Row and column sums of random 0-1 matrices. Fiona Skerman Brendan McKay. Australian National University

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1 Row and column sums of random 0-1 matrices Fiona Skerman Brendan McKay Australian National University random graphs 1

2 Models of random 0-1 Matrices (bipartite graphs) G p edges chosen with equal probability, p. G M each graph with M edges is equally likely G z each graph with given right-hand-side degrees equally likely random graphs 2

3 Models of random 0-1 Matrices (bipartite graphs) G p edges chosen with equal probability, p. u 1 v 1 u 2 kkkkkkkkkkkkkkkkkkkkkk v SSSSSSSSSSSSSSSSSSSSSS s j =deg(u j ) t k =deg(v k ) G M each graph with M edges is equally likely u 3 0 v #edges = P 0000 j u j = P k v k G z each graph with given right-hand-side (blue) degrees equally likely u m TTTTTTTTTTTTTTTTTTTTTT 0 v n v n random graphs 2

4 Figure 1: The three possible graphs with degree sequence (1,3,1,1 2,1,3) random graphs 3

5 Figure 1: The three possible graphs with degree sequence (1,3,1,1 2,1,3) The total number of bipartite graphs on (4, 3) vertices is 2 12 =4096,ofthese, 6 edges. Hence, P Gp= 1 2 (1, 3, 1, 1 2, 1, 3) = P GM=6 (1, 3, 1, 1 2, 1, 3) = =924haveprecisely random graphs 3

6 Figure 1: The three possible graphs with degree sequence (1,3,1,1 2,1,3) The total number of bipartite graphs on (4, 3) vertices is 2 12 =4096,ofthese, 6 edges. Hence, P Gp= 1 2 (1, 3, 1, 1 2, 1, 3) = P GM=6 (1, 3, 1, 1 2, 1, 3) = =924haveprecisely Number of such graphs with blue degree sequence (2, 1, 3) is = 96. Hence, P Gz=(2,1,3) (1, 3, 1, 1 2, 1, 3) = 3 96 random graphs 3

7 Motivating Theorem : Canfield, Greenhill and McKay (2008) Conditions: s j s, t k t are O(n 1/2+" ) a + b< 1 2 m = o(n 1+" ), and n = o(m 1+" ) satisfies: m, n!1 1 2 (1 ) 1+ 5m 6n + 5n 6m apple a log n i.e. (s,t) near-regular Theorem: the number of bipartite graphs with degree sequence (s,t)is mn M 1 Q j n s j Qk m t k exp Pj (s j s) 2 mn (1 ) 1 Pk (t k t) 2 mn (1 ) + O(n b ) random graphs 4

8 Motivating Theorem : Canfield, Greenhill and McKay (2008) Conditions: s j s, t k t are O(n 1/2+" ) a + b< 1 2 m = o(n 1+" ), and n = o(m 1+" ) satisfies: m, n!1 1 2 (1 ) 1+ 5m 6n + 5n 6m apple a log n i.e. (s,t) near-regular Theorem: the number of bipartite graphs with degree sequence (s,t)is mn M 1 Q j n s j Qk m t k exp Pj (s j s) 2 mn (1 ) 1 Pk (t k t) 2 mn (1 ) + O(n b ) We show: mn M 1 Q j n s j Qk m t k (with probability 1 O(e n" )) random graphs 4

9 B(s,t)=#graphswithdegreesequence(s,t) G p edges chosen with P Gp (s,t) = p M q mn M B(s,t) equal probability, p. G M each graph with P GM (s,t) = mn M M edges is equally likely 1 B(s,t) G z each graph with given P Gz (s) = Q k right-side degrees equally likely m t k 1 B(s,t) random graphs 5

10 B(s,t)=#graphswithdegreesequence(s,t) G p edges chosen with P Gp (s,t) = p M q mn M B(s,t) equal probability, p. p M mn M mn q M 1 Q j n s j Qk m t k G M each graph with P GM (s,t) = mn M M edges is equally likely mn M 1 B(s,t) 2 Q j n s j Qk m t k G z each graph with given P Gz (s) = Q k right-side degrees equally likely mn M m t k 1 Q j 1 B(s,t) n s j random graphs 5

