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Reproducing Graphs
Chris Cannings
&
Richard Southwell
Gene Networks
Genealogies
Biological/Social Networks
• Individuals <-> Vertices, Nodes• Relationships <-> Edges
Graph
• A graph G=(V,E) consists of a set V={1,2,…n} of vertices (nodes, points, individuals) and a set E={(i,j), i,j ε V} of edges (joins, lines, relationships).
• We deal here only with simple, undirected graphs i.e. there are no self-edges (i,i), no multiple edges, and edges have no direction i.e. (i,j)=(j,i).
Reproducing Graphs
• We investigate a class of models in which graphs reproduce, corresponding to a growing social/relationship network. We concentrate here initially on models without mortality, then make a few comments on age-specific-mortality. We have also studied models in which vertices are eliminated according to degree and/or payoff in a game against neighbours.
Graph Products
• Given two graphs G1=(V,E) and G2=(W,F) then a graph product G12 = G1 X G2 =(X,H) has X=V*W where * is the Cartesian product (i.e. X is the set of ordered pairs (v,w) where v ε V and
w ε W). H depends on the particular product; there will be a rule which specifies which (v1,w1) and (v2,w2) join.
Cartesian Product
• V=(v1,v2,…..,vm) and W=(w1,w2,……,wn)J={((vi,wj),(vk,wl))| definition)
• Cartesian product [vi=vk & (wj,wl) ε F] OR [(vi,vk) ε E & wj=wl]
=
Kronecker Product
• V=(v1,v2,…..,vm) and W=(w1,w2,……,wn)J={((vi,wj),(vk,wl))| definition)
• Tensor (Knonecker) product [(vi,vk) ε E & (wj,wl) ε F] Associative
• Adjacency matrix of GθH is direct product of the adjacency matrices of G and H.
=
Kronecker..Example
1
2
3
b
a c
(1,a)
(1,b)
(1,c)
(2,b) (2,a)
(2,c) (3,b)
(3,a) (3,c)
(i,α) linked to (j,β) if, and only if(i,j)εV and (α,β)εW
X
Our Models
• In our model we suppose there exists some graph at time t and this leads to another at time t+1.
• Each vertex at time t survives and gives rise to a new offspring vertex.
• Each edge at t survives.• Some subset of the possible edges
between the “new” and “old” vertices are added.
Gt+1=Fi(Gt)
• Formally we have that Gt+1 is a function of Gt, where Gt ε G the set of all simple graphs.
• The index i on Fi specifies which member of our family of models is being applied.
Our Models
• Given a network Gt=(Vt,Et) we generate Gt+1=(Vt+1,Et+1) in the following way:-Each edge (u,v) is replaced by
u1
v1
u0
v0
Always
α
β
γ
Where u1 is just u again while u0 may be regarded as an offspring of u
Our Models
• 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
1 0 0 4*
0 1 1 3
1 0 1 5
1 1 0 6
1 1 1 7*
u1
v1
u0
v0β
β
α
Our Models
u1 u0
v1v0
u1
v1
u0
v0
u1
v1
u0
v0
u1
v1
u0
v01 3
65u1
v1
u0
v02
Our Models
u1 u0
v1v0
w1 w0 w
v
u
Merger of Two Graphs
G J H
Fundamental Theorem
Theorem
• Thus we may investigate Fit(Z) where
Z=(V={0,1},E={(0,1)} is the single edge graph.
• Then apply to any G0
Our Models
u1 u0
v1v0
u1
v1
u0
v0
u1
v1
u0
v0
u1
v1
u0
v01 3
65u1
v1
u0
v02
Our Models
• 0,1,4 & 5 are Kronecker products
• 6 is Cartesian
• 2 is Comb
• 7 is Strong
• 3 is Non-standard
Model 1
1Gu1 u0
v1v0
where 1 is the Kronecker product
Model 5
u1
v1
u0
v0
1G
where 1 Is the Kronecker product
The “Knonecker” Models
• Since the adjacency matrix of the Knonecker product of two graphs is the Knonecker product of the adjacency matrices we can exploit this for models 0,1,4 & 5
The “Knonecker” Models
• The adjacency matrices for the four models 0,1,4,5 are respectively. Only that for model 1 is interesting which is essentially the bitwise AND.
11
11
10
01
10
00
11
10
Model 3
u1
v1
u0
v0
Not a standard graph product
Model 6
3Gu1
v1
u0
v0
where 3 is the Cartesian product
Model 6 u1
v1
u0
v0
1-cube
u1
v1
u0
v0
Model 6
2-cube
u1
v1
u0
v0
Model 6
3-cube
u1
v1
u0
v0
Model 6
4-cube
Model 2
u1
v1
u0
v0
5
5G
where Is the “Comb” product
0 1
u1
v1
u0
v05
Model 2
0 1
22
u1
v1
u0
v05
Model 2
0 1
22
3
33
3
u1
v1
u0
v05
Model 2
0 1
22
3
33
3
4
4
4
4 4
4
4
4
u1
v1
u0
v05
Model 2
N.B…..Binary Representation
• If at time t we have a set of nodes {v1,v2,….,vn} where each vi is a binary string then at time t+1 we have set of nodes {v11,v10,v21,v20,……..,vn1,vn0}where vi becomes vi1 and is the parent of a new node vi0.
