Some Models of Reproducing Graphs: II Age Capped Vertices

In the prequel to this paper we introduced eight reproducing graph models. The simple idea behind these models is that graphs grow because the vertices within reproduce. In this paper we make our models more realistic by adding the idea that vertices have a finite life span. The resulting models capture aspects of systems like social networks and biological networks where reproducing entities die after some amount of time. In the 1940’s Leslie introduced a population model where the reproduction and survival rates of individuals depends upon their ages. Our models may be viewed as extensions of Leslie’s model-adding the idea of network joining the reproducing individuals. By exploiting connections with Leslie’s model we are to describe how many aspects of graphs evolve under our systems. Many features such as degree distributions, number of edges and distance structure are described by the golden ratio or its higher order generalisations.


Introduction
Networks are everywhere, wherever a system can be thought of as a collection of discrete elements, linked up in some way, networks occur.With the acceleration of information technology more and more attention is being paid to the structure of these networks, and this has led to the proposal of many models [1][2][3].
In many situations networks grow-expanding in size as material is produced from the inside, not added from outside.To study network growth we introduced a class of pure reproduction models [4,5], where networks grow because the vertices within reproduce.These models can be applied to many situations where entities are introduced which derive their connections from pre existing elements.Most obviously they could be used to model social networks, collaboration networks, networks within growing organisms, the internet and protein-protein interaction networks.One of our systems (model 3) has also been introduced independently [6], proposed as a model for the growth of online social networks.
In our pure reproduction models networks grow endlessly in a deterministic fashion.This allows a rigorous analysis, but costs a degree of realism.Nature includes birth and death and entities may be destroyed for reasons of conflict, crowding or old age.In this paper we consider age; and extend our models by including vertex mortality.

The Models
In [5] we defined a set { : {0,1, ,7}} In our age capped reproduction models we think of the vertices as having ages.Graphs grow under these models exactly as before, except that vertices grow and then die when their age exceeds some pre-specified integer Q .Our new update operator , m Q T is defined so that , ( ) T G is the graph obtained by taking the graph G and performing the following process; 1) Increase the age of each vertex by one.2) Give every vertex an age zero offspring, born with connectivity dependant upon m , as above (i.e. the new graph is ( ) m F G ). 3) Remove every vertex with age greater than the age cap Q .
We are interested in the sequence } { t G of graphs which evolve from an initial structure 0 0 0 = ( , ) G V E in such a way that 1 , = ( ) We always suppose that initial vertices have age Q  .

The Number of Vertices
The number of vertices | | t G in t G is deeply connected with the golden ratio and its generalisations.The number t i n of age i vertices in t G can be conveniently described in terms of Leslie matrices.In Leslie's population model [7,8] individuals of age i have a survival rate i s and fertility rate i f .The expected number of individuals of a given age, at a given time, is kept track of via repeated multiplication of the state vector with the 'Leslie matrix' In our case 1 = .
L is a primitive matrix with characteristic polynomial and principle eigenvalue Q  (also known as the is the stable age distribution and c is a constant which depends upon the initial state of the system.Let Ld , where   [ 1]   0

Binary Strings
As we update our graphs, their vertex sets will grow, and a good way to keep track of these vertex sets is to use binary strings.Suppose v is a vertex of 0 G .When we update G we write ( ,0) v and ( ,1) v to denote v 's offspring, and v itself (respectively), in the graph 1 G .This means, for example, that (( ,0),0) v is the grand child of (( ,1), 1)  v in 2 G .We use short hand by omitting the parenthesis, so for example we write ,0),0) ((v as 00 v .An example of the evolution of model 2 is shown in Figure 1. When our age cap = Q  an initial graph 0 0 0 = ( , ) G V E will evolve in exactly the same way as in pure reproduction i.e.G F G ; this will have vertex set 0 {0,1} t V  and edge set as specified in [5].When Q is finite the situation is more complex, but our binary string notation allows us to keep track of the ages of vertices in a convenient way.
Let ab denote the concatenation of binary strings a and b and let t a denote the string obtained by concatenating a with itself t times.Suppose ( 01) 's name corresponds to an update within which denotes the set of all t length binary strings which do not contain a run of  , as is clearly the case when = 0 t .An age n vertex va in t G , with 0 v V  , will produce an G .va will also survive to become ( 1)  v a iff < n Q .Such an a must be of the form 01 n or 1 n .It follows that 1 t G  will be the subgraph of 1 ( )

