Blocking duality for p-modulus on networks✩ Nathan Albina , Pietro Poggi-Corradinia,∗, Jason Clemensb a Mathematics
Department, Kansas State University, Manhattan, KS 66506, USA Valley College, Marshall, MO 65340, USA
arXiv:1612.00435v2 [math.CO] 12 Dec 2016
b Missouri
Abstract We explore the theory of blocking duality pioneered by Fulkerson et al. in the context of p-modulus on networks. We formulate an analogue on networks to the theory of conjugate families in complex analysis. Also we draw a connection between Fulkerson’s blocking duality and the probabilistic interpretation of modulus when p = 2. Finally, we expand on a result of Lovász in the context of randomly weighted graphs. Keywords: p-modulus, blocking duality, randomly weighted graphs 2010 MSC: 90C27 1. Introduction
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10
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1.1. Modulus in the Continuum The theory of conformal modulus was originally developed in complex analysis, see Ahlfors’s comment on p. 81 of [1]. The more general theory of p-modulus grew out of the study of quasiconformal maps, which generalize the notion of conformal maps to higher dimensional real Euclidean spaces and, in fact, to abstract metric measure spaces. Intuitively, p-modulus provides a method for quantifying the richness of a family of curves, in the sense that a family with many short curves will have a larger modulus than a family with fewer and longer curves. The parameter p tends to favor the “many curves” aspect when p is close to 1 and the “short curves” aspect as p becomes large. This phenomenon was explored more precisely in [2] in the context of networks. The concept of discrete modulus on networks is not new, see for instance [8, 18, 11]. However, recently the authors have started developing the theory of p-modulus as a graph-theoretic quantity [5, 2], with the goal of finding applications, for instance to the study of epidemics [20, 10]. The concept of blocking duality explored in this paper is an analog of the concept of conjugate families in the continuum. As motivation for the discrete theory to follow, then, let us recall the relevant definitions from the continuum theory. For now, it is convenient to restrict attention to the 2-modulus of curves in the plane, which, as it happens, is a conformal invariant and thus has been carefully studied in the literature. Let Ω be a domain in C, and let E, F be two continua in Ω. Define Γ = ΓΩ (E, F ) to be the family of all rectifiable curves connecting E to F in Ω. A density is a Borel measurable function ρ : Ω → [0, ∞). We say that ρ is admissible for Γ and write ρ ∈ Adm(Γ), if Z ρ ds ≥ 1 ∀γ ∈ Γ. (1) γ
✩ Research
supported by NSF n. 1201427 and n. 1515810 author Email address:
[email protected] (Pietro Poggi-Corradini)
∗ Corresponding
Preprint submitted to Elsevier
Now, we define the modulus of Γ as Mod2 (Γ) :=
inf
ρ∈Adm(Γ)
Z
ρ2 dA.
(2)
Ω
Example 1.1 (The Rectangle). Consider a rectangle Ω := {z = x + iy ∈ C : 0 < x < L, 0 < y < H} of height H and length L. Set E := {z ∈ Ω : Re z = 0} and F := {z ∈ Ω : Re z = L} to be the leftmost and rightmost vertical sides respectively. If Γ = ΓΩ (E, F ) then, Mod2 (Γ) =
H . L
(3)
To see this, assume ρ ∈ Adm(Γ). Then for all 0 < y < H, γy (t) := t + iy is a curve in Γ, so Z
ρds =
Z
Using the Cauchy-Schwarz inequality we obtain, "Z L
ρ(t, y)dt
0
RL
In particular, L−1 ≤
0
ρ(t, y)dt ≥ 1.
0
γy
1≤
L
#2
≤L
Z
L
ρ2 (t, y)dt.
0
ρ2 (t, y)dt. Integrating over y, we get Z H ≤ ρ2 dA. L Ω
So since ρ was an arbitrary admissible density, Mod2 (Γ) ≥ H L. R 1 In the other direction, define ρ0 (z) = L 1Ω (z) and observe that Ω ρ2 dA = show that ρ0 ∈ Adm(Γ), then Mod(Γ) ≤ H L . To see this note that for any γ ∈ Γ: Z
0
L
1 1 |γ(t)|dt ˙ ≥ L L
Z
L
| Re γ(t)|dt ˙ ≥
0
HL L2
=
H L.
Hence, if we
1 (Re γ(1) − Re γ(0)) ≥ 1. L
This proves the formula (3). A famous and very useful result in this context is the notion of a conjugate family of a connecting family. For instance, in the case of the rectangle, the conjugate family Γ∗ = Γ∗Ω (E, F ) for ΓΩ (E, F ) consists of all curves that “block” or intercept every curve γ ∈ ΓΩ (E, F ). It’s clear in this case that Γ∗ is also a connecting family, namely it includes every curve connecting the two horizontal sides of Ω. In particular, by (3), we must have Mod2 (Γ∗ ) = L/H. So we deduce that Mod2 (ΓΩ (E, F )) · Mod2 (Γ∗Ω (E, F )) = 1. 20
(4)
One reason this reciprocal relation is useful is that upper-bounds for modulus are fairly easy to obtain by choosing reasonable admissible densities and computing their energy. However, lower-bounds are typically 2
25
30
35
harder to obtain. However, when an equation like (4) holds, then upper-bounds for the modulus of the conjugate family translate to lower-bounds for the given family. In higher dimensions, say in R3 , the conjugate family of a connecting family of curves consists of a family of surfaces, and therefore one must consider the concept of surface modulus, see for instance [16] and references therein. It is also possible to generalize the concept of modulus by replacing the exponent 2 in (2) with p ≥ 1 and by replacing dA with a different measure. The principal aim of this paper is to establish a conjugate duality formula similar to (4) for p-modulus on networks, which we call blocking duality. 1.2. Modulus on Networks A general framework for modulus of objects on networks was developed in [4]. In what follows, G = (V, E, σ) is taken to be a finite graph with vertex set V and edge set E. The graph may be directed or undirected and need not be simple. In general, we shall assume a weighted graph with each edge assigned a corresponding weight 0 < σ(e) < ∞. When we refer to an unweighted graph, we shall mean a graph for which all weights are assumed equal to one. The theory in [4] applies to any finite family of “objects” Γ for which each γ ∈ Γ can be assigned an associated function N (γ, ·) : E → R≥0 that measures the usage of edge e by γ. Notationally, it is convenient to consider N (γ, ·) as a row vector N (γ, ·) ∈ RE ≥0 , indexed by e ∈ E. In order to avoid pathologies, it is useful to assume that Γ is non-empty and that each γ ∈ Γ has positive usage on at least one edge. When this is the case, we will say that Γ is non-trivial. In the following it will be useful to define the quantity: Nmin := min
min
γ∈Γ e:N (γ,e)6=0
N (γ, e).
