HIERATIC

Chris Cannings Birmingham, Kick-Off 6/7 Dec. 2012

Chris Cannings

Stochastic processes, combinatorics, graph theory, algorithms IN Population genetics, human genetics, evolutionary games

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 yeast

A single cultured neurone, 2 days after planting (Shefi et.al. (2002) Phys.Rev.E, 66, 021905.)

Genealogy

Female Male

Genealogy

Genealogies

TOPICS

• 1.Genealogies & Genetics • 2.Games on Graphs…Majority Game • 3.Reproducing Graphs • 4.Growing Graphs • 5.Graphic Sequences • 6.Patterns of ESS’s 01/05/2020 © The University of Sheffield 11

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. 13 01/05/2020 © The University of Sheffield

Calculations on Genealogies

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

• Peeling 2 14 1 01/05/2020 3 © The University of Sheffield

R-functions

R

(

u

)  

R

(

v

)

Pen

(

u

,

data

)

Trans

(

u

,

v

) 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. 01/05/2020 © The University of Sheffield 15

Recurrence Relation

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.

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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. 01/05/2020 © The University of Sheffield 19

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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? 21 01/05/2020 © The University of Sheffield

ALL MAJORITY PLAYERS DYNAMICS FLASHERS

A class of “cylindrical” cubic graphs • Take two polygons of size n (here n=5).

5 4 5 1 Take a permutation P={p 1 ,p 2 ,………..p

n }, and join each i on one 3 4 3 2 1 2 polygon to p i other.

on the Example (shown) perm=(1,2,3,5,4) Denote such a graph n-CYL-P

5-CYL-{12345} 2 2 5-CYL-{12354} 1 5 5-CYL-{12453} 1 5 2

A

2 4 3 4 3 1 5 1 2

B

5 3 3 4 2 4 5-CYL_{13524} 1 1 5 2

C

4 3 3 1 1 2

D

5 4 3 3 5 4 5 4

Fixed Point Minority Game

An 8-cycle on the 3-cube

1-W W = majority player = minority player

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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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 α β  or absence of the  α, β and (indicated by 0 0 0 0 0 0 1 0 1 0 and 1). u 1 β  u 0 0 1 1 0 1 0 α 1 0 1  1 1 0 1 1 1 v 1 β v 0 Model 0 1 2 3 4 5 6 7

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. 35 01/05/2020 © The University of Sheffield

Example. Model 1, cull at age 3

• • Offspring joined to neighbours of parents.

• Next slide shows progress through time omitting isolated vertices.

1 1 Degree cap = 6 1 1 3 1 1 6 1 1

Issues

• What cycles?

• System closed under types?

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.

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

Preferential Attachment

2 N=8=M 3 1 2 4 3 Example. (weights shown at nodes) 1 3 1 2 2 1 1 2 1

2

1 1 1 1 3 1 3 N=10,M=80 1 1 1 1 3 1 2 1 1 2

The Degree Distribution

Suppose a new vertex joins to m pre-existing ones .

Pr( node has degree

k

) 

p k

 i.e.

asymptotic ally of order

k

 3

k

 2

k m

 

m

1 

k

 1   2 

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. R n 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., 01/05/2020 © The University of Sheffield 47

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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 49 01/05/2020 © The University of Sheffield

The M-model (Chrysafis & CC)

• Vertex set V and E=V*V, weight w ij each edge, s i = w i., m ij = w ij /s i.

, M=(m for ij ) 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).

Strength/weight evolution

s i



m B



i

,  1  1  

B t

,  1  1    1

w i



m B i

, 1     1  

B t

, 1     1    1 •

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-Degree Dependence

Strength

s i

and  

i

degree 

k i



A

are related  1  

A

by : •

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 network

Degree Sequence

p k



B B

 

k a

 

a

, 1 ,

a a

  3 2  

Simulation parameter values:

N=10 4 , δ=0.5, f (δ)=δ/(δ+1), n=3, m=1

Data averaged over 100 runs

Strength Distribution

p s



B B



s

 

a a

  1 , 1  ,

a a

  2  3  ,

a

 

A A

 1 Simulation parameter values:

N=10 4 , δ=0.5, f (δ)=δ/(δ+1), n=3, m=1

Data averaged over 100 runs

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. 01/05/2020 © The University of Sheffield

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. 01/05/2020 © The University of Sheffield 59

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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

Perturbed population Population (1 λ)p+λq Invaders q p = p Evolves p Population is stable wrt invasions

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 73 so u>>v if (v-u)>V/2 <-v wins  < v wins > < u wins > u-V/2 u u+V/2 © The University of Sheffield v

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 74 01/05/2020 © The University of Sheffield

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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.

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Graphs

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