Modeling how networks change under structural perturbations can yield foundational insights into network robustness, which is critical in many real-world applications. The largest connected component is a popular measure of network performance. Percolation theory provides a theoretical framework to establish statistical properties of the largest connected component of large random graphs. However, this theoretical framework is typically only exact in the large-$N$ limit, failing to capture the statistical properties of largest connected components in small networks, which many real-world networks are. We derive expected values for the largest connected component of small $G(N,p)$ random graphs from which nodes are either removed uniformly at random or targeted by highest degree and compare these values with existing theory. We also visualize the performance of our expected values compared to existing theory for predicting the largest connected component of various real-world, small graphs.

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• Attacks on networks
• Percolation phase transition
• Robustness
• Erdos-Renyi graph
• Random graphs
• Real world networks
• Percolation theory