A model for efficient dynamical ranking in networks

TL;DR

This paper tackles the problem of inferring time-varying hierarchies in networks of timestamped directed interactions. It introduces Dynamical SpringRank, a physics-inspired extension of SpringRank that couples current interactions with past ranks via a self-spring term, formalized by a dynamical Hamiltonian $H_{total}$ and solvable as linear systems with a single tunable parameter $k$. The authors provide online (DSR) and offline (OffDSR) formulations, a baseline moving-window variant, and a probabilistic generative model for synthetic data, demonstrating improved predictive performance over Elo, WHR, and TrueSkill variants across synthetic and real datasets (notably NBA) while highlighting when time information is most relevant. The approach is scalable, supports cross-validation-based hyperparameter tuning, and includes open-source implementations, offering a practical tool for dynamic ranking in diverse temporal networks.

Abstract

We present a physics-inspired method for inferring dynamic rankings in directed temporal networks - networks in which each directed and timestamped edge reflects the outcome and timing of a pairwise interaction. The inferred ranking of each node is real-valued and varies in time as each new edge, encoding an outcome like a win or loss, raises or lowers the node's estimated strength or prestige, as is often observed in real scenarios including sequences of games, tournaments, or interactions in animal hierarchies. Our method works by solving a linear system of equations and requires only one parameter to be tuned. As a result, the corresponding algorithm is scalable and efficient. We test our method by evaluating its ability to predict interactions (edges' existence) and their outcomes (edges' directions) in a variety of applications, including both synthetic and real data. Our analysis shows that in many cases our method's performance is better than existing methods for predicting dynamic rankings and interaction outcomes.

Figures (11)

• Figure 1: A visual representation of Dynamical SpringRank. Each node $i$ has rank $s_i$ at time $t$ and each edge is represented as a spring. The red springs indicate self-springs that connect past and present ranks. The black springs indicate interactions with different entities. The blue and grey nodes interact once while the grey and gold nodes interact three times. In contrast, the green node does not interact with the other entities. Arrows indicate the direction of a win in a directed interaction between two nodes.
• Figure 2: Evolution of inferred ranks over time on synthetic data. We illustrate the inferred ranks of three models over time: DSR, W-L and Elo. We also illustrate the ground truth of the synthetic ranks over time as a comparison (top left). The synthetic data is generated by setting $\beta=2.0$ and $c=0.5$. Dashed lines are ground truth ranks.
• Figure 3: Fold-by-fold evaluation on the NBA dataset. We compare the predictions of DSR to Elo, WHR and mwSR in relation to the performance metrics $\sigma_a$, $\sigma_L$ and accuracy. The black points above the diagonal represent folds where DSR outperformed its competitors; yellow points indicate equal performance and red points represent DSR loses (where it was outperformed by competitors). Numbers inside the legend are the number of trials that an algorithm outperforms the other in percentage.
• Figure 4: Evolution over time of predicted ranks for the NBA dataset. We illustrate the predicted ranks of four models over time: TS, DSR, W-L and Elo. We select a subset of 13 teams (as indicated in the legend) to highlight the behaviors of both top and bottom scoring teams. Vertical colored bands break seasons into two periods.
• Figure 5: Permutation test results on the NBA dataset: chronology matters. The histogram is generated by $1000$ random permutations to the NBA dataset, and measuring the performance of Dynamical SpringRank on these permuted datasets. The black and red dotted lines represent the results of DSR and mwSR respectively on the original, chronologically-ordered NBA dataset --- the accuracy is much higher, and the agony much lower, than the vast majority of permuted datasets. This convincingly rejects the null hypothesis that chronological order does not matter, and justifies the use of a dynamical model. In each case the $p$-value is less than $0.001$.