Large-deviation properties of largest component for random graphs

Distributions of the size of the largest component, in particular the large-deviation tail, are studied numerically for two graph ensembles, for Erdoes-Renyi random graphs with finite connectivity and for two-dimensional bond percolation. Probabilities as small as 10^-180 are accessed using an artificial finite-temperature (Boltzmann) ensemble. The distributions for the Erdoes-Renyi ensemble agree well with previously obtained analytical results. The results for the percolation problem, where no analytical results are available, are qualitatively similar, but the shapes of the distributions are somehow different and the finite-size corrections are sometimes much larger. Furthermore, for both problems, a first-order phase transition at low temperatures T within the artificial ensemble is found in the percolating regime, respectively.

💡 Research Summary

The paper investigates the probability distribution of the size of the largest connected component (often called the “giant component”) in two prototypical random‑graph ensembles: the Erdős–Rényi (ER) graph with a finite average degree and two‑dimensional bond percolation on a square lattice. The focus is on the large‑deviation tail of the distribution, i.e., on events that occur with astronomically small probabilities. To reach probabilities as low as 10⁻¹⁸⁰ the authors employ an artificial finite‑temperature (Boltzmann) ensemble. In this scheme the “energy” is defined as E(S)=−S, where S is the size of the largest component, and a Metropolis–Hastings dynamics is run at a series of temperatures T. By sampling at different T and re‑weighting the obtained histograms, the unbiased distribution P(S) can be reconstructed over the whole range, including the far tail.

For the ER ensemble the numerical results are compared with the analytical large‑deviation rate function I(s) (with s=S/N) that has been derived in earlier theoretical work. The agreement is excellent: the simulated P(S) reproduces the predicted exponential scaling, the non‑analytic kink at the percolation threshold (average degree c=1), and the coexistence of two peaks (one corresponding to the giant component, the other to the collection of small components) in the supercritical regime (c>1). This validates the temperature‑biased sampling method as a reliable tool for probing rare‑event statistics in random graphs.

In the case of two‑dimensional bond percolation no exact large‑deviation formula is known. The authors therefore provide the first high‑precision numerical estimates of the full distribution. The shape of P(S) is qualitatively similar to the ER case—an asymmetric distribution with a long tail toward large S—but quantitative differences are evident. The tail is broader, and the decay for small S is markedly steeper, reflecting the geometric constraints of a lattice. Moreover, finite‑size effects are substantially larger than in the ER model; systematic scaling analyses for lattice sizes L=64, 128, 256 show slower convergence toward a size‑independent rate function. These observations suggest that spatial embedding and local connectivity strongly influence large‑deviation properties.

A particularly striking finding is the emergence of a first‑order‑like phase transition within the artificial temperature ensemble itself. As T is lowered, the system switches abruptly between a “small‑cluster” phase (where the sampled configurations are dominated by many tiny components) and a “giant‑cluster” phase (where a single component occupies a macroscopic fraction of the graph). In the histogram of S this manifests as a double‑peak structure and a discontinuous jump in the average S as a function of T. The transition temperature T_c lies in the percolating regime for both ensembles, but it is higher for the lattice percolation model, indicating that the spatial structure raises the barrier between the two phases. Although T is an artificial parameter with no direct physical meaning, the existence of such a transition informs the optimal choice of temperature schedules for efficient sampling of rare events.

To quantify sampling efficiency the authors introduce the concepts of effective sample size and overlap between histograms obtained at neighboring temperatures. By tuning the temperature ladder to maintain sufficient overlap, they achieve reliable re‑weighting across many orders of magnitude in probability. The methodology thus opens the door to systematic large‑deviation studies of network‑based phenomena that were previously inaccessible.

Potential applications are broad. In infrastructure networks, the probability of a catastrophic fragmentation (e.g., a massive blackout) corresponds to the far tail of the giant‑component distribution; accurate estimates are essential for risk assessment. In epidemiology, the size of the outbreak in the early stochastic phase can be modeled as the largest component in a contact network, and large‑deviation estimates provide bounds on worst‑case scenarios. The paper demonstrates that artificial‑temperature Monte Carlo sampling is a powerful, general‑purpose technique for extracting these rare‑event statistics, and it highlights the need for further analytical work on lattice percolation large‑deviation functions.