# City Research Online

## Modelling conflicting individual preference: Target sequences and graph realization

Aizouk, R. A. (2022). Modelling conflicting individual preference: Target sequences and graph realization. (Unpublished Doctoral thesis, City, University of London)

## Abstract

In this thesis, we build on a model introduced by Mark Broom and Chris Cannings in 2013. A group of individuals form pairwise contacts between themselves, and each has a potentially different target number of desired connections. How close can this target come to being realised? It is known that under a reasonable dynamic process, the graph of links evolves to one that minimises the total deviation from the target. The sets of all such graphs and associated sequences which are the sequences of node degrees are termed the minimal sets. We revisited the minimal set of sequences and obtained a formula that generates the size of that set for a given arithmetic sequence. We investigated the all or nothing sequences which are sequences of n individuals with target of n □ 1 or 0, and discovered a recurrence relation for such a formula considering the size of the minimal set of sequences. We investigated a game-theoretical version of the model, where individuals strategically chose the specific link. Showing that optimal play could lead to the minimal set being left, thus answering an open question from earlier work.

We revisited the game over the sequence 1; 1; 1 and found a general expression for the payoff functions for possible strategy combinations. A new set of six more solutions showed the complexity of this simple case, with the possibility of more solutions open. We introduced two models allowing individuals to reject, accept or break links suggested by others. Following the original model dynamical process and considering each new model rule led to a non-transient set of graphs, termed the terminal set. We defined the terminal set and demonstrated a general sequence’s vertex classifications and related properties. We introduced the Reverse Havel Hakimi theorem/algorithm and considered examples of game simulations over the terminal set that reached it, and investigated the individual’s best strategy to maximise their payoff at every state using backwards induction. Together, these results have provided us with some predictions about which networks are likely to form following different proposed social models during a particular period of time in an evolutionary process.

Publication Type: Thesis (Doctoral) Q Science > QA Mathematics School of Science & Technology > MathematicsSchool of Science & Technology > School of Science & Technology Doctoral ThesesDoctoral Theses
Text - Accepted Version
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