Spanning Tree Modulus and Homogeneity of Graphs Derek Hoare 1 Brandon - - PowerPoint PPT Presentation

Spanning Tree Modulus and Homogeneity of Graphs

Derek Hoare1 Brandon Sit2 Sarah Tymochko3 Mentor: Dr. Nathan Albin

Kansas State University

1Kenyon College 2University of Portland 3College of the Holy Cross

SUMaR 2016

Background

Definition

A graph G = (V , E) is a set of vertices V and a set of edges E which identify two connected vertices.

Definition

A spanning tree of a graph G is a subgraph of G that contains no cycles and passes through every vertex in G.

Spanning Tree Example

G=(V,E) Some spanning trees on G

Notation

G = (V , E) a connected, undirected graph Γ = ΓG the set of spanning trees of G N ∈ RΓ×E

≥0

the usage matrix Nγ,e =

• 1

if e ∈ γ if e / ∈ γ ρ ∈ RE

≥0 a set of edge weights

Nρ ∈ RΓ

≥0

wρ(γ) :=

e∈E Nγ,eρe = (Nρ)γ

Example

N =   1 1 1 1 1 1 1 1 1 1 1 1  

Modulus

The primal problem

Long form

minimize

• e∈E

ρ2

e

subject to ρ ∈ RE

≥0

wρ(γ) ≥ 1 ∀γ ∈ Γ

Short form

Mod(Γ) := min

Nρ≥1 ρTρ

Notation

The dual problem

G = (V , E) a connected, undirected graph Γ = ΓG the set of spanning trees of G N ∈ RΓ×E

≥0

the usage matrix Nγ,e =

• 1

if e ∈ γ if e / ∈ γ µ ∈ PΓ :=

• µ ∈ RΓ

≥0 : µT1 = 1

• a pmf on Γ

η := N Tµ ∈ RE

≥0

ηe =

• γ∈Γ

Nγ,eµγ

Probabilistic Interpretation

µ ∈ PΓ a pmf on Γ γ a Γ-valued random variable Pµ

• γ = γ
• = µ(γ)

(µ(γ) is the probability of choosing γ) η = N Tµ ∈ RE

≥0

ηe =

• γ∈Γ

Nγ,eµγ =

• γ:e∈γ

Pµ

• γ = γ
• = Pµ
• e ∈ γ
• (ηe is the probability that e is in a random spanning tree)

Optimizing Expected Usage

The dual problem

Long form

minimize

• e∈E

Pµ

• e ∈ γ

2 subject to µ ∈ PΓ For any µ ∈ PΓ,

• e∈E

Pµ

• e ∈ γ
• =
• e∈E
• γ∈Γ

Nγ,eµγ =

• γ∈Γ

µγ

• e∈E

Nγ,e = |V | − 1

Short form

min

µ∈PΓ

• N Tµ

T N Tµ

• =

min

η=N T µ ηTη = (|V | − 1)2

|E| + |E| min

η=N T µ Var(η)

Relation to the Primal Problem

ρ∗ = η∗ (η∗)Tη∗ is the optimal set of edge weights for the primal problem. Mod(Γ) = 1 (η∗)Tη∗ is the modulus.

Definitions

Definition

A graph G is homogeneous if the optimal ρ (and consequently η) is constant.

Theorem

G is homogeneous iff ∃µ ∈ PΓ such that Pµ

• e ∈ γ
• = |V | − 1

|E| ∀e ∈ E.

Definition

A graph G is uniform if the uniform distribution on the set of spanning trees of G is an optimal probability mass function.

Example

Non-homogeneous graph

3/4 3 / 4 3 / 4 3/4

top edge must belong to every spanning tree can’t possibly use each edge 3/4 of the time

1 2 / 3 2 / 3 2/3

actual optimal η∗ choose a spanning tree with uniform probability

Example

Homogeneous graph

2 / 3 2/3 2/3 2/3 2 / 3 2/3

Can we choose µ to use each edge with equal probability? Yes! Choose uniformly from the trees above. (There are other optimal pmfs.)

Example

Homogeneous and Uniform graph

Figure: Every cycle is uniform homogeneous.

Condition for Uniform Homogeneity

Theorem

Each edge in a graph G is in the same number of spanning trees if and

• nly if G is a uniform, homogeneous graph.

Homogeneous Graphs

Research Question

What kind of graphs are homogeneous?

Definition

A graph G is d-regular if every vertex has degree d.