(5)
Hence, Γ non-trivial implies that Nmin > 0. Some examples of objects and their associated usage functions are the following. • To a walk γ = x0 e1 x1 · · · en xn we can associate the traversal-counting function N (γ, e) = number times γ traverses e. In this case N (γ, ·) ∈ ZE ≥0 . 40
• To each subset of edges T ⊂ E we can associate the characteristic function N (T, e) = e ∈ T and 0 otherwise. Here, N (γ, ·) ∈ {0, 1}E .
1T (e) = 1 if
• To each flow f we can associate the volume function N (f, e) = |f (e)|. Therefore, N (γ, ·) ∈ RE ≥0 .
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As a function of two variables, the function N can be thought of as a matrix in RΓ×E , indexed by pairs (γ, e) with γ an object in Γ and e an edge in E. This matrix N is called the usage matrix for the family Γ. Each row of N corresponds to an object γ ∈ Γ and records the usage of edge e by γ. At times will write N (Γ) instead of N , to avoid ambiguity. Note, that the families Γ under consideration may very well be infinite (e.g. families of walks), so N may have infinitely many rows. For this paper, we shall assume Γ is finite. This assumption is not quite as restrictive as it might seem. In [5] it was shown that any family Γ with an integer-valued N can be replaced, without changing the modulus, by a finite subfamily. For example, if Γ is the set of all walks between two distinct vertices, the modulus can be computed by considering only simple paths. This result implies a similar finiteness result for any family Γ whose usage matrix N is rational with positive entries bounded away from zero.
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By analogy to the continuous setting, we define a density on G to be a nonnegative function on the edge set: ρ : E → [0, ∞). The value ρ(e) can be thought of as the cost of using edge e. It is notationally useful to think of such functions as column vectors in RE ≥0 . In order to mimic (1), we define for an object γ ∈ Γ ℓρ (γ) :=
X
N (γ, e)ρ(e) = (N ρ)(γ),
e∈E
representing the total usage cost for γ with the given edge costs ρ. In linear algebra notation, ℓρ (·) is the column vector resulting from the matrix-vector product N ρ. As in the continuum case, then, a density ρ ∈ RE ≥0 is called admissible for Γ, if ℓρ (γ) ≥ 1
∀γ ∈ Γ;
or equivalently, if ℓρ (Γ) := inf ℓρ (γ) ≥ 1. γ∈Γ
In matrix notation, ρ is admissible if N ρ ≥ 1, where 1 is the column vector of ones and the inequality is understood to hold elementwise. By analogy, we define the set Adm(Γ) = ρ ∈ RE (6) ≥0 : N ρ ≥ 1
to be the set of admissible densities. Now, given an exponent p ≥ 1 we define the p-energy on densities, corresponding to the area integral from the continuum case, as X Ep,σ (ρ) := σ(e)ρ(e)p , e∈E
with the weights σ playing the role of the area element dA. In the unweighted case (σ ≡ 1), we shall use the notation Ep,1 for the energy. For p = ∞, we also define the unweighted and weighted ∞-energy respectively as 1 E∞,1 (ρ) := lim (Ep,σ (ρ)) p = max ρ(e) p→∞
e∈E
and
1
E∞,σ (ρ) := lim (Ep,σp (ρ)) p = max σ(e)ρ(e) p→∞
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e∈E
This leads to the following definition. Definition 1.2. Given a graph G = (V, E, σ), a family of objects Γ with usage matrix N ∈ RΓ×E , and an exponent 1 ≤ p ≤ ∞, the p-modulus of Γ is Modp,σ (Γ) :=
inf
ρ∈Adm(Γ)
Ep,σ (ρ)
Equivalently, p-modulus corresponds to the following optimization problem minimize
Ep,σ (ρ)
subject to
ρ ≥ 0,
Nρ ≥ 1
where each object γ ∈ Γ determines one inequality constraint. 4
(7)
Remark 1.3. (a) When ρ0 ≡ 1, we drop the subscript and write ℓ(γ) := ℓρ0 (γ). If γ is a walk, then ℓ(γ) simply counts the number of hops that the walk γ makes.
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(b) If Γ ⊂ Γ′ (here N (Γ) is the restriction of N (Γ′ ) to Γ), then Adm(Γ′ ) ⊂ Adm(Γ), so Modp,σ (Γ) ≤ Modp,σ (Γ′ ), for all 1 ≤ p ≤ ∞. This is known as monotonicity of modulus.
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−1 (c) For 1 < p < ∞ a unique extremal density ρ∗ always exists and satisfies 0 ≤ ρ∗ ≤ Nmin , where Nmin is defined in (5). Existence and uniqueness follows by compactness and strict convexity of Ep,σ , see also Lemma 2.2 of [2]. The upper bound on ρ∗ follows from the fact that each row of N contains at least one nonzero entry, which must be at least as large as Nmin . In the special case when N is integer valued, the upper bound can be taken to be 1.
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1.3. Connection to classical quantities The concept of p-modulus generalizes known classical ways of measuring the richness of a family of walks. Let G = (V, E) and two vertices s and t in V be given. We define the connecting family Γ(s, t) to be the family of all simple paths in G that start at s and end at t. To this family, we assign the usage function N (γ, e) to be 1 when e ∈ γ and 0 otherwise. Definition 1.4. A subset C ⊂ E is called a cut for the family ofPpaths Γ if for every γ ∈ Γ, there is e ∈ C such that N (γ, e) = 1. The size of a cut is measured by |C| := e∈C σ(e).
