The Structure of Level-k Phylogenetic Networks

Lille – 24/06/2009 – CPM'09

The Structure of Level-k
Phylogenetic Networks
Philippe Gambette
in collaboration with
Vincent Berry, Christophe Paul

Outline

• Phylogenetic networks
• Decomposition of level-k networks
• Construction of level-k generators
• Number of level-k generators
• Simulated level-k networks

Phylogenetic networks

level-2 split network T-Rex
network
Level-2 reticulogram
SplitsTree
minimum
synthesis spanning network
diagram
HorizStory median
network

Network TCS

Abstract or explicit networks

An explicit phylogenetic network is a phyogenetic
network where all reticulations can be interpreted as precise
biological events.

An abstract network reflects some phylogenetic signals
rather than explicitly displaying biological reticulation events.

Doolittle : Uprooting the Tree of Life, Scientific American (Feb..2000)

Abstract or explicit networks

An explicit phylogenetic network is a phyogenetic
network where all reticulations can be interpreted as precise
biological events.

An abstract network reflects some phylogenetic signals
rather than explicitly displaying biological reticulation events.

Abstract phylogenetic network of
mushroom species, www.splitstree.org

Hierarchy of network subclasses

http://www.lirmm.fr/~gambette/RePhylogeneticNetworks.php

Level-k phylogenetic networks

level 1 =

= level 0
http://www.lirmm.fr/~gambette/RePhylogeneticNetworks.php

Motivation to generalize “galled trees” (= level-1) :

Arenas, Valiente, Posada :
Characterization of
Phylogenetic Reticulate
Networks based on the
Coalescent with
Recombination, Molecular
Biology and Evolution, to
appear.


A level-k phylogenetic network N on a set X of n taxa is a
multidigraph in which:
- exactly one vertex has indegree 0 and outdegree 2: the root,
- all other vertices have either:
- indegree 1 and outdegree 2: split vertices,
- indegree 2 and outdegree ≤ 1: reticulation vertices,
- or indegree 1 and outdegree 0: leaves labeled by X,
- any blob has at most k reticulation vertices.

N r

r2
r4 All arcs are
r1 r3 oriented
downwards
a b c d e f g h i j k



N r A blob is a maximal
induced connected
subgraph with no cut
r2 arc.
r4
r1 r3
A cut arc is an arc
a b c d e f g h i j k which disconnects the
graph.



N r
A blob is a maximal
induced connected
r2 subgraph with no cut
r4 arc.
r1 r3
A cut arc is an arc
a b c d e f g h i j k which disconnects the
graph.
N has level 2.

Decomposition of level-k networks

We formalize the decomposition into blobs:

a b c d e f g h i j k a b c d e f g h i j k
N, a level-k network. N decomposed as a
tree of simple graph
patterns: generators.

Generators were introduced by van Iersel & al (Recomb 2008)
for the restricted class of simple level-k networks.

Level-k generators

A level-k generator is a level-k network with no cut arc.

G0 G1 2a 2b 2c 2d

The sides of the generator are:
- its arcs
- its reticulation vertices of outdegree 0

Decomposition theorem of level-k networks

N is a level-k network

iff
there exists a sequence (lj)jϵ[1,r] of r locations
(arcs or reticulation vertices of outdegree 0)
and a sequence (Gj)jϵ[0,r] of generators of level at most k, such that:
- N = Attachk(lr,Gr,Attachk(... Attachk(l2,G2,Attachk(l1,G1,G0))...)),
- or N = Attachk(lr,Gr,Attachk(... Attachk(l2,G2,SplitRootk(G1,G0))...)).



iff

SplitRootk(G1,G0)

G0 G1
G0 G1



iff

li is an arc of N Attachk(li,Gi,N)

N
Gi

li
Gi



iff

li is a reticulation vertex of N Attachk(li,Gi,N)

N
Gi

li
Gi



iff

This decomposition is not unique!


Unique “graph-labeled tree” decomposition:
ρ 1
1
N 3 2

h1 h2
1
h3 h4

a b c d e f g h i a b c d e f g h i

Possible applications:
- exhaustive generation of level-k networks
- counting of level-k networks

Construction of the generators
Van Iersel & al build the 4 level-2 generators by a case
analysis, generalized by Steven Kelk into an exponential
algorithm to find all 65 level-3 generators.


