Research programme · PhD work with D. Garlaschelli, R. S. Hazra, F. den Hollander and M. Mandjes

In a hard-constraint (microcanonical) ensemble, quantities such as degrees are fixed exactly. In the corresponding soft-constraint (canonical) ensemble they are imposed only in expectation, through Lagrange multipliers. The two ensembles may agree on many macroscopic observables, yet fail to be asymptotically equivalent.

This programme asks whether the largest eigenvalue $\lambda_1$ of the adjacency matrix can detect that failure. It turns out to be an informative but not universal probe: the working hypothesis is that a non-vanishing difference of expected largest eigenvalues signals breaking of ensemble equivalence, and these papers confirm it in some density regimes while identifying one where the eigenvalue misses the inequivalence entirely.

Constrained regular graphs

When all degrees are fixed at a common value, the difference between the expected largest eigenvalues of the soft- and hard-constrained ensembles has a non-zero limit,

$$ \Delta_\infty \;=\; \lim_{n\to\infty}\bigl(\mathbb{E}_{\mathrm{can}}[\lambda_1]-\mathbb{E}_{\mathrm{mic}}[\lambda_1]\bigr), $$

which depends on the density regime: $\Delta_\infty = 1-p$ in the dense regime with constant edge probability $p$, and $\Delta_\infty = 1$ when the density vanishes. In the ultra-dense regime it vanishes — there the largest eigenvalue no longer detects the inequivalence, even though it is present. By contrast, a single global constraint on the number of edges produces no such gap.

Chung–Lu random graphs

The soft-constraint model for heterogeneous expected degrees is the Chung–Lu graph. We derive a central limit theorem for its principal eigenvalue and, under a slightly stronger lower bound on the degrees, for the individual components of the principal eigenvector — under assumptions on the average degrees guaranteeing connectivity, sparsity and bounded inhomogeneity.

Configuration model

For the hard-constraint configuration model with degrees diverging uniformly on a scale between $1$ and $\sqrt{n}$, and with comparable smallest and largest degree, we compute $\mathbb{E}[\lambda_1]$ and prove a weak law of large numbers,

$$ \frac{\lambda_1}{\mathbb{E}[\lambda_1]}\;\xrightarrow{\;\mathbb{P}\;}\;1 . $$

Comparison with the corresponding soft-constrained Chung–Lu ensemble gives the asymptotic shift

$$ \mathbb{E}_{\mathrm{hard}}[\lambda_1] \;=\; \mathbb{E}_{\mathrm{soft}}[\lambda_1]-1+o(1). $$

This order-one correction persists even though the largest eigenvalue itself diverges: information about the constraint mechanism survives at a scale invisible to the leading order.

Publications

  1. P. Dionigi, D. Garlaschelli, F. den Hollander, M. Mandjes, A spectral signature of breaking of ensemble equivalence for constrained random graphs, Electronic Communications in Probability 26 (2021), 1–15. DOI · arXiv
  2. P. Dionigi, D. Garlaschelli, R. S. Hazra, F. den Hollander, M. Mandjes, Central limit theorem for the principal eigenvalue and eigenvector of Chung–Lu random graphs, Journal of Physics: Complexity 4(1), 015008 (2023). DOI · arXiv
  3. P. Dionigi, D. Garlaschelli, R. S. Hazra, F. den Hollander, Largest eigenvalue of the configuration model and breaking of ensemble equivalence, arXiv:2312.07812. arXiv