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When G is undirected, it can be thought of as an electrical network with edge conductances given by the weights σ, see [7]. In this case, given two vertices s and t in V , we write Ceff (s, t) for the effective conductance between s and t. The following result is a slight modification of the results in [2, Section 5], taking into account the definition of Nmin in (5). The theorem stated there does not hold in this context verbatim, but can be easily adapted. The only issue to take care of is the value of Nmin . Since the previous paper dealt only with families of walks, N was integer valued and, thus, Nmin could be assumed no smaller than 1. This gave rise to an inequality of the form 0 ≤ ρ∗ ≤ 1 that was used to establish a monotonicity property. When N is not restricted to integer −1 values, the bound on ρ∗ should be replaced by 0 ≤ ρ∗ ≤ Nmin (see Remark 1.3 (c)). Repeating the proof of [2, Thm. 5.2] with the corrected upper bound and rephrasing in the current context yields the following theorem. Theorem 1.5 ([2]). Let G = (V, E) be a graph P with edge weights σ. Let Γ be a nontrivial family of objects on G with usage matrix N and let σ(E) := e∈E σ(e). Then the function p 7→ Modp,σ (Γ) is continuous for 1 ≤ p < ∞, and the following two monotonicity properties hold for 1 ≤ p ≤ p′ < ∞. ′
Moreover,
p p Modp′ ,σ (Γ), Nmin Modp,σ (Γ) ≥ Nmin 1/p′ 1/p . σ(E)−1 Modp,σ (Γ) ≤ σ(E)−1 Modp′ ,σ (Γ)
• For p = ∞: 1
lim Modp,σ (Γ) p = Mod∞,1 (Γ) =
p→∞
1 . ℓ(Γ)
• For p = 1, when Γ = Γ(s, t) is a connecting family, Mod1,σ (Γ) = min{|C| : C a cut of Γ} 5
(8) (9)
• For p = 2, when G is undirected and Γ = Γ(s, t) is a connecting family, Mod2,σ (Γ) = Ceff (s, t), 85
Remark 1.6. An early version of the case p = 2 is due to Duffin [8]. The proof in [2] was guided by a very general result in metric spaces [12]. Example 1.7 (Basic Example). Let G be a graph consisting of k simple paths in parallel, each path taking ℓ hops to connect a given vertex s to a given vertex t. Assume also that G is unweighted, that is σ ≡ 1. Let Γ be the family consisting of the k simple paths from s to t. Then ℓ(Γ) = ℓ and the size of the minimum cut is k. A straightforward computation shows that Modp (Γ) =
k ℓp−1
for 1 ≤ p < ∞,
Mod∞,1 (Γ) =
1 . ℓ
In particular, Modp (Γ) is continuous in p, and limp→∞ Modp (Γ)1/p = Mod∞,1 (Γ). Intuitively, when p ≈ 1, Modp (Γ) is more sensitive to the number of parallel paths, while for p ≫ 1, Modp (Γ) is more sensitive to short walks. 90
95
100
By Theorem 1.5, the concept of modulus encompasses classical quantities such as shortest path, minimal cut, and effective conductance. For this reason, modulus has many advantages. For instance, in order to give effective conductance a proper interpretation in terms of electrical networks one needs to consider the Laplacian operator which on undirected graphs is a symmetric matrix. On directed graphs however the Laplacian ceases to be symmetric, so the electrical network model breaks down. The definition of modulus, however, does not rely on the symmetry of the Laplacian and, therefore, can still be defined and computed in this case. Moreover, modulus can measure the richness of many types of families of objects, not just connecting walk families. Here are some examples of families of objects that can be measured using modulus and that we have been actively investigating: • Spanning Tree Modulus: All spanning trees of G, [3]. • Loop Modulus: All simple cycles in G, [19] • Via Modulus: All walks that start at s, end at t, and visit u along the way, [20].
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110
There are many more families that we are either in the process or intend to investigate. For instance, families of perfect or full matchings, families of long paths (e.g., all simple paths with at least L hops), and the family of all the stars in a graph (a star is the set of edges that are incident at a vertex). 1.4. Lagrangian Duality and the Probabilistic Interpretation The optimization problem (7) is an ordinary convex program, in the sense of [17, Sec. 28]. Existence of a minimizer follows from compactness, and uniqueness holds when 1 < p < ∞ by strict convexity of the objective function. Furthermore, it can be shown that strong duality holds in the sense that a maximizer of the Lagrangian dual problem exists and has dual energy equal to the modulus. The Lagrangian dual problem was derived in detail in [2]. The Lagrangian dual was later reinterpreted in a probabilistic setting in [4]. In order to formulate the probabilistic dual, we let P(Γ) represent the set of probability mass functions (pmfs) on the set Γ. In other words, P(Γ) contains the set of vectors µ ∈ RΓ≥0 with the property that 6
115
µT 1 = 1. Given such a µ, we can define a Γ-valued random variable γ with distribution given by µ: Pµ γ = γ = µ(γ). Given an edge e ∈ E, the value N (γ, e) is again a random variable, and we represent its expectation (depending on the pmf µ) as Eµ N (γ, e) . The probabilistic interpretation of the Lagrangian dual can now be stated as follows. Theorem 1.8. Let G = (V, E) be a finite graph with edge weights σ, and let Γ be a non-trivial finite family of objects on G with usage matrix N . Then, for any 1 < p < ∞, letting q = (p − 1)/p be the conjugate exponent to p, we have 1 −p
Modp,σ (Γ)
=
min
µ∈P(Γ)
X
q −p
σ(e)
e∈E
Moreover, any optimal measure µ∗ , must satisfy
Eµ
q N (γ, e)
! 1q
.
(10)
p
Eµ∗
σ(e)ρ∗ (e) q N (γ, e) = Modp,σ (Γ)
∀e ∈ E,
where ρ∗ is the unique extremal density for Modp,σ (Γ).
120
Theorem 1.8 is a consequence of the theory developed in [4]. However, since it was only remarked on in [4], we provide a detailed proof here. P ROOF. The optimization problem (7) is a standard convex optimization problem. Its Lagrangian dual problem, derived in [2], is p p−1 X X X 1 N (γ, e)λ(γ) maximize λ(γ) − (p − 1) σ(e) (11) pσ(e) γ∈Γ
γ∈Γ
e∈E
subject to λ(γ) ≥ 0 ∀γ ∈ Γ.
