BIRTHDAY PROBLEM, MONOCHROMATIC SUBGRAPHS & THE SECOND MOMENT PHENOMENON / 23

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1 BIRTHDAY PROBLEM, MONOCHROMATIC SUBGRAPHS & THE SECOND MOMENT PHENOMENON Somabha Mukherjee 1 University of Pennsylvania March 30, 2018 Joint work with Bhaswar B. Bhattacharya 2 and Sumit Mukherjee 3 1 Department of Statistics, University of Pennsylvania 2 Department of Statistics, University of Pennsylvania 3 Department of Statistics, Columbia University

2 Outline 1 Introduction 2 The Second Moment Phenomenon 3 A More General Result 4 Sketch of Proof 5 Erdős-Renyi Random Graphs 6 Conclusion 7 Bibliography

3 The Birthday Problem Different Types of Questions on Birthday-Matching: 1 In a room, what is the approximate number of people needed to ensure that there will be s people with the same birthday, with probability at least p? (s 2 and 0 p 1 are given) 2 More generally, in a general friendship network with a large number (n) of individuals, what is the probability that there will be s friends with the same birthday? 3 In a group of n boys and n girls, what is the probability that there is a boy-girl birthday match? A General Setup To Analyze the Above Questions: Each vertex of a graph G n is colored independently of the others, and uniformly, using one of c n = 365 colors. Question 1 asks what is the approximate value of n needed to ensure that there is a monochromatic s clique in G n = K n, with probability at least p? Question 2 asks for the probability that there will be a monochromatic s clique in a general friendship-network graph G n. Question 3 asks for the probability that there will be a monochromatic edge in the complete bipartite graph G n = K n,n.

4 Our Goal To Start With: Each vertex of a graph G n is colored independently of the others, and uniformly, using one of c n ( ) colors. Our Interest: Generalizing the questions asked in the previous slide: Investigating the limiting distributional behavior of the number of monochromatic copies of a fixed, connected graph H in G n, denoted by T (H, G n ), under suitable assumptions. Bhattacharya, Diaconis and Mukherjee [1] showed that under this independent, uniform coloring scheme, the number of monochromatic edges T (K 2, G n ) of a graph G n converges in distribution to Pois(λ), under the assumption ET (K 2, G n ) λ. Natural Question: Does ET (H, G n ) λ imply that T (H, G n ) D Pois(λ)? NO! The next slide shows a counterexample.

5 Convergence of First Moment is Not Enough! Figure: The 7-star, K 1,7 K 1,n : star-graph with n + 1 vertices, also called n star. Under the independent uniform coloring scheme, suppose that for a fixed r, lim ET (K 1,r, K 1,n ) λ. n Then, T (K 1,r, G n ) D ( ) X 1 r, where X Pois((r!λ) r ) (shown in [2]).

6 Outline 1 Introduction 2 The Second Moment Phenomenon 3 A More General Result 4 Sketch of Proof 5 Erdős-Renyi Random Graphs 6 Conclusion 7 Bibliography

7 Our Poisson Convergence Result G n : growing sequence of graphs with vertices colored independently and uniformly using c n ( ) colors. H: fixed, finite, simple, connected graph. Theorem (Bhattacharya, Mukherjee, M.) If G n and H are as above, and λ > 0, then lim ET (H, Gn) = λ and lim VarT (H, Gn) = λ = T (H, Gn) D Pois(λ). n n Further, the converse is true if and only if H is a star-graph. In fact, if H is not a star-graph, then for every λ > 0, a sequence of graphs G n(h) and a sequence c n, such that T (H, G n(h)) D Pois(λ) but ET (H, G n(h)) λ. The above theorem characterizes the second moment phenomenon for the number of monochromatic subgraphs in a uniform random coloring of a graph sequence.

8 Why is the Converse not True for Non-Stars? Figure: The graph G 5(C 4) P n (H): pyramid of height n formed by merging n copies of H on V (H) 1 vertices (called base vertices), corresponding to the isomorphisms. G n (H) = P n (H) plus n disjoint copies of H. Take c n = n 1/( V (H) 1). Any copy of H in P n (H) passes through at least 2 base vertices of P n (H). So, ( ) V (H) 1 1 P(T (H, P n (H)) > 0) 0, 2 c n i.e. T (H, P n (H)) P 0. Hence, T (H, G n (H)) D Pois(1). However, lim inf ET (H, G n (H)) = lim inf ET (H, P n (H)) for every n.

