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Six degrees of graph theory: ˝ Kevin Bacon, Paul Erdos, William McKinley and me Ryan Martin rymartin@iastate.edu Assistant Professor Mathematics Department Iowa State University o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 1/4 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 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 2/4 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is . o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4 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 Rod Steiger. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4 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 Rod Steiger. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4 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. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4 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.) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 3/4 Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to Kevin Bacon. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4 Bacon number The actor’s # is the fewest number of steps it takes to connect that actor to . o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4 Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to Kevin Bacon. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4 Bacon number An actor can have inﬁnite # . o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4 Bacon number An actor can have inﬁnite # . (For example, a TV actor who appears in no movie credits.) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 4/4 Example: Kevin Costner Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4 Example: Kevin Costner is linked to because both appeared in o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4 Example: Kevin Costner is linked to because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is ??? o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4 Example: Kevin Costner is linked to because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is 1 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4 Example: Kevin Costner is linked to because both appeared in JFK (1991). # So, ’s is 1 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 5/4 Illustration: Kevin Costner x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 6/4 Illustration: Kevin Costner x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 6/4 Illustration: Kevin Costner x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 6/4 Illustration: Kevin Costner x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 6/4 E.g.: Henry “Fonz” Winkler We know that • appeared with in o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4 E.g.: Henry “Fonz” Winkler We know that • Henry Winkler appeared with Michael Keaton in Night Shift (1982) • appeared with in o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4 E.g.: 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 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4 E.g.: 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 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4 E.g.: 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) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 7/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 A chain: Henry “Fonz” Winkler x x x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 8/4 More: Henry “Fonz” Winkler But it is also true that • appeared with ??? in (2000). o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 9/4 More: Henry “Fonz” Winkler But it is also true that • appeared with in (2000). o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 9/4 More: Henry “Fonz” Winkler But it is also true that • Henry Winkler appeared with Clint Howard in Little Nicky (2000). • appeared with in (2000) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 9/4 More: 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). o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 9/4 Better: Henry “Fonz” Winkler x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4 Better: Henry “Fonz” Winkler x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4 Better: Henry “Fonz” Winkler x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4 Better: Henry “Fonz” Winkler x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4 Better: Henry “Fonz” Winkler x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4 Better: Henry “Fonz” Winkler x x x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 10/4 Can we do even better? has never appeared in a ﬁlm with . o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 11/4 Can we do even better? Kevin Bacon has never appeared in a ﬁlm with Henry Winkler. # So, ’s is o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 11/4 Can we do even better? Kevin Bacon has never appeared in a ﬁlm with Henry Winkler. So, Henry Winkler’s Kevin Bacon number is ??? o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 11/4 Can we do even better? Kevin Bacon has never appeared