It can be readily verified that strong duality holds (i.e., that the minimum in (7) equals the maximum in (11)) and that both extrema are attained. Moreover, if ρ∗ is the unique minimizer of the modulus problem and λ∗ is any maximizer of the Lagrangian dual, then the optimality conditions imply that
ρ∗ (e) =
1 pσ(e)
X
γ∈Γ
1 p−1
N (γ, e)λ∗ (γ)
.
By decomposing λ ∈ RΓ≥0 as λ = νµ with ν ≥ 0 and µ ∈ P(Γ), we can rewrite (11) as q q X X q ν N (γ, e)µ(γ) . max ν − (p − 1) min σ(e)− p ν≥0 p µ∈P(Γ) e∈E
(12)
γ∈Γ
The minimum over µ can be recognized as the minimum in (10). Let α be its minimum value. Then the p maximum over ν ≥ 0 is attained at ν ∗ := pα− q , and strong duality implies that ∗ q p ν ∗ α = α− q . Modp,σ (Γ) = ν − (p − 1) p 7
Thus, min
µ∈P(Γ)
X
e∈E
q q q σ(e)− p Eµ N (γ, e) = α = Modp,σ (Γ)− p ,
proving (10). The remainder of the theorem follows from (12):
1 p−1 ∗ X q q ν ρ∗ (e) = = α−1 σ(e)− p Eµ∗ N (γ, e) p N (γ, e)µ∗ (γ) pσ(e)
γ∈Γ
Remark 1.9. The probabilistic interpretation is particularly informative when p = 2, σ ≡ 1, and Γ is a collection of subsets of E, so that N is a (0, 1)-matrix defined as N (γ, e) = 1γ (e). In this case, this duality relation can be expressed as Mod2 (Γ)−1 = min Eµ γ ∩ γ ′ , µ∈P(Γ)
′
where γ and γ are two independent random variables chosen according to the pmf µ, and γ ∩ γ ′ is their overlap (also a random variable). In other words, computing the 2-modulus in this setting is equivalent to finding a pmf that minimizes the expected overlap of two iid Γ-valued random variables.
In the present work, we are interested in a different but closely related duality called blocking duality. 125
2. Blocking Duality and p-Modulus In this section, we review blocking duality for modulus. In order to do so, we shall focus on the set of inequalities defining Adm(Γ). 2.1. Fulkerson’s theorem
130
First, we recall some general definition. Let K be the set of all closed convex sets K ⊂ RE ≥0 that are E recessive, in the sense that K + R≥0 = K. To avoid trivial cases, we shall assume that ∅ ( K ( RE ≥0 , for K ∈ K. Definition 2.1. For each K ∈ K there is an associated blocking polyhedron, or blocker, T BL(K) := η ∈ RE ≥0 : η ρ ≥ 1, ∀ρ ∈ K .
Definition 2.2. Given K ∈ K and a point x ∈ K we say that x is an extreme point of K if x = tx1 +(1−t)x2 for some x1 , x2 ∈ K and some t ∈ (0, 1), implies that x1 = x2 = x. Moreover, we let ext(K) be the set of all extreme points of K . Definition 2.3. The dominant of a set P ⊂ RE ≥0 is the recessive closed convex set Dom(P ) = co(P ) + RE ≥0 . 135
If Γ is a finite non-trivial family of objects on a graph G, the admissible set Adm(Γ) is determined by finitely many inequalities, in particular Adm(Γ) has finitely many faces. However, Adm(Γ) is also determined by its finitely many extreme points, or “vertices”. In fact, it is well-known that Adm(Γ) is the dominant of its extreme points ext(Adm(Γ)), see [17, Theorem 18.5].
8
Definition 2.4. Suppose G = (V, E) is a finite graph and Γ is a finite non-trivial family of objects on G. We say that the family ˆ := ext(Adm(Γ)) = {ˆ Γ γ1 , . . . , γˆs } ⊂ RE ≥0 , 140
consisting of the extreme points of Adm(Γ), is the Fulkerson blocker of Γ. Also, we define the matrix ˆ ˆ ∈ RΓ×E ˆ N ˆ T , for γˆ ∈ Γ. ≥0 to be the matrix whose rows are the vectors γ Theorem 2.5 (Fulkerson [9]). Let G = (V, E) be a graph and let Γ be a non-trivial finite family of objects ˆ on G. We identify Γ with its edge-usage functions, hence we think of Γ as a finite subset of RE ≥0 . Let Γ be the Fulkerson blocker of Γ. Then ˆ = BL(Adm(Γ)); ˆ (1) Adm(Γ) = Dom(Γ)
145
ˆ = Dom(Γ) = BL(Adm(Γ)); (2) Adm(Γ) ˆˆ ⊂ Γ. (3) Γ ˆ are a subset of Γ. Combining (1) and (2) we get the In words, (3) says that the extreme points of Adm(Γ) following. Corollary 2.6. Let G = (V, E) be a graph and let Γ be a nontrivial finite family of objects on G. Identify Γ with the subset of RE ≥0 consisting of all the edge-usage functions for objects in Γ. Then, BL(BL(Adm(Γ))) = Adm(Γ)
and
BL(BL(Dom(Γ))) = Dom(Γ).
Adm(Γ) = BL (Dom(Γ))
and
BL(Adm(Γ)) = Dom(Γ).
as well as We include a proof of Theorem 2.5 for the reader’s convenience. P ROOF. We first prove (2). Suppose η ∈ BL(Adm(Γ)). Then η T ρ ≥ 1, for every ρ ∈ Adm(Γ). In particular, ˆ η T γˆ ≥ 1 ∀ˆ γ ∈ Γ. (13) ˆ Conversely, suppose η ∈ Adm(Γ), ˆ that is (13) holds. Since In other words, η ∈ Adm(Γ). ˆ + RE Adm(Γ) = co(Γ) ≥0 , ˆ and a vector z ≥ 0 such that for every ρ ∈ Adm(Γ), there is a probability measure µ ∈ P(Γ) ˆ
ρ=
|Γ| X
µ(j)ˆ γj + z.
j=1
And by (13), ˆ
T
η ρ=
|Γ| X
µ(j)η T γˆj + η T z ≥ 1.
j=1
150
So η ∈ BL(Adm(Γ)).