9 Outline 1 Introduction 2 The Second Moment Phenomenon 3 A More General Result 4 Sketch of Proof 5 Erdős-Renyi Random Graphs 6 Conclusion 7 Bibliography

10 Notations Required for Stating the General Result J t (H): (finite) set of all non-isomorphic graphs obtained by merging two copies of H in exactly t vertices (1 t V (H) ). For H = C 4, the 4 cycle, the sets J 2 (H) and J 4 (H) are illustrated below: Figure: Graphs in the set J 2(C 4) Figure: Graphs in the set J 4(C 4)

11 Our Poisson-Linear-Combination Result The following is the most general result proved by us, from which the first theorem follows as a corollary: Theorem (Bhattacharya, Mukherjee, M.) Let G n be a sequence of graphs colored uniformly with c n ( ) colors, such that: For every k [1, N(H, K V (H) )], there exists λ k 0 such that N ind (F, G n) Then lim n F H: V (F ) = V (H), N(H,F )=k V (H) 1 c n = λ k, 2 V (H) t 1 For t [2, V (H) 1] and every F J t(h), N(F, G n) = o(cn ). T (H, G n) D N(H,K V (H) ) k=1 kx k, where X k Pois(λ k ) and the collection {X k : 1 k N(H, K V (H) )} is independent.

12 Outline 1 Introduction 2 The Second Moment Phenomenon 3 A More General Result 4 Sketch of Proof 5 Erdős-Renyi Random Graphs 6 Conclusion 7 Bibliography

13 Outline of Proof of the General Theorem We followed a method of moments ( approach, but not directly. In fact, it may not r N(H,K V (H) ) be true that ET (H, G n ) r E kpois(λ k )) for every r. What we did: k=1 1 Decomposed the random variable T (H, G n) as a sum of two quantities: T + (H, G n) and T (H, G n). 2 T + (H, G n) was defined by a truncation on T (H, G n), and T (H, G n) := T (H, G n) T + (H, G n). 3 Showed that T (H, G n) converges ( in L 1 to 0. r N(H,K V (H) ) 4 Showed that ET + (H, G n) r E kpois(λ k )) for every r. k=1 In the next few slides, we sketch the proof for H = K 3. The general proof exploits an idea essentially similar to the proof for K 3, but some steps are more involved, and hard to present.

14 Proof of the Poisson Convergence Result for Triangles N(H, G n ) : Number of copies of H in G n. ET (K 3, G n ) λ = N(K3,Gn) c λ. n 2 ( 1 1 cn 2 ) ) + 2N(D, G n ) 1 c (1 1cn, n 3 VarT (K 3, G n ) = N(K 3, G n ) 1 cn 2 where D denotes the diamond, i.e. C 4 with one diagonal: Figure: The Diamond D VarT (K 3, G n ) λ and ET (K 3, G n ) λ = N(D, G n ) = o(cn). 3 For each (u, v) E(G n ), define T (u, v) to be the number of vertices in V (G n ) \ {u, v} that are adjacent to both u and v, i.e. T (u, v) := A uw (G n )A vw (G n ). w V (G n) We call T (u, v) the co-degree of u and v.

15 Remainder Term Goes to 0 in L 1 X v : Color of the vertex v V (G n ). T ɛ + (K 3, G n ) (ɛ > 0) := A uv A vw A uw 1 (T (u, v), T (v, w), T (u, w) ɛc n ) 1(X u = X v = X w ). u,v,w V (G n) ETɛ (K 3, G n ) = A uv A vw A uw 1 (T (u, v) > ɛc n or T (v, w) > ɛc n or T (u, w) > ɛc n ). 1 c 2 n 1 c 2 n = u,v,w V (G n) u,v,w V (G n) u,v V (G n) u,v V (G n) A uv A vw A uw 1 (T (u, v) > ɛc n ) A uv T (u,v) 2 ɛc 3 n A uv 2( T (u,v) 2 )+T (u,v) ɛc 3 n C 2N(D,Gn)+N(K3,Gn) ɛc 3 n = o(1), as n, for a fixed ɛ > 0.

