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Revealing physical interaction networks from statistics of collective dynamics

Mor Nitzan, Jose Casadiego, Marc Timme

arXiv:1801.05598v1physics.soc-phnlin.CD

TL;DR

Physical interaction networks are important but difficult to infer when model knowledge or detailed, ordered measurements are unavailable. This paper uses driven responses of invariant measures and compressed sensing to infer connectivity, reporting successful reconstruction across dynamics and better performance than standard baselines.

  • Problem

    Physical connectivity is often inaccessible, and existing approaches can require detailed models, simple system states, or high-resolution, synchronized, temporally ordered observations.

    Source: INTRODUCTION · p. 1

  • Method

    The method uses changes in invariant-measure centers of mass under small driving signals and compressed sensing to infer physical interactions from state statistics.

    Source: INTRODUCTION · p. 1, THEORY BASED ON TIME INVARIANTS · p. 2

  • Results

    Across a range of systems and dynamics, the strategy reconstructs physical networks and outperforms mutual-information, correlation, partial-correlation, and transfer-entropy baselines.

    Source: PERFORMANCE ON ARTIFICIAL AND REAL… · p. 3

  • Takeaways & Limitations

    The approach can infer physical connectivity from temporally disordered, widely spaced measurements collected across experiments, without detailed prior model knowledge.

    Source: INTRODUCTION · p. 1

    The method requires a well-sampled invariant measure and driving small enough for linear response yet strong enough to outweigh noise; extreme saturation or fast adaptation can constrain its use.

    Source: DISCUSSION · p. 5

Abstract

from arXiv · show

Revealing physical interactions in complex systems from observed collective dynamics constitutes a fundamental inverse problem in science. Current reconstruction methods require access to a system's model or dynamical data at a level of detail often not available. We exploit changes in invariant measures, in particular distributions of sampled states of the system in response to driving signals, and use compressed sensing to reveal physical interaction networks. Dynamical observations following driving suffice to infer physical connectivity even if they are temporally disordered, are acquired at large sampling intervals, and stem from different experiments. Testing various nonlinear dynamic processes emerging on artificial and real network topologies indicates high reconstruction quality for existence as well as type of interactions. These results advance our ability to reveal physical interaction networks in complex synthetic and natural systems.

INTRODUCTION

Physical interaction topology is essential to understanding network function, yet it is often inaccessible and difficult to infer from dynamics. The paper proposes recovering it from responses to driving without requiring detailed model knowledge or highly ordered, high-resolution data.

INTRODUCTION

  • Physical interaction networks underlie collective dynamics and function, making topology essential for identifying pathways and understanding feedback.Source: INTRODUCTION · p. 1
  • Statistical dependency networks can misrepresent physical connectivity, including when multistability yields different dependency networks for the same physical network.Source: INTRODUCTION · p. 1
  • The proposed strategy uses invariant-measure responses to small driving signals and compressed sensing, without detailed model knowledge or synchronized, ordered, high-resolution observations.It can use temporally disordered, widely spaced measurements gathered across different experiments and times for different units.Source: INTRODUCTION · p. 1

THEORY BASED ON TIME INVARIANTS

The theory relates changes in the centers of mass of driven invariant measures to network topology through the system Jacobian. It uses these relations to infer incoming interactions for each node, including when experiments are few relative to network size.

THEORY BASED ON TIME INVARIANTS

  • When recorded points sample an invariant measure well, its center of mass can be estimated by the sample mean of observed states.Source: THEORY BASED ON TIME INVARIANTS · p. 2
  • Changes in centers of mass across driving conditions are jointly determined by the driving vectors and network topology, producing linear equations involving the Jacobian.Source: THEORY BASED ON TIME INVARIANTS · p. 2
  • Reconstruction decomposes by node: nonzero Jacobian entries identify direct incoming interactions, with recovery expected when those interactions persist across relevant sampled states.Source: THEORY BASED ON TIME INVARIANTS · p. 2
  • When experiments are fewer than network nodes, an L1-norm minimum solution estimates each Jacobian row, and thresholds distinguish existing from missing interactions.Source: THEORY BASED ON TIME INVARIANTS · p. 2

PERFORMANCE ON ARTIFICIAL AND REAL NETWORK TOPOLOGIES

Tests across network types and dynamics report successful topology reconstruction and better performance than standard statistical baselines. Reconstruction quality also improves with more driving-response experiments, while compressed sensing reduces the experiment burden relative to network size.

