Electrical Networks, Hyperplane Arrangements and Matroids Bob Lutz - - PowerPoint PPT Presentation

Electrical Networks, Hyperplane Arrangements and Matroids

Bob Lutz

Mathematical Sciences Research Institute

December 2, 2019

Partially supported by NSF grants DMS-1401224, DMS-1701576 and DMS-1440140

Section I: Electrical Networks

1

What is an electrical network?

◮ A connected graph G = (V , E) (edges = wires) ◮ A set ∂V ⊆ V of at least 2 boundary nodes ◮ A (real or complex) voltage vj at every boundary node j

1

Wheatstone bridge

4 3 2 1

Star network

The Dirichlet problem

◮ Electrical current flows from higher voltages to lower voltages ◮ Consider the interior V ◦ = V \ ∂V ◮ What are the voltages at the interior nodes?

1 1

The Dirichlet solution

◮ Every wire ij ∈ E has a (real or complex) conductance cij ◮ Voltages and conductances satisfy

• j∼i

cij(vi − vj) = 0 at every interior node i ∈ V ◦ ◮ “The current across every interior node is 0” ◮ Uniquely determines the interior voltages (for generic c)

Energies

Definition

The energy dissipated by an edge ij ∈ E is eij = cij(vi − vj)2

Example

Let ∆ = (c1 + c3 + c4)(c2 + c3 + c5) − c2

• 3. We have

c1 c4 c2 c5 c3

1

      e1 e2 e3 e4 e5       = 1 ∆2       c1(c3c5 + c4(c2 + c3 + c5))2 c2(c3c4 + (c1 + c3 + c4)c5)2 c3(c2c4 − c1c5)2 c4(c2c3 + c1(c2 + c3 + c5))2 c5(c1c3 + c2(c1 + c3 + c4))2      

Question

◮ The map c → e is rational (polynomial/polynomial) ◮ What do the fibers look like? ◮ Equivalently, which interior voltages produce the energies e?

Example

Let ∆ = (c1 + c3 + c4)(c2 + c3 + c5) − c2

• 3. We have

c1 c4 c2 c5 c3

1

      e1 e2 e3 e4 e5       = 1 ∆2       c1(c3c5 + c4(c2 + c3 + c5))2 c2(c3c4 + (c1 + c3 + c4)c5)2 c3(c2c4 − c1c5)2 c4(c2c3 + c1(c2 + c3 + c5))2 c5(c1c3 + c2(c1 + c3 + c4))2      

e-harmonic functions

Definition

Fix boundary voltages v and energies e ∈ CE. A function h ∈ CV is e-harmonic on (G, v) if

• 1. There are conductances c for which h is the voltage function

for the network

• 2. The resulting energies are e.

Interesting Problem (Abrams–Kenyon 2017)

Describe the set of e-harmonic functions for a given e.

Example

Let all e ≡ 1. Fix boundary voltages 0 and 1. There are two e-harmonic functions, with conductances labeled: 5b − 5 5a − 5 5a − 5 5b − 5 5

1 b a

5a − 5 5b − 5 5b − 5 5a − 5 5

1 a b

where a = 1

2(5 −

√ 5) and b = 1

2(5 +

√ 5).

Section II: Dirichlet Arrangements

Dirichlet arrangements

Definition

The Dirichlet arrangement AG,v consists of two types of hyperplanes, corresponding to the edges of G, with coordinates indexed by interior nodes: ◮ A hyperplane xi = vj for every edge ij with j ∈ ∂V ◮ A hyperplane xi = xj for every edge ij not meeting ∂V r s

1

xr = 0 xr = 1 xs = 0 xs = 1 xr = xs

Master functions

◮ Let A be an arrangement of k hyperplanes in Cn, defined by affine functions f1, . . . , fk : Cn → C ◮ Master function of A with weights b ∈ Ck is multivalued Cn → C given by Φb(x) =

k

• i=1

fi(x)bi ◮ A critical point x ∈ Cn of Φb: ∂ ∂xj log Φb(x) =

k

• i=1

∂fi ∂xj bi fi(x) = 0 for all j = 1, . . . , n

Example

◮ Arrangement A in C2 defined by A : x = 0, y = 0, x + y − 1 = 0 ◮ Master function with weights b ∈ C3: Φb(x, y) = xb1yb2(x + y − 1)b3 ◮ One critical point:

(x, y) =

• b1

b1 + b2 + b3 , b2 b1 + b2 + b3

Why Dirichlet arrangements?

Theorem (L. 2019)

The e-harmonic functions on (G, v) are the critical points of the master function of AG,v with weights e.