16 Main Term Goes to Pois(λ) in Distribution This is the more involved part of the proof, at the heart of which lies subgraph counting arguments. We sketch the main idea below. Define ( a) collection {Z uvw : u, v, w V (G n ), distinct} of independent 1 Ber c random variables. n 2 W ɛ (G n ) := A uv A vw A uw 1 (T (u, v), T (v, w), T (u, w) ɛc n ) Z uvw. u,v,w V (G n) What did we show? ET + ɛ (H, G n ) r EW ɛ (G n ) r Cɛ(1 c α n ) for some constants C and α > 0, not depending on n or ɛ. Hence, lim lim ET + ɛ (H, G n ) r EW ɛ (G n ) r = 0. ɛ 0 n W ɛ (G n ) converges in distribution and in all moments to Pois(λ), since it has a Binomial distribution with mean converging to λ.

17 Outline 1 Introduction 2 The Second Moment Phenomenon 3 A More General Result 4 Sketch of Proof 5 Erdős-Renyi Random Graphs 6 Conclusion 7 Bibliography

18 Results for Erdős-Rényi Random Graphs Our results can be applied to Erdős-Rényi random graphs as follows. V (H) If the fixed graph H is balanced, define λ(h) = If H is unbalanced, define λ(h) = Theorem min H 1 H:α(H 1)>0 E(H). V (H) V (H 1) α(h 1) α(h 1) := E(H 1) ( V (H) 1) + E(H) ( V (H 1) 1)., where Let H be a simple connected graph, and G n G(n, p(n)) be the Erdős-Rényi random graph with p(n) (0, 1) colored with c n( ) colors, such that ET (H, G n) λ. If p(n) 0 and p(n) << n λ(h), then T (H, G n) P 0. If p(n) 0 and p(n) >> n λ(h), then T (H, G n) D Pois(λ). If p(n) = p (0, 1) is fixed, the T (H, G n) D F H: V (F ) = V (H) N(H, F )X F, ( ) where X F Pois λ Aut(H) Aut(F ) p E(F ) E(H) (H) ( V (1 p) 2 ) E(F ), independent.

19 Outline 1 Introduction 2 The Second Moment Phenomenon 3 A More General Result 4 Sketch of Proof 5 Erdős-Renyi Random Graphs 6 Conclusion 7 Bibliography

20 Summary of What We Showed: If for a connected graph H and a growing sequence of graphs G n, we have: E(T (H, G n )) λ and Var(T (H, G n )) λ as n, then T (H, G n ) d Pois(λ) (generalizing a previous result of Bhattacharya, Diaconis, Mukherjee). In fact, the above result is just a corollary of a more general Poisson linear combination convergence theorem proved by us. Weak convergence of T (H, G n ) to Pois(λ) implies conergence of the first two moments to the corresponding moments of Pois(λ) if and only if H is a star-graph. As an application, we derive the limiting distribution of T (H, G n ) when G n G(n, p). Multiple phase transitions arise as p varies from 0 to 1, depending on whether the graph is balanced or not.

21 Some Questions: What can be said about the limiting distribution of color-1 edges in the uniform coloring setup? We are in the process of characterizing this limiting distribution under some additional assumptions on the graph G n. Does the number of monochromatic / color-1 copies of connected subgraphs satisfy some large deviation principle? We have a positive answer for the case of color-1 subgraphs, although the large deviation variational problem is yet to be solved. Does the number of monochromatic copies of a connected subgraph satisfy a Central Limit Theorem under the only assumption that its expected value goes to? We have a negative answer for all non-star graphs, but do not yet know what happens for star-graphs.

22 Outline 1 Introduction 2 The Second Moment Phenomenon 3 A More General Result 4 Sketch of Proof 5 Erdős-Renyi Random Graphs 6 Conclusion 7 Bibliography

23 Reference Bhattacharya, B. B., Diaconis, P., Mukherjee, S., Universal Poisson and Normal limit theorems in graph coloring problems with connections to extremal combinatorics, Annals of Applied Probability, Vol. 27 (1), , Bhattacharya, B. B., Mukherjee, S., Limit Theorems for Monochromatic 2-Stars and Triangles arxiv: v1, 2017.