A phylogenetic application of the combinatorial graph Laplacian

1
A phylogenetic application of the combinatorial
graph Laplacian

• Eric A. Stone
• Department of Statistics
• Bioinformatics Research Center
• North Carolina State University

2
My motivation for this project

• Trees in statistics or biology
• Often a latent branching structure relating some
  observed data
• Trees in mathematics
• Always a connected graph with no cycles

3
My motivation for this project

• Trees in statistics or biology
• PROBLEM Recover properties of latent branching
  structure
• Trees in mathematics
• Always a connected graph with no cycles

4
My motivation for this project

• Trees in statistics or biology
• PROBLEM Recover properties of latent branching
  structure
• Trees in mathematics
• Characterization of observed structure by
  spectral graph theory

5
My motivation for this project

• Trees in statistics or biology
• PROBLEM Recover properties of latent branching
  structure
• Trees in mathematics
• Characterization of observed structure by
  spectral graph theory

6
Bridging the gap

• Rectifying trees and trees
• Can we use some powerful tools of spectral graph
  theory to recover latent structure?
• Natural relationship between trees and complete
  graphs?!?

7
Tree and distance matrices

• The tree with vertex set 1,,8 has distance
  matrix D
• The phylogenetic tree can only be observed at
  1,,5
• We can only observe (estimate) the phylogenetic
  portion D

The phylogenetic portion D
8
More motivation for this project

• Trees in statistics or biology
• PROBLEM Recover properties of latent branching
  structure
• Given D only, recover latent branching structure
• This is the problem of phylogenetic
  reconstruction (w/o error!)

The phylogenetic portion D
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NJ finds (2,n-2) splits from D

• A split is a bipartition of the leaf set (e.g.
  1,2,3,4,5) that can be induced by cutting a
  branch on the tree
• e.g. 1,2,3,4,5 or 1,2,5,3,4
• Neighbor-joining criterion identifies (2,n-2)
  splits through

1,2,3,4,5
1,2,5,3,4
10
A recipe for tree reconstruction from D

• Find a split
• NJ relies on theorem that guarantees (2,n-2)
  split from Q matrix
• Use knowledge of split to reduce dimension
• NJ prunes the cherry (neighboring taxa) to reduce
  leaves by one
• Iterate until tree has been fully reconstructed
• Tree topology specified by its split set

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Our narrow goal

• Find a split
• NJ relies on theorem that guarantees (2,n-2)
  split from Q matrix
• Hypothesize criterion that identifies deeper
  splits
• and prove that it actually works

12
Our solution
The phylogenetic portion D
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Our solution
The phylogenetic portion D

• Let H be the centering matrix
• Find eigenvector Y of HDH with the smallest
  eigenvalue
• The signs of the entries of Y identify a split of
  the tree

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About the matrix HDH

• Entries of HDH are Dij Di. D.j D..
• HDH is negative semidefinite
• Zero is a simple eigenvalue with unit eigenvector
• Entries of remaining eigenvalues have both and
  - entries
• HDH appears prominently in
• Multidimensional scaling
• Principal coordinate analysis

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Example of our solution

• Find eigenvector Y of HDH with the smallest
  eigenvalue
• Signs of Y identify the split 1,2,3,4,5

-0.0564
0.5793
-0.5011
-0.4636
0.4418
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A real example (data from ToL)

• Two iterations

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Our solution

• Find a split
• NJ relies on theorem that guarantees (2,n-2)
  split from Q matrix
• Hypothesize criterion that identifies deep splits
• and prove that it actually works

18
Affinity and distance

• In phylogenetics, common to consider pairwise
  distances
• In graph theory, common to consider pairwise
  affinities

Affinity-based
Distance-based
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Distance matrix ? Laplacian matrix
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The genius of Miroslav Fiedler

• G connected ? smallest eigenvalue of L, zero, is
  simple
• Smallest positive eigenvalue, ?, called algebraic
  connectivity of G
• Fiedler vectors Y satisfy LY?Y
• Fiedler cut is the sign-induced bipartition

-0.4277
-0.0223
0.4840
-0.0158
-0.3653
0.3449
0.4038
-0.4047
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The genius of Miroslav Fiedler

• G connected ? smallest eigenvalue of L, zero, is
  simple
• Smallest positive eigenvalue, ?, called algebraic
  connectivity of G
• Fiedler vectors Y satisfy LY?Y
• Fiedler cut is the sign-induced bipartition
• Fiedler cut here is
• 1,2,6,3,4,5,7,8
• Note that the cut implies a leaf split
• 1,2,3,4,5

-0.4277
-0.0223
0.4840
-0.0158
-0.3653
0.3449
0.4038
-0.4047
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Is this relevant here?

• We do not observe an 8x8 Laplacian matrix L
• All we get is a 5x5 matrix of between-leaf
  pairwise distances D
• Where is the connection to graph theory?

The phylogenetic portion D
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Recall Our solution

• Let H be the centering matrix
• Find eigenvector Y of HDH with the smallest
  eigenvalue
• The signs of the entries of Y identify a split of
  the tree

The phylogenetic portion D
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An extremely useful relationship

• Recall the centering matrix H
• The (Moore-Penrose) pseudoinverse of HDH is in
  fact -2L
• We have shown in the context of this formula
• Principal submatrices of D relate to Schur
  complements of L
• In particular, (HDH) -2L -2(L/Z) -2(W
  XZTY), where

W
X
Y
Z
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Recall Our solution

• Find eigenvector Y of HDH with the smallest
  eigenvalue
• The signs of the entries of Y identify a split of
  the tree
• The smallest eigenvalue of HDH (negative
  semidefinite) is the smallest positive eigenvalue
  of L
• In fact, L can be seen as a graph Laplacian
• And our solution, Y, is the Fiedler vector of
  that graph!
• But what does this graph look like?

