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The fundamental advantages of temporal networks

Aming Li, Sean P. Cornelius, Yang-Yu Liu, Long Wang, Albert-László Barabási

arXiv:1607.06168v1nlin.AO

TL;DR

Network temporality can disrupt paths and hinder network dynamics, raising the question of whether it offers control advantages. The paper develops an analytical framework and finds that temporal networks reach controllability faster and use far less control energy than static counterparts. These results identify temporality as a source of control flexibility rather than only an impediment.

  • Problem

    The paper asks whether temporal networks, whose intermittent links can disrupt paths and hinder dynamics, have control advantages over static counterparts.

  • Method

    The paper develops an analytical framework for analyzing controllability, control energy, and trajectories in temporal networks and their static counterparts.

  • Results

    Temporal networks reach controllability faster, require orders of magnitude less control energy, and have more compact control trajectories than static counterparts.

  • Takeaways & Limitations

    Temporality can provide control flexibility that is unattainable in static networks, making temporality an asset rather than an obstacle to control.

Abstract

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Despite the traditional focus of network science on static networks, most networked systems of scientific interest are characterized by temporal links. By disrupting the paths, link temporality has been shown to frustrate many dynamical processes on networks, from information spreading to accessibility. Considering the ubiquity of temporal networks in nature, we must ask: Are there any advantages of the networks' temporality? Here we develop an analytical framework to explore the control properties of temporal networks, arriving at the counterintuitive conclusion that temporal networks, compared to their static (i.e. aggregated) counterparts, reach controllability faster, demand orders of magnitude less control energy, and the control trajectories, through which the system reaches its final states, are significantly more compact than those characterizing their static counterparts. The combination of analytical, numerical and empirical results demonstrates that temporality ensures a degree of flexibility that would be unattainable in static networks, significantly enhancing our ability to control them.

I. INTRODUCTION

Network science has traditionally emphasized static networks, although many natural and social systems are better represented by intermittently existing temporal links. Because temporality disrupts signal-carrying paths and slows or impedes network processes, it may also limit controllability.

  • I. INTRODUCTION: Static networks provide permanent connections, whereas temporal networks contain links that exist only intermittently.Examples include brief chemical reactions, short-duration social interactions, and other time-varying links.
  • I. INTRODUCTION: Temporality has been associated with slower synchronization and information diffusion, impeded exploration and navigation, and greater accessibility barriers.
  • I. INTRODUCTION: Controllability means driving a system with input signals to any desired final state in finite time.
  • I. INTRODUCTION: Controllability typically requires continuous paths carrying input signals from driver nodes to the rest of the network.Static networks offer such paths permanently, while temporal networks do not guarantee complete instantaneous paths.

II. DYNAMICS ON TEMPORAL NETWORKS

The paper models a temporal network as an ordered sequence of network snapshots with changing weighted topologies and linear dynamics within each snapshot. Driver nodes and snapshot timing are constrained so that topology changes, rather than chosen switching, determine the temporal effects.

  • II. DYNAMICS ON TEMPORAL NETWORKS: A temporal network is an ordered sequence of M separate networks on the same N nodes.
  • II. DYNAMICS ON TEMPORAL NETWORKS: Each snapshot m has a weighted adjacency matrix A_m and lasts for duration τ_m.
  • II. DYNAMICS ON TEMPORAL NETWORKS: Within each snapshot, the system follows canonical linear time-invariant dynamics over that snapshot’s time interval.
  • II. DYNAMICS ON TEMPORAL NETWORKS: The state x_i(t) represents node i, while B_m identifies driver nodes controlled through p independent inputs u_m(t).The framework uses state variables such as metabolite concentrations as an example.
  • II. DYNAMICS ON TEMPORAL NETWORKS: The same driver-node set is used across snapshots, and the order and timing of topology changes are not controllable.Influence over the system is therefore confined to the control inputs.
  • II. DYNAMICS ON TEMPORAL NETWORKS: Although each snapshot has linear dynamics, switching among snapshots makes the overall dynamics nonlinear and places the system within switched-system dynamics.Such systems can exhibit multiple limit cycles, chaos, and Zeno-like behavior.

III. TIME TO CONTROL

The controllable space of a temporal network can grow as its topology changes, allowing temporal sequences to become fully controllable in fewer snapshots than aggregated static networks. Across empirical networks and aggregation windows, this temporal advantage increases with higher temporality.

  • III. TIME TO CONTROL: A temporal network is controllable when the controllable spaces contributed by its snapshots span the full state space R^N.For identical snapshots, the condition reduces to the classic Kalman rank condition.
  • III. TIME TO CONTROL: Changing network structure can expand the controllable space, enabling calculation of the snapshot count S_t needed for full controllability.The corresponding static comparison uses S_s, the number of snapshots aggregated to obtain a controllable static network.
  • III. TIME TO CONTROL: S_t = 2 snapshots suffices for the illustrated temporal sequence, whereas S_s = 3 aggregated snapshots are required for its static counterpart.
  • III. TIME TO CONTROL: Across aggregation windows, temporal networks require fewer snapshots for control than static counterparts, with the advantage increasing as Δt decreases.Smaller Δt produces sparser, more disjoint snapshots and represents higher temporality.
  • III. TIME TO CONTROL: 71 time steps (19.72 hours) are required for face-to-face interactions at Δt = 10^3, compared with 185 aggregated snapshots over two days for a controllable static network.
  • III. TIME TO CONTROL: Randomized empirical networks also satisfy S_t < S_s, indicating that the observed control advantage does not depend on the specific temporal ordering or overall network structure.The authors conclude that temporality alone is sufficient to improve controllability.

