HIERATIC Chris Cannings - PowerPoint Presentation

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HIERATIC

Chris

Cannings

Birmingham, Kick-Off 6/7 Dec. 2012Slide2

Chris

Cannings

Stochastic processes,

combinatorics

,

graph theory, algorithms

IN

Population genetics, human genetics, evolutionary gamesSlide3

John

Haslegrave

Random graphs,

extremal

problems on graphs and trees.

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GRAPHS

G=(V,E) where V is a set (vertices) &

E \in (V*V) a set of edges.

Begin with a few examples.

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Protein-protein

interactions in yeastSlide7

A single cultured neurone, 2 days after planting

(Shefi et.al. (2002) Phys.Rev.E, 66, 021905.)Slide8

Genealogy

Female

MaleSlide9

GenealogySlide10

GenealogiesSlide11

TOPICS

1.Genealogies & Genetics

2.Games on Graphs…Majority Game

3.Reproducing Graphs

4.Growing Graphs

5.Graphic Sequences6.Patterns of ESS’s

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1. Genealogies AND Genetics

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Genealogies and Genetics

The vertices in a genealogy have genotypes. If we have alleles a1,a2,…,

ak

then we have types

ai.aj and the

passage of alleles obey Mendel’s laws. An ai.aj individual contributes an ai OR an

aj to each offspring independently.

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Calculations on Genealogies

Given some observations and a model calculate the Likelihood(model | data).

Peeling

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1

2

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R-functions

where R(.) is the “likelihood” for the genealogy “peeled” to “v”, Pen(.,.) is the penetrance i.e. mapping from genotype to phenotype, Trans(.,.) as per

Mendelian

law.

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Recurrence Relation

,

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How

many such assignments are there to

a genealogy

?

We can derive recurrence relations as we add families. Example Gr built up of nuclear families,

Then number of configurations (when k=2) increases as

λ

=(11+(177)

0.5

)/2 so rate per individual about 2.3.Slide17

Repeated Double First Cousins

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2. GAMES ON GRAPHS

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Games on Graphs

G=(V,E). Vertex

i

at time (t+1) plays f(S(

i

)) where S(i

) is the set of strategies of i’s neighbours.Threshold Game. Strategies B and W. If vertex i

has more than w(i) B neighbours at time t then it plays B at time (t+1). Such games converge to fixed point or two-cycle.

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Majority Game

Example. Majority Game. Two strategies B and W.

Play at time t+1 strategy played by majority of neighbours at time t.

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Example. Game on 3-cube

There are 23 different (up to permutation) configurations of B and W on a 3-cube. It is easy to specify the dynamics.

What can we say more generally?

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ALL MAJORITY

PLAYERS DYNAMICS

FLASHERSSlide23

A class of “cylindrical” cubic graphs

Take two polygons of size n (here n=5).

5

1

2

3

4

5

1

2

3

4

Take a permutation

P={p

1

,p

2

,………..

p

n

},

and join each

i

on one

polygon to p

i

on the

other.

Example (shown)

perm=(1,2,3,5,4)

Denote such a graph n-CYL-PSlide24

1

2

3

4

5

2

3

4

5

1

1

2

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5

2

3

4

5

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5

1

B

A

C

D

5-CYL-{12345}

5-CYL-{12354}

5-CYL-{12453}

5-CYL_{13524}Slide25

Fixed Point Minority GameSlide26

An 8-cycle on the 3-cube

= majority player

= minority player

W

1-WSlide27
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Hypercubes

Majority game on hypercube. Can we characterise the fixed points and two-cycles?

For the 5-cube there is a characterisation of the fixed points which allows there specification (10 distinct patterns).

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3.REPRODUCING GRAPHS

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Reproducing Graphs

Every vertex is duplicated, old edges persist and certain edges added. Here just parent-offspring joined.

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Parent

OffspringSlide32

Reproducing Graphs

In fact the resulting graph is the union of result from each edge separately.

Accordingly we can study the process starting from a single edge.

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Reproducing Graphs

We fix the presence or absence of the edges labelled

α

,

β

and (indicated by 0 and 1).

α

β

Model

0

0

0

0

0

0

1

1

0

1

0

2

0

1

1

3

1

0

0

4

1

0

1

5

1

1

0

6

1

1

1

7

u

1

v

1

u

0

v

0

β

β

αSlide34

Graph Products

These 8 cases are equivalent to certain graph products (and some new ones).

We (

Southwell

&

Cannings) have derived the degree dist., numbers of vertices and edges, chromatic number, distance structure and

automorphisms of these 8 models.

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Culling

We add “culling” by age, by degree, by fitness, corresponding to age, crowding and game payoffs.

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Example. Model 1, cull at age 3

Offspring joined to neighbours of parents.

Next slide shows progress through time omitting isolated vertices.Slide37

1

1

1

1

1

6

1

1

1

3

Degree cap = 6 Slide38

Issues

What cycles?

System closed under types?Slide39

4. GROWING GRAPHS

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Preferential Attachment

Simon(1954) (Barabasi and Albert(1999)) introduced an interesting model for the growth of a random network, “preferential attachment”.