Van Iersel & al give a simple case analysis for level-2.

We give rules to build level-(k+1) from level-k generators.

Construction rules of level-k+1 generators
from level k-generators:

N e2
e1
h1 h2

Rule R1 :

h1 h2 h1 h2 h1 h2 h1 h2
h3 h3 h3 h3
R1(N,h1,h2) R1(N,h1,e2) R1(N,e2,e2) R1(N,e1,e2)

Construction rules of level-k+1 generators
from level k-generators:

N e2
e1
h1 h2

Rule R2 :

h1 h3 h2 h3 h
h3
h2 h1 h1 2

R2(N,h1,e2) R2(N,e1,e1) R2(N,e2,e1)

Problem!
Some of the level-k+1 generators obtained from level-k
generators are isomorphic!

h1 h2 h1 h2
h3 h3
R1(2b,h1,e2) R1(2b,h2,e1)

Problem!
Some of the level-k+1 generators obtained from level-k
generators are isomorphic!

h1 h2 h1 h2
h3 h3
R1(N,h1,e2) R1(N,h2,e1)

→ difficult to count!

Upper bound
R1 and R2 can be applied at most on all pairs of sides
A level-k generator has at most 5k slides:
gk+1 < 50 k² gk

Upper bound:
gk < k!² 50k

Theoretical corollary:
There is a polynomial algorithm to build the set of level-
k+1 generators from the set of level-k generators.

Practical corollary:
g4 < 28350
→ it is possible to enumerate all level-4 generators.

Number of level-k generators

It is possible to enumerate all level-4 generators.

Isomorphism of graphs of bounded valence:
polynomial
(Luks, FOCS 1980)

Practical algorithm?
Simple backtracking exponential algorithm sufficient for
level 4 :
go through both graphs from their root in parallel and
identify their vertices: O(n2n-h)
→ g4 = 1993
→ g5 > 71000

http://www.lirmm.fr/~gambette/ProgramGenerators

Lower bound
Lower bound:
gk ≥ 2k-1

There is an exponential number of generators!

Idea:
Code every number between 0 and 2k-1-1 by a level-k
generator.

Lower bound
Lower bound:
gk ≥ 2k-1


Idea:
generator.
0 1
k=1
0 1

Lower bound
Lower bound:
gk ≥ 2k-1


Idea:
generator.

0 1 2 3

k=2
0 0 1 1

0 1 0 1

Simulated level-k networks

Simulate 1000 phylogenetic networks using the coalescent
model with recombination.
Arenas, Valiente, Posada 2008
Program Recodon

How many are level-1,2,3... networks?

nb of networks nb of networks
1000 1000

900 900

800 800

700 n=10,r=1 700 n=50,r=1
n=10,r=2 n=50,r=2
600 600
n=10,r=4 n=50,r=4
500 n=10,r=8 500 n=50,r=8
n=10,r=16 n=50,r=16
400 400
n=10,r=32 n=50,r=32
300 300

200 200

100 100

0 0
12

24

36

12

24

36
0
4
8

16
20

28
32

40
44
48
52
56
60
64
68
72
76
80
84
88
92
96
100

0
4
8

16
20

28
32

40
44
48
52
56
60
64
68
72
76
80
84
88
92
96
100
level level


Simulated level-k networks

Simulate 1000 phylogenetic networks using the coalescent
model with recombination.

Link between level and number of reticulations:

level
9

5

0 number of
reticulations
0 5 9


Summary on the level parameter

Advantages:
• natural structure for all explicit phylogenetic networks
• global tree-structure used algorithmically
• finite graph patterns to represent blobs: generators

Limits:
• number of generators exponential in the level
• complex structure of generators
• when recombination is not local, level doesn't help

Questions?
Thank you for your attention! TreeCloud T-Rex

TreeCloud

Split network and reticulogram of the 50 most frequent
words in CPM titles, hyperlex cooccurrence distance.

TreeCloud available at http://treecloud.org - Slides available at http://www.lirmm.fr/~gambette/RePresentationsENG.php

The Structure of Level-k Phylogenetic Networks

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