This paper proposes a novel algorithm to retroactively compute the evolution of edge signals from a given sequence of partial observations from topological structures, a concept referred to as evolution backcasting. Our backcasting algorithm exploits the spatio-temporal dependencies present in the real-world edge signals using the simplicial vector autoregressive (S-VAR) model. The proposed algorithm jointly estimates the S-VAR filter coefficients and recovers missing data from the partial observations. Subsequently, the algorithm capitalizes on the learned S-VAR model and the reconstructed signals to execute the backcasting of edge signal evolution. Using traffic and water distribution networks as case studies, we showcase the superior capabilities of our algorithm compared with baseline alternatives.

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An online topology estimation algorithm for nonlinear structural equation models (SEM) is proposed in this paper, addressing the nonlinearity and the non-stationarity of real-world systems. The nonlinearity is modeled using kernel formulations, and the curse of dimensionality associated with the kernels is mitigated using random feature approximation. The online learning strategy uses a group-lasso-based optimization framework with a prediction-corrections technique that accounts for the model evolution. The proposed approach has three properties of interest. First, it enjoys node-separable learning, which allows for scalability in large networks. Second, it offers privacy in SEM learning by replacing the actual data with node-specific random features. Third, its performance can be characterized theoretically via a dynamic regret analysis, showing that it is possible to obtain a linear dynamic regret bound under mild assumptions. Numerical results with synthetic and real data corroborate our findings and show competitive performance w.r.t. state-of-the-art alternatives.@en

Vector autoregressive (VAR) model is widely used to model time-varying processes, but it suffers from prohibitive growth of the parameters when the number of time series exceeds a few hundreds. We propose a simplicial VAR model to mitigate the curse of dimensionality of the VAR models when the time series are defined over higher-order network structures such as edges, triangles, etc. The proposed model shares parameters across the simplicial signals by leveraging the simplicial convolutional filter and captures structure-aware spatio-temporal dependencies of the time-varying processes. Targetting the streaming signals from the real-world nonstationary networks, we develop a group-lasso-based online strategy to learn the proposed model. Using traffic and water distribution networks, we demonstrate that the proposed model achieves competitive signal prediction accuracy with a significantly less number of parameters than the VAR models.@en

An online algorithm for missing data imputation for networks with signals defined on the edges is presented. Leveraging the prior knowledge intrinsic to real-world networks, we propose a bi-level optimization scheme that exploits the causal dependencies and the flow conservation, respectively via <italic>(i)</italic> a sparse line graph identification strategy based on a group-Lasso and <italic>(ii)</italic> a Kalman filtering-based signal reconstruction strategy developed using simplicial complex (SC) formulation. The advantages of this first SC-based attempt for time-varying signal imputation have been demonstrated through numerical experiments using EPANET models of both synthetic and real water distribution networks.

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