Reproducing the first and second moment of empirical degree distributions

TL;DR

This work tackles the limitation of linear Exponential Random Graphs in capturing variance of empirical degree distributions. It introduces a canonical, fitness-induced non-linear extension of the two-star model (fit2SM) that reproduces both the first and second moments of the degree distribution and remains computationally efficient by using two global parameters and a fixed-point solution. Through extensive tests on the eMID interbank network across multiple time scales, fit2SM demonstrates improved reproduction of degree variance, spectral properties, and network structure compared with UBCM and dcGM, while offering robust generative capabilities for early-warning signals. The results suggest a practical, minimal non-linear ERG for real-world networks and provide a framework to infer unobserved higher-order statistics from limited information, with implications for systemic risk and epidemic threshold analyses.

Abstract

The study of probabilistic models for the analysis of complex networks represents a flourishing research field. Among the former, Exponential Random Graphs (ERGs) have gained increasing attention over the years. So far, only linear ERGs have been extensively employed to gain insight into the structural organisation of real-world complex networks. None, however, is capable of accounting for the variance of the empirical degree distribution. To this aim, non-linear ERGs must be considered. After showing that the usual mean-field approximation forces the degree-corrected version of the two-star model to degenerate, we define a fitness-induced variant of it. Such a `softened' model is capable of reproducing the sample variance, while retaining the explanatory power of its linear counterpart, within a purely canonical framework.

Figures (10)

• Figure 1: Analysis of eMID during the 31st week (left panel) and the 42nd week (right panel) of the year 2004. Graphical representation of the agreement between the sample variance $\text{Var}[\bm{k}]$ (black, dashed, vertical line) and its expected value $\langle\text{Var}[\bm{k}]\rangle$ under the UBCM (red), the dcGM (yellow) and the fit2SM (brown): while the UBCM steadily overestimates it and the dcGM either overestimates or underestimates it, the fit2SM correctly reproduces it. Each ensemble distribution is well approximated by a Gaussian whose parameters match the corresponding average and standard deviation.
• Figure 2: Quantile-quantile plots comparing the observed degrees and the expected ones under the UBCM (red), the dcGM (yellow) and the fit2SM (brown), across five aggregations of the eMID interbank market (from top-left to bottom-right: daily, weekly, monthly, quarterly and yearly). These snapshots are selected by drawing one calendar year at random (here, $2001$) and, conditionally on it, drawing at random one quarter (here, Q3), one month (here, April), one ISO week (here, week 2) and one trading day (here, $2001$-$11$-$21$). As it can be appreciated, the dcGM and the fit2SM perform very similarly, and both satisfactorily, in reproducing the degree sequence induced by the corresponding snapshot.
• Figure 3: Left panel: distributions of the spectral radius induced by $M=10^3$ configurations sampled from the Chung-Lu model defined by the topology of the eMID snapshot corresponding to the day $2010$-$07$-$19$, according to the UBCM (red), the dcGM (yellow) and the fit2SM (brown). The three models were solved for each of the $M=10^3$, generated configurations and the corresponding ensembles explicitly sampled in order to obtain an estimation of $\pi_1$: what we find is that $\lambda_1^\text{dcGM}<\pi_1\lesssim\lambda_1^\text{fit2SM}<\lambda_1^\text{UBCM}$. In words, while the UBCM overestimates the spectral radius of the generative model (black, dashed, vertical line) and the dcGM underestimates it, the fit2SM correctly reproduces it. Each ensemble distribution is well approximated by a Gaussian whose parameters match the corresponding average and standard deviation. Right panel: violin plots (summarising the distributions) of the relative error in reproducing the empirical spectral radius $\lambda_1^{\text{obs}}$, across snapshots and temporal aggregations. The fit2SM (brown) yields smaller errors i) than the dcGM (yellow) at all time-scales; ii) than the UBCM (red) at the daily and weekly time-scales.
• Figure 4: For each snapshot and aggregation level, we show the violin plots (summarising the distributions) of the relative error $\delta\text{BIC}_{model}$, defined in eq. \ref{['eq:bic']}. Upon remembering that a null value of such an index identifies the best-performing model on the considered snapshot, let us notice that the score induced by the fit2SM (brown) concentrates near zero - hence such a model dominates over the competing ones - at finer aggregations, while the UBCM (red) is the winner at coarser aggregations.
• Figure 5: Left panel: relative improvement of the fit2SM versus the relative error in estimating the number of two-stars with respect to the UBCM. Right panel: relative improvement of the fit2SM versus the relative error in estimating the number of two-stars with respect to the dcGM. A trend appear: the larger the error in estimating the total number of two-stars, the more the fit2SM outperforms the other models, an evidence suggesting that a poorer estimation of the total number of two-stars corresponds to a worse statistical fit of the eMID structure. This effect is more pronounced for sparser configurations, on which both the UBCM and the dcGM tend to commit larger errors - while the UBCM steadily overestimates $S$, the dcGM either overestimate or underestimate it. Each point is a network at a given level of aggregation.