[Paper Review] Stochastic blockmodel approximation of a graphon: Theory and consistent estimation

Edoardo M. Airoldi, Thiago B. Costa+1 more|arXiv (Cornell University)|Nov 7, 2013

Complex Network Analysis Techniques26 references116 citations

TL;DR

This paper proposes a computationally efficient stochastic blockmodel approximation (SBA) algorithm to consistently estimate a graphon from observed network data. By approximating the graphon with a piecewise constant function based on node similarity in latent positions, the method achieves almost sure consistency in mean absolute error as network size grows, under piecewise Lipschitz assumptions.

ABSTRACT

Non-parametric approaches for analyzing network data based on exchangeable graph models (ExGM) have recently gained interest. The key object that defines an ExGM is often referred to as a graphon. This non-parametric perspective on network modeling poses challenging questions on how to make inference on the graphon underlying observed network data. In this paper, we propose a computationally efficient procedure to estimate a graphon from a set of observed networks generated from it. This procedure is based on a stochastic blockmodel approximation (SBA) of the graphon. We show that, by approximating the graphon with a stochastic block model, the graphon can be consistently estimated, that is, the estimation error vanishes as the size of the graph approaches infinity.

Motivation & Objective

• Address the challenge of non-parametric inference on graphons from observed network data.
• Develop a computationally efficient method for estimating the underlying graphon function.
• Establish theoretical consistency of the estimation procedure under realistic smoothness assumptions.
• Provide a practical algorithm that achieves consistent estimation without requiring canonical labeling of nodes.

Proposed method

• Approximate the graphon using a two-dimensional step function, equivalent to a stochastic blockmodel with diminishing block sizes as n increases.
• Define node similarity via L1 distances between row and column slices of the graphon, using observed adjacency matrices to estimate these distances.
• Cluster nodes into blocks based on estimated similarity, with block size controlled by a threshold Δn that decreases with n.
• Estimate the graphon value in each block as the empirical average of edge probabilities within that block.
• Use concentration inequalities to bound estimation error, leveraging the piecewise Lipschitz condition on the graphon.
• Establish consistency by showing that the mean absolute error (MAE) and mean squared error (MSE) converge to zero in probability and in expectation as n → ∞.

Experimental results

Research questions

• RQ1 Can a computationally efficient algorithm consistently estimate a graphon from observed network data?
• RQ2 Does approximating a graphon via a stochastic blockmodel yield consistent estimation as network size increases?
• RQ3 How does the estimation error behave under piecewise Lipschitz continuity of the graphon?
• RQ4 What is the rate of convergence of the estimation error in terms of network size?
• RQ5 Can the method achieve consistency even when node labels are permuted, due to the block-based structure?

Key findings

• The SBA algorithm achieves almost sure consistency: the mean absolute error (MAE) of the estimated graphon converges to zero in probability as n → ∞.
• The expected mean absolute error converges to zero, implying consistency in expectation under the piecewise Lipschitz assumption.
• The method outperforms existing approaches like USVT in simulation studies, with better computational complexity.
• The estimation error is bounded by O(√Δn + Δn), where Δn is a tuning parameter that decreases with n, ensuring convergence.
• The consistency result holds even when the estimated graphon is defined up to node permutations, as the error is measured relative to the true generating graphon.
• The proof relies on concentration inequalities and control of block sizes to ensure that the fraction of nodes in small blocks vanishes as n increases.