# Optimal Localization of Diffusion Sources in Complex Networks

## TL;DR

This paper presents a new framework combining controllability theory and compressive sensing for optimal source localization in complex networks, enabling efficient and robust identification of diffusion sources with minimal measurements.

## Contribution

It introduces a theoretical framework that determines the minimal number of messenger nodes needed for source localization and applies compressive sensing for sparse signal reconstruction.

## Key findings

- Sources in dense networks are easier to locate.
- A single messenger node suffices in certain undirected networks with weak noise.
- The framework enhances understanding and offers practical tools for network source localization.

## Abstract

Locating sources of diffusion and spreading from minimum data is a significant problem in network science with great applied values to the society. However, a general theoretical framework dealing with optimal source localization is lacking. Combining the controllability theory for complex networks and compressive sensing, we develop a framework with high efficiency and robustness for optimal source localization in arbitrary weighted networks with arbitrary distribution of sources. We offer a minimum output analysis to quantify the source locatability through a minimal number of messenger nodes that produce sufficient measurement for fully locating the sources. When the minimum messenger nodes are discerned, the problem of optimal source localization becomes one of sparse signal reconstruction, which can be solved using compressive sensing. Application of our framework to model and empirical networks demonstrates that sources in homogeneous and denser networks are more readily to be located. A surprising finding is that, for a connected undirected network with random link weights and weak noise, a single messenger node is sufficient for locating any number of sources. The framework deepens our understanding of the network source localization problem and offers efficient tools with broad applications.