Information Propagation in Multilayer Systems with Higher-Order Interactions across Timescales

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Information Propagation in Multilayer Systems with Higher-Order Interactions across Timescales

Giorgio Nicoletti and Daniel Maria Busiello

Phys. Rev. X 14, 021007 – Published 8 April 2024

Abstract

Complex systems are characterized by multiple spatial and temporal scales. A natural framework to capture their multiscale nature is that of multilayer networks, where different layers represent distinct physical processes that often regulate each other indirectly. We model these regulatory mechanisms through triadic higher-order interactions between nodes and edges. In this work, we focus on how the different timescales associated with each layer impact their reciprocal effective couplings. First, we rigorously derive a decomposition of the joint probability distribution of any dynamical process acting on such multilayer networks. By inspecting this probabilistic structure, we unravel the general principles governing how information propagates across timescales, elucidating the interplay between mutual information and causality in multiscale systems. In particular, we show that feedback interactions, i.e., those representing regulatory mechanisms from slow to fast variables, generate mutual information between layers. On the contrary, direct interactions, i.e., from fast to slow layers, can propagate this information only under certain conditions that depend solely on the structure of the underlying higher-order couplings. We introduce the mutual information matrix for multiscale observables to capture these emergent functional couplings. We apply our results to study archetypal examples of biological signaling networks and effective environmental dependencies in stochastic processes. Our framework generalizes to any dynamics on multilayer networks, paving the way for a deeper understanding of how the multiscale nature of real-world systems shapes their information content and complexity.

Received 19 December 2023
Revised 20 February 2024
Accepted 8 March 2024

DOI:https://doi.org/10.1103/PhysRevX.14.021007

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Stochastic processes

Multilayer & multiplex networks Multiple time scale dynamics

Information theory

Condensed Matter, Materials & Applied PhysicsInterdisciplinary PhysicsNonlinear DynamicsPhysics of Living SystemsStatistical Physics & ThermodynamicsNetworks

Authors & Affiliations

Giorgio Nicoletti ^1,2,3,† and Daniel Maria Busiello ^4,*

¹ECHO Laboratory, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
²Laboratory of Interdisciplinary Physics, Department of Physics and Astronomy “Galileo Galilei”, University of Padova, Padova, Italy
³Department of Mathematics “Tullio Levi-Civita”, University of Padova, Padova, Italy
⁴Max Planck Institute for the Physics of Complex Systems, Dresden, Germany

^*Corresponding author: busiello@pks.mpg.de
^†giorgio.nicoletti@epfl.ch

Popular Summary

Complex systems are characterized by many interconnected dynamical processes taking place at different temporal scales. In recent years, multilayer networks have emerged as a powerful tool to capture this fundamental structure, shared by many biological and nonbiological systems ranging from biochemistry to ecology. A key feature is that interactions across timescales are typically not pairwise, because the dynamics in a layer may be influenced by the state of one or more other layers, creating higher-order dependencies. Understanding how these interlayer regulatory mechanisms shape the functionality of complex systems is a challenging question with profound real-world implications. We build a novel information-theoretic framework to unravel how processes occurring at different timescales are coupled together at the functional level by sharing information.

In our framework, the shared information is generated between specific timescales and propagates along the multilayer structure. We unveil the foundational principles governing these phenomena, allowing us to determine when causal connections translate into effective couplings at the information level. In particular, our approach identifies the most relevant interactions contributing to the functional operations of any complex system, independently of the underlying dynamics. Chemical reaction networks, brain activity, and ecological population dynamics are only a few examples in which the relations between processes at different timescales are the fundamental drivers of their properties.

This work will shed new light on how the multiscale nature of complex real-world systems shapes the effective communication between their many degrees of freedom, ultimately driving their emergent behaviors.

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Vol. 14, Iss. 2 — April - June 2024