11 Introduce 3 binomial probability spaces B p, B M, B z. All defined in terms of the probability space I p. I p (s,t) : s independent binomials (n, p) t independent bimonials (m, p) random graphs 6

12 Introduce 3 binomial probability spaces B p, B M, B z. All defined in terms of the probability space I p. I p (s,t) : s independent binomials (n, p) t independent bimonials (m, p) B p P Bp (s,t):=p Ip (s,t P j s j = P k t k) B M P BM (s,t):=p Ip (s,t P j s j = P k t k = M) B z P Bz (s) :=P Ip (s,t t = z) random graphs 6

13 Main Result For any acceptable M, p or z we have: p-model P Gp (s,t) = edge-model P GM (s,t) = O(tiny) + (1 + o(1))p BM (s,t) half-model P Gz (s) = O(tiny) + (1 + o(1))p Bz (s,t) random graphs 7

14 Main Result For any acceptable M, p or z we have: p-model P Gp (s,t) = O(tiny) + (1 + o(1)) R 1 0 K p(p 0 )P Bp 0 (s, t)dp 0 where, K p (p 0 ):= mn pq 1/2 exp mn (p pq p0 ) 2 edge-model P GM (s,t) = O(tiny) + (1 + o(1))p BM (s,t) half-model P Gz (s) = O(tiny) + (1 + o(1))p Bz (s,t) random graphs 7

15 Calculations graph half -model, G z bipartite half -model, B z P Gz (s) =O(tiny) + (1 + o(1))p Bz (s,t) B(s,t):= 1 mn Y n Y m 1 exp 1 2 M j s j k t k Pj (s! j s) 2 1 mn (1 ) Pk (t! k t) 2 + O(n b ) mn (1 ) We will show E P k (s! j s) 2 =1+O(n b ) mn (1 ) w.h.p. Pk (s j s) 2 E P k (s! j s) 2 = O(n b ) mn (1 ) w.h.p...using Doob s Martingale Process random graphs 8

16 Doob s Martingale Process technique used to show that a random variable is highly concentrated about its mean random graphs 9

17 Doob s Martingale Process technique used to show that a random variable is highly concentrated about its mean Steps in Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (P i )) prove the X i s form a martingale with respect to the filter. bound the magnitude of successive di erences X i X i 1 for each 1 apple i apple n. apply Azuma-Hoe ding Theorem random graphs 9

18 Doob s Martingale Process - Example! Concentration of Chromatic Number The chromatic number of a random graph on n vertices is highly likely to be near the average chromatic number over all random graphs with n vertices. (Shamir and Spencer 1 and later Luczak 2 ) 1 E. Shamir, J. Spencer 1987: Sharp conentration of the chromatic number on random graphs, Combinatorica 7 (1) T. Luczak 1989, A note on the sharp concentration of the chromatic number of random graphs, Combinatorica 11 (3) (1991) p random graphs 10

19 Doob s Martingale Process - Example! Concentration of Chromatic Number The chromatic number of a random graph on n vertices is highly likely to be near the average chromatic number over all random graphs with n vertices. (Shamir and Spencer 1 and later Luczak 2 ) Random graph, chosen how? Take n vertices, each edge is selected independently with probability p. 1 E. Shamir, J. Spencer 1987: Sharp conentration of the chromatic number on random graphs, Combinatorica 7 (1) T. Luczak 1989, A note on the sharp concentration of the chromatic number of random graphs, Combinatorica 11 (3) (1991) p random graphs 10

20 Doob s Martingale Process - Example! Given a particular graph G 2 G(n, p) wecompare X n := (G) X 0 := E[ (H) H 2 G(n, p)] (G) toe[ (H),H 2 G(n, p)]. Partition! Vertex uncovering Define H i G if H and G agree on every edge that is incident with one of the first i vertices G 0 G G = G 1 G 3 X i := E[ (H) H i G] random graphs 11