9
Note that η ∈ BL(Adm(Γ)) if and only if the value of the following linear program is greater or equal 1. ηT ρ N ρ ≥ 1, ρ ≥ 0,
minimize subject to
(14)
where N is the usage matrix for Γ. The Lagrangian in this case is L(ρ, λ, ν) := η T ρ + λT (1 − N λ) − ν T ρ = λT 1 + ρT (η − N T λ − ν), with ρ ∈ RE , λ ∈ RΓ≥0 and ν ∈ RE ≥0 . In particular, the dual problem is maximize
λT 1
subject to
N T λ ≤ η, λ ≥ 0.
(15)
Splitting λ = νµ, with ν ≥ 0 and µ ∈ P(Γ), we can rewrite this problem as maximize
ν
subject to
νN T µ ≤ η, µ ∈ P(Γ).
(16)
By strong duality, η ∈ BL(Adm(Γ)) if and only if there is ν ≥ 1 and µ ∈ P(Γ) so that η ≥ νN T µ Namely, η ∈ BL(Adm(Γ)) implies that η ≥ N T µ, so η ∈ Dom(Γ). Conversely, if η ∈ Dom(Γ), then ˆˆ are the extreme points of ν = 1 works and η ∈ BL(Adm(Γ)). So we have proved (2). In particular, since Γ ˆ and Adm(Γ) ˆˆ ˆ = Dom(Γ), Dom(Γ) = Adm(Γ) ˆ ˆ ⊂ Γ, hence (3) is proved as well. we conclude that Γ ˆ and find that To prove (1), we apply (2) to Adm(Γ) ˆˆ ˆ = Adm(Γ) BL(Adm(Γ)) ⊃ Adm(Γ), ˆˆ ˆ the extreme points of ⊂ Γ. Also, by (3) applied to Γ, where the last inclusion follows from (3), since Γ ˆˆ ˆ Adm(Γ) are a subset of Γ and therefore they are a subset of the vertices of Adm(Γ). This implies that ˆˆ ⊂ Adm(Γ). So we have BL(Adm(Γ)) ˆ = Adm(Γ). Adm(Γ) ˆ Moreover, by (2) applied to Γ, we get that ˆ = Dom(Γ). ˆ BL(Adm(Γ)) 155
So (1) is proved as well.
10
2.2. Blocking duality for p-modulus Theorem 2.7. Let G = (V, E) be a graph and let Γ be a nontrivial finite family of objects on G with usage matrix N . Let the exponent 1 < p < ∞ be given, with q = (p − 1)/p its conjugate exponent. For any set of q weights σ ∈ RE ˆ as σ ˆ (e) := σ(e)− p , for all e ∈ E. >0 define the dual set of weights σ Then 1 ˆ q1 = 1. Modp,σ (Γ) p Modq,ˆσ (Γ) (17) ˆ are unique and are related as follows: Moreover, the optimal ρ∗ ∈ Adm(Γ) and η ∗ ∈ Adm(Γ) ρ∗ (e) = Also
σ ˆ (e)η ∗ (e)q−1 ˆ Modq,ˆσ (Γ)
∀e ∈ E.
(18)
ˆ = 1. Mod1,σ (Γ) Mod∞,σ−1 (Γ)
(19)
Remark 2.8. The case for p = 2 is essentially contained in [14, Lemma 2], although not stated in terms of modulus and with a different proof. However, it is worth stating it separetely. Namely, ˆ = 1. Mod2,σ (Γ) Mod2,σ−1 (Γ) And σ(e)ρ∗ (e) = Mod2,σ (Γ)η ∗ (e) 160
∀e ∈ E,
in this case. ˆ Hölder’s inequality implies that P ROOF. For all ρ ∈ Adm(Γ) and η ∈ Adm(Γ), X X 1≤ ρ(e)η(e) = σ(e)1/p ρ(e) σ(e)−1/p η(e) e∈E
e∈E
≤
X
p
σ(e)ρ(e)
e∈E
so
!1/p
X
q
σ ˆ (e)η(e)
e∈E
!1/q
(20) ,
ˆ 1/q ≥ 1. Modp,σ (Γ)1/p Modq,ˆσ (Γ)
(21)
ˆ −1 and let η ∗ ∈ Adm(Γ) ˆ be the minimizer for Modq,ˆσ (Γ). ˆ Then (21) implies Now, let α := Modq,ˆσ (Γ) that p 1 (22) Modp,σ (Γ) ≥ α q = α q−1 . Define ∗
ρ (e) := α Note that Ep,σ (ρ∗ ) =
X
e∈E ∗
σ ˆ (e) ∗ q η (e) σ(e)
σ(e)ρ∗ (e)p = αp
1/p
X
= αˆ σ (e)η ∗ (e)q/p .
(23) 1
σ ˆ (e)η ∗ (e)q = αp−1 = α q−1 .
e∈E
Thus, if we can show that ρ ∈ Adm(Γ), then (22) is attained and ρ∗ must be extremal for Modp,σ (Γ). In particular, (17) would follow. Moreover, (18) is another way of writing (23).
11
P ˆ First, consider To see that ρ∗ ∈ Adm(Γ), we will verify that e∈E ρ∗ (e)η(e) ≥ 1 for all η ∈ Adm(Γ). ∗ η = η . In this case X X ρ∗ (e)η ∗ (e) = α σ ˆ (e)η ∗ (e)q = 1. e∈E
e∈E
ˆ be arbitrary. Since Adm(Γ) ˆ is convex, we have that (1 − θ)η ∗ + θη ∈ Adm(Γ) ˆ for Now let η ∈ Adm(Γ) all θ ∈ [0, 1]. So, using Taylor’s theorem, we have X q α−1 = Eq,ˆσ (η ∗ ) ≤ Eq,ˆσ ((1 − θ)η ∗ + θη) = σ ˆ (e) [(1 − θ)η ∗ (e) + θη(e)] e∈E
−1
=α
+ qθ
X
∗
q−1
σ ˆ (e)η (e)
(η(e) − η ∗ (e)) + O(θ2 )
e∈E −1
=α
−1
+α
qθ
X
ρ∗ (e) (η(e) − η ∗ (e)) + O(θ2 ).
