Mining higher-order triadic interactions

TL;DR

This work addresses the challenge of higher-order triadic interactions that cannot be captured by pairwise networks. It introduces the Triadic Perceptron Model (TPM), which embeds triadic regulation via a triadic Laplacian ${\bf L}^{(\text{T})}$ and perceptron-like couplings $J_{ij}(\mathbf{X})$, demonstrating that triadic effects modulate the end-to-end mutual information $MI(X,Y)$ between edge endpoints. Building on this, the Triadic Interaction Mining (TRIM) algorithm mines triadic interactions from time-series data by examining how the conditional mutual information $MI_z(m)$ varies with a regulator variable $Z$, using two null models and an entropic score $S$ to assess significance. Validation on TPM data shows robust detection of true triads, and application to AML gene-expression data reveals biologically relevant triadic interactions, with many triples involving AML-associated genes. Overall, the framework provides a principled way to infer higher-order interactions with potential impact across biology, climate, and finance, supported by an openly available Python package.

Abstract

Complex systems often involve higher-order interactions which require us to go beyond their description in terms of pairwise networks. Triadic interactions are a fundamental type of higher-order interaction that occurs when one node regulates the interaction between two other nodes. Triadic interactions are found in a large variety of biological systems, from neuron-glia interactions to gene-regulation and ecosystems. However, triadic interactions have so far been mostly neglected. In this article, we propose {the Triadic Perceptron Model (TPM)} that demonstrates that triadic interactions can modulate the mutual information between the dynamical state of two linked nodes. Leveraging this result, we formulate the Triadic Interaction Mining (TRIM) algorithm to extract triadic interactions from node metadata, and we apply this framework to gene expression data, finding new candidates for triadic interactions relevant for Acute Myeloid Leukemia. Our work reveals important aspects of higher-order triadic interactions that are often ignored, yet can transform our understanding of complex systems and be applied to a large variety of systems ranging from biology to climate.

Figures (15)

• Figure 1: (Panel a) A triadic interaction occurs when a node $Z$, called a regulator node, regulates (either positively or negatively) the interaction between two other nodes $X$ and $Y$. The regulated edge can be conceptualized as a factor node (shown here as a cyan diamond). (Panel b) A network with triadic interactions can be seen as a network of networks formed by a simple structural network and by a bipartite regulatory network between regulator nodes and regulated edges (factor nodes).
• Figure 2: The TRIM algorithm identifies triples of nodes $X$, $Y$, and $Z$ involved in a putative triadic interaction, starting from the knowledge of the structural network and the dynamical variables associated with its nodes. For each putative triple of nodes involved in a triadic interaction (panel (a)) which belong to a network whose structure and dynamics is known (panel (b)), we study the functional behavior of the conditional mutual information $MIz$ (panel (c)), and assess the significance of the observed modulations of $MIz$ with respect to a null model (panel (d)). Given a predetermined confidence level, we can use these statistics to identify significant triadic interactions (panel (e)). This procedure can be extended to different triples of the network, thereby identifying the triadic interactions present in it (panel (f)).
• Figure 3: We consider a network with $N=10$ nodes, $L=12$ edges, and $\hat{L}=5$ triadic interactions (panel (a)). Panels (b) and (c) display the effect of triadic interactions on the Mutual Information profile $MIz$. Panel (b) shows $MIz$ for the triple $[4,9,5]$ involved in a positive triadic interaction. Panel (c) shows $MIz$ for the triple $[1,2,6]$ that is not involved in a triadic interaction. In all panels simulations were run to $t_{\textup{max}}=4,000$ with a timestep of $dt=10^{-2}$. For the analysis we consider $40,000$ time steps. The parameters of the model are: $\alpha=0.05,\hat{T}=10^{-3}, \Gamma=10^{-2}$, $w^+=8, w^-=0.5$, number of bins $P=400$.
• Figure 4: Representative results for triples of nodes involved in triadic interactions in the continuous model with triadic interactions. Results for the triple $[4,9,5]$ of the network in Figure 3 of the main text, which is triadic, are shown. The joint distributions of variables $X$ and $Y$ conditional on the values of $Z$ are shown in panel (a). Panel (b) shows the behavior of $MIz$ as a function of the values of $z_m$, which clearly departs from the constant behavior expected in absence of triadic interactions. Panel (c) presents the decision tree for fitting the $MIz$ functional behavior and determining the range of values of $Z$ for which the most significant differences among the joint distributions of the variables $X$ and $Y$ conditioned on $Z$ are observed. The parameters used are the same as in Figure \ref{['arxiv_2:fig:non-tri']}.
• Figure 5: We consider the network in Figure \ref{['arxiv_2:fig:non-tri']}(a). The time series obtained by integrating the stochastic dynamics of the proposed dynamical model for triadic interactions (Eq.(\ref{['arxiv_2:eq:dyn']})) are analyzed with the TRIM algorithm. Panel (a) displays the Receiver Operating Characteristic curve (ROC curve) obtained by running TRIM with $P=400$ bins and $\mathcal{N}=10^3$ realizations of the null model on these synthetic time series, using $\Theta_{\Sigma}$ to score for different parameters values indicated in the legend. Panel (b) displays the corresponding Precision-Recall curve (PR curve) obtained by running TRIM with the same parameters. The timeseries are simulated up to a maximum time $t_{max}=4000$ with a $dt=10^{-2}$. For the analysis, we consider $40,000$ time steps (see the SI for details). The parameter of the model are: $\hat{T}=10^{-3}$, $w^+=8$, $w^-=0.5$, and $\alpha$ and $\Gamma$ as indicated in the figure legend.