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Figure 1
Information between two layers with states $x_{1}$ and $x_{2}$ is influenced by the their timescale ratio $τ_{1} / τ_{2}$ . (a) Sketch of the system with the node $B_{1}$ influencing the link $A_{2} \to B_{2}$ . (b) Mutual information $I_{12}$ between $x_{1}$ and $x_{2}$ as a function of $τ_{1} / τ_{2}$ and the triadic interaction strength $c$ . (c) At fixed $c = 10$ , $I_{12}$ vanishes when $x_{1}$ is faster than $x_{2}$ . When $τ_{1} / τ_{2} \to \infty$ , the regulating layer is slower, and $I_{12}$ converges to a nonzero value. The master equation of the system can be solved exactly (solid black line), simulated via a Gillespie algorithm [50] (blue dots), and solved in a timescale separation regime (dashed blue line).
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Figure 2
Sketch of a multilayer system with intralayer pairwise interactions and triadic higher-order interactions across layers. We name interlayer interactions going from a faster layer to a slower one as “direct” interactions, as they follow the ordering induced by the timescales. On the contrary, we refer to those that go toward faster layers as “feedback” interactions, as they provide a slow modulation to fast processes.
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Figure 3
Accuracy of the timescale separation solution and mutual information in a five-layer system with triadic interactions. (a) Scheme of the interactions, defined in Eq. (3). For simplicity, we set feedback interactions as $F_{μ ν}^{k, i \to j} = F_{μ ν}^{eq} δ_{j k} + F_{μ ν}^{cr} (1 - δ_{j k})$ , and similarly for direct interactions $D_{μ ν}^{k, i \to j}$ , with $F_{μ ν}^{eq} = 10$ , $D_{μ ν}^{eq} = 0.1 = F_{μ ν}^{cr}$ , and $D_{μ ν}^{cr} = 5$ (see Appendix pp4). (b) Simulated trajectories in each layer with $τ_{μ + 1} / τ_{μ} = 0.1$ , so that each layer is an order of magnitude faster than the previous one, using a Gillespie algorithm. (c) Comparison of the joint probability estimated from simulated trajectories at different separations between the timescales of each layer. Gray bars represent the analytical solution in Eq. (6), which is well approximated already for $τ_{μ + 1} / τ_{μ} = 0.1$ . (d) MIMMO associated with this system when $τ_{μ + 1} / τ_{μ} \to 0$ . (e) Mutual information between each pair of layers is shaped by the pathways among them defined by direct and feedback interactions.
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Figure 4
The principles of information propagation in multilayer networks. (a) A three-layer system with a higher-order feedback interaction from layer 3 to 1 and a direct link from 2 to 3. Each layer has two nodes, as in the previous examples. (b) The slow layer acts as an information source for the fast one so that $I_{13} \neq 0$ in the MIMMO, while the direct link does not generate information. (c) A stronger feedback interaction $f$ increases the mutual information between the two layers. (d) The same system as in panel (a), with the direct link from 1 to 2. (e) All the elements of the MIMMO are nonzero, since it exists a minimal propagation path $3 \to 1 \to 2$ . The direct link propagates the information created in 1. (f) All the entries of the MIMMO depend on $f$ , and they vanish as $f$ goes to zero, i.e., no information is created. (g) A four-layer system with feedback interactions from 4 to 2 and 2 to 1, and a direct link from 1 to 3. While for $f \neq 0$ all the entries of the MIMMO are nonzero, when $f = 0$ , the information created in 1 by 2 cannot be propagated to 3, since the dynamics of $x_{3}$ is faster than $x_{2}$ . As a consequence, only $I_{12} \neq 0$ .
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Figure 5
Shared higher-order interactions generate information among random walks in disjoint network components. (a) An archetypal system where one layer has two disjoint components $(A_{1}, B_{2})$ and $(A_{2}, B_{2})$ , with two random walks evolving with timescales $τ_{1} = τ_{2} = τ$ . The links of both these components are influenced by another layer $E$ through a triadic interaction $c$ that models an environmental-like dynamics with timescale $τ_{E}$ influencing two separate degrees of freedom $x_{1}$ and $x_{2}$ . (b) In the limit $τ_{E} ≫ τ$ , the shared environment induces a nonzero mutual information between the two disjoint random walks $I_{12}$ . The black curve shows $I_{12}$ in the limit $c \to \infty$ , while blue dots show the corresponding Gillespie simulation. The two random walks are conditionally independent in this limit, so their mutual information can be decomposed exactly (gray dashed line) and converges to the Shannon entropy of the environment $E$ (blue dashed line).
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Figure 6
Information in birth-and-death processes modeling multiscale signaling networks. (a),(b) Two biochemical architectures known to implement biological adaptation and habituation, encompassing a receptor $R$ , a storage $S$ , and a readout population $U$ . Inhibitory links, which increase the death rate of a population, are indicated with a bar at the end. (c),(d) Both architectures display similar dynamical behavior in the presence of a repeated input to the receptor, with the readout average $⟨ U ⟩$ decreasing and $⟨ S ⟩$ increasing at each signal (gray shaded area). (e) The two architectures display important differences at the information level across layers. In the first, the receptor and the storage share no mutual information, and the regulation of $S$ acts independently of the signal. (f) In the second, the storage is also informationally coupled to the receptor; hence the regulatory mechanism is tangled with the external signal and all entries of the MIMMO are nonzero. In this figure, we set the maximum population of all species, i.e., the nodes of each layer, to $M_{R} = 15$ , $M_{S} = 10$ , and $M_{U} = 20$ .
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