Missing links prediction: comparing machine learning with physics-rooted approaches

TL;DR

This work systematically compares white-box, physics/rooted (entropy-based) approaches with black-box machine-learning methods for link prediction on incomplete networks. By evaluating gravity-like, configuration/fitness-based, and maximum-entropy ERG frameworks against a Gradient Boosting Tree model on the World Trade Web and eMID datasets, the study shows that entropy-based methods, particularly when combining endogenous (topology) and exogenous (GDP, distances) features, can match or exceed ML performance while offering faster and more interpretable predictions. The findings highlight that the usefulness of ML hinges on data richness and feature availability, and that structural information alone can rival data-hungry ML models in certain regimes, especially for predictive tasks in economic networks. Overall, the paper provides practical guidance on when to prefer white-box approaches vs ML for link prediction in real-world networks, and clarifies the roles of exogenous versus endogenous information in shaping link formation.

Abstract

An active research line within the broader field of network science is the one concerning link prediction. Close in scope to network reconstruction, link prediction targets specific connections with the aim of uncovering the missing ones, as well as predicting those most likely to emerge in the future, from the available information. In this paper, we consider two families of methods, i.e. those rooted in statistical physics and those based upon machine learning: the members of the first family identify missing links as the most probable non-observed ones, the probability coefficients being determined by solving maximum-entropy benchmarks over the accessible network structure; the members of the second family, instead, associate the presence of single edges to explanatory node-specific variables. Running likelihood-based models such as the Configuration Model, or one of its many fitness-based variants, in parallel with the Gradient Boosting Decision Tree algorithm reveals that the former's accuracy is comparable to (and sometimes slightly higher than) the latter's. Such a result confirms that white-box algorithms are viable competitors to the currently available black-box ones, being computationally faster and more interpretable than the latter.

Figures (9)

• Figure 1: Visual representation of the framework adopted to carry out link prediction in parisi_entropy-based_2018. Panel a: the red, solid lines indicate the empirical edges while the red, dotted lines indicate the non-existent edges; the green, solid lines indicate the edges we have direct access to and the information about which can be employed to predict the ones indicated by the green, dotted lines; the aim of a link prediction exercise is understanding which of the non-observed links is actually non-existent (indicated by the light blue, dotted lines) and which is genuinely missing (indicated by the light blue, solid lines). Panel b: the three subpanels depict a slightly different framework, within which link prediction is carried out by employing node-specific features only - in a way that is reminiscent of network reconstruction exercises ialongo_reconstructing_2022.
• Figure 2: Performance of the models, measured in terms of TPR, for the years $1990$, $1995$ and $2000$ of the WTW. Each panel collects methods that rely on the same set of features (see the colour legend). The upper bar in each panel, marked with a diagonal pattern, corresponds to the black-box GBDT. We have randomly selected the $10\%$ of links $10$ times to populate $\mathcal{E}^{miss}$ and generate $\mathcal{E}^{obs}=\mathcal{E}\setminus\mathcal{E}^{miss}$; each statistical indicator has, then, been averaged over such samples, the standard deviation being represented by a horizontal, black bar. When considering 'exogenous' features, each instance of the GBDT outperforms its white-box counterpart. The two classes of models, instead, perform in a comparable way when coming to consider 'endogenous' features such as the degrees. The only white-box model outperforming the corresponding instance of the GBDT is the CMD, taking as input a combination of 'endogenous' and 'exogenous' features, i.e. the degrees and the geographic distances.
• Figure 3: Performance of the models, measured in terms of JI, for the years $1990$, $1995$ and $2000$ of the WTW. Each panel collects methods that rely on the same set of features (see the colour legend). The upper bar in each panel, marked with a diagonal pattern, corresponds to the black-box GBDT. We have randomly selected the $10\%$ of links $10$ times to populate $\mathcal{E}^{miss}$ and generate $\mathcal{E}^{obs}=\mathcal{E}\setminus\mathcal{E}^{miss}$; each statistical indicator has, then, been averaged over such samples, the standard deviation being represented by a horizontal, black bar. When considering 'exogenous' features, each instance of the GBDT outperforms its white-box counterpart. The two classes of models, instead, perform in a comparable way when coming to consider 'endogenous' features such as the degrees. The only white-box model outperforming the corresponding instance of the GBDT is the CMD, taking as input a combination of 'endogenous' and 'exogenous' features, i.e. the degrees and the geographic distances.
• Figure 4: Performance of the models, measured in terms of ACC, for the years $1990$, $1995$ and $2000$ of the WTW. Each panel collects methods that rely on the same set of features (see the colour legend). The upper bar in each panel, marked with a diagonal pattern, corresponds to the black-box GBDT. We have randomly selected the $10\%$ of links $10$ times to populate $\mathcal{E}^{miss}$ and generate $\mathcal{E}^{obs}=\mathcal{E}\setminus\mathcal{E}^{miss}$; each statistical indicator has, then, been averaged over such samples, the standard deviation being represented by a horizontal, black bar. When considering 'exogenous' features, each instance of the GBDT outperforms its white-box counterpart. The two classes of models, instead, perform in a comparable way when coming to consider 'endogenous' features such as the degrees. The only white-box model outperforming the corresponding instance of the GBDT is the CMD, taking as input a combination of 'endogenous' and 'exogenous' features, i.e. the degrees and the geographic distances.
• Figure 5: Panels a: performance of the models, measured in terms of ROC curves, for the years $1990$, $1995$ and $2000$ of the WTW. Each panel collects methods that rely on the same set of features. We have randomly selected the $10\%$ of links $10$ times to populate $\mathcal{E}^{miss}$ and generate $\mathcal{E}^{obs}=\mathcal{E}\setminus\mathcal{E}^{miss}$ (the curves relative to each indicator are, in fact, $10$ partially overlapping curves corresponding to each realization). The 'X' markers indicate the TPR and FPR values obtained by selecting a number of missing links equal to $|\mathcal{E}^{miss}|$. Panel b: each statistical indicator has been averaged over the $10$ realizations sample, the standard deviation being represented by a vertical, black bar. The AUROC, depicted for all years (darker shades correspond to more recent years), enlarges when moving from less structured models, like the GM, to more structured ones, like the CMD.