Placement matters in making good decisions sooner: the influence of topology in reaching public utility thresholds

Social systems are increasingly being modelled as complex networks, and the interactions and decision making of individuals in such systems can be modelled using game theory. Therefore, networked game theory can be effectively used to model social dynamics. Individuals can use pure or mixed strategies in their decision making, and recent research has shown that there is a connection between the topological placement of an individual within a social network and the best strategy they can choose to maximise their returns. Therefore, if certain individuals have a preference to employ a certain strategy, they can be swapped or moved around within the social network to more desirable topological locations where their chosen strategies will be more effective. To this end, it has been shown that to increase the overall public good, the cooperators should be placed at the hubs, and the defectors should be placed at the peripheral nodes. In this paper, we tackle a related question, which is the time (or number of swaps) it takes for individuals who are randomly placed within the network to move to optimal topological locations which ensure that the public utility satisfies a certain utility threshold. We show that this time depends on the topology of the social network, and we analyse this topological dependence in terms of topological metrics such as scale-free exponent, assortativity, clustering coefficient, and Shannon information content. We show that the higher the scale-free exponent, the quicker the public utility threshold can be reached by swapping individuals from an initial random allocation. On the other hand, we find that assortativity has negative correlation with the time it takes to reach the public utility threshold. We find also that in terms of the correlation between information content and the time it takes to reach a public utility threshold from a random initial assignment, there is a bifurcation: one class of networks show a positive correlation, while another shows a negative correlation. Our results highlight that by designing networks with appropriate topological properties, one can minimise the need for the movement of individuals within a network before a certain public good threshold is achieved. This result has obvious implications for defence strategies in particular.