Learning the effective order of a hypergraph dynamical system.

Dynamical systems on hypergraphs can display a rich set of behaviors not observable for systems with pairwise interactions. Given a distributed dynamical system with a putative hypergraph structure, an interesting question is thus how much of this hypergraph structure is actually necessary to faithfully replicate the observed dynamical behavior. To answer this question, we propose a method to determine the minimum order of a hypergraph necessary to approximate the corresponding dynamics accurately. Specifically, we develop a mathematical framework that allows us to determine this order when the type of dynamics is known. We use these ideas in conjunction with a hypergraph neural network to directly learn the dynamics itself and the resulting order of the hypergraph from both synthetic and real datasets consisting of observed system trajectories.

Consensus dynamics on temporal hypergraphs.

We investigate consensus dynamics on temporal hypergraphs that encode network systems with time-dependent, multiway interactions. We compare these consensus processes with dynamics evolving on projections that remove the temporal and/or the multiway interactions of the higher-order network representation. For linear average consensus dynamics, we find that the convergence of a randomly switching time-varying system with multiway interactions is slower than the convergence of the corresponding system with pairwise interactions, which in turn exhibits a slower convergence rate than a consensus dynamics on the corresponding static network. We then consider a nonlinear consensus dynamics model in the temporal setting. Here we find that in addition to an effect on the convergence speed, the final consensus value of the temporal system can differ strongly from the consensus on the aggregated, static hypergraph. In particular, we observe a first-mover advantage in the consensus formation process: If there is a local majority opinion in the hyperedges that are active early on, then the majority in these first-mover groups has a higher influence on the final consensus value-a behavior that is not observable in this form in projections of the temporal hypergraph.

Multibody interactions and nonlinear consensus dynamics on networked systems.

Multibody interactions can reveal higher-order dynamical effects that are not captured by traditional two-body network models. In this work, we derive and analyze models for consensus dynamics on hypergraphs, where nodes interact in groups rather than in pairs. Our work reveals that multibody dynamical effects that go beyond rescaled pairwise interactions can appear only if the interaction function is nonlinear, regardless of the underlying multibody structure. As a practical application, we introduce a specific nonlinear function to model three-body consensus, which incorporates reinforcing group effects such as peer pressure. Unlike consensus processes on networks, we find that the resulting dynamics can cause shifts away from the average system state. The nature of these shifts depends on a complex interplay between the distribution of the initial states, the underlying structure, and the form of the interaction function. By considering modular hypergraphs, we discover state-dependent, asymmetric dynamics between polarized clusters where multibody interactions make one cluster dominate the other.

Joint reconstruction and classification of tumor cells and cell interactions in melanoma tissue sections with synthesized training data.

PURPOSE: Cancers are almost always diagnosed by morphologic features in tissue sections. In this context, machine learning tools provide new opportunities to describe tumor immune cell interactions within the tumor microenvironment and thus provide phenotypic information that might be predictive for the response to immunotherapy. METHODS: We develop a machine learning approach using variational networks for joint image denoising and classification of tissue sections for melanoma, which is an established model tumor for immuno-oncology research. The manual annotation of real training data would require substantial user interaction of experienced pathologists for each single training image, and the training of larger networks would rely on a very large number of such data sets with ground truth annotation. To overcome this bottleneck, we synthesize training data together with a proper tissue structure classification. To this end, a stochastic data generation process is used to mimic cell morphology, cell distribution and tissue architecture in the tumor microenvironment. Particular components of this tool are random placement and rotation of a large number of patches for presegmented cell nuclei, a stochastic fast marching approach to mimic the geometry of cells and texture generation based on a color covariance analysis of real data. Here, the generated training data reflect a large range of interaction patterns. RESULTS: In several applications to histological tissue sections, we analyze the efficiency and accuracy of the proposed approach. As a result, depending on the scenario considered, almost all cells and nuclei which ought to be detected are actually marked as classified and hardly any misclassifications occur. CONCLUSIONS: The proposed method allows for a computer-aided screening of histological tissue sections utilizing variational networks with a particular emphasis on tumor immune cell interactions and on the robust cell nuclei classification.