# Structural Balance and Random Walks on Complex Networks with Complex Weights

arXiv: https://arxiv.org/abs/2307.01813 · HTML: https://mlsanity.com/paper/2307.01813/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 55 references that are in the map.
- First posted on arXiv 4 Jul 2023.

## Builds on (most cited first, 11 of 11)

- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Random walks and diffusion on networks](https://mlsanity.com/paper/1612.03281/) (2016)
- [MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian](https://mlsanity.com/paper/2209.00546/) (2022)
- [Hermitian matrices for clustering directed graphs: insights and applications](https://mlsanity.com/paper/1908.02096/) (2019)
- [Magnetic eigenmaps for community detection in directed networks](https://mlsanity.com/paper/1606.07359/) (2016)
- [The Slashdot Zoo: Mining a Social Network with Negative Edges](https://mlsanity.com/paper/1710.11395/) (2017)
- [Complex networks with complex weights](https://mlsanity.com/paper/2212.06257/) (2022)
- [Spreading and Structural Balance on Signed Networks](https://mlsanity.com/paper/2212.10158/) (2022)
- [Complex Networks from Classical to Quantum](https://mlsanity.com/paper/1702.08459/) (2017)
- [Cheeger constants, structural balance, and spectral clustering analysis for signed graphs](https://mlsanity.com/paper/1411.3530/) (2014)
- [Extracting Complements and Substitutes from Sales Data: A Network Perspective](https://mlsanity.com/paper/2103.02042/) (2021)

## Built on by (most cited first, 3 of 3)

- [Matrix-weighted networks for modeling multidimensional dynamics](https://mlsanity.com/paper/2410.05188/) (2024)
- [Permutation-Invariant Spectral Learning via Dyson Diffusion](https://mlsanity.com/paper/2510.08535/) (2025)
- [Complex-Weighted Convolutional Networks: Provable Expressiveness via Complex Diffusion](https://mlsanity.com/paper/2511.13937/) (2025)

## Abstract (arXiv)

Complex numbers define the relationship between entities in many situations. A canonical example would be the off-diagonal terms in a Hamiltonian matrix in quantum physics. Recent years have seen an increasing interest to extend the tools of network science when the weight of edges are complex numbers. Here, we focus on the case when the weight matrix is Hermitian, a reasonable assumption in many applications, and investigate both structural and dynamical properties of the complex-weighted networks. Building on concepts from signed graphs, we introduce a classification of complex-weighted networks based on the notion of structural balance, and illustrate the shared spectral properties within each type. We then apply the results to characterise the dynamics of random walks on complex-weighted networks, where local consensus can be achieved asymptotically when the graph is structurally balanced, while global consensus will be obtained when it is strictly unbalanced. Finally, we explore potential applications of our findings by generalising the notion of cut, and propose an associated spectral clustering algorithm. We also provide further characteristics of the magnetic Laplacian, associating directed networks to complex-weighted ones. The performance of the algorithm is verified on both synthetic and real networks.