PERFORMANCE ON ARTIFICIAL AND REAL NETWORK TOPOLOGIES

C A B

Separately identifying activating and inhibiting interactions improves reconstruction quality, and the approach also performs on the Drosophila circadian clock. Evaluation shows reconstruction quality improves with more experiments and is robust to noise and network size.

C A B

  • The evaluation indicates that the experiments needed for AUC > 0.95 increase sublinearly with network size and change only weakly with noise.AUC also increases as noise decreases and as the number of experiments increases.Source: Figure 2 · p. 3
  • Separately identifying activating and inhibiting interactions improves reconstruction quality over jointly identifying existing and missing links.The figure also shows quality increasing with the number of driving-response experiments.Source: Figure 3 · p. 3, C A B · p. 4
  • For the Drosophila circadian clock, reconstruction quality increases with more driving-response experiments and changes only weakly with noise.The figure compares noiseless and noisy dynamics.Source: C A B · p. 4

DISCUSSION

The discussion presents invariant-measure reconstruction as applicable to varied sampling conditions and partial observations, while identifying stationarity and suitable driving as important conditions. It also connects the approach to biological network reconstruction.

DISCUSSION

  • The method relies on sampled invariant measures, sufficiently small driving for linear responses, and driving large enough to outweigh noise and finite-sampling effects.No fine-tuning of driving signals is needed.Source: DISCUSSION · pp. 4–5
  • Reconstruction from one observed variable per multidimensional unit remains robust, with only a slight quality reduction relative to observing all three variables.Quality also increases with the number of driving-response experiments.Source: Figure 4 · p. 4
  • Extreme perfect saturation and fast adaptation can constrain applicability; slow adaptation relative to measurement allows the system to remain effectively stationary.In such cases, shifts in averages can be observed within the same framework.Source: DISCUSSION · p. 5
  • The approach can use unordered state samples, varying sampling intervals, and data from multiple experiments performed under the same conditions.The method uses statistical information from dynamical data rather than known temporal order.Source: DISCUSSION · p. 5
  • The authors identify gene-expression measurements across different genes and experiments as a potential setting for inferring physical, not only correlative, connectivity.The cited examples include simultaneous system-wide measurements at one time point and measurements of a few genes over several time points.Source: DISCUSSION · p. 5

Goodwin oscillators

The controlled model uses networks of three-variable Goodwin oscillators with nonlinear local dynamics and diffusive interactions.

Goodwin oscillators

  • Each Goodwin oscillator has variables x_i, y_i, and z_i, with local nonlinearity in f(z_i) and diffusive coupling g(y_i, y_j) = y_i − y_j.The simulations set a_i = b_i = c_i = 0.4, v_o = K_m = 1, and n = 17.Source: Goodwin oscillators · p. 5

Network topologies

The evaluation spans random network models and the Drosophila circadian clock. The reconstruction uses compressed sensing to infer sparse physical connections from a relatively small number of driving-response measurements.

Network topologies

  • For generic evaluations (Figs. 2 to 4), the study used random networks of N = 50 Goodwin oscillators with regular incoming degree 4, plus an Erdős Rényi network of N = 50 genetic regulators with edge probability p = 0.01.Source: Network topologies · p. 5
  • The real biological topology evaluated is the Drosophila circadian clock network.The paper also uses an Erdős–Rényi random network of genetic regulators.Source: Network topologies · p. 5
  • Compressed sensing reconstructs each unit’s physical connections from averaged driving signals and a driving-response matrix, with M ≪ N.For sufficiently sparse connections and a matrix satisfying specified conditions, the problem is posed as L1-norm convex optimization with robust, stable solution guarantees.Source: Network topologies · p. 5

SUPPLEMENTARY MATERIALS

The supplementary materials provide derivations, evaluation details, model descriptions, and figures covering reconstruction performance across network and driving conditions.

Notes S1–S8

  • Notes S1–S8 cover the invariant-based reconstruction derivation, sampled-density error estimates, evaluation, and performance across network structures and dynamics.They include homogeneous and heterogeneous networks, systems near fixed points, chaotic systems, and comparisons with standard baselines.Source: SUPPLEMENTARY MATERIALS · p. 5

Figures S1–S5

  • Figures S1–S5 address sample-mean approximation, experiment counts, topology robustness, genetic regulators, and chaotic Rössler oscillators.Source: SUPPLEMENTARY MATERIALS · p. 5

Notes S9–S13 and figures S6–S10

  • Notes S9–S13 and figures S6–S10 cover interaction types, missing information, driving conditions, and comparisons with other approaches.The listed figures include comparisons against transfer entropy and reconstruction quality under varying hidden-unit fractions and driving strengths.Source: SUPPLEMENTARY MATERIALS · p. 5
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