• All our models can be specified in terms of the nodes as binary strings and logical operations defining the edges.
Binary Representation
• Model 1. Kronecker product of
0
1
n
=
G[V={x in {0,1}n},E={(x,y) s.t. (xi,yi) n.e. (0,0)}
Properties
• 1. Chromatic Number • 2. Number of Vertices
and Edges• 3. Distance Structure• 4. Degree Distribution• 5. Automorphism• 6. Generating all
graphs as subgraphs.
)(G
Invariant
Chromatic number
0
1
2
3
4
5
6
7Goes up by 1Goes up but doesn’tmore than double
Model 7u1
v1
u0
v0
)(2)(1)( 1 ttt GGG
Since model 7 “contains” model 6
Equality achieved by complete graph
Number of Vertices & Edges• No of nodes Vt doubles• No of edges Et
so
t
t
t
t
V
E
V
E
20
21
1
1
)4,0(),2,1(),3,1(),3,0(),(
)21(
)2/()2(00
and
where
VEE tttt
for Models 1,2,3,4,5. For model 3 second term is linear.
The Distance Structure
• Denote distance (shortest path) between vertices u and v by d(u,v),the diameter (max distance) by D(G), and the number of pairs of vertices with distance x as Nt(x).
• We demonstrate our methods wrt Model 2.
• If u in Vt then u0 and u1 are the resulting offspring and parent vertices.
The Distance Structure Model 2
• u & v in Vt and d(u,v)=d then
• d(u0,u1)=1, d(u0,v0)=d+2,
• d(u0,v1)=d(u1,v0)=d+1,
• d(u1,v1)=d.
• We can then deriveNt+1(0)=2Nt(0); Nt+1(1)=Nt(0)+Nt(1); Nt+1(2)=2Nt(1)+Nt(2) &Nt+1(k)=Nt(k-2)+2Nt(k-1)+Nt(k) for k>2
u1
v1
u0
v02
The Distance Structure Model 2
• We have also number of distances Lt =4t(V0)2+2tV0 and the total distance Lt
*=4tL0*+22t-1(N0(0))2-(22t-1-2t-1)N0(0)
• From this we derive an expression for the average distance dt=Lt
*/Lt->c+t for large t which we can also obtain more directly by considering a random pair of vertices at time t+1 in terms of those at t. In fact we can also show that asymptotically the variance of the average distance is f(G0)+3t/2.
Degree Distribution *,1 & 5
• Adjacency matrix of G H is Kronecker product of the adjacency matrices of G and H.
i.e. here
so degrees are direct products of
giving 2t 1’s, tCi 2i’s and 2t 2t ‘s respectively.
11
11&
01
11,
01
10
2,2&1,2,1,1
0 1 5
Degree Distribution Model 6
• Starting from a single edge we have just a hypercube at time t so all nodes of degree 2t
u1
v1
u0
v0
Degree “Dist” Model 3
• A node u of degree d gives rise to an offspring u0 of degree (x+1) and a survivor u1 of degree (2x+1). Thus (1) >(2,3)->(3,4,5,7)->(4,5,6,7,8,9,11,15)->(5,6,7,8,9,9,10,11,12,13,15,16,17,19,23,31) totals 1,5,19,65,211,……,3n-2n,….
u1
v1
u0
v0
frequency
degree
Degree Distribution Model 3
(1) after 18 updates, nodes of degree < 5,000
(1) After 18 updates, nodes of degree <500
Age Culling
• Suppose now that we associate with each node an index which we regard as its age.
• If at time t node u in V is age s then at timet+1 u1 is of age s+1 and u0 is of age 0.