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Schur complementation of a vertex

• The vertices adjacent to 8 become adjacent to
  each other

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Schur complementation of the interior

• The graph described by L is fully connected
• All cuts yield connected subgraphs ? No help from
  Fiedler

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Recap thus far

• Given matrix D of pairwise distances between
  leaves
• Find eigenvector Y of HDH with the smallest
  eigenvalue
• Claim The signs of the entries of Y identify a
  split of the tree
• Y shown to be a Fiedler vector of the Laplacian
  L
• But graph of L is fully connected, has no
  apparent structure
• Thus Fiedler says nothing about signs of entries
  of Y
• But claim requires signs to be consistent with
  structure of the tree

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Recap thus far

• Thus Fiedler says nothing about signs of entries
  of Y
• But claim requires signs to be consistent with
  structure of the tree
• How does L inherit the structure of the tree?

NO
NO
YES
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The quotient rule inspires a Schur tower
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The quotient rule inspires a Schur tower

• How does this help?

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Cutpoints and connected components

• A point of articulation (or cutpoint) is a point
  r?G whose deletion yields a subgraph with ?2
  connected components
• Cutpoints 6,7,8
• Shown 1, 2, 3,4,5,7,8 are
  connectedcomponents at 6
• The cutpoints of a tree are its internal nodes

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The key observation (i.e. theorem)

• Let L be the Laplacian of a graph G with some
  cutpoint v
• Let Lv be the Laplacian of Gv obtained by
  Schur complement at v
• Then the Fiedler cut Gv identifies a split of G
• Here the Fiedler cut of G6 is
  1,2,5,8,3,4,7
• Including 6 in 1,2,5,8 defines two connected
  components in G

0.0570

-
-0.4129

0.5828
0.0380

-
?
-0.3439
G
G6
0.4660

-0.3870
-
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The quotient rule inspires a Schur tower
L
L

• How does this help?
• ? Look at Schur paths to graph with Laplacian L

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The punch line

• The graph with Laplacian L can be obtained in
  three ways
• The Fiedler cut of G6,7,8 must split G6,7 and
  G6,8 and G7,8

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The punch line

• The graph with Laplacian L can be obtained in
  three ways
• The Fiedler cut of G6,7,8 must split G6,7 and
  G6,8 and G7,8

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Recall Example

• Find eigenvector Y of HDH with the smallest
  eigenvalue
• Signs of Y identify the split 1,2,3,4,5

-0.0564
0.5793
-0.5011
-0.4636
0.4418
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The punch line

• The graph with Laplacian L can be obtained in
  three ways
• The Fiedler cut of G6,7,8 must split G6,7 and
  G6,8 and G7,8
• This implies that the cut splits the progenitor
  graph G!

1,2,6,3,4,5,7,8
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Our solution actually works

• Let H be the centering matrix
• Find eigenvector Y of HDH with the smallest
  eigenvalue
• The signs of the entries of Y identify a split of
  the tree

The phylogenetic portion D
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A recipe for tree reconstruction

• Find a split
• NJ relies on theorem that guarantees (2,n-2)
  split from Q matrix
• We have a theorem that guarantees splits from
  HDH matrix
• Use knowledge of split to reduce dimension
• NJ prunes the cherry (neighboring taxa) to reduce
  leaves by one
• We use a divisive method that reduces to pairs of
  subtrees
• Iterate until tree has been fully reconstructed
• Tree topology specified by its split set

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Reconstruction from the inside out
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Connections with Classical MDS and PCoA

• Classical solution to multidimensional scaling
• a.k.a. Principal coordinate analysis
• Recipe for dimension reduction given distance
  matrix D
• Construct matrix A from D entrywise x ? -x2/2
• Double centering B HAH
• Find k largest eigenvalues ?i of B with
  corresponding eigenvectors Xi
• Coordinates of point Pr given by row r of
  eigenvector entries
• ? k 1 with sqrt of tree distance equivalent to
  our approach

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Phylogenetic ordination

• PCoA on sequence data with k 3
• For appropriate distance, C1 (x-axis) guaranteed
  to split taxa at 0
• Our results support popular use of PCoA
• Provided that the right distance is considered

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Conclusion I

• Natural connection between matrix of pairwise
  distances and the Laplacian of a complete graph

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Conclusion II

• Structure of tree embedded in complete graph and
  recoverable via spectral theory
• Notion of Fiedler cut extends concept to
  Fiedler split
• Inheritance propagated through Schur tower

NO
NO
YES
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Conclusion III

• Results inspire fast divisive tree reconstruction
  method

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Conclusion IV

• Provides guidance and justification for
  ordination approach

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Acknowledgements

• Alex Griffing (NCSU Bioinformatics)
• Carl Meyer (NCSU Math)
• Amy Langville (CoC Math)

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Cutpoints and Perron components

• Each connected component identifies a principal
  submatrix
• Each such principal submatrix is inverse positive
• Implies that the inverse has a Perron value that
  is simple
• The Perron component is that with the largest
  Perron value

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Cutpoints and Perron components
INVERSE PRINCIPAL SUBMATRICES
? 1
? .5
PERRON COMPONENT
? 7.49
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The key observation

• Take Schur complement of L at cutpoint, e.g. 6
• Consider Fiedler vector of derived Laplacian
• Signs of entries outside Perron component are
  positive ()
• Signs of entries inside Perron component
  indeterminate (/-)

/-
/-
INVERSE PRINCIPAL SUBMATRICES

/-
? 1
/-
SCHUR GRAPH AT 6
? .5

/-
PERRON COMPONENT
? 7.49