On Balancing Sparsity with Reliable Connectivity in Distributed Network Design with Random K-out Graphs

In several applications in distributed systems, an important design criterion is ensuring that the network is sparse, i.e., does not contain too many edges, while achieving reliable connectivity. Sparsity ensures communication overhead remains low, while reliable connectivity is tied to reliable communication and inference on decentralized data reservoirs and computational resources. A class of network models called random K-out graphs appear widely as a heuristic to balance connectivity and sparsity, especially in settings with limited trust, e.g., privacy-preserving aggregation of networked data in which networks are deployed. However, several questions remain regarding how to choose network parameters in response to different operational requirements, including the need to go beyond asymptotic results and the ability to model the stochastic and adversarial environments. To address this gap, we present theorems to inform the choice of network parameters that guarantee reliable connectivity in regimes where nodes can be finite or unreliable. We first derive upper and lower bounds for probability of connectivity in random K-out graphs when the number of nodes is finite. Next, we analyze the property of r-robustness, a stronger notion than connectivity that enables resilient consensus in the presence of malicious nodes. Finally, motivated by aggregation mechanisms based on pairwise masking, we model and analyze the impact of a subset of adversarial nodes, modeled as deletions, on connectivity and giant component size - metrics that are closely tied to privacy guarantees. Together, our results pave the way for end-to-end performance guarantees for a suite of algorithms for reliable inference on networks.