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The block-cutpoint tree characterization of a covering polynomial of a graph Robert Ellis (IIT) James Ferry, Darren Lo (Metron, Inc.) Dhruv Mubayi (UIC) AMS Fall Central Section Meeting November 6, 2010 Random Intersection Model B*(n,m,p) • Introduced: Karoński, Scheinerman, SingerCohen `99 • Bipartite graph models collaboration – Activity nodes – Participant nodes • Random Intersection Graph B*(n,m,p) – Bipartite edges arise independently with constant probability – Unipartite projection onto participant nodes m: number of “movies” Bipartite graph n: number of “actors” Unipartite projection Collaboration graph: who’s worked with whom Slide 2 Expected sugbraph count vs. E-R E[XH] Erdős–Rényi n = 1000 pER = 0.002 Subgraph H Random Intersection n = 1000 m = 100 p = 0.0045 Yields pER = 0.002 60-cycle Erdős–Rényi Random Intersection 1.3 1500 11 4 ´ 105 2 ´ 1015 2 ´ 1038 1.2 2 ´ 1010 3 ´ 10- 6 1700 E [X H (G( N , pER )]£ E éêëX H (B ( N , M , p)ù ú û * for p = 1- (1- pER )1 m Slide 3 Erdős–Rényi vs. RI Model as m → ∞ • m = “number of movies” • pER = 0.028 (edge probability) Erdős–Rényi G(n,pER) model 1) RC 10,000) 100) 10) 1000) RI model (m = 2) B*(N,M,p) Slide 4 Erdős–Rényi vs. RI Model as m → ∞ • Theorem [Ferry, Mifflin]. For a fixed expected number of edges pER n2 , and any graph G with n vertices, the probability of G being generated by the Random Intersection model approaches the probability of G being generated by the Erdős–Rényi model as m → ∞. • Formula for rate of convergence: 32 æ ö æ ö æ ö n PrRC G ( ) RI - 1/ 2 - 1 ÷ (n - 2)e (G ) 1 + c (G ) 1 ÷ ççlog 1 ÷ ÷ ÷ = 1 + ççç2 çç ÷m + O m ( ) 3 ÷ 3 ÷ç ÷ ÷ ÷ ÷ç çè çè3ø PrER (G ) pER pER 1 p øè ER ø • [(Independently) Fill, Scheinerman, Singer-Cohen `00] With m=nα, α>6, total variation distance for probability of G goes to zero as n → ∞. Slide 5 RI model in the constant-μ limit Idea: Let m → ∞ and fix the expected number of movies per actor at constant μ=pm. This allows simplified asymptotic probabilities for random intersection graphs on a fixed number of nodes. • Probability formulas are from edge clique covers • Most probable graphs have block-complete structure • Least probable graphs have connections to Turán-type extremal graphs Slide 6 Edge clique covers • Unipartite projection corresponds to an edge clique cover • The projection-induced cover encodes collaboration structure B B* Unipartite projection Hidden collaboration perspective: • Given B*(n,m,p)=G, we can infer which clique covers are most likely • This reveals the most likely hidden collaboration structure that produced G G B* likeliest B Slide 7 Covering polynomial of G s ( ) X X “size” of clique cover S wt (G ) min wt ( ) “weight” of S, G wt ( ) s ( ) s(G; x ) ai x i ai = #least-wt covers of size i i Projection wt=4, s=6 wt=4, s=7 (2 ways) wt=5 (not least-weight) Thus wt(G) = 4 and s(G; x) = x6 + 2x7. Slide 8 Fixed graphs in the constant-μ limit Theorem [Lo]. Let n be fixed and let G be a fixed graph on n vertices. In the constant-m limit, s (G ; ) Pr (n, m, p ) G wt( G ) 1 O m 1 m * • Lower weight graphs are more likely • If G has a lower weight supergraph H, G is more likely to appear as a subgraph of H than as an induced graph Theorem. Let n be fixed and let G be a fixed graph on n non-isolated vertices with j cut-vertices. In the constant-m limit, Pr ( n, m, p ) G * n j m j i 1 bi rank ( G ) 1 O m 1 where the bi are block degrees of the cut-vertices of G, and f b ( x) = is the bth Touchard polynomial. å ïìï bïü í ïý i= 0 ïïîï i ïïþ ï b Slide 9 xi Example subgraph probability Let H be rank(H) = 7 n(H)=8 b1= 3 2 cut-vertices; 4 blocks b2= 2 Block-cutpoint tree of H 3 s 1 2 1 3 s 1 s 3 Pr (8, m, p) H * 1 Stirling numbers bi count partitions s of bi blocks into s “movies” 6 1 3 2 1 m7 1 Om 1 Slide 10 Block-cutpoint tree → Least-weight supergraphs 1. Select an unvisited cut-vertex. 2. Partition incident blocks, merge, and make block-complete. 3. Update block-cutpoint tree. 4. Repeat 1 until all original cut-vertices are visited. Slide 11 An extremal graph weight conjecture Conjecture [Lo]. Let G have n vertices. Then wt (G ) n 2 4 with equality iff there exists a bipartition V(G)= A B such that: •A= n 2 •B= n 2 •The complete (A,B)-bipartite graph is a subgraph of G •Either A or B is an independent set. Slide 12 Related simpler questions Conjecture. Every K4-free graph G on n vertices and 2 n 4 m edges has at least m edge-disjoint K3’s. Theorem [Győri]. True for G with chromatic number at most 3. Theorem. True when G is K4-free and 2 n 4 m t (n,3) k , where k≤n2/84+O(1). Slide 13