Yeganeh Alimohammadi

Yeganeh Alimohammadi, Senem Işık, Amin Saberi

Small-world networks, known for high local clustering and short path lengths, are a fundamental structure in many real-world systems, including social, biological, and technological networks. We apply the theory of (marked) local convergence (also known as Benjamini-Schramm convergence) to derive the limiting behavior of the local structures for two commonly studied small-world network models: the Watts-Strogatz and the Kleinberg models. Establishing local convergence enables us to show that key network measures, such as clustering coefficient, PageRank, greedy maximal independent set, number of spanning trees and tree entropy, converge as network size increases, with their limits determined by the graph's local structure. Additionally, this framework facilitates the estimation of global phenomena, such as the size of the giant component under bond percolation and the closely related properties, the size of the epidemic and information cascades, using local information from small neighborhoods. Furthermore, we observe a critical change in the behavior of the limit exactly when the parameter governing long-range connections in the Kleinberg model crosses the threshold where decentralized search remains efficient, offering a new perspective on why decentralized algorithms fail in certain regimes.

Yeganeh Alimohammadi, Luana Ruiz, Amin Saberi

We propose a theoretical framework for training Graph Neural Networks (GNNs) on large input graphs via training on small, fixed-size sampled subgraphs. This framework is applicable to a wide range of models, including popular sampling-based GNNs, such as GraphSAGE and FastGCN. Leveraging the theory of graph local limits, we prove that, under mild assumptions, parameters learned from training sampling-based GNNs on small samples of a large input graph are within an $ε$-neighborhood of the outcome of training the same architecture on the whole graph. We derive bounds on the number of samples, the size of the graph, and the training steps required as a function of $ε$. Our results give a novel theoretical understanding for using sampling in training GNNs. They also suggest that by training GNNs on small samples of the input graph, practitioners can identify and select the best models, hyperparameters, and sampling algorithms more efficiently. We empirically illustrate our results on a node classification task on large citation graphs, observing that sampling-based GNNs trained on local subgraphs 12$\times$ smaller than the original graph achieve comparable performance to those trained on the input graph.