Reducibility of higher-order networks from dynamics

TL;DR

An information-theoretic framework is proposed, accounting for how structure affect diffusion behaviors, quantifying the entropic cost and distinguishability of higher-order interactions to assess their reducibility to lower-order structures while preserving relevant functional information.

Abstract

Empirical complex systems can be characterized not only by pairwise interactions, but also by higher-order (group) interactions influencing collective phenomena, from metabolic reactions to epidemics. Nevertheless, higher-order networks' apparent superior descriptive power -- compared to classical pairwise networks -- comes with a much increased model complexity and computational cost, challenging their application. Consequently, it is of paramount importance to establish a quantitative method to determine when such a modeling framework is advantageous with respect to pairwise models, and to which extent it provides a valuable description of empirical systems. Here, we propose an information-theoretic framework, accounting for how structure affect diffusion behaviors, quantifying the entropic cost and distinguishability of higher-order interactions to assess their reducibility to lower-order structures while preserving relevant functional information. Empirical analyses indicate that some systems retain essential higher-order structure, whereas in some technological and biological networks it collapses to pairwise interactions. With controlled randomization procedures, we investigate the role of nestedness and degree heterogeneity in this reducibility process. Our findings contribute to ongoing efforts to minimize the dimensionality of models for complex systems.

Figures (18)

• Figure 1: Functional reducibility of higher-order networks. Illustration of our method with an example hypergraph. Given (a) an original hypergraph with interactions of orders up to $d_{\rm max}$ ($=3$, here), (b) we use higher-order diffusion processes to probe the system at a chosen diffusion time $\tau$ (e.g. small) that acts as a topological scale. (c) We then compute the cost function, a trade-off between information loss and model complexity, of the same hypergraph, but considering orders only up to $d$. We determine the optimal order $d_{\rm opt}$ as the one that minimizes the cost function. Finally, (d) we reduce the original hypergraph to an optimal version by considering orders up to $d_{\rm opt}$.
• Figure 2: The cost function is the sum of information loss and model complexity based on Kullback-Leibler divergence. We illustrate the terms in the cost function defined in \ref{['eq:message_length']}, on an example random simplicial complex: (a) information loss $D_{\rm KL}(\bm{\rho}_{\tau}^{[d_{\rm max}]}|\bm{\rho}_{\tau'}^{[d]})$, (b) model complexity $C ( \bm{\rho}_{\tau}^{[d]})$, and (c) their sum, the cost function $\mathcal{L}(\bm{\rho}_{\tau}^{[d_{\rm max}]}|\bm{\rho}_{\tau'}^{[d]})$. The minimum of the cost function is indicated by the vertical line. Parameters were set to $N=100$ nodes and wiring probabilities $p_d = 50 / N^d$ at order $d$ with $d_{\rm max}=4$. See SI \ref{['sec:sm:other_definition']} for results based on Jensen-Shannon divergence.
• Figure 3: Analytical reducibility of hyperrings. (a) Eigenvalues of the multiorder Laplacians up to order $d=1, 2$, for a hyperring with $N=100$ nodes. (b) Eigenvalues of the corresponding density matrices, for $\tau=0.01$ (blue), $0.1$ (orange), and $1$ (green), for $d=1$ (dashed) and $d=2$ (solid). (c) Difference between the cost function at order $d=2$ and $d=1$ as a function of $\tau$: it is negative (positive) when $d_{\rm opt}=2$ ($=1$), i.e. the hypergraph is irreducible (reducible, beige shade). Vertical dashed lines indicate the short ($\tau_{\rm short} = 1/\lambda^{[2]}_{\rm max}$) and long ($\tau_{\rm long} = 1/\lambda^{[2]}_{\rm min}$) timescales of the full system.
• Figure 4: Reducibility of random higher-order networks. Cost function in (a) random simplicial complexes and (b) random hypergraphs. Cost function for (c) a single random simplicial complex, and then after randomly shuffling (d) 50%, and (e) all of its hyperedges, corresponding to a random hypergraph. From (c) to (e), the nestedness between hyperedges of different orders decreases. Each point corresponds to one of 100 realizations. Box plots show median (center line), interquartile range (box), and the non-outlier range (whiskers). The $d_{\rm opt}$ minimizing the cost function is indicated by the vertical line. Parameters were set to $N=100$ nodes and wiring probabilities $p_d = 50 / N^d$ at order $d$ with $d_{\rm max}=4$.
• Figure 5: Sixty empirical datasets show different levels of reducibility. We show the cost function against the largest order considered in (a) sfhh-conference, (b) dawn, and (c) congress-bills. (d) Reducibility $\chi$ for all 60 datasets colored by category and (e) its overall distribution. The number of datasets in each category is reported in parenthesis. Box plots show median (center line), interquartile range (box), and the non-outlier range (whiskers). All reducibility values are reported in \ref{['tab:datasets']}.