Task-Aware Connectivity Learning for Incoming Nodes Over Growing Graphs

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.