Combinatorial Types of Tropical Eigenvector

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1 Combinatorial Types of Tropical Eigenvector arxiv: Ngoc Mai Tran Department of Statistics, UC Berkeley Joint work with Bernd Sturmfels

2 2 / 13 Tropical eigenvalues and eigenvectors Max-plus: (R,, ): a b = max{a, b}, a b = a + b. Tropical torus: TP n 1 = R n \(1,..., 1) Matrix A R n n. Eigenpair (λ, x) R TP n 1 of A: A x = λ x Example = 2 = (1) (2)

3 How to compute? Eigenvalue (Cuninghame-Green 79) The eigenvalue solves the LP: Minimize λ subject to a ij + x j λ + x i for all 1 i, j n. Dual program: Maximize n i,j=1 a ijp ij subject to p ij 0 for 1 i, j n, n i,j=1 p ij = 1 and n j=1 p ij = n k=1 p ki for all 1 i n. C n := the n(n 1)-dimensional polytope of feasible dual solutions. Vertices of C n = uniform distributions on cycles. Strong duality the eigenvalue function λ A : A λ(a) = support function of C n. Cones of linearity of λ A are those in the normal fan of C n. λ(a) is unique, and is the max normalized cycle in the graph of A. 3 / 13

4 4 / 13 How about the eigenvector? Piecewise linear: yes. Unique: No. Cones of linearity? Graph interpretation? Fan? Graph interpretation: Define B := A ( λ(a)). If for some critical node i, y j = B ji = max B(P ji ) path P ji then y is an eigenvector of A. The tropical eigenspace is the tropical convex hull of such vectors. Get tropical multiples if critical nodes belong to the same cycle.

5 5 / 13 Example revisited A = , λ(a) = 2, B = Critical cycles of A are (1, 2) and (3) Columns 1 and 2 of B are tropical multiples of the same vector. Eigenspace is the tropical line segment between columns 1 and 3 of B.

6 6 / 13 Eigenvector cones Theorem[Sturmfels-T]: There exists a refinement of the eigenvalue cone into eigenvector cones such that: On each open cone, x(a) is unique, and the eigenpair map (λ A, x A ) : A (λ(a), x(a)) is represented by a unique linear function Each cone is linearly isomorphic to R n R n(n 1) 0. The cones are in bijection with connected functions on [n] = {1, 2,..., n} The number of the cones is: n n! k=1 (n k)! nn k 1. These cones do not form a fan.

7 7 / 13 Sketch of proof Recall: y is an eigenvector of A iff y j = B ji = max Pji B(P ji ) for some critical vertex i. Therefore, Refinement: unique critical cycle unique eigenvector. When the eigenvector map is linear, the path achieving the max is unique Restriction of maximal paths are also maximal. This implies: Eigenvector cones are in bijection with connected functions. To specify a connected function: only need n(n 1) inequalities. It remains to show that every connected function is realizable (induction).

8 8 / 13 Lack of fan property This is due to the discontinuity of the map A x(a) when switching between eigenvalue cones of disjoint critical cycle Figure: The simplicial complex Σ 3 of connected functions φ : [3] [3]. Fixed-point free φ are on the left and functions with φ(3) = 3 on the right.

9 9 / 13 Application: convergence of Perron eigenvector to its tropical limit (Gaubert 90): If X R n n is elementwise positive, then as k ρ(x (k) ) 1/k λ max, (X) v(x (k) ) 1/k x max, (X) if x(x) is unique Same context as max-plus when take elementwise log. Can use cones to study the convergence rate. Worse convergence at the boundary. Open problem: what happens at the interesting boundaries?

10 10 / 13 Application: random tropical matrices Let G be a graph on n nodes, its edges have iid lengths. What is the length of the maximal mean cycle? Equivalently: let G be a random matrix with iid entries. What is the induced distribution over the eigenvalue cones? Gaussian case: equivalent to computing spherical volume of the cones.

11 11 / 13 Applications to ranking The subspace 2 R n of skew-symmetric matrices (A = A T ) has applications to ranking by pairwise comparison: A ij = how much i wins over j. A ji = A ij = how much i loses over j. Tropical ranking (Elsner and van den Driessche 06): Interpret x(a) as the score vector: x(a) i = score of i. Recall: the eigenpair satisfies the LP: Minimize λ subject to a ij (x i x j ) λ for all 1 i, j n. Matrix (x ij ) = (x i x j ): consistent (Saaty 78). Interpretation: λ is the l distance in 2 R n from A to the subspace of consistent matrices. The eigenpolytope lies inside the set of all possible l minimizers. Open question - ED10: When are the two polytopes equal?

12 12 / 13 Thank you! Figure: Eigencones in 2 R 4.

Combinatorial types of tropical eigenvectors

Bull. London Math. Soc. 45 (2013) 27 36 C 2012 London Mathematical Society doi:10.1112/blms/bds058 Combinatorial types of tropical eigenvectors Bernd Sturmfels and Ngoc Mai Tran Abstract The map which