# Steering Opinion Dynamics Through Control of Social Networks

Understanding the dynamics of opinion formation and control in social networks is a critical area of research. This study, supervised by Susana Gomes and Marie-Therese Wolfram, explores the manipulation of collective behavior through various models including ODE, agent-based, and stochastic analysis. It delves into the interaction of individual opinions within populations and the utilization of control theory to influence consensus towards a target opinion. The research also investigates the role of social networks in shaping opinions and the application of control strategies to steer group behavior effectively.

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## Presentation Transcript

**Steering opinion dynamics**through control of social networks Andrew Nugent Supervised by Susana Gomes and Marie-Therese Wolfram**1. Opinion Dynamics**Interacting particle system Collective behaviour Collective optimisation ODE models Agent-based models SDE models Stochastic analysis Control theory Network science PDE models Kinetic theory**1. Opinion Dynamics**Interacting particle system Collective behaviour Collective optimisation ODE models Agent-based models SDE models Stochastic analysis Control theory Network science PDE models Kinetic theory**1. Opinion Dynamics**Individual s opinion Interaction function Population size**3. Control theory**Goal: For given initial conditions, find a control that brings the whole population to consensusat a preselected target opinion x*.**3. Control theory**Goal: For given initial conditions, find a control that brings the whole population to consensusat a preselected target opinion x*.**3. Control theory**Goal: For given initial conditions, find a control that brings the whole population to consensusat a preselected target opinion x*.**4. Controllability**1. For what combination of initial conditions and targets does such a control exist? 2. If it exists, how do we find it? 3. If we can find such a control, how well does it work? 4. If we can't find such a control, how close can we get?**4. Controllability**Problem 1: Opinion fragmentation prevents consensus. Problem 2: The target opinion is not inside the opinion interval.**4. Controllability**Proposition: If the population reaches consensus and the target opinion stays inside the range of opinions, the population will reach consensus at the target. Which means: These two problems are essentially the only two problems.**4. Controllability**Proposition: If the population reaches consensus and the target opinion stays inside the range of opinions, the population will reach consensus at the target. Which means: These two problems are essentially the only two problems. When can we guarantee that the population reaches consensus?**4. Controllability**Proposition: If the population reaches consensus and the target opinion stays inside the range of opinions, the population will reach consensus at the target. Which means: These two problems are essentially the only two problems. When can we guarantee that the population reaches consensus? Remember that example from earlier?**4. Controllability**Lemma (paraphrased): By removing all edges connecting an individual to the rest of the population as quickly as possible, we can limit their overall opinion change. Corollary (paraphrased): If the control acts sufficiently quickly, we can prevent individuals crossing over the target opinion.**4. Controllability**Proposition: If the population reaches consensus and the target opinion stays inside the range of opinions, the population will reach consensus at the target. Which means: These two problems are essentially the only two problems. Corollary derived from worst-case scenario. Existing consensus results.**4. Controllability**Theorem (paraphrased): Assume the following The target opinion is in the interior of the initial opinions. The initial opinions are not already clustered. The control can create and remove edges anywhere within the network. Then, if the control acts sufficiently quickly, consensus can be achieved at the desired target opinion.**4. Controllability**Theorem (paraphrased): Assume the following The target opinion is in the interior of the initial opinions. The initial opinions are not already clustered. The control can create and remove edges anywhere within the network. Then, if the control acts sufficiently quickly, consensus can be achieved at the desired target opinion.**Summary**1. Opinion Dynamics 2. Social networks 3. Control theory 4. Controllability**Steering opinion dynamics**through control of social networks Andrew Nugent Supervised by Susana Gomes and Marie-Therese Wolfram