How Edges Connect Vertices of Different Ages
To keep track of the number of edges of t G it helps to consider how vertices of different ages link to one another.Let S denote the Leslie matrix with all survival rates set at one and all fertility rates set at zero.Let F denote the Leslie matrix with all survival rates set at zero and all fertility rates set at one (note ).Let us define the age sampling vector of a vertex to be such that i X is the number of neighbours it has of age i .
Applying the Q m T , update will cause an age Q n  vertex to have an offspring with age sampling vector and also, provided Q n < , this vertex will also survive the Q m T , update to become a parent with age sampling vector Equations ( 1) and (2) describe how the age sampling vector of a vertex determines the age sampling vector of itself and its offspring on the next time step.Repeatedly using these equations allows us to understand how the history of a vertex relates to its connectivity.The sequence of zeros and ones in a tell us the sequence of birth and survival stages which lead to the creation of a vertex Since the graphs we are concerned with are undirected we have , j i,  .The average asymptotic rates of increase of the minimal and maximal degrees for the different models are given in Table 1.We use the term average because, under some models, these extremal degrees increase at varying rates dependant upon the time modulo 1  Q .These rates where found by determining which binary string describes a vertex with maximal (or minimal) degree and using Equations ( 1) and (2) .For example, suppose the initial graph 0 G is age zero, and holds a vertex v with maximal degree, ( ) , will have maximal degree in t G .This vertex will have age sampling vector   .
The degree of the vertex with this form will increase by   steps, and so it follows that the average asymptotic rates of increase of the maximal degree when

Table 1. A table showing the average asymptotic growth rates of the minimal and maximal degrees under the different models m . The notation LIN( x ) indicates that the extremal degrees increase linearly (as opposed to exponentially) with time with gradient x .
m growth rate of the minimal degree growth rate of the maximal degree

Connectivity, Degrees and Distances in Specific Models
In this section we will focus on reproduction mechanisms with {1, 2,3,5, 6, 7} m  , one after another, and discuss the development of: connected components, number of edges, degree distributions, average path length and diameter.We do not discuss the dynamics when = 0 m or = 4 m because they are relatively uninteresting.Before we discuss the specifics it is worth pointing out an effect that occurs under many models.We say that a graph is age mixed when each of it edges connect a pair of vertices with different ages.
then t G will be age mixed.The reason is that when 0 =  offspring are not born connected to one another.So when all of the initial vertices will be dead, and t G will never again produce linked vertices with the same age.
Saying that t G is age mixed has many implications, for example it means that t G has chromatic number Q  because its vertices may be coloured according to their ages.

Suppose
= 1 m and we begin with a connected graph.t G will typically consist of a growing connected component and lots of isolated vertices.
In the special case when > =1 t Q , updates do not cause the connected part of t G 's structure to change.The reason is that t G is age mixed and every new born vertex either has a dead parent, which it replaces, or no surviving neighbours.Suppose 0 Any vertex ua of t G will be isolated iff a holds a run of 1  Q consecutive zeros.To see this note that theorem 1, together with results from [5], imply that vb will be a neighbour of ua iff 1001100 Y  at the next time step.Following similar reasoning one can see that, for generic Q , we have the difference equation; Regarding the edges, Equations ( 3) and ( 4) lose their dependance upon t i n and , > =0 , . Given these considerations we can reduce ( 3) and ( 4) to the following system of linear difference equations: = .
We can cast this system as a matrix difference equation which describes the evolution of The matrix which describes how (7) so it appears at the high end, as if the distribution obeys a geometric law.Whilst it seems there is some pattern in the degree distribution at the high degrees, the behaviour of the distribution of the lower degree vertices is more mysterious.For example it appears that when 1 > Q there will be less degree 1 vertices than degree 2 vertices when t is large.
Global notions of distance (such as diameter) do not really make sense when 1 = m because the structure is disconnected, with many isolated vertices.

