In pairing model, find (asymptotically) expected change in Yi in one step, as function of Yj ‘s. Writing the expected change as a derivative gives a differential equation:
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
d.e. for independent set algorithm
yi 0 = f (y , y , ... , y d )
where yi ( x ) approximates Yi ( t)/ n at time t, x = t / n . (Details omitted!)
The d.e. method includes general results for showing that a.a.s. the Yi stay close to the scaled solutions of the d.e.:
Yi (t) = nyi (t/ n ) + o(n ) a.a.s.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Conclusion of simple algorithm
Let x be the solution of yi (x) = 0. Then a.a.s. the process lasts for Xx n + o(n ) steps. Thus (G) > x n + o(n ) a.a.s. d 3 We find for that x = (1 / 2)(1 (d 1) 2 / (d 2 )) d x 3 0.3750 4 0.3333
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Degree greedy algorithm
Give priority to vertices with minimum degree in the ever-shrinking graph. d x the upper bounds again 3 0.4327 0.4554 4 0.3901 0.4163 (Analysis requires extra bells and whistles.) The case d=3 also obtained by Frieze and Suen (analytically as 6 log(1.5) - 2) analysing the same algorithm another way.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Colouring
( n, ) = 3 Easy exercise that G a.a.s. (Hints: Brooks Thm, and short odd cycles.) Greedy algorithm: assign colours randomly to vertices: a.a.s. requires d + 1 colours.
Better algorithm: higher priority to vertices with more colours already on their neighbours. Achlioptas and Moore showed in this way that P( ( n, ) = 3) > c + o(1 ) c > 0 G for some .
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA Colouring - even better algorithm Modified better algorithm: first colour the short odd cycles, then proceed as before. This shows above with c = 1 (Shi & Wormald). ( n, ) = 3 So G a.a.s. ( n, ) = 3 4 Similarly, we get G or a.a.s., ( n, ) = 4 G a.a.s., etc. Analysis for upper bounds by d.e. method using bells, whistles and flashing lights. Lower bounds proved earlier (Molloy & Reed) using expectation.
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
Unsolved Problems
Conjecture that a random d-regular graph with an even number of vertices a.a.s. has a perfect 1-factorisation (d ≥ 3).
Does a random d-regular directed graph a.a.s. have d edge-disjoint Hamilton cycles (d ≥ 3)?
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
More unsolved Problems
Is the uniform random d-regular graph contiguous to the algorithmically defined model (add edges at random subject to maximum degree d)?
Is a random 5-regular graph a.a.s. 3-colourable?
Jan 4, 2005 LECTURE 5 Jan 4, 2005 ATLANTA MAA SHORT COURSE ON RANDOM GRAPHS ATLANTA MAA
References
“Recent publications” 16, 17, 27, 36, 54, 66 on my web page:
http://www.math.uwaterloo.ca/~nwormald/abstracts.ht
Jan 4, 2005 LECTURE 5 Jan 4, 2005