# Input design for the optimal control of networked moments

**Authors:** Philip Solimine, Anke Meyer-Baese  
**arXiv:** [2106.05265](https://arxiv.org/abs/2106.05265)  
**Date:** 2021-06-09  
**Summary page:** https://www.summarizepaper.com/en/paper/2106.05265v4/

## Key points

- The paper focuses on optimal control of the mean and variance of a network state vector.
- An algorithm is developed that uses projected gradient descent to optimize the placement of control inputs subject to constraints on the state that must be achieved within a given time threshold.
- The goal is to design an input that moves the moment at minimum cost.
- Solutions related to the eigenvalues of the system's Gramian matrices are found to solve the state-selection problem for various variants of the first and second moment, which are then nested into projected gradient descent to design optimal inputs.
- The context for this research lies in social networks where information spreads rapidly, and opinions can be manipulated by strategic influencers.
- The paper builds upon previous studies exploring strategies for controlling opinion dynamics in social networks, including spread of misinformation and adversarial perturbations.
- Several relevant works in this area are cited, including Mostagir et al., Liu et al., Acemoglu et al., Vohra, Gaitonde et al., Galeotti et al., Golub and Jackson, Candogan et al., Gao et al. and Li et al.
- Overall, this paper presents a novel approach to optimizing control inputs for moments in networked environments with potential implications for understanding how information spreads through social networks and how strategic influencers can manipulate opinions.

## Summary

This paper focuses on the optimal control of the mean and variance of a network state vector. The authors develop an algorithm that utilizes projected gradient descent to optimize the placement of control inputs, subject to constraints on the state that must be achieved within a given time threshold. The goal is to design an input that moves the moment at minimum cost. To solve the state-selection problem for various variants of the first and second moment, the authors find solutions related to the eigenvalues of the system's Gramian matrices. They then nest this state selection into projected gradient descent to design optimal inputs. The context for this research lies in social networks where information spreads rapidly, and opinions can be manipulated by strategic influencers. Previous studies have explored strategies for controlling opinion dynamics in social networks, including spread of misinformation and adversarial perturbations. This paper builds upon these studies by focusing on optimal control of moments in a networked environment. The authors cite several relevant works in this area, including Mostagir et al. 's study on society susceptibility to manipulation, Liu et al. 's work on opinion dynamics control in social networks, Acemoglu et al. 's research on spread of misinformation in social networks, Vohra's study on strategic influencers and belief shaping, Gaitonde et al. 's work on adversarial perturbations of opinion dynamics in networks, Galeotti et al. 's research on discord and harmony in networks, Golub and Jackson's study on homophily affecting learning speed and best-response dynamics, Candogan et al. 's work on social learning under platform influence with consensus and persistent disagreement , Gao et al. 's research on target control of complex networks , Li et al. 's study on target control of directed complex networks with multiple leaders or followers . Overall, this paper presents a novel approach to optimizing control inputs for moments in networked environments. Its findings could have implications for understanding how information spreads through social networks and how strategic influencers can manipulate opinions.