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Toward a decentralized design framework for discrete-time distributed algorithms via their emergent dynamics

Jinwook Heo, Hyungbo Shim, Jin Gyu Lee

arXiv:2610.29656v1eess.SYcs.SY

TL;DR

Distributed algorithms for heterogeneous agents need a way to design local updates for desired collective behavior, while the earlier multi-step approach is vulnerable to asynchronous perturbations. This paper designs simultaneous explicit or implicit updates to realize prescribed blended dynamics, with approximation guarantees and a finite-iteration trade-off under stated conditions.

  • Problem

    For heterogeneous agents, designing local updates for desired collective behavior is challenging, and the earlier multi-step method is vulnerable to asynchronous perturbations.

    Source: §1 · p. 1

  • Method

    The paper derives conditions for designing local parameters of simultaneous explicit or implicit updates to realize prescribed blended dynamics.

    Source: §1 · p. 2, §2.1 · p. 4

  • Results

    Under a stability condition on the blended dynamics, agent states eventually remain close to its trajectory over an infinite horizon, and the guaranteed iteration count can be minimized at finite k under stated conditions.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The framework is illustrated through four applications, and the paper reports that it can support initialization-free operation without requiring stable individual node dynamics.

    Source: §1 · p. 2, §6 · p. 11

    The weighted least-squares analysis does not cover rank-deficient A, where solutions and the stated equilibrium need not be unique.

    Source: §4.3 · p. 10

Abstract

from arXiv · show

This paper develops a procedure for designing discrete-time distributed algorithms to achieve a desired collective behavior. Each agent combines its node and coupling updates at every iteration, in either explicit or implicit form. For this structure, we derive necessary and sufficient conditions on the local parameters for realizing a desired continuous-time blended dynamics. These conditions yield a decentralized procedure for computing the parameters. A Tikhonov-type approximation result relates the resulting agent trajectories to the blended dynamics on a rescaled time axis. Under a stability condition imposed only on the blended dynamics, we further establish uniform-in-time approximation bounds. We also characterize the trade-off between approximation accuracy and the iteration count guaranteed by these bounds. The framework is illustrated through four applications in distributed computation and estimation.

1 Introduction

The paper replaces synchronized multi-step updates with simultaneous node and coupling updates, then designs their local parameters to produce prescribed blended dynamics. It establishes approximate synchronization and analyzes how approximation accuracy trades off against the iteration count guaranteed by its bounds.

1 Introduction

  • The proposed structure performs node and coupling updates simultaneously at every iteration, in explicit or implicit form.The forms differ in whether the node function uses xi[n] or xi[n + 1]; 1/k spreads the node-update effect across iterations.Source: §1 · pp. 1–2
  • For sufficiently large finite k, each agent’s state eventually remains close to the blended-dynamics trajectory over an infinite horizon, under a stability condition on the blended dynamics.The analysis first identifies a synchronized trajectory with individually scaled states as k →∞.Source: §1 · p. 2
  • Necessary and sufficient conditions characterize when local parameters realize prescribed blended dynamics, enabling a decentralized procedure to compute them.The target uses desired weights and scaling factors.Source: §1 · p. 2
  • The guaranteed iteration count can be minimized at a finite k under stated conditions, reflecting a trade-off between tighter approximation bounds and weaker per-iteration node updates.Increasing k reduces the bounds, but the 1/k factor may increase the iterations needed for prescribed accuracy.Source: §1 · p. 2
  • Unlike prior discrete-time synchronization studies focused mostly on linear models with structural assumptions, this framework considers nonlinear node dynamics without requiring stable individual nodes or specific initial conditions.The initialization-free property supports plug-and-play operation when agents join or leave the network.Source: §1 · p. 2
  • The framework is illustrated in four applications: network-size estimation, reciprocal left-eigenvector estimation, weighted least squares, and PageRank estimation.In the asynchronous PageRank simulation, estimates remain close to desired scores, whereas those from the multi-step method do not.Source: §1 · p. 2

2 Analysis of Emergent Collective Behaviors

For the specified distributed updates, the agents’ large-k behavior is described by continuous-time blended dynamics on a rescaled time axis. If those blended dynamics are contractive, the approximation to each agent’s individually scaled trajectory is uniform in time.

