Structural Balance and Random Walks on Complex Networks with Complex Weights
This work addresses the challenge of analyzing complex-weighted networks, which is incremental as it builds on concepts from signed graphs and applies to domains like quantum physics and spectral clustering.
The authors tackled the problem of extending network science tools to complex-weighted networks by introducing a classification based on structural balance and analyzing spectral properties and random walk dynamics, showing that local consensus is achieved asymptotically in structurally balanced graphs while global consensus occurs in strictly unbalanced ones, with performance verified on synthetic and real networks.
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.