e∈E
Since this inequality must hold for arbitrarily small θ > 0, it follows that X X ρ∗ (e)η(e) ≥ ρ∗ (e)η ∗ (e) = 1, e∈E
e∈E
and the proof is complete. 2.3. The cases p = 1 and p = ∞ Now we turn our attention to establishing the duality relationship in the cases p = 1 and p = ∞. Recall that by Theorem 1.5, 1 1 , lim Modp,σ (Γ) p = Mod∞,1 (Γ) = p→∞ ℓ(Γ) 165
where ℓ(Γ) is defined to be the smallest element of the vector N 1. In order to pass to the limit in (17), we need to establish the limits for the second term in the left-hand side product. Lemma 2.9. Under the assumptions of Theorem 2.7, 1
ˆ q = Mod1,1 (Γ) ˆ and lim Modq,ˆσ (Γ)
q→1
ˆ 1q = Mod∞,σ−1 (Γ), ˆ lim Modq,ˆσ (Γ)
(24)
q→∞
where σ −1 (e) = σ(e)−1 . Γ×E ˆ respectively. Let σ ∈ RE×E be the diagonal P ROOF. Let N , Nˆ ∈ R≥0 be the usage matrices for Γ and Γ ˜ its associated family in RE . Note that matrix with entries σ(e, e) = σ(e), and define N˜ = Nˆ σ, with Γ ≥0 ˆ if and only if σ −1 η ∈ Adm(Γ). ˜ Moreover, for every η ∈ Adm(Γ), ˆ η ∈ Adm(Γ) q X X η(e) = Eq,σ (σ −1 η), Eq,ˆσ (η) = σ ˆ (e)η(e)q = σ(e) σ(e) e∈E
which implies that
e∈E
ˆ = Modq,σ (Γ). ˜ Modq,ˆσ (Γ) 12
Taking the limit as q → 1 and using Theorem 1.5 shows that 1
ˆ q = Mod1,σ (Γ) ˜ = lim Modq,ˆσ (Γ)
q→1
min
ˆ η∈Adm(Γ)
X
σ(e)
e∈E
η(e) σ(e)
ˆ = Mod1,1 (Γ).
Taking the limit as q → ∞ and using Theorem 1.5 shows that ˆ 1q = Mod∞,1 (Γ) ˜ = lim Modq,ˆσ (Γ)
q→∞
min
max
ˆ e∈E η∈Adm(Γ)
η(e) σ(e)
ˆ = Mod∞,σ−1 (Γ).
Taking the limit as p → 1 in Theorem 2.7 then gives the following theorem. Theorem 2.10. Under the assumptions of Theorem 2.7, ˆ = 1. Mod1,σ (Γ) Mod∞,σ−1 (Γ) 170
(25)
Note that taking the limit as p → ∞ simply yields the same result for the unweighted case. 3. Blocking Duality for Families of Objects 3.1. Duality for 1-modulus Suppose that G = (V, E, σ) is a weighted graph, with weights σ ∈ RE ≥0 , and Γ is a non-trivial, finite family of subsets of E, where N be the corresponding usage matrix. In this case we can equate each γ ∈ Γ E E with the vector 1γ ∈ RE ≥0 , so we think of Γ as living in {0, 1} ⊂ R≥0 . Recall that Mod1,σ (Γ) is the value of the linear program: minimize
σT ρ
subject to
ρ ≥ 0,
Nρ ≥ 1
(26)
Since it’s a linear program, strong duality holds, and the dual problem is maximize
λT 1
subject to
λ ≥ 0,
N T λ ≤ σ.
(27)
We think of (27) as a (generalized) max-flow problem, given the weights σ. That’s because the condition N T λ ≤ σ says that for every e ∈ E X λ(γ) ≤ σ(e). γ∈Γ
e∈γ
However, to think of (26) as a (generalized) min-cut problem, we would need to be able to restrict the densities ρ to some given subsets of E. That’s exactly what the Fulkerson blocker does. Proposition 3.1. Suppose G = (V, E) is a finite graph and Γ is a family of subsets of E with Fulkerson ˆ Then for any set of weights σ ∈ RE , blocker family Γ. >0 Mod1,σ (Γ) = min σ(ˆ γ ). ˆ γ ˆ ∈Γ
175
ˆ there is a choice of σ ∈ RE such that γˆ is the unique solution of (28). Moreover, for every γˆ ∈ Γ ≥0 13
(28)
P ROOF. By Theorem 2.5(1)
ˆ Adm(Γ) = Dom(Γ)
So if σ ∈ RE >0 is a given set of weights, then, by (26), Mod1,σ (Γ) is the value of the linear program minimize
σT ρ
subject to
ˆ ρ ∈ Dom(Γ).
(29)
ˆ namely for an object γˆ ∈ Γ. ˆ Therefore, In particular, the optimal value is attained at a vertex of Dom(Γ), ˆ the optimization can be restricted to Γ.
