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Adding Random Edges to Dense Graphs

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Adding Random Edges to Dense Graphs Adding random edges to dense graphs Ryan Martin [email protected] Zeev Nehari Visiting Assistant Professor Mathematics Department Carnegie Mellon University Adding random edges to dense graphs – p.1/46 Joint Work This talk is based on joint work with Tom Bohman, Carnegie Mellon University Alan Frieze, Carnegie Mellon University Michael Krivilevich, Tel Aviv University Adding random edges to dense graphs – p.2/46 We link two actors together if they appeared together in the same movie. (They must be together on a cast list at the IMDb.) Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. Adding random edges to dense graphs – p.3/46 We link two actors together if they appeared together in the same movie. (They must be together on a cast list at the IMDb.) Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is . Adding random edges to dense graphs – p.3/46 We link two actors together if they appeared together in the same movie. (They must be together on a cast list at the IMDb.) Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. This is false. It is Christopher Lee. Adding random edges to dense graphs – p.3/46 (They must be together on a cast list at the IMDb.) Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. We link two actors together if they appeared together in the same movie. Adding random edges to dense graphs – p.3/46 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. We link two actors together if they appeared together in the same movie. (They must be together on a cast list at the IMDb.) Adding random edges to dense graphs – p.3/46 Bacon number Kevin Bacon. An actor can have infinite (For example, a TV actor who appears in no movie credits.) Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to Kevin Bacon. Adding random edges to dense graphs – p.4/46 Bacon number Kevin Bacon. An actor can have infinite (For example, a TV actor who appears in no movie credits.) Bacon number The actor’s # is the fewest number of steps it takes to connect that actor to . Adding random edges to dense graphs – p.4/46 (For example, a TV actor who appears in no movie credits.) Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to Kevin Bacon. An actor can have infinite Kevin Bacon number. Adding random edges to dense graphs – p.4/46 Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to Kevin Bacon. # An actor can have infinite . (For example, a TV actor who appears in no movie credits.) Adding random edges to dense graphs – p.4/46 For example, Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is Example: Kevin Costner Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). Adding random edges to dense graphs – p.5/46 For example, Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is Example: Kevin Costner is linked to because both appeared in Adding random edges to dense graphs – p.5/46 is linked to because both appeared in Example: Kevin Costner For example, Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is ??? Adding random edges to dense graphs – p.5/46 is linked to because both appeared in Example: Kevin Costner For example, Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is 1 Adding random edges to dense graphs – p.5/46 is linked to because both appeared in Example: Kevin Costner For example, Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). # So, ’s is 1 Adding random edges to dense graphs – p.5/46 Illustration: Kevin Costner Adding random edges to dense graphs – p.6/46 Illustration: Kevin Costner Adding random edges to dense graphs – p.6/46 Illustration: Kevin Costner Adding random edges to dense graphs – p.6/46 Illustration: Kevin Costner Adding random edges to dense graphs – p.6/46 appeared with in appeared with in appeared with in Example: Henry “Fonz” Winkler We know that appeared with in Adding random