21 Martingales Definition of a martingale is: E[X i+1 ]=X i (and finite expectation). Show the X i form a Martingale! So we must prove E E( (H) H i+1 G] H i G) = E( (G) H i G] This is equivalent to the Tower Property, i.e. E[E[A B,C] C] =E[A C] and so we have our martingale! random graphs 12

22 Doob s Martingale Process - Example! Steps in Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. random graphs 13

23 Doob s Martingale Process - Example! Steps in Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. defined vertex uncovering partitions based on the equivalence relation i. random graphs 13

24 Doob s Martingale Process - Example! Steps in Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. defined vertex uncovering partitions based on the equivalence relation i. define X i = E(X (P i )) X random graphs 13

25 Doob s Martingale Process - Example! Steps in Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. defined vertex uncovering partitions based on the equivalence relation i. define X i = E(X (P i )) X prove the X i s form a martingale with respect to the filter (tower property) X. random graphs 13

26 Doob s Martingale Process - Example! Steps in Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. defined vertex uncovering partitions based on the equivalence relation i. define X i = E(X (P i )) X prove the X i s form a martingale with respect to the filter (tower property) X. bound the magnitude of successive di erences X i X i 1 for each 1 apple i apple n. random graphs 13

27 Doob s Martingale Process - Example! Steps in Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. defined vertex uncovering partitions based on the equivalence relation i. define X i = E(X (P i )) X prove the X i s form a martingale with respect to the filter (tower property) X. bound the magnitude of successive di erences X i X i 1 for each 1 apple i apple n. X i X i 1 apple 1 X apply Azuma-Hoe ding Theorem Azuma-Hoe ding Inequality (where {X k } is a martingale s.t. X k X k 1 apple c k ). 2 P X n X 0 apple exp 2 P t. k=1 c2 k random graphs 13

28 Doob s Martingale Process - Example! Steps in Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. defined vertex uncovering partitions based on the equivalence relation i. define X i = E(X (P i )) X prove the X i s form a martingale with respect to the filter (tower property) X. bound the magnitude of successive di erences X i X i 1 for each 1 apple i apple n. X i X i 1 apple 1 X apply Azuma-Hoe ding Theorem X Azuma-Hoe ding Inequality (where {X k } is a martingale s.t. X k X k 1 apple c k ). 2 P X n X 0 apple exp 2 P t ) P (G) E( ) apple exp k=1 c2 k 2 2n random graphs 13

29 Genaralised Doob s Martingale Process Also to show a random variable is highly concentrated about its mean random graphs 14

30 Genaralised Doob s Martingale Process Also to show a random variable is highly concentrated about its mean Useful when good bounds hold only with high probability. random graphs 14

31 Genaralised Doob s Martingale Process Also to show a random variable is highly concentrated about its mean Useful when good bounds hold only with high probability. random graphs 14

32 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (F i )) prove the X i s form a martingale with respect to the filter. random graphs 15

33 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (F i )) prove the X i s form a martingale with respect to the filter. Define good and bad nodes in the partition such that: random graphs 15

34 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (F i )) prove the X i s form a martingale with respect to the filter. Define good and bad nodes in the partition such that: bound (strongly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for G in good nodes. random graphs 15

35 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (F i )) prove the X i s form a martingale with respect to the filter. Define good and bad nodes in the partition such that: bound (strongly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for G in good nodes. bound (perhaps weakly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for all G. random graphs 15

36 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (F i )) prove the X i s form a martingale with respect to the filter. Define good and bad nodes in the partition such that: bound (strongly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for G in good nodes. bound (perhaps weakly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for all G. the probability of bad nodes is small. random graphs 15

37 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (F i )) prove the X i s form a martingale with respect to the filter. Define good and bad nodes in the partition such that: bound (strongly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for G in good nodes. bound (perhaps weakly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for all G. the probability of bad nodes is small. apply Generalised Azuma-Hoe ding Theorem random graphs 15