180
185
Remark 3.2. When Γ is a family of subsets of E, it is customary to say that Γ has the max-flow-minˆ is also a family of subsets of E. For more details we refer to the cut property, if its Fulkerson blocker Γ discussion in [13, Chapter 3]. 3.2. Connecting families Let G be an undirected graph and let Γ = Γ(s, t) be the family of all simple paths connecting two distinct nodes s and t, i.e., the st-paths in G. The Fulkerson blocker in this case is the family Γcut (s, t) of all minimal st-cuts. Recall that an st-cut is called minimal if it contains no other st-cuts as a strict subset. To see this note that (27) in this case is exactly the Max-Flow problem. Moreover, the Max-Flow-MinCut Theorem implies that (26) is always attained by a minimal st-cut, and therefore Proposition 3.1 shows ˆ is a minimal st-cut. Conversely, if γˆ is a minimal st-cut, then we can define σ to that every element of Γ be very small on γˆ and large otherwise, so that γˆ is the unique solution of the Min-Cut problem. Therefore, ˆ Γcut (s, t) = Γ. Moreover, the duality 1 ˆ 1q = 1 Modp,σ (Γ) p Modq,ˆσ (Γ) can be viewed as a generalization of the max-flow min-cut theorem. To see this, consider the limiting case given by Theorem 2.10: ˆ = 1. Mod1,σ (Γ) Mod∞,σ−1 (Γ) (30) As discussed above, Mod1,σ (Γ) takes the value of the minimum st-cut with edge weights σ. With a little work, the second modulus in (30), can be recognized as the reciprocal of the corresponding max flow probˆ can also be transformed lem. Using the standard trick for ∞-norms, the modulus problem Mod∞,σ−1 (Γ) into a linear program taking the form minimize
t
subject to
σ(e)−1 η(e) ≤ t ∀e ∈ E ˆη ≥ 1 η ≥ 0, N
ˆ and, therefore, by Theorem 2.5(2), must The minimum must occur somewhere on the boundary of Adm(Γ) take the form X X λ(γ) ≥ 0, λ(γ) = 1. η(e) = λ(γ)1γ (e) γ∈Γ
γ∈Γ
14
In other words, the minimum occurs at a unit st-flow η, and the problem can be restated as t 1 η(e) ≤ σ(e) ∀e ∈ E t η a unit st-flow
minimize subject to
190
The minimum is attained when 1t η is a maximum st-flow respecting edge capacities σ(e); the value of such a flow is 1/t, recovering the max-flow min-cut theorem. 3.3. Spanning tree modulus When Γ is the set of spanning trees on an unweighted, undirected graph G with N (γ, ·) = ˆ can be interpreted as the set of (weighted) feasible partitions [6]. Fulkerson blocker Γ
195
1γ (·), the
Definition 3.3. A feasible partition P of a graph G = (V, E) is a partition of the vertex set V into two or more subsets, {V1 , . . . , VkP }, such that each of the induced subgraphs G(Vi ) is connected. The corresponding edge set, EP , is defined to be the set of edges in G that connect vertices belonging to different Vi ’s. The results of [6] imply the following theorem. Theorem 3.4. Let G = (V, E) be a simple, connected, unweighted, undirected graph and let Γ be the family of spanning trees on G. Then the Fulkerson blocker of Γ is the set of all vectors 1 1E . kP − 1 P
200
ranging over all feasible partitions P . 4. Blocking Duality and the Probabilistic Interpretation At the end of Section 1.4 it was claimed that blocking duality was closely related to Lagrangian duality. In this section, we make this connection explicit. Theorem 4.1. Let G = (V, E, σ) be a graph and Γ a finite family of objects on G with Fulkerson blocker ˆ For a given 1 < p < ∞, let µ∗ be an optimal pmf for the minimization problem in (10) and let η ∗ be Γ. ˆ Then, in the notation of Section 1.4, optimal for Modq,ˆσ (Γ). η ∗ (e) = Eµ∗ N (γ, e) . ˆ can be written as the sum of a convex combination of the vertices of Adm(Γ) ˆ P ROOF. Every η ∈ Adm(Γ) E ˆ and a nonnegative vector. In other words, η ∈ Adm(Γ) if and only if there exists µ ∈ P(Γ) and η0 ∈ R≥0 such that η = N T µ + η0 . Or, in probabilistic notation, X η(e) = N (γ, e)µ(γ) + η0 (e) = Eµ N (γ, e) + η0 (e). γ∈Γ
For such an η, Eq,ˆσ (η) =
X
q
σ(e)− p η(e)q ≥
X
e∈E
e∈E
15
q q σ(e)− p Eµ N (γ, e)
205
with equality holding if and only if η0 = 0. This implies that the optimal η ∗ must be of the form η ∗ = N T µ′ = Eµ′ N (γ, ·) for some µ′ ∈ P(Γ). Now, let µ∗ be any optimal pmf for (10) and let η ′ = N T µ∗ . As a convex combination of the rows of ˆ Moreover, by optimality of µ∗ , N , η ′ ∈ Adm(Γ). X q X q q q Eq,ˆσ (η ′ ) = σ(e)− p Eµ′ N (γ, e) = Eq,ˆσ (η ∗ ). σ(e)− p Eµ∗ N (γ, e) ≤ e∈E
e∈E
′ ∗ ∗ T ∗ ˆ But, since 1 < q < ∞, the minimizer for Modq,ˆσ (Γ) is unique and, therefore, η = η . So η = N µ = Eµ∗ N (γ, ·) as claimed.
5. Randomly weighted graphs
210
In this section we explore the main arguments in [14] and recast them in the language of modulus. The goal is to study graphs G = (V, E, σ) where the weights σ ∈ RE >0 are random variables and compare modulus computed on G to the corresponding modulus computed on the deterministic graph EG := (V, E, Eσ). Theorem 5.3 below is a reformulation of Theorem 7 in [14], which generalized Theorem 2.1 in [15]. First we recall a lemma from Lovász’s paper. Lemma 5.1 ([14, Lemma 9]). Let W ∈ RE >0 be a random variable with survival function S(t) := P (W ≥ t) ,
for t ∈ RE ≥0 .
If S(t) is log-concave, then the survival function of mine∈E W (e) is also log-concave and W satisfies !−1 X 1 . (31) E min W (e) ≥ e∈E E(W (e)) e∈E
215
Property (31) is satisfied if for instance the random variables {W (e)}e∈E are independent and distributed as exponential variables Exp(λ(e)), i.e., so that P(W (e) > t) = min{exp(−λ(e)t), 1}. It is useful to collect some properties of random variables with log-concave survival functions. Proposition 5.2. Let W ∈ RE >0 be a random variable with log-concave survival function. Then the following random variables also have log-concave survival function: (a) CW , where C = Diag(c(·)) with c ∈ RE >0 . ∗
220
∗
E (b) W ∗ , where E ∗ ⊂ E, and W ∗ ∈ RE >0 is the projection of W onto R>0 .
P ROOF. We define S(t) := P(W ≥ t) for t ∈ RE ≥0 . For (a), note that log P (CW ≥ t) = log S(C −1 t), which is the composition of a concave function with an affine function. Likewise (b) follows by composing a concave function with a projection. Theorem 5.3. Let G = (V, E, σ) be a simple finite graph. Assume the σ is a random variable in RE >0 with the property that its survival function is log-concave. Let Γ be a finite non-trivial family of objects on G, with Nmin defined as in (5). Then E Mod1,σ (Γ) ≥ Nmin Mod2,Eσ (Γ). 16
ˆ be the Fulkerson blocker of Γ. Let ρ∗ be extremal for Mod2,Eσ (Γ) and η ∗ be extremal for P ROOF. Let Γ ˆ Also let µ∗ ∈ P(Γ) be an optimal measure, then we know that Mod2,(Eσ)−1 (Γ). η ∗ (e) =
X Eσ(e)ρ∗ (e) = µ∗ (γ)N (γ, e) = Eµ∗ (N (γ, e)) , Mod2,E(σ) (Γ)
∀e ∈ E.