in a ﬁlm with Henry Winkler. So, Henry Winkler’s Kevin Bacon number is 2 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 11/4 In sum: Henry “Fonz” Winkler x @@@@hhhhhh x hx @@ @@ @ @@ h hhh xh x hh h x @@@ h @ h@@ o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 12/4 In sum: Henry “Fonz” Winkler x @@@@hhhhhh x hx @@ @@ @ @@ h hhh xh x hh h x @@@ h @ h@@ o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 12/4 In sum: Henry “Fonz” Winkler x @@@@hhhhhh x hx @@ @@ @ @@ h hhh xh x hh h x @@@ h @ h@@ o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 12/4 Experimental data # of actors 0 1 1 1806 2 145024 3 395126 4 95497 5 7451 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 13/4 Experimental data # of actors 0 1 # of actors 1 1806 6 933 2 145024 7 106 3 395126 8 13 4 95497 5 7451 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 13/4 What about the high numbers? As we said before, there are actors with inﬁnite # . o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 14/4 What about the high numbers? As we said before, there are actors with inﬁnite # . # The actors with large are obscure and the reason why is fairly obvious. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 14/4 Kevin Bacon not so special Most successful actors follow the same pattern: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 15/4 Kevin Bacon not so special Most successful actors follow the same pattern: For every pair of successful actors, they are connected by a path of length ≤ 5 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 15/4 Kevin Bacon not so special Most successful actors follow the same pattern: For every pair of successful actors, they are connected by a path of length ≤ 5 Why? o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 15/4 Our model We will represent actors by vertices o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4 Our model We will represent actors by vertices x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4 Our model We will represent actors by vertices x and connect them with edges o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4 Our model We will represent actors by vertices x and connect them with edges x x if they appeared in the same ﬁlm. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4 Our model We will represent actors by vertices x and connect them with edges x x if they appeared in the same ﬁlm. This is a graph. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 16/4 Model parameters • There are n actors. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4 Model parameters • There are n actors. • Fix a constant d. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4 Model parameters • There are n actors. • Fix a constant d. • We will begin with an arbitrary graph H. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4 Model parameters • There are n actors. • Fix a constant d. • We will begin with an arbitrary graph H. • In H, each actor is connected to at least dn other actors. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4 Model parameters • There are n actors. • Fix a constant d. • We will begin with an arbitrary graph H. • In H, each actor is connected to at least dn other actors. • The constant d can be extremely tiny: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4 Model parameters • There are n actors. • Fix a constant d. • We will begin with an arbitrary graph H. • In H, each actor is connected to at least dn other actors. • The constant d can be extremely tiny: 0.1 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4 Model parameters • There are n actors. • Fix a constant d. • We will begin with an arbitrary graph H. • In H, each actor is connected to at least dn other actors. • The constant d can be extremely tiny: 0.1, 0.01 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4 Model parameters • There are n actors. • Fix a constant d. • We will begin with an arbitrary graph H. • In H, each actor is connected to at least dn other actors. • The constant d can be extremely tiny: 0.1, 0.01, 0.000001 It just needs to be independent of n. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 17/4 Random casting We add f (n) random casting connections. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 18/4 Random casting We add f (n) random casting connections. What does random mean? o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 18/4 Random edges Let N be the number of pairs with no connection between them (non-edges). We can create m new random edges in two ways: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4 Random edges Let N be the number of pairs with no connection between them (non-edges). We can create m new random edges in two ways: • For every set of m unconnected pairs, choose one set uniformly at random. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4 Random edges Let N be the number of pairs with no connection between them (non-edges). We can create m new random edges in two ways: • For every set of m unconnected pairs, choose one set uniformly at random. • Connect a previously unconnected pair, independently, with probability m/N . o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4 Random edges Let N be the number of pairs with no connection between them (non-edges). We can create m new random edges in two ways: • For every set of m unconnected pairs, choose one set uniformly at random. • Connect a previously unconnected pair, independently, with probability m/N (coin ﬂips). o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4 Random edges Let N be the number of pairs with no connection between them (non-edges). We can create m new random edges in two ways: • For every set of m unconnected pairs, choose one set uniformly at random. • Connect a previously unconnected pair, independently, with probability m/N (coin ﬂips). The average number of new connections is m. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 19/4 The models For our purposes, these produce the same results. The question: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 20/4 The models For our purposes, these produce the same results. The question: What is the longest distance between any pair of actors ? o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 20/4 The models For our purposes, these produce the same results. The question: What is the longest distance between any pair of actors (diameter)? o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 20/4 The theorem Theorem. If f (n) → ∞ as n → ∞, then Pr (diam ≤ 5) → 1 Important points: • Recall f (n) is the number of random connections. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4 The theorem Theorem. If f (n) → ∞ as n → ∞, then Pr (diam ≤ 5) → 1 Important points: • Recall f (n) is the number of random connections. • “5” doesn’t depend on d at all. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4 The theorem Theorem. If f (n) → ∞ as n → ∞, then Pr (diam ≤ 5) → 1 Important points: • Recall f (n) is the number of random connections. • “5” doesn’t depend on d at all. • f (n) can be very small: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4 The theorem Theorem. If f (n) → ∞ as n → ∞, then Pr (diam ≤ 5) → 1 Important points: • Recall f (n) is the number of random connections. • “5” doesn’t depend on d at all. • f (n) can be very small: √ n o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4 The theorem Theorem. If f (n) → ∞ as n → ∞, then Pr (diam ≤ 5) → 1 Important points: • Recall f (n) is the number of random connections. • “5” doesn’t depend on d at all. • f (n) can be very small: √ n, log n o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4 The theorem Theorem. If f (n) → ∞ as n → ∞, then Pr (diam ≤ 5) → 1 Important points: • Recall f (n) is the number of random connections. • “5” doesn’t depend on d at all. • f (n) can be very small: √ √ n, log n, log log log n o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 21/4 Qualifying the model This is a nice result on a pretty good model. The model only assumes some density conditions and a little bit of randomness. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 22/4 The proof To prove the theorem, you need the Regularity Lemma. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 23/4 The proof To prove the theorem, you need the Regularity Lemma. The Regularity Lemma is ’s powerful and complicated graph theoretic tool. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 23/4 The proof To prove the theorem, you need the Regularity Lemma. The Regularity Lemma is Endre Szemerédi’s powerful and complicated graph theoretic tool. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 23/4 Best possible? The theorem is “tight”: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 24/4 Best possible? The theorem is “tight”: If there aren’t an inﬁnite number of edges added, then some H’s will be disconnected. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 24/4 What about closer connections? • To get diam ≤ 4, you need random connections. • To get diam ≤ 3, you need random connections. • To get diam ≤ 2, you need random connections. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 25/4 What about closer connections? • To get diam ≤ 4, you need c1 log n random connections. • To get diam ≤ 3, you need c1 log n random connections. • To get diam ≤ 2, you need random connections. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 25/4 What about closer connections? • To get diam ≤ 4, you need c1 log n random connections. • To get diam ≤ 3, you need c1 log n random connections. • To get diam ≤ 2, you need c2 n log n random connections. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 25/4 Applying this knowledge To have it be very likely that everyone is connected by a path of no more than 5 acquaintances, just arrange a few random meetings. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 26/4 Applying this knowledge To have it be very likely that everyone is connected by a path of no more than 5 acquaintances, just arrange a few random meetings. Think about people at Central College. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 26/4 Central College cliques ' $ xhhx 2x h t x x 22 e ¥e t ¡¡ e 222 x 2 t ¥ e x x i ¡ & x hx ¤¤ l ¡ t e ¥ $ hh l i e e $ l t g x & x¤ ¤ v i¡ t l g e & l x 3 3f ¤ 3 v &@ x 3 x f¤ v x @ @3 l t t @ e & ¨ x e ¨¨i e i ¨ eix & % Mathematicians o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 27/4 Central College cliques ' ' $ $ xhhx x 3x x h x t 2 3 d x 222 ¥e x x 2 h h @x 2 e 3 h 44 t ¡¡ e 3 x x 2 ¢ t hd @h ¢ x 4 @ x ¥ e 4 x i ¡ & x hx e ¤¤ l ¡ t e ¥ $ hh r li r e ¥ e @ x rr x e $ r e ¥ l t g x i¡ & x¤ x ¤ v l t g ¤ @ @e @ e¥ e & l x 3 33 f e e v x &@ x l 3 x f¤ e x l g hhx v l v x@ e@ @ h @ @h ¨ e @3 ¨ t t v & @ lt e v x¨ x v x$ e ¨¨ i $ g¨ x e x 2 lt e x 2 e i ¨ eix 2 t & & % % Mathematicians Social Scientists o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 27/4 More Central College cliques ' $ x xx x x x x But for us to get 4 xh x 4 diameter ≤ 5, we 4 4 h h h x do need each per- x xx x son to know at x 22 x least dn others be- x2 fore we add few & % random edges. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 28/4 More Central College cliques ' $ x xx x x x x But for us to get 4 xh x 4 diameter ≤ 5, we 4 4 h h h x do need each per- x x x x son to know at x 22 x least dn others be- x2 fore we add few & % random edges. Computer Scientists o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 28/4 The cliché The cliché states that every pair of people is separated by at most six degrees of separation. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 29/4 The cliché The cliché states that every pair of people is separated by at most six degrees of separation. In fact, it is FIVE degrees of separation and there’s an actual proof! o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 29/4 ˝ Erdos number One of the most proliﬁc mathematicians of the 20th century was o Paul (Pál) Erd˝ s March 26, 1913-September 20, 1996 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 30/4 ˝ Erdos number One of the most proliﬁc mathematicians of the 20th century was o Paul (Pál) Erd˝ s March 26, 1913-September 20, 1996 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 30/4 ˝ Erdos number project The Erd˝ s number project is concerned with the o o distance of mathematicians from Paul Erd˝ s. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 31/4 ˝ Erdos number project # The project is concerned with the o distance of mathematicians from Paul Erd˝ s. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 31/4 ˝ Erdos number project # The project is concerned with the o distance of mathematicians from Paul Erd˝ s. Two mathematicians are connected if they co-authored a paper together and that paper appears in Mathematical Reviews, accessible by MathSciNet. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 31/4 Most proliﬁc authors • : 1401 papers (Erd˝ s number ) o o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4 Most proliﬁc authors • : 1401 papers (Erd˝ s number ) o • Drumi Bainov: 782 (Erd˝ s number ) o o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4 Most proliﬁc authors • : 1401 papers (Erd˝ s number ) o • Drumi Bainov: 782 (Erd˝ s number ) o • Leonard Carlitz: 730 (Erd˝ s number ) o o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4 Most proliﬁc authors • : 1401 papers (Erd˝ s number ) o • Drumi Bainov: 782 (Erd˝ s number ) o • Leonard Carlitz: 730 (Erd˝ s number ) o • Lucien Godeaux: 644 (Erd˝ s number o ) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4 Most proliﬁc authors • : 1401 papers (Erd˝ s number ) o • Drumi Bainov: 782 (Erd˝ s number ) o • Leonard Carlitz: 730 (Erd˝ s number ) o • Lucien Godeaux: 644 (Erd˝ s number o ) • Saharon Shelah: 600 (Erd˝ s number ) o o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4 Most proliﬁc authors • : 1401 papers (Erd˝ s number 0) o • Drumi Bainov: 782 (Erd˝ s number 4) o • Leonard Carlitz: 730 (Erd˝ s number 2) o • Lucien Godeaux: 644 (Erd˝ s number ∞) o • Saharon Shelah: 600 (Erd˝ s number 1) o o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 32/4 ˝ Erdos number statistics • wrote 1401 papers in Math Reviews. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4 ˝ Erdos number statistics • wrote 1401 papers in Math Reviews. • There are 337, 000 vertices (authors) in the graph. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4 ˝ Erdos number statistics • wrote 1401 papers in Math Reviews. • There are 337, 000 vertices (authors) in the graph. • There are about 496, 000 edges. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4 ˝ Erdos number statistics • wrote 1401 papers in Math Reviews. • There are 337, 000 vertices (authors) in the graph. • There are about 496, 000 edges. • Average number of authors per paper: 1.45 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4 ˝ Erdos number statistics • wrote 1401 papers in Math Reviews. • There are 337, 000 vertices (authors) in the graph. • There are about 496, 000 edges. • Average number of authors per paper: 1.45 • Average number of papers per author: 6.87 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 33/4 Experimental data # 0 1 1 502 2 5713 3 26422 4 62136 5 66157 6 32280 7 10431 o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 34/4 Experimental data # # 0 1 8 3214 1 509 9 953 2 6984 10 262 3 26422 11 94 4 62136 12 23 5 66157 13 4 6 32280 14 7 7 10431 15 1 (Most recent data) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 34/4 Experimental data # # 0 1 8 3214 1 509 9 953 2 6984 10 262 3 26422 11 94 4 62136 12 23 5 66157 13 4 6 32280 14 7 7 10431 15 1 (R. G. Kamalov) (Most recent data) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 34/4 The unknown mathematician x Dl D l D l D l D l D l D l D l D l x x lx D l D x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4 The unknown mathematician x Dl D l D l D l D l D l D l D l D l x x lx D l D x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4 The unknown mathematician x Dl D l D l D l D l D l D l D l D l x x lx D l D x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4 The unknown mathematician x Dl D l D l D l D l D l D l D l D l x x lx D l D x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4 The unknown mathematician x Dl D l D l D l D l D l D l D l D l x x lx D l D x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4 The unknown mathematician x Dl D l D l D l D l D l D l D l D l x x lx D l D x o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 35/4 Computer networks Graphs model much more serious stuff. I.e., • computer networks, • shipping routes, • distribution networks. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 36/4 Network question In networks we are concerned with one particular quantity: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 37/4 Network question In networks we are concerned with one particular quantity: connectivity: A connected graph is k-connected if removing any set of k − 1 vertices (and all relevant edges) leaves the graph connected. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 37/4 Same model • n computers o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 38/4 Same model • n computers • in H, each computer is connected to ≥ dn others o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 38/4 Same model • n computers • in H, each computer is connected to ≥ dn others • add f (n) random connections o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 38/4 Same model • n computers • in H, each computer is connected to ≥ dn others • add f (n) random connections Of course, we want high connectivity with as little randomness as possible. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 38/4 Connectivity theorem Theorem. Let k be a function of n that is ≪ n. Let H have the property that each vertex is connected to at least dn other vertices. • If f (n) ≫ k, then the graph becomes k-connected, with high probability. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 39/4 Connectivity theorem Theorem. Let k be a function of n that is ≪ n. Let H have the property that each vertex is connected to at least dn other vertices. • If f (n) ≫ k, then the graph becomes k-connected, with high probability. • If d < 1/2, there is an H0 such that for every k ≪ n, f (n) = k − 1 ensures that the graph fails to be k-connected, with high probability. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 39/4 Bottom line A way to interpret this theorem is: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 40/4 Bottom line A way to interpret this theorem is: If you need k-connectivity, o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 40/4 Bottom line A way to interpret this theorem is: If you need k-connectivity, then you need to add a little more (asymptotically) random edges than k. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 40/4 Bottom line A way to interpret this theorem is: If you need k-connectivity, then you need to add a little more (asymptotically) random edges than k. If fewer than k random edges are added, k-connectivity does not necessarily occur. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 40/4 Worst case What is that H0 ? x x x x x x x x 4 e 4 4 e 4 4 e 4 4 e 4 x e x x 4 e x x 4 e x x 4 e x 4 ¡¡ 4 ¡¡ 4 ¡¡ 4 ¡¡ 4 ¡ 4 4 ¡ 4 4 ¡ 4 4 ¡ 4 4 xx x 4 ex x 4 ex x 4 ex e 4 e 4 e 4 e 4 ¡ e 4 ¡ ¡ ¡ x x x x x x x x H0 = ¡4 e 4 ¡ 4 e 4 ¡4 e 4 ¡ e 4 4 x e ¡ x x 4 e¡ x x 4 e¡ x x 4 e¡ x 4 4 4 4 4 ¡ 4 4 ¡ 4 4 ¡ 4 4 ¡ 4 4 xx x 4 ex x 4 ex x 4 ex e 4 e 4 e 4 e 4 ¡ e 4 ¡ ¡ ¡ Disjoint cliques give the worst case. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 41/4 Other properties We’ve used this model to investigate other properties: • Hamilton cycle o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 42/4 Other properties We’ve used this model to investigate other properties: • Hamilton cycle • Small cliques as subgraphs o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 42/4 Other properties We’ve used this model to investigate other properties: • Hamilton cycle • Small cliques as subgraphs • Chromatic number o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 42/4 It’s just killing you, isn’t it? Let us return to the Kevin Bacon question. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 43/4 It’s just killing you, isn’t it? Let us return to the Kevin Bacon question. We want to ﬁnd actors with an # • inﬁnite and # • with =8 . o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 43/4 Inﬁnite Kevin Bacon number # is someone with inﬁnite . Thomas Alva Edison only appeared in one movie (a brief documentary) and was the only actor. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 44/4 Inﬁnite Kevin Bacon number # is someone with inﬁnite . Thomas Alva Edison only appeared in one movie (a brief documentary) and was the only actor. Not soon coming to DVD: o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 44/4 Inﬁnite Kevin Bacon number # is someone with inﬁnite . Thomas Alva Edison only appeared in one movie (a brief documentary) and was the only actor. Not soon coming to DVD: Mr. Edison at Work in His Chemical Laboratory (1897). o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 44/4 Kevin Bacon number 7 Joseph Wheeler appeared in two ﬁlms: • General Wheeler and Secretary of War Alger at Camp Wikoff (1898), a short documentary, in which he appeared with Russell Alexander Alger (Kevin Bacon number 6) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 45/4 Kevin Bacon number 7 Joseph Wheeler appeared in two ﬁlms: • General Wheeler and Secretary of War Alger at Camp Wikoff (1898), a short documentary, in which he appeared with Russell Alexander Alger (Kevin Bacon number 6) • Surrender of General Toral (1898), again, a short documentary, with William Rufus Shafter. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 45/4 Kevin Bacon number 8 William Rufus Shafter also appeared in two ﬁlms: • Surrender of General Toral (1898) with Joseph Wheeler. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 46/4 Kevin Bacon number 8 William Rufus Shafter also appeared in two ﬁlms: • Surrender of General Toral (1898) with Joseph Wheeler. • Major General Shafter (1898) as the only credited cast member. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 46/4 Conclusion # • Russell Alexander Alger has =6, o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 47/4 Conclusion # • Russell Alexander Alger has =6, # • Joseph Wheeler has =7, and o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 47/4 Conclusion # • Russell Alexander Alger has =6, # • Joseph Wheeler has =7, and # • William Rufus Shafter has =8. o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 47/4 The chain 8 William Rufus Shafter was in Surrender of General Toral (1898) with Joseph Wheeler o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 8 William Rufus Shafter was in Surrender of General Toral (1898) with Joseph Wheeler 7 Joseph Wheeler was in General Wheeler and Secretary of War Alger at Camp Wikoff (1898) with Russell Alexander Alger o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 6 Russell Alexander Alger was in President McKinley’s Inspection of Camp Wikoff (1898) with President William McKinley o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 6 Russell Alexander Alger was in President McKinley’s Inspection of Camp Wikoff (1898) with President William McKinley 5 President William McKinley was in President McKinley Taking the Oath (1901) with U. S. Senator Marcus Hanna o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 6 Russell Alexander Alger was in President McKinley’s Inspection of Camp Wikoff (1898) with President William McKinley 5 President William McKinley was in President McKinley Taking the Oath (1901) with U. S. Senator Marcus Hanna (R-OH) o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 4 U. S. Senator Marcus Hanna (R-OH) was in Opening of the Pan-American Exposition Showing Vice President Roosevelt Leading the Procession (1901) with President Theodore Roosevelt o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 4 U. S. Senator Marcus Hanna (R-OH) was in Opening of the Pan-American Exposition Showing Vice President Roosevelt Leading the Procession (1901) with President Theodore Roosevelt 3 President Theodore Roosevelt was in Womanhood, the Glory of the Nation (1917) with Walter McGrail o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 2 Walter McGrail was in Dick Tracy vs. Crime Inc. (1941) with Wally Rose o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 2 Walter McGrail was in Dick Tracy vs. Crime Inc. (1941) with Wally Rose 1 Wally Rose was in Murder in the First (1995) with o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 The chain 2 Walter McGrail was in Dick Tracy vs. Crime Inc. (1941) with Wally Rose 1 Wally Rose was in Murder in the First (1995) with o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 48/4 Thanks Thank you for allowing me to speak today. rymartin@iastate.edu http://orion.math.iastate.edu/rymartin/ o Six degrees of graph theory:Kevin Bacon, Paul Erd˝ s, William McKinley and me – p. 49/4

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