Aspects of Model 2
Introducing an age cap into the 2 = m model leads to fascinating self replicative behaviour.Whatever graph we begin with we end up with a set of special tree graphs that grow up and break into more tree graphs.Let t Q S denote the graph obtained by starting with an age zero isolated vertex and evolving updating it with 2,Q T , t times.This graph will have vertex set t Q W and a pair of vertices b a, will be adjacent iff {1, 2, , } , over n time steps, will be age-isomorphic to the graph n S  which grew out of the initial vertex, over n time steps (by age-isomorphic we mean there is a one to one mapping, from one vertex set to the other, which preserves the adjacency, non-adjacency and ages of the vertices).
More generally = and removing the oldest vertex, is a tree, the removal of 1 1  Q causes the graph to break into numerous components, namely Each of these connected components will evolve in the same manner-growing until the age of its central vertex exceeds Q , at which point it will fragment into yet more of these special trees.
Any initial graph will evolve to become a set of these trees after = ( , , , ) is equivalent to the transpose of the Leslie matrix L .It follows that when t is large t n C will increase at a rate of Q  and the probability that a random connected component is age-isomorphic to i S  will be ( 1) =0 ( 1) The number of edges is described by the equations: and so the number of edges in t G increases at a rate of Q  asymptotically.
We can gain the asymptotic form of the degree distribution of t G .First note that the graph i S  has i 2 vertices.The number of degree k vertices in i S  will be that a randomly selected vertex of t G will be of degree k will be equal to [the probability that a randomly selected vertex of t G be-longs to a connected component isomorphic to i S  ] times [the probability that a randomly selected vertex of i S  will have degree k ], summed over all {0,1, , } i Q   .For large t the probability that a randomly selected vertex of t G belongs to a connected component isomor- The probability that a randomly selected vertex of i S  will be of degree k will be Hence as t   we have ( ) Suppose t is large.
Once again we do not discuss distances because global notions of distance do not really make sense upon graphs which constantly disconnect.

Aspects of Model 3
Growth model 3 produces complicated structures; we can say a little about their connectivity using reasoning like that used when , i  , and we will hence have that: In the 1 = m case the number of edges increase asymptotically at a rate of Q case is similar except that the number of edges is bolstered by the number of vertices t i n , which increases at a lesser rate of Q  .For large t the effect of these additional edges is hence negligible and the number of edges again increases at a rate of Q  .
Like the 1 = m case computer simulations again suggest that when n t < 1  we have we have and when 2 When we introduce mortality our graphs seem to get longer.Diameter and average path length become greater.This is a result of the death of old vertices (which tend to be more central), this decreases the ease with which one can travel between the extremities.
Let the average length is given by   0,0 = .
When 3 = m it seems as if both the average length and diameter of t G increase linearly with t whenever Q is finite.In the special case where and (provided v u  ), ( 0, 0) then after the update we have ( 1, 0) . If u and v are both of age one then after the update . The diameter of t G will increase by two every two time steps and moreover the system obeys the equations These equations imply that which means that when t is large, the average length increases linearly with .15. 10 This implies that the curve which describes t l , for generic Q is bounded below by a constant (because of the  = Q case, see [5]) and bounded above by a straight line.

Aspects of Model 5
In this case, when 0 G is age zero, t G may be obtained by replacing each vertex v of 0 G with a cluster v C of [ 1]   2 Q t f   isolated vertices, and then connecting each vertex of u C to each vertex of v C whenever u and v where adjacent in 0 G .It follows that Equations ( 3) and ( 4) which describes the development of the edges may be cast as the matrix equation The matrix involved is clearly the Kronecker product of L with itself, it is hence primitive with principle eigenvalue 2 Q  .It follows that the asymptotic growth rate of the number of edges will be 2 Q  .
Suppose our initial graph is connected, non-trivial and age zero.t G can be obtained by replacing each vertex by a cluster of [ 1]   2 , in the initial graph gives rise to ordered pairs, spaced by distance k , in t G .In addition to this, every cluster adds 2. 1 to the total distance, by the fact that every pair of distinct vertices within a given cluster will be spaced by distance 2. It follows that the total distance of t G will be   This means (irrespective of Q ) that when t is large the average length approaches the constant The diameter of t G will be the maximum of the diameter of 0 G and 2 .

