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1. Core motifs predict dynamic attractors in combinatorial threshold-linear networks

   https://doi.org/10.1371/journal.pone.0264456  

   Parmelee, Caitlyn; Moore, Samantha; Morrison, Katherine; Curto, Carina (March 2022, PLOS ONE)

   Kryven, Ivan (Ed.)

   Combinatorial threshold-linear networks (CTLNs) are a special class of inhibition-dominated TLNs defined from directed graphs. Like more general TLNs, they display a wide variety of nonlinear dynamics including multistability, limit cycles, quasiperiodic attractors, and chaos. In prior work, we have developed a detailed mathematical theory relating stable and unstable fixed points of CTLNs to graph-theoretic properties of the underlying network. Here we find that a special type of fixed points, corresponding to core motifs , are predictive of both static and dynamic attractors. Moreover, the attractors can be found by choosing initial conditions that are small perturbations of these fixed points. This motivates us to hypothesize that dynamic attractors of a network correspond to unstable fixed points supported on core motifs. We tested this hypothesis on a large family of directed graphs of size n = 5, and found remarkable agreement. Furthermore, we discovered that core motifs with similar embeddings give rise to nearly identical attractors. This allowed us to classify attractors based on structurally-defined graph families. Our results suggest that graphical properties of the connectivity can be used to predict a network’s complex repertoire of nonlinear dynamics. 

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2. MagNet: A Neural Network for Directed Graphs

   Zhang, Xitong; He, Yixuan; Brugnone, Nathan; Perlmutter, Michael; Hirn, Matthew (January 2021, Advances in neural information processing systems)

   Ranzato, M.; Beygelzimer, A.; Dauphin, Y.; Liang, P.S.; Vaughan, J. Wortman (Ed.)

   The prevalence of graph-based data has spurred the rapid development of graph neural networks (GNNs) and related machine learning algorithms. Yet, despite the many datasets naturally modeled as directed graphs, including citation, website, and traffic networks, the vast majority of this research focuses on undirected graphs. In this paper, we propose MagNet, a GNN for directed graphs based on a complex Hermitian matrix known as the magnetic Laplacian. This matrix encodes undirected geometric structure in the magnitude of its entries and directional information in their phase. A charge parameter attunes spectral information to variation among directed cycles. We apply our network to a variety of directed graph node classification and link prediction tasks showing that MagNet performs well on all tasks and that its performance exceeds all other methods on a majority of such tasks. The underlying principles of MagNet are such that it can be adapted to other GNN architectures. 

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3. Learning Graph Structure from Convolutional Mixtures

   Wasserman, Max; Sihag, Saurabh; Mateos, Gonzalo; Ribeiro, Alejandro (June 2023, Transactions on machine learning research)

   Machine learning frameworks such as graph neural networks typically rely on a given, fixed graph to exploit relational inductive biases and thus effectively learn from network data. However, when said graphs are (partially) unobserved, noisy, or dynamic, the problem of inferring graph structure from data becomes relevant. In this paper, we postulate a graph convolutional relationship between the observed and latent graphs, and formulate the graph structure learning task as a network inverse (deconvolution) problem. In lieu of eigendecomposition-based spectral methods or iterative optimization solutions, we unroll and truncate proximal gradient iterations to arrive at a parameterized neural network architecture that we call a Graph Deconvolution Network (GDN). GDNs can learn a distribution of graphs in a supervised fashion, perform link prediction or edge-weight regression tasks by adapting the loss function, and they are inherently inductive as well as node permutation equivariant. We corroborate GDN’s superior graph learning performance and its generalization to larger graphs using synthetic data in supervised settings. Moreover, we demonstrate the robustness and representation power of GDNs on real world neuroimaging and social network datasets. 

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4. Hidden Hypergraphs, Error-Correcting Codes, and Critical Learning in Hopfield Networks

   https://doi.org/10.3390/e23111494  

   Hillar, Christopher; Chan, Tenzin; Taubman, Rachel; Rolnick, David (November 2021, Entropy)

   In 1943, McCulloch and Pitts introduced a discrete recurrent neural network as a model for computation in brains. The work inspired breakthroughs such as the first computer design and the theory of finite automata. We focus on learning in Hopfield networks, a special case with symmetric weights and fixed-point attractor dynamics. Specifically, we explore minimum energy flow (MEF) as a scalable convex objective for determining network parameters. We catalog various properties of MEF, such as biological plausibility, and then compare to classical approaches in the theory of learning. Trained Hopfield networks can perform unsupervised clustering and define novel error-correcting coding schemes. They also efficiently find hidden structures (cliques) in graph theory. We extend this known connection from graphs to hypergraphs and discover n-node networks with robust storage of 2Ω(n1−ϵ) memories for any ϵ>0. In the case of graphs, we also determine a critical ratio of training samples at which networks generalize completely. 

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5. Graph Sparsifications using Neural Network Assisted Monte Carlo Tree Search

   Chiu, Alvin; Ghosh, Mithun; Ahmed, Reyan; Jun, Kwang-Sung; Kobourov, Stephen; Goodrich, Michael T (November 2023, ArXiv)

   Graph neural networks have been successful for machine learning, as well as for combinatorial and graph problems such as the Subgraph Isomorphism Problem and the Traveling Salesman Problem. We describe an approach for computing graph sparsifiers by combining a graph neural network and Monte Carlo Tree Search. We first train a graph neural network that takes as input a partial solution and proposes a new node to be added as output. This neural network is then used in a Monte Carlo search to compute a sparsifier. The proposed method consistently outperforms several standard approximation algorithms on different types of graphs and often finds the optimal solution. 

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