The Small World. problem and so .. 6 much more! 666 degrees of graph theory. Ryan Martin [email protected] Mathematics Department Iowa State University Joint Work This talk is based on joint work with Tom Bohman, Carnegie Mellon University Alan Frieze, Carnegie Mellon University Michael Krivelevich, Tel Aviv University ▽The Small World problem and so much more! – p.1/56 Joint Work This talk is based on joint work with Tom Bohman, Carnegie Mellon University Alan Frieze, Carnegie Mellon University Michael Krivelevich, Tel Aviv University André Kézdy, University of Louisville Maria Axenovich ▽The Small World problem and so much more! – p.1/56 Joint Work This talk is based on joint work with Tom Bohman, Carnegie Mellon University Alan Frieze, Carnegie Mellon University Michael Krivelevich, Tel Aviv University André Kézdy, University of Louisville Maria Axenovich, 412 Carver The Small World problem and so much more! – p.1/56 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. ▽The Small World problem and so much more! – p.2/56 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is ▽The Small World problem and so much more! – p.2/56 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 Dennis Hopper. ▽The Small World problem and so much more! – p.2/56 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 Dennis Hopper. ▽The Small World problem and so much more! – p.2/56 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. ▽The Small World problem and so much more! – p.2/56 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.) The Small World problem and so much more! – p.2/56 Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to ▽The Small World problem and so much more! – p.3/56 Bacon number # The actor’s is the fewest number of steps it takes to connect that actor to Kevin Bacon. ▽The Small World problem and so much more! – p.3/56 Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to ▽The Small World problem and so much more! – p.3/56 Bacon number An actor can have infinite Kevin Bacon number. ▽The Small World problem and so much more! – p.3/56 Bacon number # An actor can have infinite . (For example, a TV actor who appears in no movie credits.) The Small World problem and so much more! – p.3/56 Example: Kevin Costner Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). ▽The Small World problem and so much more! – p.4/56 Example: Kevin Costner is linked to because both appeared in ▽The Small World problem and so much more! – p.4/56 Example: Kevin Costner Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is ??? ▽The Small World problem and so much more! – p.4/56 Example: Kevin Costner Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is 1 ▽The Small World problem and so much more! – p.4/56 Example: Kevin Costner Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). # So, ’s is 1 The Small World problem and so much more! – p.4/56 Illustration: Kevin Costner x x ▽The Small World problem and so much more! – p.5/56 Illustration: Kevin Costner x x ▽The Small World problem and so much more! – p.5/56 Illustration: Kevin Costner x x ▽The Small World problem and so much more! – p.5/56 Illustration: Kevin Costner x x The Small World problem and so much more! – p.5/56 E.g.: Keanu “Woah!” Reeves We know that appeared with in ▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997) appeared with in ▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997) appeared with in ▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997) Al Pacino appeared with Christopher Walken in Gigli (2003) appeared with in ▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997) Al Pacino appeared with Christopher Walken in Gigli (2003) Christopher Walken appeared with Courtney Love in Basquiat (1996) appeared with in ▽The Small World problem and so much more! – p.6/56 E.g.: Keanu “Woah!” Reeves We know that Keanu Reeves appeared with Al Pacino in The Devil's Advocate (1997) Al Pacino appeared with Christopher Walken in Gigli (2003) Christopher Walken appeared with Courtney Love in Basquiat (1996) Courtney Love appeared with Kevin Bacon in Trapped (2002) The Small World problem and so much more! – p.6/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x ▽The Small World problem and so much more! – p.7/56 A chain: Keanu “Neo” Reeves x x x x x The Small World problem and so much more! – p.7/56 More: Keanu “Constantine” Reeves But it is also true that appeared with ??? in . ▽The Small World problem and so much more! – p.8/56 More: Keanu “Constantine” Reeves But it is also true that appeared with in . ▽The Small World problem and so much more! – p.8/56 More: Keanu “Constantine” Reeves But it is also true that Keanu Reeves appeared with Clint Howard in Parenthood (1989). appeared with in ▽The Small World problem and so much more! – p.8/56 More: Keanu “Constantine” Reeves But it is also true that Keanu Reeves appeared with Clint Howard in Parenthood (1989). Clint Howard appeared with Kevin Bacon in My Dog Skip (2000). The Small World problem and so much more! – p.8/56 Better: Keanu “Excellent!” Reeves x x x ▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves x x x ▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves x x x ▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves x x x ▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves x x x ▽The Small World problem and so much more! – p.9/56 Better: Keanu “Excellent!” Reeves x x x The Small World problem and so much more! – p.9/56 Can we do even better? has never appeared in a film with . ▽The Small World problem and so much more! – p.10/56 Can we do even better? Kevin Bacon has never appeared in a film with Keanu Reeves. # So, ’ is ▽The Small World problem and so much more! – p.10/56 Can we do even better? Kevin Bacon has never appeared in a film with Keanu Reeves. So, Keanu Reeves’ Kevin Bacon number is ??? ▽The Small World problem and so much more! – p.10/56 Can we do even better? Kevin Bacon has never appeared in a film with Keanu Reeves. So, Keanu Reeves’ Kevin Bacon number is 2 The Small World problem and so much more! – p.10/56 In sum: Keanu “Bogus” Reeves ((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x ▽The Small World problem and so much more! – p.11/56 In sum: Keanu “Bogus” Reeves ((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x ▽The Small World problem and so much more! – p.11/56 In sum: Keanu “Bogus” Reeves ((hh ((( hhh ((( hhh ((( hhh PP x PP x PPhh (( x hhhh(((( x x x The Small World problem and so much more! – p.11/56 Experimental data # of actors 0 1 1 1806 2 145024 3 395126 4 95497 5 7451 ▽The Small World problem and so much more! – p.12/56 Experimental data # of actors 0 1 # of actors 1 1806 6 933 2 145024 7 106 3 395126 8 13 4 95497 5 7451 The Small World problem and so much more! – p.12/56 What about the high numbers? As we said before, there are actors with infinite # .