Aspects of Model 6
In this case t G will be a connected graph that can be obtained by taking the t dimensional hypercube graph and removing some vertices.The next graph in a sequence can be obtained by fusing together previous structures.For example when can be obtained by taking the disjoint union of t G and to the subgraph of t G induced upon its age zero vertices (such an isomorphism always exists) and adding an edge from each v vertex of . The age one vertices of will be those which came from .This implies we can split the sums to get making substitutions we find When t is large the minimal degree of t G becomes large-implying that the average degree also becomes large.This implies and so the asymptotic growth rate of the number of edges will be Q  .
Determination of the degree distribution when 6 = m appears to be a difficult problem.Although some progress can be made when 1 = Q the resulting formulae are long and complicated.
With respect to distances it appears that the diameter and average length of t G increase linearly when t is large.We can show explicitly that this is the case when .This implies that the system obeys the equations: These equations imply that as   t the average length increases linearly with The reasoning behind this is very similar to that when . The diameter of t G will increase by one every time step once the graph becomes zero spanning.

Aspects of Model 7
When our initial graph 0 G is age zero t G may be obtained by replacing each vertex v of 0 G with a complete graph vertices, and then connecting each vertex of u K to each vertex of v K whenever u and v where adjacent in 0 G .It follows that With respect to the edges this case is similar to the 5 = m case, except that there is an extra dependence upon t i n caused by the presence of edges linking offspring to their parents.
ordered pairs, spaced by distance k , in t G .In addition to this, every cluster adds . 1 to the total distance, by the fact that every pair of distinct vertices within a given cluster will be spaced by distance 1.It follows that the total distance of t G will be   Interestingly this means that when t is large t l loses its dependance upon Q and approaches the constant . The diameter of t G will be equal to the diameter of 0 G .

Discussion
We have discussed many properties of age capped models, however many open problems remain.These include describing degree distribution when {1,3,6}  m and demonstrating the linearity of average length when {3,6}  m (for generic Q ).There are many directions in which our models may be expanded.As highlighted by theorem 1, our models may be regarded as an extension of pure reproduction models by adding restrictions upon the language of binary strings which the vertices can possess.Many other restrictions could be considered, e.g.forbidding the subword 0 1 1    Q (which would correspond to saying vertices of age Q > become infertile).Our models can be viewed as an extension of Leslie's population model, introducing the idea of a network which connects the reproducing individuals.We will further develop this connection by considering the evolution of generic Leslie matrices (so that individuals of a given age can have differing numbers of offspring and chances of survival).Taking this approach and considering connectivity as stochastic (so that ,   and  are probabilities, rather then binary integers) should yield models which directly simulate the development of animal social networks and other phenomena.
This paper demonstrates how our original reproducing graph models can be generalised in different directions whilst remaining analytically tractable.Perhaps the main reason these models are amenable to analysis is that the growth of one part of a graph is not influenced by the structure of another.This spatial independence allows one to understand the evolution of generic structures by studying the evolution of simple ones.
There are many extensions of these models that it would be interesting to consider.In the future papers we will discuss the fascinating dynamics which can ensue when game theory is incorporated into these models.In this case we lose the spatial independence and dynamics of immense complexity become possible.It is also possible to extend many of the results here to cases where individuals produce several offspring-connected up in different ways.This kind of generalisation allows one model how the social networks of specific types of organisms grow in a more direct way.

1 
functions m F which map graphs to graphs.( ) m F G is the graph obtained by simultaneously giving each of G 's vertices an offspring vertex and then adding edges according to some rule.The connections given to offspring depend upon the binary representation  of m (i.e.offspring are connected to their parent's neighbour's offspring.


is the Kronecker delta.The n step Fibonacci numbers [ ] n i f are natural generalisations of the famous Fibonacci numbers [9] which can be generated by repeatedly multiplying 0 d by L .When 0 G is age zero (i.e.all its vertices have zero age)

Figure 1 .
Figure 1.A depiction of the evolution of model 2 when = 2 Q , starting with an isolated vertex named p .
this vertex survives as a parent and gets older by one.It follows that( 01 ) we can use induction with t to prove the result  .
no neighbour vb can actually exist.
changes is primitive with principle eigenvalue Q understand the self replicative behaviour in t such that either u is of age i , v is of age j or u is of age j , v is of age i .When 1 = Q Suppose our initial graph is connected, non-trivial and age zero.tG can be obtained by replacing each vertex by a complete graph on[ 1]