Evolving games and evolving random topologies

The first part of the project is related to the study of evolutionary games on random graphs. An evolutionary game on a graph is a process that evolves in rounds. Every node of the graph corresponds to an agent who plays a game against each one of its neighbouring nodes. This game is determined by a certain payoff matrix that is the same for all nodes.
Thus, at each round each node accumulates a total payoff. If the node is better off by changing strategy (assuming that its neighbouring nodes retain their strategies), then it does so.

In this project, we will consider the evolution of the above process on binomial random graphs, where every connection between two nodes is present with some probability p. The aim is to show that after a certain number of steps consensus is achieved, that is, all nodes will adopt the same strategy.

Furthermore, our intention is to extend our research to the case of modularity of random graphs on the hyperbolic plane. The latter is a model of random networks whose typical properties are ubiquitous properties of networks that emerge as a result of human activity or in nature. Such networks also appear to have a certain community structure. That is, there are parts of them that are denser in comparison to the entire network and correspond to certain communities. The community structure is captured by a quantity that is called modularity. Our aim is to study the modularity of random graphs on the hyperbolic plane. In turn, this will pave the way towards the detailed study of communities in these random networks.