IV. CONTROL ENERGY

Temporal networks require less control energy than corresponding static networks because changing topology can avoid energetically expensive directions in state space. The paper formulates minimum-energy control through an effective temporal-network energy structure and compares it with static networks.

  • IV. CONTROL ENERGY: The paper develops a control formalism for the minimum input energy required to drive a system from x_0 to x_f.The energy is expressed using the difference between the desired final state and the uncontrolled natural final state.
  • IV. CONTROL ENERGY: The vector d captures the desired-state displacement, while the N × N matrix W_eff encodes the temporal network’s energy structure.For identical snapshots, W_eff reduces to the controllability gramian W, yielding the static-network control energy.
  • IV. CONTROL ENERGY: Temporal networks have average control energies many orders of magnitude smaller than corresponding static networks.
  • IV. CONTROL ENERGY: 130 orders of magnitude separate static and temporal control energy at Δt = 10^-6.
  • IV. CONTROL ENERGY: 274 orders of magnitude are saved when the technological network’s true temporal structure is accounted for.
  • IV. CONTROL ENERGY: Temporal topology can avoid expensive state-space directions by applying control when the network dynamics make the energy cost acceptable.Static networks lack this option when the required direction is energetically costly.

V. LOCALITY OF THE CONTROL TRAJECTORIES

The paper evaluates locality by measuring control-trajectory length between initial and final states. Temporal networks produce substantially shorter trajectories than static counterparts, while length still grows linearly with state-space distance.

  • Trajectory locality: Control-trajectory length L measures the path traveled through state space from x0 to xf.The length depends on the initial state and destination state.
  • Trajectory locality: For x0 = 0, both static and temporal networks show L increasing linearly with δ = ∥xf − x0∥.
  • Temporal versus static networks: About five orders of magnitude shorter trajectories characterize temporal networks than their static counterparts for any δ.
  • Temporal versus static networks: 29 orders of magnitude separate the temporal and static control trajectories for the technological network.
  • Interpretation: Temporality allows the state x(t) to move from x0 to xf without wandering far into phase space.Static networks can require highly non-local trajectories, whereas temporal dynamics offer greater locality.

VI. DISCUSSION

The discussion contrasts practical control difficulties in static networks with results showing that temporal structure improves control time, energy, and trajectory compactness. The authors interpret temporality as a source of dynamical flexibility and a control asset.

  • Control challenges: Static networks can be difficult to control in practice because heterogeneous degree distributions require a high fraction of nodes to be directly controlled.
  • Control challenges: Static control faces a tradeoff between driver-node number and control time, while available trajectories may also be non-local.
  • Motivation: Temporal networks are presented as a promising solution to the need for natural and technological systems to reach and maintain desired states.
  • Main findings: Temporal networks show simultaneous reductions in time to control, control energy, and trajectory lengths, with effects exceeding many orders of magnitude.
  • Interpretation: Temporality exploits favorable dynamics across subsnapshots, providing a flexibility described as the “best of all worlds.”
  • Interpretation: The authors find that nonlinearity induced by temporality can be an asset rather than an obstacle to control.
  • Static limit: When all snapshots are identical, the temporal formulation reduces to the static case and the classic Kalman rank condition.
  • Controllable spaces: The relationship between temporal and static controllable spaces is not determinate: either space may strictly contain the other.

Appendix B: Description of empirical data sets

The appendix describes empirical temporal-network datasets spanning human contacts, student contacts, ant interactions, protein interactions, and technological packet exchanges. Networks are represented through snapshots whose number depends on the chosen time window.

  • Dataset types: The empirical collection includes four data types: human, animal, protein, and technological networks.
  • Human contact data: The ACM Hypertext 2009 dataset records face-to-face interactions among 113 conference attendees over about 2.5 days.
  • Human contact data: Student-contact data records contacts among 126 students in three classes over four days in December 2011.
  • Animal interaction data: The ant dataset contains 1,911 antenna-body interactions among 89 ants over 1,438 seconds.
  • Protein data: The protein dataset uses Saccharomyces cerevisiae interactions and constructs dynamic networks from gene-expression activity across cellular, molecular, and biological domains.
  • Technological data: The technological datasets contain packet exchanges among 25 wireless radios in mobile ad-hoc networks experiencing denial-of-service attacks.Each dataset spans 900 seconds, with the attack beginning after approximately 300 seconds.
  • Snapshot construction: Snapshot duration ∆t is varied across datasets, producing different numbers of snapshots; Table I reports N and maximum M.Other network attributes are provided in supplementary figures.