• We suppose that after reproduction a node of age Q+1 dies (i.e. is deleted from the graph)
Model 6 Age Culled
u1
v1
u0
v0
Grows as hypercube with pure reproduction
0 1
Model 6 with age-cap=1 u1
v1
u0
v0
Ages
1 2
Model 6 with age-cap=1 u1
v1
u0
v0
Ages
1 2
0 0
Model 6 age-cap = 1 u1
v1
u0
v0
1 2
0 0
Model 3 age-cap=1 u1
v1
u0
v0
1 2
0 0
Model 6 age-cap=1 u1
v1
u0
v0
1
0 0
Model 6 age-cap=1u1
v1
u0
v0
2
1 1
Mother-Daughter with age cap 1
2
1 1
0
0 0
Model 6 age-cap=1u1
v1
u0
v0
2
1 1
0
0 0
Model 6 age-cap=1u1
v1
u0
v0
2
1 1
0
0 0
Model 6 age-cap=1u1
v1
u0
v0
1 1
0
0 0
Model 6 age-cap=1u1
v1
u0
v0
2 2
1
1 1
Model 6 age-cap=1u1
v1
u0
v0
2 2
1
1 1
0 0
0
0 0
Model 6 age-cap=1 u1
v1
u0
v0
2 2
1
1 1
0 0
0
0 0
Model 6 age-cap=1 u1
v1
u0
v0
2 2
1
1 1
0 0
0
0 0
Model 6 age-cap=1 u1
v1
u0
v0
1
1 1
0 0
0
0 0
Model 6 age-cap=1u1
v1
u0
v0
Model 2
• This is the easiest case to treat since the graph grows trees on each original individual. Here we can start with a single node
Model 2
• Now when we cull at any given age Q+1 we obtain copies of all the trees “of ages” 0,1,2,3,……,Q
• If there are nit trees of age i at time t then we get ni,t+1=ni-1,t + nQ,t i=1,2,…Q so we have nt+1= Ant where nt is the column vector of the ni,t’s.
0
u1
v1
u0
v05
Model 2 age-cap=2
1
u1
v1
u0
v05
Model 2 age-cap=2
1 0
u1
v1
u0
v05
Model 2 age-cap=2
1 0
u1
v1
u0
v05
Model 2 age-cap=2
2 1
u1
v1
u0
v05
Model 2 age-cap=2
2 1
00
u1
v1
u0
v05
Model 2 age-cap=2
2 1
00
u1
v1
u0
v05
Model 2 age-cap=2
3 2
11
u1
v1
u0
v05
Model 2 age-cap=2
3 2
11
0
00
0
u1
v1
u0
v05
Model 2 age-cap=2
3 2
11
0
00
0
u1
v1
u0
v05
Model 2 age-cap=2
3 2
11
0
00
0
u1
v1
u0
v05
Model 2 age-cap=2
2
11
0
00
0
u1
v1
u0
v05
Model 2 age-cap=2
3
22
1
11
1
u1
v1
u0
v05
Model 2 age-cap=2
3
22
1
11
1
0
0
0
0
0
0
0
u1
v1
u0
v05
Model 2 age-cap=2
3
22
1
11
1
0
0
0
0
0
0
0
u1
v1
u0
v05
Model 2 age-cap=2
3
22
1
11
1
0
0
0
0
0
0
0
u1
v1
u0
v05
Model 2 age-cap=2
22
1
11
1
0
0
0
0
0
0
0
u1
v1
u0
v05
Model 2 age-cap=2
Tree “Dist” Model 2 age-cap=3
• The matrix A is similar to a Leslie matrix L with guaranteed survival except that ai,j=lQ-i,Q-2i+j
e.g. {rate λ max root of λ4= λ3 + λ2 + λ +1, λ(4)}
1100
1010
1001
1000
A
0100
0010
0001
1111
L
Leslie Matrix
nn
nn
nnn
s
s
s
s
s
bbbbbb
)1(
)1)(2(
23
12
01
12210
...
...
...
...
............
other elements all 0
Distribution of age(i,j) edges
• It is convenient to introduce a direction
i+1
j+1
0
0
i
j
Distribution of age(i,j) edges
tt
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
33
32
31
30
23
22
21
20
13
12
11
10
03
02
01
00
1
33
32
31
30
23
22
21
20
13
12
11
10
03
02
01
00
000001000000000
0000001000000000
0000000100000000
000000000000
0000000001000000
0000000000100000
0000000000010000
000000000000
0000000000000100
0000000000000010
0000000000000001
000000000000
000000000000
000000000000
000000000000
Distribution of age(i,j) edges
• et+1=Let where
000
000
000
B
B
B
AAAA
L
000
000
000
A
0100
0010
0001
B
Distribution of age(i,j) edges
• We can prove that the system is irreducible (though if α=0 need to refine the state space to exclude (i,i) states).
• Essentially a generalisation of the Lesley matrix notion.
• Eigenvalues related to generalisations of the “golden ratio” (tribonacci, etc.)
Example. Model 1, cull at age 3
• Offspring joined to neighbours of parents.•
• Next slide shows progress through time omitting isolated vertices.
1 1 1
1
161
1
1
3
Degree cap = 6
References
• Southwell & Cannings, Some models of graph reproduction; 1 Pure Reproduction. (to appear) AM.2 Age Capped Vertices (to appear) AM3 Game Based Reproduction (to appear) AM
• Jordan & Southwell. Further properties of reproducing graphs (to appear) AM
• AM http://www.scirp.org/journal/am/
Applied Mathematics
• Editor in Chief ….CC
Editorial Board includes
Mark Broom David Greenhalgh
http://www.scirp.org/journal/am/
Other Results
• For age-culled,
degree distribution diameter
average distance
Further Work• 1) Culling by degree
• 2) Stochastic/non-synchronous
• 3) Conflict with neighbours to determine survivor &/or reproduction
