Spectral Estimation of Conditional Random Graph Models for Large-Scale Network Data

Generative models for graphs have been typically committed to strong prior assumptions concerning the form of the modeled distributions. Moreover, the vast majority of currently available models are either only suitable for characterizing some particular network properties (such as degree distribution or clustering coefficient), or they are aimed at estimating joint probability distributions, which is often intractable in large-scale networks. In this paper, we first propose a novel network statistic, based on the Laplacian spectrum of graphs, which allows to dispense with any parametric assumption concerning the modeled network properties. Second, we use the defined statistic to develop the Fiedler random graph model, switching the focus from the estimation of joint probability distributions to a more tractable conditional estimation setting. After analyzing the dependence structure characterizing Fiedler random graphs, we evaluate them experimentally in edge prediction over several real-world networks, showing that they allow to reach a much higher prediction accuracy than various alternative statistical models.

💡 Research Summary

The paper tackles two fundamental shortcomings of most existing graph generative models: (1) they rely on strong parametric assumptions that tie the model to specific network properties such as degree distribution or clustering coefficient, and (2) they attempt to estimate the full joint probability distribution over all possible graphs, a task that becomes intractable as the number of nodes grows into the thousands or millions. To overcome these issues, the authors introduce a novel, non‑parametric graph statistic derived from the Laplacian spectrum of a graph. By focusing on the entire set of Laplacian eigenvalues—particularly the second smallest eigenvalue, known as the Fiedler value—they capture global structural information (connectivity, community boundaries, diffusion characteristics) without committing to any predefined distributional form.

Using this spectral statistic, they define the Fiedler Random Graph (FRG) model. Instead of modeling the joint distribution (P(G)), FRG models the conditional probability of each edge given the spectrum of the graph with that edge removed:

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