Appendix C: Data randomization and null models

The appendix represents temporal data as contact sequences and tests four null models that separately reverse time, permute timestamps, randomize edges, or combine edge and time randomization.

  • Contact sequences represent interactions as triplets containing time and the interacting nodes.
  • Time Reversal reverses the temporal order of contacts and can alter interaction durations, especially in high-resolution data.
  • Randomly Permuted Times shuffles contact timestamps while preserving link sources and targets, thereby destroying temporal patterns and time correlations.
  • Randomized Edges rewires edge pairs while avoiding self-loops, preserving whole-network interaction durations but often changing node-level contact counts and durations.
  • The four null models are applied to empirical temporal networks, and their effects on aggregated-network degree and component counts are reported in supplementary figures.

Appendix D: Relationship between St and Ss

The appendix defines static networks as duration-weighted aggregations of temporal snapshots and compares the snapshot counts required for temporal and static controllability. It also provides analytical control-energy formulations that recover the static case when snapshots are identical.

  • The associated static network is formed by taking the element-wise average of snapshot adjacency matrices weighted by their durations.
  • The mechanism illustrated in Fig. 2A is described as the rule rather than the exception, with real temporal networks almost always more controllable than static equivalents.
  • St and Ss count snapshots needed for temporal and static controllability, with values of M indicating failure even after the final or aggregated snapshot.
  • The minimum-energy control problem is expressed as a quadratic program and solved analytically through the effective Gramian Weff = SWST.
  • When all snapshots are identical, the temporal-network energy expression reduces to the corresponding static-network result.

Appendix F: Analysis of the control energy

The appendix analyzes how the effective Gramian’s spectrum determines control energy and checks the result using synthetic temporal networks built from independently randomized connected snapshots with stabilizing self-loops.

  • The worst-case control direction aligns with the smallest effective-Gramian eigenvalue, whose scale is expected to represent average control energy.
  • The minimum effective-Gramian eigenvalue is generally greater in temporal networks, so their control energy is usually smaller, often much smaller, than in static equivalents.
  • Average control energy is typically much less in temporal networks than in static equivalents, despite potentially differing worst-case directions.
  • ⟨E⟩ decreases as ⟨E⟩ ∼ ∆t^-γ for small ∆t before reaching a plateau.
  • Self-loops with weight am stabilize each snapshot’s standalone dynamics, while the theory also works for unstable dynamics.

Appendix H: Locality of the optimal control trajectories for temporal networks

The appendix evaluates the locality of optimal control trajectories across low-dimensional examples and network settings. Temporal trajectories remain shorter than static trajectories across tested initial states and control distances, with a large technological-network contrast.

  • The analysis visualizes control trajectories for temporal and static systems in two to three dimensions to assess control non-locality.
  • For fixed control distance δ = 10^-3, nonzero initial states can make static trajectory length depend strongly on excursions through other orthants of phase space.
  • Additional examples with two and five snapshots, multiple final states, and three-dimensional systems preserve the reported trajectory-length comparison.
  • Temporal-network trajectory lengths are always less than static-network lengths, independently of the initial state and control distance.
  • The discrete numerical approximation uses tstep = 0.025, and its limitation vanishes as tstep → 0.
  • For the technological network, the maximum state component is used because it dominates the corresponding trajectory length.
  • 10^35 versus 10^64: the reported Li* orders for temporal and static technological networks support more local temporal trajectories.

Appendix I: Supplementary Figures

The supplementary figures test whether temporal networks’ controllability advantages persist across aggregation windows, interaction durations, driver-node counts, network types, and trajectory settings. They also show that temporal and static controllability can differ in either direction theoretically, while temporal control often requires less energy and shorter trajectories.

  • Controllability comparisons: No theoretically determinate relation exists between temporal and static controllability: either network can be controllable while the other is not.The comparison assumes one input corresponds to exactly one driver node.
  • Robustness of faster control: Temporal networks reach controllability faster across a wide range of aggregation windows in conference, ant-interaction, and student-contact networks.The supplementary analysis compares the required snapshot counts St and Ss as ∆t varies.
  • Robustness of faster control: Temporal networks still reach controllability faster when ant contacts are assigned finite 20-second durations and time is rescaled by a factor of 60.For ant interactions, St is not bigger than Ss under this duration-sensitive construction.
  • Robustness of faster control: The faster-controllability result persists when the driver-node set represents 80% of network size, including cases where static networks remain uncontrollable under 20% control.This tests whether the comparison depends on the number of driver nodes.
  • Control energy: The minimum eigenvalue of Weff dominates average control energy because the eigenvalues vary over many orders of magnitude; temporal networks have a much larger λmin.The worst-case direction therefore largely determines the average energy comparison.
  • Control energy: Temporal networks require less control energy than static networks across additional synthetic parameter settings and in a real 1-ip6 network.In the real network, energy decreases as either the number of driver nodes or ∆t increases.
  • Control trajectories: Supplementary trajectory examples show more localized temporal control paths in two- and three-dimensional systems, including cases with small control distances.The trajectory studies compare temporal networks with static counterparts for two and five snapshots.
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