We study the problem of optimal power allocation in a single-hop ad hoc wireless network. In solving this problem, we depart from classical purely model-based approaches and propose a hybrid method that retains key modeling elements in conjunction with data-driven components. More precisely, we put forth a neural network architecture inspired by the algorithmic unfolding of the iterative weighted minimum mean squared error (WMMSE) method, that we denote by unfolded WMMSE (UWMMSE). The learnable weights within UWMMSE are parameterized using graph neural networks (GNNs), where the time-varying underlying graphs are given by the fading interference coefficients in the wireless network. These GNNs are trained through a gradient descent approach based on multiple instances of the power allocation problem. We show that the proposed architecture is permutation equivariant, thus facilitating generalizability across network topologies. Comprehensive numerical experiments illustrate the performance attained by UWMMSE along with its robustness to hyper-parameter selection and generalizability to unseen scenarios such as different network densities and network sizes.

Unfolding WMMSE Using Graph Neural Networks for Efficient Power Allocation

Spatial Models Of Parliamentary Voting

Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond

/pdf/sombor-index-and-degree-related-properties-of-simplicial-3e2y80vy.pdf

Sombor index and degree-related properties of simplicial networks

We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture: namely, permutation equivariance, orientation equivariance, and simplicial awareness. The first two properties respectively account for the fact that the node indexing and the simplex orientations in a simplicial complex are arbitrary. The last property encodes the desirable feature that the output of the neural network depends on the entire simplicial complex and not on a subset of its dimensions. Based on these properties, we propose a simple convolutional architecture, rooted in tools from algebraic topology, for the problem of trajectory prediction, and show that it obeys all three of these properties when an odd, nonlinear activation function is used. We then demonstrate the effectiveness of this architecture in extrapolating trajectories on synthetic and real datasets, with particular emphasis on the gains in generalizability to unseen trajectories.

/pdf/principled-simplicial-neural-networks-for-trajectory-5pz5ixs7xd.pdf

Principled Simplicial Neural Networks for Trajectory Prediction

Networks and network processes have emerged as powerful tools for modeling social interactions, disease propagation, and a variety of additional dynamics driven by relational structures. Recently, neural networks have been generalized to process data on graphs, thus being able to learn from the aforementioned network processes achieving cutting-edge performance in traditional tasks such as node classification and link prediction. However, these methods have all been formulated in a way suited only to data on the nodes of a graph. The application of these techniques to data supported on the edges of a graph, namely flow signals, has not been explored in detail. To bridge this gap, we propose the use of the so-called Hodge Laplacian combined with graph neural network architectures for the analysis of flow data. Specifically, we apply two graph neural network architectures to solve the problems of flow interpolation and source localization.

HodgeNet: Graph Neural Networks for Edge Data

Networks and data supported on graphs have become ubiquitous in the sciences and engineering. This paper studies the ‘blind’ community detection problem, where we seek to infer the community structure of a graph model given the observation of independent graph signals on a set of nodes whose connections are unknown. We model each observation as filtered white noise, where the underlying network structure varies with every observation. These varying network structures are modeled as independent realizations of a latent planted partition model (PPM), justifying our assumption of a constant underlying community structure over all observations. Under certain conditions on the graph filter and PPM parameters, we suggest simple algorithms for determining (i) the number of latent communities and (ii) the associated partitions of the PPM. We then prove statistical guarantees in the asymptotic and non-asymptotic sampling cases. Numerical experiments on real and synthetic data demonstrate the efficacy of these algorithms.

Exact Blind Community Detection From Signals on Multiple Graphs

We study the blind centrality ranking problem, where our goal is to infer the eigenvector centrality ranking of nodes solely from nodal observations, i.e., without information about the topology of the network. We formalize these nodal observations as graph signals and model them as the outputs of a network process on the underlying (unobserved) network. A simple spectral algorithm is proposed to estimate the leading eigenvector of the associated adjacency matrix, thus serving as a proxy for the centrality ranking. A finite rate performance analysis of the algorithm is provided, where we find a lower bound on the number of graph signals needed to correctly rank (with high probability) two nodes of interest. We then specialize our general analysis for the particular case of dense Erdős-Renyi graphs, where existing graph-theoretical results can be leveraged. Finally, we illustrate the proposed algorithm via numerical experiments on synthetic and real-world networks, with special emphasis on how the network features influence the performance.

T. Mitchell Roddenberry

Papers

Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond

Principled Simplicial Neural Networks for Trajectory Prediction

HodgeNet: Graph Neural Networks for Edge Data

Exact Blind Community Detection From Signals on Multiple Graphs

Blind Inference of Centrality Rankings from Graph Signals