Note on von Neumann and R´ enyi entropies of a graph Jephian C.-H. Lin Department of Mathematics, Iowa State University April 9, 2017 Graduate Student Combinatorics Conference 2017 Entropies of a graph 1/15 Department of Mathematics, Iowa State University
Joshua Michael Leslie Lockhart Dairyko Hogben Michael Simone David Young Severini Roberson Entropies of a graph 2/15 Department of Mathematics, Iowa State University
Entropy Let p = ( p 1 , p 2 , . . . , p n ) be a probability distribution, meaning n � p i = 1 and p i ≥ 0 . i =1 The Shannon entropy of p is n 1 � S ( p ) = p i log 2 . p i i =1 For a given α ≥ 0 with α � = 1, the R´ enyi entropy is � n � 1 � H α ( p ) = 1 − α log 2 p α . i i =1 Entropies of a graph 3/15 Department of Mathematics, Iowa State University
1 The function x log 2 x Entropies of a graph 4/15 Department of Mathematics, Iowa State University
Convexity and Jensen’s inequality If f is a convex function, then Jensen’s inequality says � n � 1 1 � � f ( p i ) ≤ f p i . n n i =1 Let p = ( 1 n , . . . , 1 1 n ). Since x log 2 x ≥ 0 is convex, 0 ≤ S ( p ) ≤ S ( p ) for all p . � maximized by ( 1 n , 1 n , . . . , 1 n ) , Therefore, S ( p ) is minimized by (1 , 0 , . . . , 0) . Entropy measures mixedness. Entropies of a graph 5/15 Department of Mathematics, Iowa State University
Convexity and Jensen’s inequality If f is a convex function, then Jensen’s inequality says � n � 1 1 � � f ( p i ) ≤ f p i . n n i =1 Let p = ( 1 n , . . . , 1 1 n ). Since x log 2 x ≥ 0 is convex, 0 ≤ S ( p ) ≤ S ( p ) for all p . � maximized by ( 1 n , 1 n , . . . , 1 n ) , Therefore, S ( p ) is minimized by (1 , 0 , . . . , 0) . Entropy measures mixedness. Entropies of a graph 5/15 Department of Mathematics, Iowa State University
Density matrix A density matrix M is a (symmetric) positive semi-definite matrix with trace one. Every density matrix has the spectral decomposition n n M = QDQ ⊤ = � � λ i v i v ⊤ i = λ i E i , i =1 i =1 where λ i ≥ 0 and � n i =1 λ i = tr( M ) = 1. Each of E i is of rank one and trace one; such a matrix is called a pure state in quantum information. A density matrix is a convex combination of pure states with probability distribution ( λ 1 , . . . , λ n ). Entropies of a graph 6/15 Department of Mathematics, Iowa State University
Density matrix A density matrix M is a (symmetric) positive semi-definite matrix with trace one. Every density matrix has the spectral decomposition n n M = QDQ ⊤ = � � λ i v i v ⊤ i = λ i E i , i =1 i =1 where λ i ≥ 0 and � n i =1 λ i = tr( M ) = 1. Each of E i is of rank one and trace one; such a matrix is called a pure state in quantum information. A density matrix is a convex combination of pure states with probability distribution ( λ 1 , . . . , λ n ). Entropies of a graph 6/15 Department of Mathematics, Iowa State University
Density matrix of a graph Let G be a graph. The Laplacian matrix of G is a matrix L with d i if i = j , L i , j = − 1 if i ∼ j , 0 otherwise. Any Laplacian matrix is positive semi-definite and has n � tr( L ) = d i = 2 | E ( G ) | =: d G . i =1 1 The density matrix of G is ρ ( G ) = d G L . Entropies of a graph 7/15 Department of Mathematics, Iowa State University
Entropies of a graph Let G be a graph and ρ ( G ) its density matrix. Then spec( ρ ( G )) is a probability distribution. The von Neumann entropy of a graph G is S ( G ) = S (spec( ρ ( G ))); the R´ enyi entropy of a graph G is H α ( G ) = H α (spec( ρ ( G ))). Proposition d Gi Let G 1 , . . . , G k be disjoint graphs, c i = , and � k i =1 d Gi c = ( c 1 , . . . , c k ) . Then � k � � ˙ = c 1 S ( G 1 ) + · · · + c k S ( G k ) + S ( c ) . S i =1 G i Entropies of a graph 8/15 Department of Mathematics, Iowa State University
Union of graphs Theorem (Passerini and Severini 2009) If G 1 and G 2 are two graphs on the same vertex set and E ( G 1 ) ∩ E ( G 2 ) = ∅ , then S ( G 1 ∪ G 2 ) ≥ c 1 S ( G 1 ) + c 2 S ( G 2 ) , d Gi where c i = d G 1 + d G 2 . In particular, for a graph G and e ∈ E ( G ), then d G S ( G + e ) ≥ d G + 2 S ( G ) . Entropies of a graph 9/15 Department of Mathematics, Iowa State University
