MATERIAL BEING ADDED IN NEAR FUTURE

## Graphic Deviations

Joint project with Mark Broom City University, London.

A sequence {x1,x2,x3,........,xn} where x_i is a positive integer is said to be GRAPHIC if there is a simple graph (i.e. one with no multiple edge or self-edges) whoes n vertex degrees are precisely x1,x2,.....,xn. Suppose one has a set of entities associated with each is some integer range, [mi,Mi] for the ith. This range specifies where the entity would wish to fix its degree. Suppose there is some simple graph G(t), at time t. A vertex is selected at random, the ith say. If the degree of i is in [mi,Mi] no change is made. If i < mi then some j to which i is not currently linked is selected at random and the edge (i,j) added. If i > Mi then some j to which i is linked is selected at random and the edge (i,j) deleted. Thus we have a Markov chain.

In our first paper (Broom and Cannings Submitted for publication)we discuss the basic theory.

Abstract.

We consider an arbitrary sequence of integers There is growing interest in modeling populations using graph theory where individuals are represented by vertices and interactions between pairs by edges. Suppose in some population each individual has a preferred range for their number of links to other individuals. There is then, for any graph on those individuals, a deviation from the preferred values. As individuals adjust their links according to their preferred range the graph evolves towards some set of graphs which achieve the minimum possible deviation. The study of this process raises a new question in graph theory. If we are given a sequence of \$n\$ non-negative integers how can we find the graphs which achieve the minimum deviation from that sequence as its degrees. This extends the classical problem regarding what sequence are "graphic", that is can be the degrees of a simple graph, to issues regarding arbitrary sequences. In this context we shall demonstrate how a variation on the Havel-Hakimi algorithm can supply the value of the minimum possible deviation, and how consideration of the Ruch-Gutman condition and the Ferrer diagram can yield the complete set of graphs achieving this minimum.

In our second paper (Broom and Cannings, In Press)we discuss the application of the model to biological populations via the Markov chain.

Abstract.

Historically most evolutionary models have considered infinite populations with no structure. Recently more realistic evolutionary models have been developed using evolutionary graph theory, which considered the evolution of structured populations. The structures involved in these populations are typically fixed, however, and real populations change their structure over both long and short time periods. In this paper we consider the dynamics of such a population structure. The timescales involved are sufficiently short that no individuals are born or die, but the links between individuals are in a constant state of flux, being actively governed by the preferences of the members of the population. The process is modelled using a Markov chain over the possible structures. We find that under the specified process the population evolves to a closed class of structures, and we show a method to find the stationary distribution on this class. We also consider some special cases of interest.

## Parker's Model

Evolutionary Conflicts with Mutations, 1. Parker's Model. Parker's Model has no ESS. The object of this project is to study the behaviour of a population playing Parker's Model with reward V and some set ([0,m] or {0,1,2,...,m-1}) of possible plays, when individuals playing these strategies arise through repeated mutation. The aim is to specify the stationary distribution of such a population.Three papers are in preparation (1) considering general aspects of the problem, including specification of patterns of dominance and (2) specifying the stationary distribution in the special class where V is in (2,4], the strategy set is {0,1,2,....,m-1} and mutations occur infrequently and uniformly over the possible values.

The Stationary Distribution of a Class of Markov Chains. (2013) Applied Mathematics, 4, 769-773

(3) Suppose there are n possible plays, and there is infrequent mutation with equal rates for all the plays. Then the system will at any time have only two plays one of which will rapidly displace the other. We investigate when every play will be represented in the stationary distribution of the resulting Markov chain. The conditions are related to Catalan numbers and Dcke words.

Combinatorics of Parker's model. Dynamic Games and Applications {To appear}

## Best Response Games on Regular Graphs

Joint work with Richard Southwell

,The Chinese University of Hong Kong.

With the growth of the internet it is becoming increasingly important to understand how the behaviour of players is affected by the topology of the network interconnecting them. Many models which involve networks of interacting players have been proposed and best response games are amongst the simplest. In best response games each vertex simultaneously updates to employ the best response to their current surroundings. We concentrate upon trying to understand the dynamics of best response games on regular graphs with many strategies. When more than two strategies are present highly complex dynamics can ensue. We focus upon trying to understand exactly how best response games on regular graphs sample from the space of possible cellular automata. To understand this issue we investigate convex divisions in high dimensional space and we prove that almost every division of (k-1) dimensional space into k convex regions includes a single point where all regions meet. We then find connections between the convex geometry of best response games and the theory of alternating circuits on graphs. Exploiting these unexpected connections allows us to gain an interesting answer to our question of when cellular automata are best response games.

Best Response Games on Regular Graphs. (2013) Applied Mathematics, 4, 950-962

## Trinets

Joint work with Richard Southwell

and

Jianwei Huang

## Exploring the space of substitution systems

Joint work with Richard Southwell

Southwell R and Cannings C. (2013) Exploring the space of rewrite system. Complex Systems, 22,?-?.

## Random Walks and Preferential Attachment

Joint work with Jonathan Jordan School of Maths. and Stats., University of Sheffield.

## Geiringer's Theorem

Joint work with Boris Mitavskiy et al. The University of Aberystwyth

## Pure Nash Equilibria in Co-ordination Games

Joint work with Rob Cannings.

In the congestion game with monotonically decreasing payoffs with increasing congestion Milchtaich (Games and Economic Behaviour, 13, 111-124 (1996)) proved that there is always a pure Nash equilibrium. We consider the co-ordination game with increasing payoffs with increasing co-ordination (see Southwell, ). We prove, by way of counterexamples that, except when there are only two strategies, there is no guarrantee that a pure Nash equilibrium exist. The model introduced to provide a counterexample is generalised to provide futher models where under miopic behaviour we obtains cycles corresponding to the orbits under a cyclic group action. (Cannings C and Cannings R, (2013) Absence of pure Nash equilibria in a class of co-ordination games. (In Press), Statististic, Optimization & Information Computing.

## Majority/minority games on graphs.

Joint work with John Haslegrave, SoMaS, University of Sheffield.