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.
Recent progress in generative AI, especially diffusion models, has demonstrated significant utility in text-to-image synthesis. Particularly in healthcare, these models offer immense potential in generating synthetic datasets and training medical students. However, despite these strong performances, it remains uncertain if the image generation quality is consistent across different demographic subgroups. To address this critical concern, we present the first comprehensive study on the fairness of medical text-to-image diffusion models. Our extensive evaluations of the popular Stable Diffusion model reveal significant disparities across gender, race, and ethnicity. To mitigate these biases, we introduce FairDiffusion, an equity-aware latent diffusion model that enhances fairness in both image generation quality as well as the semantic correlation of clinical features. In addition, we also design and curate FairGenMed, the first dataset for studying the fairness of medical generative models. Complementing this effort, we further evaluate FairDiffusion on two widely-used external medical datasets: HAM10000 (dermatoscopic images) and CheXpert (chest X-rays) to demonstrate FairDiffusion's effectiveness in addressing fairness concerns across diverse medical imaging modalities. Together, FairDiffusion and FairGenMed significantly advance research in fair generative learning, promoting equitable benefits of generative AI in healthcare.
Random walks play an important role in probing the structure of complex networks. On traditional networks, they can be used to extract community structure, understand node centrality, perform link prediction, or capture the similarity between nodes. On signed networks, where the edge weights can be either positive or negative, it is non-trivial to design a random walk which can be used to extract information about the signed structure of the network, in particular the ability to partition the graph into communities with positive edges inside and negative edges in between. Prior works on signed network random walks focus on the case where there are only two such communities (strong balance), which is rarely the case in empirical networks. In this paper, we propose a signed network random walk which can capture the structure of a network with more than two such communities (weak balance). The walk results in a similarity matrix which can be used to cluster the nodes into antagonistic communities. We compare the characteristics of the so-called strong and weak random walks, in terms of walk length and stationarity. We show through a series of experiments on synthetic and empirical networks that the similarity matrix based on weak walks can be used for both unsupervised and semi-supervised clustering, outperforming the same similarity matrix based on strong walks when the graph has more than two communities, or exhibits asymmetry in the density of links. These results suggest that other random-walk based algorithms for signed networks could be improved simply by running them with weak walks instead of strong walks.