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Temporal networks: slowing down diffusion by long lasting interactions
Naoki Masuda, Konstantin Klemm, Víctor M. Eguíluz
TL;DR
Temporal interactions unfold sequentially, raising the question of how they alter diffusive processes relative to permanent aggregate networks. The paper analyzes effective temporal spectra and finds slower diffusion, including for individual sequences, with noncommuting interactions implicated in discrepancies.
Problem
The paper asks how sequential interactions change diffusive dynamics relative to permanent aggregate interactions, which matter for processes including synchronization and random walks.
Source: p. 1Method
The authors analyze effective matrices for piecewise-constant interaction sequences and compare their spectra with aggregate-network matrices.
Source: pp. 1–2Results
At least 72% of the original slow/fast eigenvalue ratio is retained, with the bound attained by stars, while temporal diffusive dynamics is slower than aggregate dynamics.
Source: pp. 1–2Takeaways & Limitations
Noncommuting interactions are implicated in discrepancies between temporal and aggregate dynamics and between different temporal sequences sharing an aggregate network.
Source: p. 4Individual-sequence eigenvalues can vary substantially, and the relationship between correlated interaction patterns and individual-sequence behavior remains for future work.
Source: p. 4
Abstract
from arXiv · showhide
Interactions among units in complex systems occur in a specific sequential order thus affecting the flow of information, the propagation of diseases, and general dynamical processes. We investigate the Laplacian spectrum of temporal networks and compare it with that of the corresponding aggregate network. First, we show that the spectrum of the ensemble average of a temporal network has identical eigenmodes but smaller eigenvalues than the aggregate networks. In large networks without edge condensation, the expected temporal dynamics is a time-rescaled version of the aggregate dynamics. Even for single sequential realizations, diffusive dynamics is slower in temporal networks. These discrepancies are due to the noncommutability of interactions. We illustrate our analytical findings using a simple temporal motif, larger network models and real temporal networks.
Supplemental Material for Temporal networks: slowing down diffusion by long lasting interactions
The supplement extends the spectral framework to nonlinear dynamics on temporal networks and tests whether effective-coupling eigenvalues support master-stability predictions. In the illustrated star networks, temporal coupling shifts the synchrony threshold, while stability loss at strong coupling is governed by the largest-magnitude eigenvalue.
Supplemental Material
- The supplement extends the framework to identical-node reaction-diffusion systems, studying chaotic Rössler oscillators with three coupled degrees of freedom and time-dependent diffusive coupling.The coupling strength σ is tunable.Source: Supplemental Material for Temporal… · p. 6
- Effective-coupling eigenvalues provide the master-stability input; σ enters the temporal evolution operator through τσ, and aggregate eigenvalues map to effective eigenvalues.The supplement states that this eigenvalue-based input is correct for the master-stability approach.Source: Supplemental Material for Temporal… · p. 6
- For an N=3 star, stable-synchrony onset shifts from σ=3.0 at τ=0 to σ≈3.9 at τ=0.2, and the effective eigenvalue approximately captures the stability threshold.Figure S1 replots the transition against the effective eigenvalue; increasing τ also smooths the transition rather than collapsing the full stability curves.Source: Supplemental Material for Temporal… · p. 6, Figure S1 · p. 7
- For Rössler stability, too-weak and too-strong coupling can both destabilize synchrony; at the strong-coupling boundary, the largest-magnitude eigenvalue matters rather than the spectral gap.The supplement identifies the largest-magnitude eigenvalue as the relevant quantity for loss of stability at excessive coupling.Source: Supplemental Material for Temporal… · pp. 6–7