edges to dense graphs – p.7/46 appeared with in appeared with in Example: Henry “Fonz” Winkler We know that Henry Winkler appeared with Michael Keaton in Night Shift (1982) appeared with in Adding random edges to dense graphs – p.7/46 appeared with in Example: Henry “Fonz” Winkler We know that Henry Winkler appeared with Michael Keaton in Night Shift (1982) Michael Keaton appeared with Kim Basinger in Batman (1989) appeared with in Adding random edges to dense graphs – p.7/46 Example: Henry “Fonz” Winkler We know that Henry Winkler appeared with Michael Keaton in Night Shift (1982) Michael Keaton appeared with Kim Basinger in Batman (1989) Kim Basinger appeared with Mickey Rourke in 9 1/2 Weeks (1986) appeared with in Adding random edges to dense graphs – p.7/46 Example: Henry “Fonz” Winkler We know that Henry Winkler appeared with Michael Keaton in Night Shift (1982) Michael Keaton appeared with Kim Basinger in Batman (1989) Kim Basinger appeared with Mickey Rourke in 9 1/2 Weeks (1986) Mickey Rourke appeared with Kevin Bacon in Diner (1982) Adding random edges to dense graphs – p.7/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 Illustration: Henry “Fonz” Winkler Adding random edges to dense graphs – p.8/46 appeared with in More with Henry “Fonz” Winkler But it is also true that appeared with in Adding random edges to dense graphs – p.9/46 More with Henry “Fonz” Winkler But it is also true that Henry Winkler appeared with Clint Howard in Little Nicky (2000) appeared with in Adding random edges to dense graphs – p.9/46 More with Henry “Fonz” Winkler But it is also true that Henry Winkler appeared with Clint Howard in Little Nicky (2000) Clint Howard appeared with Kevin Bacon in My Dog Skip (2000) Adding random edges to dense graphs – p.9/46 Even more: Henry “Fonz” Winkler Adding random edges to dense graphs – p.10/46 Even more: Henry “Fonz” Winkler Adding random edges to dense graphs – p.10/46 Even more: Henry “Fonz” Winkler Adding random edges to dense graphs – p.10/46 Even more: Henry “Fonz” Winkler Adding random edges to dense graphs – p.10/46 Even more: Henry “Fonz” Winkler Adding random edges to dense graphs – p.10/46 Even more: Henry “Fonz” Winkler Adding random edges to dense graphs – p.10/46 So, ’s is Can we do better? has never appeared in a film with Adding random edges to dense graphs – p.11/46 Can we do better? Kevin Bacon has never appeared in a film with Henry Winkler # So, ’s is Adding random edges to dense graphs – p.11/46 Can we do better? Kevin Bacon has never appeared in a film with Henry Winkler So, Henry Winkler’s Kevin Bacon number is ??? Adding random edges to dense graphs – p.11/46 Can we do better? Kevin Bacon has never appeared in a film with Henry Winkler So, Henry Winkler’s Kevin Bacon number is 2 Adding random edges to dense graphs – p.11/46 .12/46 p – aphs r g dense to edges ¤ andom r ¤ ¡ inkler ¤ ¡ Adding ¤ ¡ W ¤ ¡ ¤ ¡ £ ¡ £ ¡ £ ¡ onz” £ ¡ £ £ ¡ “F £ ¡ £ ¡ ¡ £ ¡ £ ¡ £ ¡ £ ¢¡ Henry £ £ £ of £ £ Diagram .12/46 p – aphs r g dense to edges ¤ andom r ¤ ¡ inkler ¤ ¡ Adding ¤ ¡ W ¤ ¡ ¤ ¡ £ ¡ £ ¡ £ ¡ onz” £ ¡ £ £ ¡ “F £ ¡ £ ¡ ¡ £ ¡ £ ¡ £ ¡ £ ¢¡ Henry £ £ £ of £ £ Diagram .12/46 p – aphs r g dense to edges ¤ andom r ¤ ¡ inkler ¤ ¡ Adding ¤ ¡ W ¤ ¡ ¤ ¡ £ ¡ £ ¡ £ ¡ onz” £ ¡ £ £ ¡ “F £ ¡ £ ¡ ¡ £ ¡ £ ¡ £ ¡ £ ¢¡ Henry £ £ £ of £ £ Diagram # of actors 7 134 8 24 9 1 10 1 Experimental data # of actors 0 1 1 1614 2 122988 3 316970 4 73111 5 5553 6 729 Adding random edges to dense graphs – p.13/46 Experimental data # of actors 0 1 # of actors 1 1614 7 134 2 122988 8 24 3 316970 9 1 4 73111 10 1 5 5553 6 729 Adding random edges to dense graphs – p.13/46 # The actors with large are obscure and the reason why is fairly obvious. What about those high numbers? As we said before, there are actors with infinite # . Adding random edges to dense graphs – p.14/46 What about those high numbers? As we said before, there are actors with infinite # .
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