Mathematical Theory of Opinion Dynamics with Applications

Opinion dynamics can be modeled by using agent-based simulations, where agents in a population are characterized by binary opinions on a number of different issues. They engage in pairwise interactions, whereby if the agreement level is high, the interlocutor is recognized as an ``ally" and the individual will flip one of their opinions to coincide with the interlocutor; if the agreement is low, they will switch away from the interlocutor. While it is usually assumed that all issues in the opinion vector are equally important, in chapter \ref{ch1} we investigate how breaking this symmetry influences the dynamics. We find that the model outcomes can be predicted by a single Agreement-Disagreement Score (ADS) in $[-1,1]$. ADS characterizes how likely individuals in the population are to regard an interlocutor as an ally; low-ADS (very ``cautious") populations tend to converge to a two-faction system with exponentially high convergence times, while high-ADS (very ``trusting") populations tend to converge to a single-faction system relatively fast. In heterogeneous populations characterized by individual issue weighting, individuals that are more ``trusting" are more likely to join the majority group compared to those that are more ``cautious". In the presence of an influencer, for ADS both near $-1$ and $1$, a single faction tends to emerge, but in the former case it coincides with the influencer's opinions, while in the latter case it is the opposite. Time to fixation is also affected by the presence of an influencer, especially for negative-ADS populations, where it no longer experiences such a large increase near $-1$. One can say that an influencer unifies the population to align with the source of influence if ADS$>0$ and to disagree with it if ADS$<0$, and consensus is reached relatively fast for both extremely ``trusting" and extremely ``cautious" populations. In chapter \ref{ch2} we introduce a system of ordinary differential equations (ODEs) which govern the behavior of the discrete stochastic model studied in chapter \ref{ch1}. We find a neutrally stable solution to the system which, when it is the only stable solution, provides a mathematical description for the extremely long times to fixation observed in the stochastic model under certain conditions. Chapter \ref{ch3} turns to a real world example as we analyze the so-called re-tweet network of tweets concerning the HPV vaccine on the social network Twitter. Community detection algorithms provide a way to split the nodes of a network into two or more communities. Using a label propagation method, survey data, and a construction of cultural consensus for survey response data we find a natural division in this network into two distinct communities, one of which is pro-HPV vaccine and the other is anti-HPV vaccine