Source-linked AI summary
Inferring Network Topology from Complex Dynamics
Srinivas Gorur Shandilya, Marc Timme
TL;DR
The paper asks how hidden network topology can be inferred when unit dynamics are observable but coupling structure is not. It develops a direct analytical method that uses a generic observed trajectory and known dynamical forms, then shows reconstruction across diverse dynamics and under substantial noise, while relying on those known functional forms.
Problem
Network topology and coupling strengths often cannot be directly measured even when individual-unit dynamics are observable.
Method
The method evaluates known intrinsic and coupling functions along observed dynamics to solve directly for topology, coupling strengths, and parameters appearing linearly in the equations.
Results
The method reconstructs network structure across diverse collective dynamics and simultaneously recovers topology and dynamical parameters despite substantial additive noise.
Takeaways & Limitations
A sufficiently long observation can replace external intervention for reconstructing network topology and coupling strengths across unrestricted collective dynamics.
Takeaways & Limitations
The method presupposes knowledge of the functional form of the intrinsic and interaction dynamics, and some examples rely on topology occurring in a parameter-free dynamical dimension.
Abstract
from arXiv · showhide
Inferring network topology from dynamical observations is a fundamental problem pervading research on complex systems. Here, we present a simple, direct method to infer the structural connection topology of a network, given an observation of one collective dynamical trajectory. The general theoretical framework is applicable to arbitrary network dynamical systems described by ordinary differential equations. No interference (external driving) is required and the type of dynamics is not restricted in any way. In particular, the observed dynamics may be arbitrarily complex; stationary, invariant or transient; synchronous or asynchronous and chaotic or periodic. Presupposing a knowledge of the functional form of the dynamical units and of the coupling functions between them, we present an analytical solution to the inverse problem of finding the network topology. Robust reconstruction is achieved in any sufficiently long generic observation of the system. We extend our method to simultaneously reconstruct both the entire network topology and all parameters appearing linear in the system's equations of motion. Reconstruction of network topology and system parameters is viable even in the presence of substantial external noise.
1. Background
The paper addresses inferring hidden network topology from observed unit dynamics, extending beyond methods requiring fixed points, limit cycles, synchronization, or external interventions. It introduces a direct, intervention-free reconstruction method for arbitrary topology from generic collective dynamics.
- Network topology and coupling strengths are often unmeasured even when individual-unit dynamics can be observed.
- Earlier approaches use fixed-point perturbations, multiple experiments, synchronization, or control signals to constrain candidate network structures.
- The proposed method reconstructs arbitrary network topology directly from a single observation of generic collective dynamics without external intervention.
2. Theory of Direct Reconstruction from Dynamical Trajectories
The theory evaluates known dynamical equations along an observed trajectory, converting unknown input couplings into an overdetermined linear reconstruction problem. Solving each unit and dimension separately yields the complete network, with Rössler examples spanning synchronous, asynchronous, periodic, and chaotic states.
- The method is tested on Rössler networks exhibiting synchronous or asynchronous, periodic or chaotic collective dynamics.
- The network consists of units with arbitrary known intrinsic and interaction functions coupled through a directed graph of unknown connectivity.
- Evaluating the equations at multiple times produces linear equations whose unknowns are the coupling strengths for each unit.
- Each dynamical dimension is treated separately because the corresponding equations are uncoupled across dimensions.
- When the number of observations exceeds the number of units, the coupling vector is estimated by minimizing the Euclidean error of the overdetermined system.
- The reconstructed coupling vectors for all units form the complete network, with the minimum-norm solution implementable through standard mathematical software.
3. Performance for Different Collective Dynamics
The method reconstructs networks across periodic, chaotic, synchronous, and asynchronous collective dynamics, with quality improving for longer or more informative observations. It also scales sublinearly with network size and remains viable for simultaneous topology and parameter reconstruction under substantial noise.
- Different collective dynamics: The method reconstructs Rössler network topology across periodic synchronous, chaotic synchronous, periodic asynchronous, and chaotic asynchronous dynamics.The local parameters are unknown but are unnecessary for topology reconstruction in this example because topology parameters occur in equations without other unknown parameters.
- Different collective dynamics: Synchronized systems can still be reconstructed from transient trajectories even when the coupling function is uniformly zero at synchrony.Using whole trajectories in nonsynchronous cases produces lower errors than using only transients toward synchrony.
- Observation quality: Reconstruction quality typically increases with sampling rate and observation time, approaching Q0.95 = 1 at lower sampling rates when trajectories are longer.Figure 3 averages results over 50 networks for observation times T = 1, 5, 10, and 20.
- Observation quality: Irregular collective dynamics can reach high reconstruction quality at lower sampling frequencies because observations at separated times are less correlated and provide more relevant information.Thus, quality depends on temporal information content, not only on the total number of sampled equations.
- Scaling and robustness: The minimum observation time grows sublinearly with network size, with Figure 4 suggesting Tq,α,ω ∝ N^γ and γ ≈ 0.5.For q = 0.98, α = 0.95, and ω = 200, the authors report that observation cost does not grow prohibitively quickly.
- Scaling and robustness: Topology and all local dynamical parameters can be reconstructed for heterogeneous Lorenz oscillators despite substantial additive noise.The example uses noise amplitude η = 0.5, chosen to drastically alter the deterministic dynamics.
4. Conclusion
The paper presents a direct, intervention-free method for reconstructing network topology and coupling strengths from observed dynamics, including deterministic and noisy systems. It applies across diverse dynamical states and provides an explicit analytic inverse solution, while suggesting further efficiency improvements for sparse networks.
- The method reconstructs connection topology and coupling strengths from sufficiently long observations of deterministic and noisy network dynamics.It assumes the functional form of the evolution equations is known.
- It requires no external intervention and is not restricted to fixed points, periodic orbits, synchronous states, or other specific motions.The authors report reconstruction across asynchronous chaotic states and transient states approaching global synchrony.
- The theory provides an explicit analytic solution to the inverse problem of finding network structure from dynamical observations.The solution directly restates the differential equations governing the dynamics.
- The method scales sublinearly with network size and appears robust to substantial added noise.The authors present it as a complement to existing reconstruction methods based on more complex techniques or interventions.
- Future work could use ℓ1 minimization for more efficient reconstruction of sparse networks and select time-series points to reduce the inverse problem.The authors also suggest choosing maximally linearly independent points to obtain an exactly determined system.