38 Gen. Doob s Martingale Process - Statement! Given a particular graph G 2 G z we compare P j (s j n) 2 (G) to P j E (s j n) 2. X M := P j (s j(g) n) 2 X 0 := E[ P j (s j(h) n) 2,H 2 G z )] Why? random graphs 16

39 Gen. Doob s Martingale Process - Statement! Given a particular graph G 2 G z we compare P j (s j n) 2 (G) to P j E (s j n) 2. X M := P j (s j(g) n) 2 X 0 := E[ P j (s j(h) n) 2,H 2 G z )] Why? To show graph half -model, G z bipartite half -model, B z P Gz (s) =O(tiny) + (1 + o(1))p Bz (s,t) We will show (s j n) 2 E (s j n) 2 mn (1 )! = O(n b ) w.h.p...using Doob s Martingale Process random graphs 16

40 Gen. Doob s Martingale Process - Statement! Random graph, chosen how? All locally ordered graphs with blue degree sequence z are equally likely. Call this Half -model of random Bipartite Graphs, G z. 1 Defn: locally ordered The edges incident with v k are labelled 1, 2,...,t k. D 2 QQQQQQQQQQQQQQQQQQQQQ 1 2 DDDDDDDDDDDDDDDDDDDDDDDD mmmmmmmmmmmmmmmmmmmmm QQQQQQQQQQQQQQQQQQQQQ 2 Equivalent probability space to sampling over non-labelled graphs with blue degree sequence z random graphs 17

41 Gen. Doob s Martingale Process - Partition! Given graph G 2 G z we compare P j (s j(g) n) 2 to E[ P j (s j(h) n) 2,H 2 G z )]. X M := P j (s j(g) n) 2 X 0 := E[ P j (s j(h) n) 2,H 2 G z )] Edge uncovering Define H i G if H and G agree on first l edges, ordered from the right. G 1 {,,, G 2,, G =,,,,, } G 3 X i := E[ P j (s j(h) n) 2 H i G] random graphs 18

42 ,,,,,,,,,,,,,,,,,,,,,,,,,, random graphs 19

43 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Figure 3: random graphs 20

44 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, random graphs 21

45 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, random graphs 22

46 (not shown),,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, random graphs 23

47 (not included in figure),,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, random graphs 24

48 (not included in figure),,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, random graphs 26

49 (not included in figure),,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, random graphs 27

50 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (P i )) X prove the X i s form a martingale with respect to the filter. X Define good and bad nodes in the partition such that: random graphs 28

51 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (P i )) X prove the X i s form a martingale with respect to the filter. X Define good and bad nodes in the partition such that: bound (strongly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for G in good nodes. bound (perhaps weakly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for all G. the probability of bad nodes is small. apply Generalised Azuma-Hoe ding Theorem random graphs 28

52 (not included in figure) X 0 =,,,,,,,,,,,,,,,,,,,,,,,,,, 4.3 X 1 =,,,,,,,, 4.3 X 2 =,, 6.3,, 3.3,, 3.3 X 3 = random graphs 29

53 (not included in figure),,,,,,,,,,,,,,, X 0 =,,,,,,,,,,, 4.3 bad X 1 =,,,,,,,, 4.3 good X 2 =,, 6.3,, 3.3,, 3.3 X 3 = random graphs 30

54 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (P i )) X prove the X i s form a martingale with respect to the filter. X Define good and bad nodes in the partition such that: random graphs 31

55 Genaralised Doob s Martingale Process Steps in Gen. Doob s martingale process on prob space (,, P) create a filter (P 0 ) (P 1 )... (P n )viapartitions P 0,...,P n of. define X i = E(X (P i )) X prove the X i s form a martingale with respect to the filter. X Define good and bad nodes in the partition such that: bound (strongly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for G in good nodes. bound (perhaps weakly) the magnitude of X i (G) X i 1 (G) for each 1 apple i apple n for all G. the probability of bad nodes is small. apply Generalised Azuma-Hoe ding Theorem random graphs 31

