Introduction to the min cost homomorphism problem for undirected and directed graphs
Spanning Tree Modulus and Homogeneity of Undirected Graphs · Brandon Sit: University of Portland...
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Spanning Tree Modulus and Homogeneity of Undirected Graphs Derek Hoare, Brandon Sit, & Sarah Tymochko Mentor: Nathan Albin KSU SUMaR 2016 I NTRODUCTION What is Spanning Tree Modulus? • Spanning tree modulus measures the “richness” of a family of spanning trees on a network • ρ ∈ R ≥0 a set of edge weights • N is the usage matrix for each edge in each span- ning tree • Mod(T ) := min N ρ≥1 ρ T ρ Probabilistic Interpretation [1] • μ ∈ P (T ) is a pmf on spanning trees • T is a random variable representing a spanning tree such that μ(T ) is the probability of choosing T • η e = P(e ∈ T ) is the probability that an edge is in a random spanning tree minimize ∑ e∈E P μ (e ∈ T ) 2 subject to μ ∈ P (T ) • μ * is an optimal pmf on T • η * is the optimal usage of edges in spanning trees Uniform • μ 0 is the uniform pmf on T • A graph is uniform if μ 0 is an optimal pmf Homogeneous • ρ ≡ (| V |- 1) -1 is always admissible • A graph is homogeneous if ρ * ≡ (| V |- 1) -1 • If a graph is homogeneous, Mod(T )= |E | (| V |- 1) 2 = ⇒ η * = | V |- 1 |E | Regular, Connected • A graph is d -regular if every node has degree d • A graph is k -connected if a graph cannot be discon- nected by removing fewer than k nodes C ONTACT I NFORMATION Derek Hoare: Kenyon College [email protected] Brandon Sit: University of Portland [email protected] Sarah Tymochko: College of the Holy Cross [email protected] Dr. Nathan Albin: Kansas State University [email protected] D -R EGULAR , K -C ONNECTED 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: 1-connected, 3-regular graph, labeled with η values Numerical Experiments Small Graphs (4-20 nodes) • 96.8% of graphs were homogeneous • 97.9% of homogeneous graphs had d = k • 0% of non-homogeneous graphs had d = k Large Graphs (50-100 nodes) • 99.7% of graphs were homogeneous • 99.9% of homogeneous graphs had d = k • 0% of non-homogeneous graphs had d = k (Note: Out of 10,000 d-regular, k-connected graphs of each size. All graphs had 3 ≤ k ≤ 15) D EFLATION 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 . 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 0 on T G so that μ 0 is supported on T 0 and μ 0 (T H ∪ T G\H ) := μ H (T H )μ G\H (T G\H ) Then μ 0 is optimal for T G . (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 2: Example of Deflation R ESULTS 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. Theorem: For k ≥ 3, a k -regular, k -connected graph is homogeneous. It is also proven that as d gets large, d -regular graphs are almost surely d -connected [2]. Our theorem then proves that as d gets large, d -regular graphs are almost surely homogeneous. Figure 3: Every cycle is uniform homogeneous. Figure 4: The complete bipartite graph K n,n is ho- mogeneous. F UTURE R ESEARCH • Find necessary conditions for a graph to be homo- geneous. • Look at applications to network security questions, specifically in a network broadcast. R EFERENCES [1] N. Albin and P. Poggi-Corradini. Minimal subfamilies and the probabilistic interpretation for modulus on graphs. Jour- nal of Analysis, to appear. https://arxiv.org/abs/1605.08462. [2] B. Bollobás. Random graphs. Academic Press, Inc. [Har- court Brace Jovanovich, Publishers], London, 1985. ACKNOWLEDGEMENTS This research was conducted under mentor Nathan Al- bin at Kansas State University SUMaR under support of NSF grant number DMS-1262877 and NSF grant number DMS-1515810.
Transcript of Spanning Tree Modulus and Homogeneity of Undirected Graphs · Brandon Sit: University of Portland...
