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

Spanning Tree Modulus and Homogeneity of Graphs Derek Hoare 1 Brandon Sit 2 Sarah Tymochko 3 Mentor: Dr. Nathan Albin Kansas State University 1 Kenyon College 2 University of Portland 3 College 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 = Γ G the set of spanning trees of G N ∈ R Γ × E the usage matrix ≥ 0 � 1 if e ∈ γ N γ, e = ∈ γ 0 if e / ρ ∈ R E ≥ 0 a set of edge weights N ρ ∈ R Γ ≥ 0 w ρ ( γ ) := � e ∈ E N γ, e ρ e = ( N ρ ) γ

Example   1 1 0 0 1 1 N = 0 1 1 1 1 0   1 0 1 1 0 1

Modulus The primal problem Long form � ρ 2 minimize e e ∈ E ρ ∈ R E subject to ≥ 0 w ρ ( γ ) ≥ 1 ∀ γ ∈ Γ Short form N ρ ≥ 1 ρ T ρ Mod(Γ) := min

Notation The dual problem G = ( V , E ) a connected, undirected graph Γ = Γ G the set of spanning trees of G N ∈ R Γ × E the usage matrix ≥ 0 � 1 if e ∈ γ N γ, e = ∈ γ 0 if e / � µ ∈ R Γ ≥ 0 : µ T 1 = 1 � µ ∈ P Γ := a pmf on Γ � η := N T µ ∈ R E η e = N γ, e µ γ ≥ 0 γ ∈ Γ

Probabilistic Interpretation µ ∈ P Γ a pmf on Γ γ a Γ-valued random variable � � γ = γ = µ ( γ ) ( µ ( γ ) is the probability of choosing γ ) P µ η = N T µ ∈ R E ≥ 0 � � � � � � η e = N γ, e µ γ = γ = γ = P µ e ∈ γ P µ γ : e ∈ γ γ ∈ Γ ( η e is the probability that e is in a random spanning tree)

Optimizing Expected Usage The dual problem Long form � 2 � � minimize e ∈ γ P µ e ∈ E µ ∈ P Γ subject to For any µ ∈ P Γ , � � � � � � e ∈ γ � N γ, e µ γ = N γ, e = | V | − 1 P µ = µ γ e ∈ E e ∈ E γ ∈ Γ γ ∈ Γ e ∈ E Short form η = N T µ η T η = ( | V | − 1) 2 � T � � � N T µ N T µ min = min + | E | min η = N T µ Var( η ) | E | µ ∈P Γ

Relation to the Primal Problem η ∗ ρ ∗ = ( η ∗ ) T η ∗ is the optimal set of edge weights for the primal problem. 1 Mod(Γ) = ( η ∗ ) T η ∗ 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 ∈ P Γ such that = | V | − 1 � � e ∈ γ ∀ e ∈ E . P µ | 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 1 4 3 3 2 / / / / 3 4 2 3 3/4 2/3 actual optimal η ∗ top edge must belong to every spanning tree choose a spanning tree with can’t possibly use each edge uniform probability 3 / 4 of the time

Example Homogeneous graph 3 2 / / 2 3 2/3 2/3 2/3 2/3 Can we choose µ to use each Yes! Choose uniformly from edge with equal probability? 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 only 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 4 7 7 4 4 7 7 4 4 4 4 4 4 1 7 7 7 7 7 7 4 4 7 7 4 4 7 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 = 15 32 ≤ 1 | E | 2

Homogeneous Graphs Component Theorem Let G = ( V G , E G ) be an undirected multigraph. Let η ∗ = N T µ ∗ be the optimal expected edge usage for spanning tree modulus, and let η min be its minimum value. Then there exists a connected, vertex-induced subgraph H = ( V H , E H ) of G such that all of the following hold. 1 E H is non-empty. 2 η ( e ) = η min for all e ∈ E H . 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 µ H and µ G \ H be optimal pmfs for T H and T G \ H respectively. Define the pmf µ ′ on T G so that µ ′ is supported on T ′ and µ ′ ( T H ∪ T G \ H ) := µ H ( T H ) µ G \ H ( T G \ H ) Then µ ′ is optimal for T G .

Small Deflation Example Figure: Cycle on 6 nodes with added triangle Figure: Cycle on 5 nodes

Small Deflation Example (a) P = 1 (b) P = 1 (c) P = 1 3 3 3 (d) P = 1 (e) P = 1 (f) P = 1 (g) P = 1 (h) P = 1 5 5 5 5 5

Deflation Example (a) η min = 0 . 167 (b) η min = 0 . 167 (c) η min = 0 . 222 (d) η min = 0 . 286 (e) η min = 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 E A

Applications E E A A A η E = 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 of random linear network coding. IEEE Communications Letters , to appear. http://arxiv.org/abs/1508.03664v1 .

Questions?

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