56 Agraphisnear-regular if for each red vertex u j, s j s <n 1/2+3"/2. Node is toxic if P G? G not near-regular G in node ) >n 1/2. GOOD and BAD Node is bad if it is toxic or has a toxic sibling. Otherwise the node is good. random graphs 32

57 Agraphisnear-regular if for each red vertex u j, s j s <n 1/2+3"/2. Node is toxic if P G? G not near-regular G in node ) >n 1/2. GOOD and BAD Node is bad if it is toxic or has a toxic sibling. Otherwise the node is good. BOUNDS random graphs 32

58 Agraphisnear-regular if for each red vertex u j, s j s <n 1/2+3"/2. Node is toxic if P G? G not near-regular G in node ) >n 1/2. GOOD and BAD Node is bad if it is toxic or has a toxic sibling. Otherwise the node is good. BOUNDS X i (G) X i+1 (G) <n 1/2+2" for all G in good nodes. random graphs 32

59 Agraphisnear-regular if for each red vertex u j, s j s <n 1/2+3"/2. Node is toxic if P G? G not near-regular G in node ) >n 1/2. GOOD and BAD Node is bad if it is toxic or has a toxic sibling. Otherwise the node is good. BOUNDS X i (G) X i+1 (G) <n 1/2+2" for all G in good nodes. X i (G) X i+1 (G) < 2n +2forallG random graphs 32

60 Agraphisnear-regular if for each red vertex u j, s j s <n 1/2+3"/2. Node is toxic if P G? G not near-regular G in node ) >n 1/2. GOOD and BAD Node is bad if it is toxic or has a toxic sibling. Otherwise the node is good. BOUNDS X i (G) X i+1 (G) <n 1/2+2" for all G in good nodes. X i (G) X i+1 (G) < 2n +2forallG P i P G? (bad nodes in partition P i ) <e n7"/6. random graphs 32

61 Near-c-Lipschitz with an exceptional probability X P X k X k 1 c k apple k Generalised Azuma s Inequality 3 For non-negative c 1,...,c t, suppose that random variables X 0,...,X t form a near-c-lipschitz martingale. Then we have: 2 P X t X 0 apple exp 2 P t + k=1 c2 k 3 Chung and Lu, Concentration Inequalities and Martingale Inequalities, Internet Mathematics, Vol. 3, No. 1: random graphs 33

62 Near-c-Lipschitz with an exceptional probability X P X k X k 1 c k apple k Generalised Azuma s Inequality 3 For non-negative c 1,...,c t, suppose that random variables X 0,...,X t form a near-c-lipschitz martingale. Then we have: 2 P X t X 0 apple exp 2 P t + k=1 c2 k Result c k = n 1/2+2" 8k P X j (s j (G) n) 2 E[ X j (s j (G) n) 2 2 ] apple exp + o(tiny) 2n 3/2+5" 3 Chung and Lu, Concentration Inequalities and Martingale Inequalities, Internet Mathematics, Vol. 3, No. 1: random graphs 33

63 Calculations graph half -model, G z bipartite half -model, B z P Gz (s) =O(tiny) + (1 + o(1))p Bz (s,t) B(s,t):= 1 mn Y n Y m 1 exp 1 2 M j s j k t k Pj (s! j s) 2 1 mn (1 ) Pk (t! k t) 2 + O(n b ) mn (1 ) We have shown Pk (s j s) 2 E P k (s j s) 2 mn (1 )! = O(n b ) w.h.p...using Doob s Martingale Process random graphs 34

64 Main Result (again) For any acceptable M, p or z we have: p-model P Gp (s,t) = O(tiny) + (1 + o(1)) R 1 0 K p(p 0 )P Bp 0 (s, t)dp 0 where, K p (p 0 ):= mn pq 1/2 exp mn (p pq p0 ) 2 edge-model P GM (s,t) = O(tiny) + (1 + o(1))p BM (s,t) half-model P Gz (s) = O(tiny) + (1 + o(1))p Bz (s,t) random graphs 35