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Complex systems across various domains can be naturally modeled as signed networks with positive and negative edges. In this work, we design a new class of signage models and show how to select the model parameters that best fit real-world datasets using maximum likelihood. 

Knot mosaics are a model of a quantum knot system. A knot mosaic is a m-by-n grid where each location on the grid may contain any of 11 possible tiles such that the final layout has closed loops. Oh et al. proved a recurrence relation of state matrices to count the number of m-by-n knot mosaics. Our contribution is to use ALLSAT solvers to count knot mosaics and to experimentally try different ways to encode the AT MOST ONE constraint in SAT. We plan to use our SAT method as a tool to list knot mosaics of interest for specific classes of knots. 

Signed networks (networks with positive and negative edges) commonly arise in various domains from molecular biology to social media. The edge signs -- i.e., the graph signage -- represent the interaction pattern between the vertices and can provide insights into the underlying system formation process. Generative models considering signage formation are essential for testing hypotheses about the emergence of interactions and for creating synthetic datasets for algorithm benchmarking (especially in areas where obtaining real-world datasets is difficult).In this work, we pose a novel Maximum-Likelihood-based optimization problem for modeling signages given their topology and showcase it in the context of gene regulation. Regulatory interactions of genes play a key role in the process of organism development, and when broken can lead to serious organism abnormalities and diseases. Our contributions are threefold: First, we design a new class of signage models for a given topology, and, based on the parameter setting, we discuss its biological interpretations for gene regulatory networks (GRNs). Second, we design algorithms computing the Maximum Likelihood -- depending on the parameter setting, our algorithms range from closed-form expressions to MCMC sampling. Third, we evaluated the results of our algorithms on synthetic datasets and real-world large GRNs. Our work can lead to the prediction of unknown gene regulations, novel biological hypotheses, and realistic benchmark datasets in the realm of gene regulation.