Start with a few nodes, add a new node and link to the existing nodes with probabilities proportional to the current degrees of those nodes.Slide41

Preferential Attachment

Suppose we start with two joined nodes, and at each stage add a new node and a

single edge

which joins the new node to one of the old edges with probability proportional to the degree of that node.

We consider the complete set of possible realisations generated in the following way Slide42

Preferential Attachment

Example. (weights shown at nodes)

3

1

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1

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2

1

1

1

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2

2

1

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3

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1

1

1

1

2

2

3

2

2

1

1

1

N=8=M

N=10,M=80Slide43

The Degree Distribution

Suppose a new vertex joins to

m

pre-existing ones

.Slide44

Preferential Attachment

Trouble is that the probabilities require a global property (total degree would be enough then sequential).

Saramaki

and

Kaski

suggested that one could pick a vertex at random and then carry out a random walk. Limit has P(vertex

i)=degree i / total degree.

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Random Walk

Saramaki

&

Kaski

claimed that any random walk would deliver preferential attachment.

Jordan & Cannings

have proved that a 1 step random walk has Ln is the proportion of leaves.

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Polya’s Urn-Friedman’s Urn

An urn has r red and b blue balls

Polya

. Draw a ball at random return 2 of that colour, then

R

n

(number of red balls) converges to a beta distribution with parameters depending on r and b.Friedman. Draw a ball at random and return with one of opposite colour. Rn

converges to 0.5 almost surely.

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Random Walk

Consider a bipartite graph, let

R

n

be the number of red vertices (i.e. one part of the graph) and the rest are blue. Select a vertex at random and make s steps. Is s is even join to colour proportional to number of that colour and so add a vertex of the opposite colour, s odd add vertex of same colour as that selected.,

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Random Walk

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Random Walk

For the BA model on a bipartite graph think of each edge as coloured red at the red set end and blue at the blue set end. Now picking a vertex with PA is like picking a half edge at random and then adding the other half. Thus

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The

M-model (

Chrysafis

& CC)

Vertex set V and E=V*V, weight

w

ij for each edge, si

= wi., mij

=

w

ij

/

s

i

.

, M=(mij

) is stochastic matrix for random walk. Pick node according to strength distribution then add weight according to the edge weights (models the “economic”effect of local development). Slide51
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Simulation parameter values:

Network size:

N=10,000

δ=0.5

f

(

δ)=δ/(δ+1)

Propagation steps:

n=3

No of edges per new node:

m=2

Data averaged over 100 runs

Strength/weight evolutionSlide53

Strength-Degree Dependence

Simulation parameter values:

Network size:

N=10

4

δ=0.5

f

(

δ)=δ/(δ+1)

Propagation steps:

n=3

No of edges per new node:

m=1

Data from single random realization of the networkSlide54

Degree Sequence

Simulation parameter values:

N=10

4

,

δ=0.5

, f

(

δ)=δ/(δ+1)

,

n=3, m=1

Data averaged over 100 runsSlide55

Strength Distribution

Simulation parameter values:

N=10

4

,

δ=0.5

, f

(

δ)=δ/(δ+1)

, n=3, m=1

Data averaged over 100 runsSlide56

5.GRAPHIC SEQUENCES

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Graphic Deviations

A sequence {d1,d2,……

dk

} is said to graphic if there exists a simple graph whose degrees are exactly the

di’s

.

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Degree Preference

Population of k individuals. For individual

i

there is a range [

mi,Mi

] where it wishes its degree to lie.At each time t pick an individual, i

say. If the degree i \in [mi,Mi] do nothing, if i

< mi add a random edge, if i>Mi remove a random edge.

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Degree Preference

There are various tests for a sequence being graphic, Havel-

Hakimi

,

Erdos-Galais

, etc. We (Mark Broom and I) have extended these to allow one to calculate the minimum deviation (total degree), and proved that the set of graphs which achieve this score is connected under the possible transitions.

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Degree Preference

The process is Markov. Within the set of “minimal” states we have reversibility so have detailed balance. However still difficult to calculate the stationary probabilities except in fairly simple cases.

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6.PATTERNS OF ESS’S

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Evolutionary Conflicts

What strategies persist in biological/social populations

?

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Evolutionarily Stable Strategies

p

q

=

(1-

λ

)p+

λ

q

p

Evolves

Population

Invaders

Perturbed population

p

Population is stable wrt invasionsSlide67

Exclusion Results

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Exclusion Results

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Parker’s Model

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Parker’s Model

If we have just u and v in the population then the payoff matrix is

so u>>v if (v-u)>V/2

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u-V/2 u

u+V

/2

<-v wins -

 <u wins> < v wins > < u wins >

vSlide74

Parker’s Model

Suppose then that V=2

+

so x eliminates y

iff

x>y+1 and suppose we have x and y from S={0,2,……,m-1} then as mutations arise randomly and uniformly over S we have a Markov Chain

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Parker’s Model

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Stationary Distribution

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Stationary Distribution

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V ne 2+

Algorithm for any m

Continuous [0,m

] is open.

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7.MODULES IN GRAPH DYNAMICS.

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7. Modules on Graphs

One way of identifying a “module” in a graph is to find subsets of V such that if

i

in V then

i

has more neighbours in V than in not-V. Obviously this is related to the majority game.

An alternative approach is that of Irons and Monk who looked for dynamic modules

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Graphs

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