Measures of net effects in signed social and ecological networks

TL;DR

This work introduces a unified framework for measuring net and indirect effects in signed, weighted, and directed networks by defining the net effects matrix $\mathrm{Net}(\mathbf{A}) = \sum_{\mathscr{l}=1}^{\infty} \mathbf{A}^{\mathscr{l}}$ and its plus-one variant $\mathrm{Net}_{+1}(\mathbf{A}) = (\mathbf{I}-\mathbf{A})^{-1}$ under the condition $\rho(\mathbf{A})<1$. It develops three convergence-guaranteeing rescalings (global, column, and row) that preserve signs and relative weights and connect to classical centralities such as Katz centrality and PageRank, including reverse PageRank via transposed formulations. In ecology, it introduces the alpha matrix $\boldsymbol{\alpha}$ and beta matrix $\boldsymbol{\beta}$ to recast interactions in non-dimensional forms, links net effects to press-perturbation responses, and defines a collectivity parameter $\phi = \rho(\boldsymbol{\alpha}^\dagger)$; it further demonstrates that negative incoming net effects with respect to $\frac{1}{n}\boldsymbol{\beta}^\dagger$ predict higher extinction risk under generalized Lotka-Volterra dynamics. In social networks, the framework reinterprets existing measures (e.g., PN centrality) as instances of net effects and shows full net effects better capture influence and power dynamics in real-world signed networks like Sampson’s monastery. Overall, the paper provides practical tools for quantifying direct and indirect influences across domains and offers guidance on rescaling choices and aggregation directions to tailor metrics to specific research questions.

Abstract

With improvements in data resolution and quality, researchers can now represent complex systems as signed, weighted, and directed networks. In this article, we introduce a framework for measuring net and indirect effects without simplifying these information-rich networks. It captures both direct and indirect interactions, the effect of the whole network on a node, and conversely, the effect of a node on the entire network, while accommodating the complexity of signed, weighted, and directed links. Our taxonomy unifies and extends existing approaches and measures from network science, computational social science, and ecological networks. We demonstrate its value in ecological systems, where net and indirect effects are critical yet difficult to quantify. Using generalized Lotka-Volterra dynamics, we find a strong correlation between negative net effects and species extinction. We further apply the framework to a real-world social network, where it identifies informative rankings that illuminate influence propagation and power dynamics.

Figures (4)

• Figure 1: Overview of the fundamental notions developed for net effects. Arrow color and thickness represent the sign and weight of the links.
• Figure 2: Average extinction probability, in random ecological networks with different numbers of species and connectances, of the species with minimum incoming net effect, i.e. $\mathrm{Net}\left(\frac{1}{n}\boldsymbol{\beta}^{\dagger}, * \leftarrow \mathrm{all} \right)$, a species with negative incoming net effect, and any species. For each pair of number of species and connectance, we simulated 1000 interaction networks whose weights were drawn from $\mathcal{U}(-1,1)$.
• Figure 3: Comparison of the monks ranking according to the net effect truncated at walks of length 3 calculated in liu2020, left column, with the ranking when all walks are taken into account, right column. The right column better captures the actual ranking based on the finally expelled monks (in red).
• Figure 4: (a) Ranking of the monks given by their incoming net effects and (b) evolution of their popularity as modeled by replicator dynamics.