Opinion dynamics on co-evolving complex networks

The study of social phenomena identifies humans as the essential entity and attempts to describe human behaviour. Probably, the opinions an individual holds represent the most important factors behind the human behaviour. Human beings take actions according to their beliefs therefore, the investigation of the mechanism of opinion formation assumes great interest in the study of social phenomena. This mechanism depends on large number of variables and looking for a mathematical model to describe such a complex process might seem pretentious. However, many models have been proposed since the late 50's which have provided many insights into the process of opinion formation. This research field gained more attention with the advent of social media. Individuals have access to a huge amount of information and social media allow for intense interaction. This is clearly beneficial however such a massive exchange of opinions can favour the spreading of counterfactual rumours and the emergence of radicalized groups. Examples of such groups are the Flat Earth Society or the anti-vax movement. Interactions among individuals can be naturally captured through a graph. The population taken into consideration, usually regarded as the set of agents, is mapped onto the nodes of the graph while edges represent the mutual interplay among agents. With such a description, the powerful tools of graph theory can be applied to gain insights into those phenomena. The literature presents various models to describe the process of opinion formation. In this work the adaptive voter model proposed by Durrett et al. is considered. The starting point is represented by a graph in which each node has a discrete opinion (0 or 1) and there are two possible processes which determine the evolution of the graph, chosen according to a Bernoulli choice at each iteration. Firstly, an edge connecting two nodes holding different opinions (called discordant edge) is selected. With probability α the “classical” voter model is performed namely, one of the two end nodes copies the opinion of its neighbour. Instead, with probability 1-α, the structure of the graph itself is modified. The link is disconnected from one node and randomly attached to one holding the same opinion (“rewire-to-same” version) or any other in the graph (“rewire-to-random”). The fact that the graphs changes in response to the opinions of the agents is called adaptivity and is of primary interest in this field. In this work have been investigated both the behaviour of the fraction of nodes in the minority opinion when an absorbing state is reached, and the time needed to reach consensus. Regarding the first, a phase transition is observed in function of the rewiring, as already known in the literature. We have tried to change the starting network, forcing it to have a community structure. It was expected a reduction in the threshold which leads to fragmentation. However, the model responded in the same way as a random graph without communities, showing insensitivity towards communities. Whereas, regarding the time to consensus, it has been observed that by only slightly modifying the updating rule the time to consensus dramatically changes. In the instance we have studied the time to consensus becomes so high it results difficult to observe convergence even on modest networks. A situation of persistent disagreement initiates, hindering convergence.