# Controllability of Bandlimited Graph Processes Over Random Time Varying Graphs

## TL;DR

This paper studies how to control network states over randomly changing graphs by leveraging bandlimited graph signals, proposing methods to ensure expected controllability and minimize deviations in stochastic environments.

## Contribution

It introduces the concept of controllability in the mean for bandlimited graph processes on random time-varying graphs and develops control strategies accounting for topological randomness.

## Key findings

- Control strategies effectively drive expected network states towards targets.
- Graph filters can remove out-of-band frequency content caused by control.
- Proposed methods perform well on synthetic and social network models.

## Abstract

Controllability of complex networks arises in many technological problems involving social, financial, road, communication, and smart grid networks. In many practical situations, the underlying topology might change randomly with time, due to link failures such as changing friendships, road blocks or sensor malfunctions. Thus, it leads to poorly controlled dynamics if randomness is not properly accounted for. We consider the problem of controlling the network state when the topology varies randomly with time. Our problem concerns target states that are bandlimited over the graph; these are states that have nonzero frequency content only on a specific graph frequency band. We thus leverage graph signal processing and exploit the bandlimited model to drive the network state from a fixed set of control nodes. When controlling the state from a few nodes, we observe that spurious, out-of-band frequency content is created. Therefore, we focus on controlling the network state over the desired frequency band, and then use a graph filter to get rid of the unwanted frequency content. To account for the topological randomness, we develop the concept of controllability in the mean, which consists of driving the expected network state towards the target state. A detailed mean squared error analysis is performed to quantify the statistical deviation between the final controlled state on a particular graph realization and the actual target state. Finally, we propose different control strategies and evaluate their effectiveness on synthetic network models and social networks.