(32)
γ∈Γ
To avoid dividing by zero let E ∗ := {e ∈ E : η ∗ (e) > 0} and let Γ∗ := {γ ∈ Γ : µ∗ (γ) > 0}. Note that, if e 6∈ E ∗ , then X 0 = η ∗ (e) = µ∗ (γ)N (γ, e), γ∈Γ∗
hence N (γ, e) = 0 for all γ ∈ Γ∗ . Therefore, for any ρ ∈ Adm(Γ) and γ ∈ Γ∗ , X N (γ, e)ρ(e) = ℓρ (γ) ≥ 1.
(33)
e∈E ∗
Now, fix an arbitrary ρ ∈ Adm(Γ). Then, by (32), E1,σ (ρ) ≥ Mod2,Eσ (Γ)
X
σ(e)ρ(e)
e∈E ∗
Note that X
σ(e)ρ(e)
1 Eµ∗ (N (γ, e)) Eσ(e)ρ∗ (e)
X X σ(e) 1 Eµ∗ (N (γ, e)) = N (γ, e)ρ(e) µ∗ (γ) ∗ Eσ(e)ρ (e) Eσ(e)ρ∗ (e) ∗ ∗ γ∈Γ
e∈E ∗
(34)
≥
e∈E
X
µ∗ (γ)
X
µ∗ (γ)
X
µ∗ (γ)
γ∈Γ∗
≥
γ∈Γ∗
min∗
e∈E N (γ,e)6=0
X σ(e) N (γ, e)ρ(e) ∗ Eσ(e)ρ (e) ∗ e∈E
min∗
σ(e) , Eσ(e)ρ∗ (e)
min∗
σ(e) Eσ(e)ρ∗ (e)
e∈E N (γ,e)6=0
where the last inequality follows by (33). Minimizing in (34) over ρ ∈ Adm(Γ) we find Mod1,σ (Γ) ≥ Mod2,Eσ (Γ)
γ∈Γ
e∈E N (γ,e)6=0
Note that for each γ ∈ Γ∗ , by Proposition (5.2) (a) and (b) and Lemma 5.1, the scaled random variables X(e) :=
σ(e) Eσ(e)ρ∗ (e)
for e ∈ E ∗ with N (γ, e) 6= 0,
have the property that
E min∗ X(e) ≥ e∈E N (γ,e)6=0
X
e∈E ∗ N (γ,e)6=0
−1
1 E(X(e)) 17
=
X
e∈E ∗ N (γ,e)6=0
−1
ρ∗ (e)
.
Moreover, by (5),
X
−1
e∈E ∗ N (γ,e)6=0
ρ∗ (e)
≥ Nmin
X
e∈E ∗ N (γ,e)6=0
−1
N (γ, e)ρ∗ (e)
.
Finally, by complementary slackness, since γ ∈ Γ∗ , we have µ∗ (γ) > 0, hence X N (γ, e)ρ∗ (e) = 1. e∈E ∗ N (γ,e)6=0
This gives the claim. Theorem 5.3 has some interesting consequences for p-modulus on randomly weighted graphs. First, recall from [2, Lemma 6.5] that the map σ 7→ Modp,σ (Γ) is concave for 1 ≤ p < ∞. In particular, if σ ∈ RE >0 is a random variable, then by Jensen’s inequality: E Modp,σ (Γ) ≤ Modp,Eσ (Γ). 225
(35)
The following corollary gives a lower bound. Corollary 5.4. Let G = (V, E, σ) be a simple finite graph. Assume σ is a random variable in RE >0 with log-concave survival function. Let Γ be a finite non-trivial family of objects on G with Nmin defined as in (5). Then, for 1 ≤ p ≤ 2, p Nmin Modp,Eσ (Γ)2 . (36) E Modp,σ (Γ) ≥ Eσ(E) P ROOF. When 1 < p ≤ 2 we have, by (9), Mod2,Eσ (Γ) ≥ Eσ(E)1−2/p Modp,σ (Γ)2/p . So by Theorem 5.3 we get E Mod1,σ (Γ) ≥ Nmin Eσ(E)1−2/p Modp,Eσ (Γ)2/p .
(37)
Letting p → 1 and by continuity in p (Theorem 1.5) we get E Mod1,σ (Γ) ≥
Nmin Mod1,Eσ (Γ)2 Eσ(E)
Moreover, estimating the 1-modulus in terms of p-modulus, using (9) a second time, and then applying Hölder’s inequality gives 1/p (38) E Mod1,σ (Γ) ≤ E σ(E)1/q Modp,σ (Γ)1/p ≤ Eσ(E)1/q E (Modp,σ (Γ))
Combining (37) and (38) gives (36).
18
Remark 5.5. By combining (35) with (36) we find that, for 1 ≤ p ≤ 2, Modp,Eσ (Γ) ≤
Eσ(E) . p Nmin
−1 This is not a contradiction because this inequality is always satisfied, since the constant density ρ ≡ Nmin is always admissible.
230
235
Corollary 5.4 leads one to wonder what lower bounds can be established for E Mod2,σ (Γ) when σ is allowed to vanish and its survival function is not necessarily log-concave. For instance, it would be interesting to study what happens when the weights σ(e) are independent Bernoulli variables, namely when G is an Erd˝osRényi graph. The situation there is complicated by the fact that the family Γ will change with every new sample of the weights σ. For instance, the family of all spanning trees will be different for different choices of σ. References [1] A HLFORS , L. V. Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, 1973. [2] A LBIN , N., B RUNNER , M., P EREZ , R., P OGGI -C ORRADINI , P., AND W IENS , N. Modulus on graphs as a generalization of standard graph theoretic quantities. Conform. Geom. Dyn. 19 (2015), 298–317. http://dx.doi.org/10.1090/ecgd/287.
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