Example

Let a = 1

10(5 −

√ 5) and b = 1

10(5 +

√ 5) with all energies 1:

1 a or b b or a (a, b) (b, a)

So what?

Critical points of master functions are well studied! ◮ Interior point methods: logarithmic barrier functions and analytic centers ◮ Algebraic statistics: maximum likelihood estimation . . . especially for certain Dirichlet arrangements. ◮ Real algebraic geometry: solution of B. and M. Shapiro conjecture on Wronskians ◮ Quantum integrable systems: Bethe ansatz in the Gaudin model

Current flow

◮ Real-valued e-harmonic functions induce current flows ◮ Edges directed from higher voltages to lower ◮ # nonzero current flows = # e-harmonic functions = # bounded chambers

Example

In the running example:

1 b a 1 a b

where a = 1

10(5 −

√ 5) and b = 1

10(5 +

√ 5).

Counting bounded chambers

◮ Chromatic polynomial χG counts proper vertex colorings of G ◮ The beta invariant is β(G) = |χ′

G(1)|

◮ β(G) = 0 iff G is disconnected by removing single vertex

Theorem (L. 2019)

Construct G from G by adding edges between all boundary

• vertices. For generic energies e (including all positive) we have

β( G) (|∂V | − 2)! = # current flows # e-harmonic functions # bounded chambers

Visibility arrangements

Definition

Let P be a convex n-polytope in Rn. The visibility arrangement vis(P) is the set of affine spans of the top-dimensional faces of P

Visibility sets

Chambers of vis(P) Sets of top-dimensional faces of P visible from different points in Rn

Order polytopes

Definition

Let P be a finite poset. The order polytope O(P) is the convex polytope in RP of all order-preserving functions P → [0, 1]

Example

If every pair in P = {x1, . . . , xn} is incomparable, then O(P) = [0, 1]n is the unit hypercube in Rn

Example

If P = {x1, . . . , xn} is totally ordered, then O(P) is an n-simplex in Rn

Visibility arrangements of order polytopes

Proposition (Stanley 2015)

Let P be a finite poset. The visibility arrangement vis(O(P)) of the order polytope of P is a Dirichlet arrangement AG,v. Hasse diagram of P

1

(G, v)

• G

Counting visibility sets

Theorem (L. 2019)

The number of visibility sets of the order polytope O(P) is 1 2α( G), where α( G) is the number of acyclic orientations of

• G. Of the

visibility sets, all but β( G) are visible from far away, where β( G) is the beta invariant.

Section III: Network Duals & Matroid Quotients

Graphic matroids

◮ Graphic matroids M(G) are a fundamental class of matroids ◮ Recall: circuits of M(G) are (edge sets of) cycles of G ◮ Circuits of dual matroid M∗(G) are bonds of G a d b e c Cycles = abc, abde, cde Bonds = ab, ace, bce, de

Cellularly embedded graphs

◮ Let Σ be a compact surface ◮ Suppose G ֒ → Σ such that every face is a 2-cell ◮ Can define a geometric dual G ∗

Planar duality

Theorem (Whitney 1932)

If G is planar, then M∗(G) ∼ = M(G ∗). In other words, Bonds of G = Cycles of G ∗

Generalization

◮ Ranks of M∗(G) and M(G ∗) differ by 2 − χ(Σ), so no isomorphism in general ◮ There is a quotient map M ։ N if every circuit of M is a union of circuits of N ◮ Quotient maps are the bijective morphisms of matroids

Theorem (Richter–Shank 1984)

If G ֒ → Σ, then there is a quotient map M∗(G) ։ M(G ∗).

Network duals

◮ Let Σ be a compact surface with boundary ◮ Suppose G ֒ → Σ is cellularly embedded with ∂V ⊂ ∂Σ and no face having > 1 boundary component ◮ Given D = (G, ∂V ), can define dual network D∗

Dirichlet matroids

◮ Dirichlet arrangement AG,v defines matroid M(D) depending

• nly on D = (G, ∂V )

◮ Circuits of M(D) keep track of cycles and paths between boundary nodes ◮ M(D) arises from almost-balanced biased graphs and complete principal truncations of graphic matroids |∂V | = 2 M(D) ∼ = M( G) is graphic D = star network M(D) ∼ = U2,|V | is uniform

Quotient map for networks

Theorem (L. 2019)

If D ֒ → Σ, then there is a quotient map M∗(D) ։ M(D∗). ◮ Ranks differ by |∂V | − χ(Σ) − 1 ◮ Recover graphic case when |∂V | = 2 ◮ Neither M∗(D) nor M(D∗) is graphic/cographic in general!

Thanks

Thank you!