Many combinatorial optimization problems can be phrased in the language of constraint satisfaction problems. We introduce a graph neural network architecture for solving such optimization problems. The architecture is generic; it works for all binary constraint satisfaction problems. Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger). We experimentally evaluate our approach for a variety of problems, including Maximum Cut and Maximum Independent Set. Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific problems.

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We systematically evaluate the (in-)stability of state-of-the-art node embedding algorithms due to randomness, i.e., the random variation of their outcomes given identical algorithms and graphs. We apply five node embeddings algorithms-HOPE, LINE, node2vec, SDNE, and GraphSAGE-to synthetic and empirical graphs and assess their stability under randomness with respect to (i) the geometry of embedding spaces as well as (ii) their performance in downstream tasks. We find significant instabilities in the geometry of embedding spaces independent of the centrality of a node. In the evaluation of downstream tasks, we find that the accuracy of node classification seems to be unaffected by random seeding while the actual classification of nodes can vary significantly. This suggests that instability effects need to be taken into account when working with node embeddings. Our work is relevant for researchers and engineers interested in the effectiveness, reliability, and reproducibility of node embedding approaches.

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In this paper we propose a graph neural network architecture solving the AC power flow problem under realistic constraints. While the energy transition is changing the energy industry to a digitalized and decentralized energy system, the challenges are increasingly shifting to the distribution grid level to integrate new loads and generation technologies. To ensure a save and resilient operation of distribution grids, AC power flow calculations are the means of choice to determine grid operating limits or analyze grid asset utilization in planning procedures. In our approach we demonstrate the development of a framework which makes use of graph neural networks to learn the physical constraints of the power flow. We present our model architecture on which we perform unsupervised training to learn a general solution of the AC power flow formulation that is independent of the specific topologies and supply tasks used for training. Finally, we demonstrate, validate and discuss our results on medium voltage benchmark grids.