Graph filters are a staple tool for processing signals over graphs in a multitude of downstream tasks. However, they are commonly designed for graphs with a fixed number of nodes, despite real-world networks typically grow over time. This topological evolution is often known up to a stochastic model, thus, making conventional graph filters ill-equipped to withstand such topological changes, their uncertainty, as well as the dynamic nature of the incoming data. To tackle these issues, we propose an online graph filtering framework by relying on online learning principles. We design filters for scenarios where the topology is both known and unknown, including a learner adaptive to such evolution. We conduct a regret analysis to highlight the role played by the different components such as the online algorithm, the filter order, and the growing graph model. Numerical experiments with synthetic and real data corroborate the proposed approach for graph signal inference tasks and show a competitive performance w.r.t. baselines and state-of-the-art alternatives.

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Simplicial convolutional filters can process signals defined over levels of a simplicial complex such as nodes, edges, triangles, and so on with applications in e.g., flow prediction in transportation or financial networks. However, the underlying topology expands over time in a way that new edges and triangles form. For example, in a transportation network, a new connection between two locations is newly built, or in a currency exchange market, two currencies can be exchanged without an intermediate currency that can be understood as a new edge between them. To handle the streaming nature of data, we propose an online prediction for edge flows which generalizes to other higher-order simplicial signals. This is achieved by updating the filter coefficients via an online gradient descent with a provable sub-linear regret relative to the simplicial filter optimized over the whole sequence of edge flows. The update of the filter coefficients associated with the lower and upper Hodge Laplacians can be uncoupled in general. We test the online edge flow prediction on an expanding synthetic simplicial complex and a coauthorship complex showing a close performance to the offline counterpart.@en

Current spatiotemporal learning methods for complex data exploit the graph structure as an inductive bias to restrict the function space and improve data and computation efficiency. However, these methods work principally on graphs with a fixed size, whereas in several applications there are expanding graphs where new nodes join the network; e.g., new sensors joining a sensor network or new users joining a recommender system. This paper focuses on the non-trivial extension of spatiotemporal methods to this setting, where now it is key to jointly capture both the topological and signal dynamics. Specifically, it considers a graph vector autoregressive (GVAR) model for multivariate time series. The GVAR is a multivariate linear model that leverages a bank of graph filters allowing scalability and data efficiency. To account for the dynamic nature of the graphs, the filters’s parameters are learned on-the-fly via adaptive gradient descent with provable sub-linear regret. Numerical results on both synthetic and real data corroborate the proposed method.@en

Data processing over graphs is usually done on graphs of fixed size. However, graphs often grow with new nodes arriving over time. Knowing the connectivity information of these nodes, and thus, the expanded graph is crucial for processing data over the expanded graph. In its absence, its inference and the subsequent data processing become essential. This paper provides contributions along this direction by considering task-driven data processing for incoming nodes without connectivity information. We model the incoming node attachment as a random process dictated by the parameterized vectors of probabilities and weights of attachment. The attachment is driven by the existing graph topology, the corresponding graph signal, and an associated processing task. We consider two such tasks, one of interpolation at the incoming node, and that of graph signal smoothness. We show that the model bounds implicitly the spectral perturbation between the nominal topology of the expanded graph and the drawn realizations. In the absence of connectivity information our topology, task, and data-aware stochastic attachment performs better than purely data-driven and topology driven stochastic attachment rules, as is confirmed by numerical results over synthetic and real data.

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Performing signal processing over graphs requires knowledge of the underlying fixed topology. However, graphs often grow in size with new nodes appearing over time, whose connectivity is typically unknown; hence, making more challenging the downstream tasks in applications like cold start recommendation. We address such a challenge for signal interpolation at the incoming nodes blind to the topological connectivity of the specific node. Specifically, we propose a stochastic attachment model for incoming nodes parameterized by the attachment probabilities and edge weights. We estimate these parameters in a data-driven fashion by relying only on the attachment behaviour of earlier incoming nodes with the goal of interpolating the signal value. We study the non-convexity of the problem at hand, derive conditions when it can be marginally convexified, and propose an alternating projected descent approach between estimating the attachment probabilities and the edge weights. Numerical experiments with synthetic and real data dealing in cold start collaborative filtering corroborate our findings.@en