Spanning Tree Modulus and Homogeneity of Undirected GraphsDerek Hoare, Brandon Sit, & Sarah Tymochko
Mentor: Nathan AlbinKSU SUMaR 2016
INTRODUCTIONWhat is Spanning Tree Modulus?• Spanning tree modulus measures the “richness” of
a family of spanning trees on a network• ρ ∈ R≥0 a set of edge weights• N is the usage matrix for each edge in each span-
ning tree• Mod(T ) := min
N ρ≥1ρT ρ
Probabilistic Interpretation [1]• µ ∈ P (T ) is a pmf on spanning trees• T is a random variable representing a spanning tree
such that µ(T ) is the probability of choosing T• ηe = P(e ∈ T ) is the probability that an edge is in a
random spanning tree
minimize ∑e∈E
Pµ (e ∈ T )2
subject to µ ∈ P (T )
• µ∗ is an optimal pmf on T• η∗ is the optimal usage of edges in spanning trees
Uniform• µ0 is the uniform pmf on T• A graph is uniform if µ0 is an optimal pmf
Homogeneous• ρ≡ (|V |−1)−1 is always admissible• A graph is homogeneous if ρ∗ ≡ (|V |−1)−1
• If a graph is homogeneous,
Mod(T ) =|E|
(|V |−1)2 =⇒ η∗ =|V |−1|E|
Regular, Connected• A graph is d-regular if every node has degree d• A graph is k-connected if a graph cannot be discon-
nected by removing fewer than k nodes
CONTACT INFORMATION
Derek Hoare:Kenyon [email protected]
Brandon Sit:University of [email protected]
Sarah Tymochko:College of the Holy [email protected]
Dr. Nathan Albin:Kansas State [email protected]
D-REGULAR, K-CONNECTED
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Figure 1: 1-connected, 3-regular graph, labeled with η values
Numerical ExperimentsSmall Graphs (4-20 nodes)• 96.8% of graphs were homogeneous• 97.9% of homogeneous graphs had d = k• 0% of non-homogeneous graphs had d = k
Large Graphs (50-100 nodes)• 99.7% of graphs were homogeneous• 99.9% of homogeneous graphs had d = k• 0% of non-homogeneous graphs had d = k
(Note: Out of 10,000 d-regular, k-connected graphs of each size. All graphs had 3≤ k ≤ 15)
DEFLATIONComponent Theorem:Let G = (VG,EG) be an undirected multigraph. Let η∗ = N Tµ∗ be the optimal expected edge usage for spanning treemodulus, and let ηmin be its minimum value. Then there exists a connected, vertex-induced subgraph H = (VH ,EH) of Gsuch 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.
Deflation Theorem:Let G be a non-homogeneous, undirected multigraph, and let H be a vertex-induced subgraph satisfying the conditionsof the Component Theorem. Let µH and µG\H be optimal pmfs for TH and TG\H respectively. Define the pmf µ′ on TGso that µ′ is supported on T ′ and
µ′(TH ∪TG\H) := µH(TH)µG\H(TG\H)
Then µ′ is optimal for TG.
(a) ηmin = 0.167 (b) ηmin = 0.167 (c) ηmin = 0.222 (d) ηmin = 0.286 (e) ηmin = 0.500
Figure 2: Example of Deflation
RESULTSTheorem:Each edge in a graph G is in the same number of spanningtrees if and only if G is a uniform homogeneous graph.
Theorem:For k≥ 3, a k-regular, k-connected graph is homogeneous.
It is also proven that as d gets large, d-regular graphs arealmost surely d-connected [2]. Our theorem then provesthat as d gets large, d-regular graphs are almost surelyhomogeneous.
Figure 3: Every cycle isuniform homogeneous.
Figure 4: The completebipartite graph Kn,n is ho-mogeneous.
FUTURE RESEARCH• Find necessary conditions for a graph to be homo-
geneous.• Look at applications to network security questions,
specifically in a network broadcast.
REFERENCES[1] N. Albin and P. Poggi-Corradini. Minimal subfamilies and
the probabilistic interpretation for modulus on graphs. Jour-nal of Analysis, to appear. https://arxiv.org/abs/1605.08462.
[2] B. Bollobás. Random graphs. Academic Press, Inc. [Har-court Brace Jovanovich, Publishers], London, 1985.
ACKNOWLEDGEMENTSThis research was conducted under mentor Nathan Al-bin at Kansas State University SUMaR under support ofNSF grant number DMS-1262877 and NSF grant numberDMS-1515810.
