Lec01 network flows

Lecture 1: Network Flows

Wai-Shing Luk (陆伟成)

Fudan University

2012年 8月 11日

W.-S. Luk (Fudan Univ.) Lecture 1: Network Flows 2012年 8月 11日 1 / 13

Basic Elements of Network

Definition

[Network] A network is a collection of finite-dimensional vector spaces of

nodes and edges

V = {v1, v2, · · · , vN}, where |V | = N.

E = {e1, e2, e3, · · · , eM} where |E | = M

which satisfies 2 requirements:

1 The boundary of each edge is comprised of the union of nodes

2 The intersection of any edges is either empty or a boundary node of

both edges.


Orientation

Definition (Orientation)

An orientation of an edge is an ordering of its boundary node {s, t}, where

s is called a source/initial node

t is called a target/terminal node

Definition (Coherent)

Two orientations to be the same is called coherent


Incidence Matrix AT

Definition (Incidence matrix)

A(i , j) =

+1 if vj is the initial node of ei

−1 if vj is the terminal node of ei

0 if vj is not the boundary node of ei


Chain (τ)

An edge (node) chain τ is an M (N)-tuple of scalar which assigns a

coefficient to each edge (node), where M (N) is the number of

distinct edges (nodes) in the network.

A chain may be viewed as an (oriented) indictor vector representing a

set of edges (nodes).

Example

[0, 0, 1, 1, 1], [0, 0, 1,−1, 1]


Discrete Boundary Operator ∂

The incidence matrix natually maps edges into their corresponding

boundary elements:

β = AT τ

A chain is called a cycle if it is in the nullspace of the boundary

operator, i.e.

AT τ = 0

A chain β is called a boundary of τ if it is in the range of the

boundary operator


Co-boundary Operator d

Definition (Co-boundary)

The co-boundary (or differential) operator

d = ∂∗ = (AT )∗ = A

Note

Nullspace of A is #components of a graph


Generalized Stokes’ Theorem

Conventional (integration):∫S

d ω̃ =

∮∂Sω̃

Discrete (pairing)

[τ,Aω] = [AT τ, ω]

where

τi =

1 if ei ∈ S ,

0 otherwise .


Fundamental Theorem of Calculus

Conventional (integration):∫ b

af (t)dt = F (b)− F (a)

Discrete (pairing)

[τ1,Ac0] = [AT τ1, c0]

2 4 3 5

7

c0

-1 -1 1

1

AT τ1

2 -1 2

4Ac0

1 1 1

1τ1

7

7


Divergence and Flow

Definition (Divergence)

div x = AT x

Definition (Flow)

x is called a flow if∑

div x = 0, where all positive entries of (div x) are

called sources and negative entries are called sinks.

Definition (Circulation)

A network is called a circulation if there is no source or sink. In other

words, div x = 0


Tension and Potential

Definition (Tension)

A tension (in co-domain) y is a differential of a potential u, i.e. y = Au.

Theorem (Tellgen’s)

Flow and tension are bi-orthogonal (isomorphic).

Proof.

0 = [AT x , u] = (AT x)Tu = xT (Au) = xT y


Path (P)

Definition (Path indicator vector)

A path indicator vector τ of P that

τi =

1 if ei ∈ P ,

0 otherwise .

Theorem

[total tension y on P] = [total potential on the boundary of P].

Proof.

yT τ = (Au)T τ = uT (AT τ) = uT (∂P) .


Cut (Q)

Definition (Cut)

Two node sets S and S ′ (the complement of S , i.e. V − S). A cut Q is an

edge set, denoted by [S , S ′]−. A cut indicator vector q (oriented) of Q is

defined as Ac where

ci =

1 if vi ∈ S ,

0 otherwise .

Theorem (Stokes’ theorem!)

[total divergence of x on S] = [total x across Q].

Proof.

(div x)T c = (AT x)T c = xT (Ac) = xTq .


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