Adding an edge can decrease the von Neumann entropy K 2 , n − 2 K 2 , n − 2 + e 1 S ( K 2 , n − 2 ) ∼ 1 + ( n − 3) · 2 n − 4 log 2 (2 n − 4) 1 S ( K 2 , n − 2 + e ) ∼ 1 + ( n − 3) · 2 n − 3 log 2 (2 n − 3) Entropies of a graph 10/15 Department of Mathematics, Iowa State University
Extreme values of the von Neumann entropy � maximized by ( 1 n , 1 n , . . . , 1 n ) , Recall S ( p ) is minimized by (1 , 0 , . . . , 0) . � maximized by K n , For graphs on n vertices, S ( G ) is minimized by K 2 ˙ ∪ ( n − 2) K 1 . Conjecture (DHLLRSY 2017) For connected graphs on n vertices, the minimum von Neumann entropy is attained by K 1 , n − 1 . Entropies of a graph 11/15 Department of Mathematics, Iowa State University
Computational results and possible approaches By Sage, S ( K 1 , n − 1 ) ≤ S ( G ) for all connected graphs G on n ≤ 8 vertices, and S ( K 1 , n − 1 ) ≤ S ( T ) for all trees T on n ≤ 20 vertices. The conjecture is still open, but we can prove assymptotically. Idea: The R´ enyi entropy H α ( p ) ր S ( p ) as α ց 1. In particular, H 2 ( G ) ≤ S ( G ) . Entropies of a graph 12/15 Department of Mathematics, Iowa State University
Computational results and possible approaches By Sage, S ( K 1 , n − 1 ) ≤ S ( G ) for all connected graphs G on n ≤ 8 vertices, and S ( K 1 , n − 1 ) ≤ S ( T ) for all trees T on n ≤ 20 vertices. The conjecture is still open, but we can prove assymptotically. Idea: The R´ enyi entropy H α ( p ) ր S ( p ) as α ց 1. In particular, H 2 ( G ) ≤ S ( G ) . Entropies of a graph 12/15 Department of Mathematics, Iowa State University
Computational results and possible approaches By Sage, S ( K 1 , n − 1 ) ≤ S ( G ) for all connected graphs G on n ≤ 8 vertices, and S ( K 1 , n − 1 ) ≤ S ( T ) for all trees T on n ≤ 20 vertices. The conjecture is still open, but we can prove assymptotically. Idea: The R´ enyi entropy H α ( p ) ր S ( p ) as α ց 1. In particular, H 2 ( G ) ≤ S ( G ) . Entropies of a graph 12/15 Department of Mathematics, Iowa State University
What’s nice about H 2 ( G )? Let M = ρ ( G ). Then by definition, the R´ enyi entropy H 2 ( G ) is � n � d 2 1 � � � λ 2 = − log 2 (tr M 2 ) = log 2 G 1 − 2 log 2 . i i d 2 d G + � i i =1 Theorem (DHLLRSY 2017) d 2 ≥ 2 n − 2 G 2 n − 2 , then S ( G ) ≥ H 2 ( G ) ≥ S ( K 1 , n − 1 ) . If d G + � n n i =1 d 2 n i � � It is known that � n i =1 d 2 2 m i ≤ m n − 1 + n − 2 . By some computation, almost all graphs have S ( G ) ≥ S ( K 1 , n − 1 ) when n → ∞ . Entropies of a graph 13/15 Department of Mathematics, Iowa State University
Conclusion Whether S ( G ) ≥ S ( K 1 , n − 1 ) for all G or not remains open. Conjecture (DHLLRSY 2017) For every connected graph G on n vertices and α > 1 , H α ( G ) ≥ H α ( K 1 , n − 1 ) . We are able to show H 2 ( G ) ≥ H 2 ( K 1 , n − 1 ) for every connected graphs on n vertices. Thank You! Entropies of a graph 14/15 Department of Mathematics, Iowa State University
Conclusion Whether S ( G ) ≥ S ( K 1 , n − 1 ) for all G or not remains open. Conjecture (DHLLRSY 2017) For every connected graph G on n vertices and α > 1 , H α ( G ) ≥ H α ( K 1 , n − 1 ) . We are able to show H 2 ( G ) ≥ H 2 ( K 1 , n − 1 ) for every connected graphs on n vertices. Thank You! Entropies of a graph 14/15 Department of Mathematics, Iowa State University
D. de Caen. An upper bound on the sum of squares of degrees in a graph. Discrete Math. , 185:245–248, 1998. M. Dairyko, L. Hogben, J. C.-H. Lin, J. Lockhart, D. Roberson, S. Severini, and M. Young. Note on von Neumann and R´ enyi entropies of a graph. Linear Algebra Appl. , 521:240–253, 2017. F. Passerini and S. Severini. Quantifying complexity in networks: the von Neumann entropy. Int. J. Agent Technol. Syst., 1:58–68, 2009. Entropies of a graph 15/15 Department of Mathematics, Iowa State University
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