Definition

A graph G is k-connected if G cannot be disconnected by removing fewer than k vertices.

Homogeneous Graphs

Research Question

What kind of graphs aren’t homogeneous? Features of graphs that make them non-homogeneous

A “bridge” between sections of the graph Edges that are used more often than others

Non-Homogeneous Graphs

Example of a graph with a bridge:

4 7 4 7 4 7 4 7 4 7 4 7 4 7 4 7 4 7 4 7

1

4 7 4 7 4 7 4 7

Figure: 1-connected, 3-regular graph, labeled with η values

Non-Homogeneous Graphs

Example of a graph with edges used more frequently than others in spanning trees:

1 2 1 2

Figure: Bi-connected, 4-regular graph

If this graph was homogeneous, η∗ ≡ |V |−1

|E|

= 15

32 ≤ 1 2

Homogeneous Graphs

Component Theorem

Let G = (VG, EG) be an undirected multigraph. Let η∗ = N Tµ∗ be the

• ptimal expected edge usage for spanning tree modulus, and let ηmin be its

minimum value. Then there exists a connected, vertex-induced subgraph H = (VH, EH) of G such that all of the following hold.

1 EH is non-empty. 2 η(e) = ηmin for all e ∈ EH. 3 Every spanning tree T in the support of µ∗ restricts to a spanning

tree of H.

Homogeneous Graphs

Deflation Theorem

Let G be a non-homogeneous, undirected multigraph, and let H be a vertex-induced subgraph satisfying the conditions of the Component

• Theorem. Let µH and µG\H be optimal pmfs for TH and TG\H
• respectively. Define the pmf µ′ on TG so that µ′ is supported on T ′ and

µ′(TH ∪ TG\H) := µH(TH)µG\H(TG\H) Then µ′ is optimal for TG.

Small Deflation Example

Figure: Cycle on 6 nodes with added triangle Figure: Cycle on 5 nodes

Small Deflation Example

(a) P = 1

3

(b) P = 1

3

(c) P = 1

3

(d) P = 1

5

(e) P = 1

5

(f) P = 1

5

(g) P = 1

5

(h) P = 1

5

Deflation Example

(a) ηmin = 0.167 (b) ηmin = 0.167 (c) ηmin = 0.222 (d) ηmin = 0.286 (e) ηmin = 0.500

Figure: Example of Deflation

Homogeneous Graphs

Theorem

For k ≥ 2, if a graph is k-connected and k-regular, then it is also homogeneous.

Theorem [2]

For a random connected k-regular graph G, as the number of nodes n → ∞, G is almost surely k-connected

Example: k-regular, k-connected

Figure: A 4-connected 4-regular graph.

Applications

Network Security Problem [3]

If Alice is sending a message to Bob, and Eve is an eavesdropper, on an edge between Alice and Bob, what is the probability that Eve is able to intercept the message? Alice sends N coded pieces of the message A recipient needs K coded pieces of the message to recover the entire message (where K ≤ N) What if a link between Alice and a recipient fails? Then what is the probability?

Applications

Where Does Modulus Fit In?

Provides the best possible communication plan. Uses the edges as evenly as possible, decreasing the chance that Eve can intercept enough information to decode the message. Under an optimal pmf on the spanning trees, Eve’s probability of interception is approximately η(e).

Why Homogeneous Graphs?

If we have a homogeneous graph, then there is a very low probability that Eve can gain all of the information.

Applications

A

Applications

A E

Applications

A E A E A ηE = 2

3, which is Eve’s probability of interception.

Future Research

Work more on the network security application Find necessary conditions for homogeneity

Acknowledgements

Thank you to the following for making this research possible:

• Dr. Albin

Jason Clemens

• Dr. Korten and Dr. Yetter

SUMaR REU Support for this project has been provided by NSF grant DMS-1262877 and NSF grant DMS-1515810

References

[1] N. Albin and P. Poggi-Corradini. Minimal subfamilies and the probabilistic interpretation for modulus on graphs. Journal of Analysis, to appear. https://arxiv.org/abs/1605.08462. [2] B. Bollob´

• as. Random graphs. Academic Press, Inc. [Harcourt Brace

Jovanovich, Publishers], London, 1985. [3] A. Khan, A. Tassi, and I. Chatzigeorgiou. Rethinking the intercept probability

• f random linear network coding. IEEE Communications Letters, to appear.

http://arxiv.org/abs/1508.03664v1.

Questions?

Thank You!