2 Analysis of Emergent Collective Behaviors

  • The analysis assumes a strongly connected graph, nonnegative b_i, positive c_i, at least one positive b_i, and a parameter matrix W with spectral radius one.Together, the graph and parameter conditions make W primitive.Source: §2.1 · p. 3
  • The continuous-time blended dynamics is the k-independent reduced system that describes collective behavior as the sampling time shrinks.The explicit and implicit reduced updates resemble Euler discretizations with sampling time tending to zero as k grows.Source: §2.1 · p. 4
  • For large k, agents approach individually scaled synchronization, and their shared trajectory follows the continuous-time blended dynamics on a rescaled time axis.The paper interprets this as a discrete-time Tikhonov-type approximation, with 1/k as the small perturbation parameter.Source: §2.1 · p. 4

2.2 Uniform convergence under stability of (7)

  • The uniform-in-time guarantee requires stability only of the blended dynamics, not of every individual agent’s dynamics.The paper defines this stability through a contraction condition on the blended dynamics.Source: §2.2 · p. 5
  • When the blended dynamics are contractive, the approximation error remains bounded uniformly over time and can be made arbitrarily small by increasing k.The sufficient thresholds are k > k̄ + k* for explicit updates and k > k* for implicit updates.Source: §2.2 · p. 5

3 Decentralized Design of the Algorithm

The design section specifies a desired continuous-time blended system and derives decentralized local parameters that realize it. It then gives special-case updates and analyzes how k trades approximation accuracy against the iteration count needed to maintain that accuracy.

3.1 Decentralized computation of ai, bi, ci

  • Necessary and sufficient conditions characterize when local parameters realize the prescribed blended dynamics, and a decentralized procedure computes suitable parameters using local information.The design assigns desired weights and scaling factors, then uses local quantities and neighbor information to implement the update.Source: §3.1 · p. 6

3.2 Special cases

  • The three special cases are summarized as design-parameter choices for the general decentralized design.The table’s caption identifies its scope as parameter choices in the special cases.Source: §3.2 · p. 6, Table 1 · p. 7

3.2.1 Averaged blended dynamics over undirected graphs

  • For undirected connected graphs, the averaged-dynamics update needs only each agent’s local degree to select its parameter.The weights, scaling factors, and normalized left-eigenvector entries cancel from this special-case update.Source: §3.2.1 · p. 7

3.2.2 Weighted blended dynamics over strongly connected digraphs

  • For strongly connected digraphs, the weighted-dynamics update compensates for graph asymmetry using the reciprocal of the normalized left-eigenvector entry.The paper relates obtaining that quantity to a distributed estimator.Source: §3.2.2 · p. 7

3.2.3 Scaled blended dynamics over strongly connected digraphs

  • For scaled PageRank dynamics on strongly connected digraphs, agents need only their local out-degree because the desired scaling cancels from the implemented update.The resulting scaled PageRank score can be locally rescaled to recover the PageRank score.Source: §3.2.3 · pp. 7–8

3.3 Optimization of the transient behavior

  • The finite-iteration error bound combines synchronization error with discretization error and has transient terms that depend on initial conditions.The worst-case bound ranges over prescribed initial sets and optimizes over the auxiliary dynamics’ initial value.Source: §3.3 · p. 8
  • The asymptotic error bound decreases continuously with k, defining a unique threshold k_ϵ above which it is below the target ϵ.The finite-iteration bound need not decrease with k at a fixed iteration.Source: §3.3 · p. 8
  • Under the stated conditions, the guaranteed iteration count diverges as k approaches either endpoint, so it attains a minimum at a finite interior k.The paper notes that this analysis uses a possibly conservative bound and that an example shows the same tendency.Source: §3.3 · p. 8

4 Applications

The applications use designed blended dynamics to build decentralized estimators and solvers from local updates. They cover network size and Laplacian left-eigenvector information, weighted least squares, and PageRank.

4.1 Network size estimation

  • For network size, agents eventually stay close enough to N to recover it exactly by rounding, once the approximation error is below 0.5.The sufficient iteration count N0.5(k) is minimized at a finite k > 2.Source: §4.1 · p. 9
  • Agents can estimate 1/ζi for any k > 0 without a global lower bound on k, and the construction can also estimate the left eigenvector ζ without specific initialization.The reciprocal estimator uses the local algorithm while canceling the unknown ζi; the alternative construction has ζ as its unique equilibrium.Source: §4.2 · pp. 9–10

4.3 Weighted least-squares solver

  • With exact 1/ζi values, the distributed WLS update eventually stays close to the unique solution when k is sufficiently large, for a strongly connected digraph.The analyzed case assumes A has full column rank; rank-deficient problems are not covered.Source: §4.3 · p. 10

4.4 PageRank estimation

  • For PageRank, the scaled state approaches ri/dout_i, so rescaling it gives an estimate of ri; the update combines node and coupling updates at every iteration.The construction uses a strongly connected digraph and local transmissions to out-neighbors.Source: §4.4 · p. 10

5 Simulations

The simulations illustrate online estimation alongside WLS and compare explicit and implicit WLS updates, then test PageRank estimation under asynchronous node schedules. The implicit WLS update remains convergent across the tested k values, while the proposed PageRank estimates stay close to scores when the multi-step method does not.

5 Simulations

  • The online 1/ζi estimates converge while the WLS solution error decreases in the concurrent implementation.The simulation uses kζ = 1 for the estimator and k = 100 for WLS.Source: §5.1 · p. 11
  • The implicit WLS update remains convergent at k = 20, 64, and 100, while the explicit update diverges at k = 20, oscillates with growing amplitude at k = 64, and approaches implicit behavior as k increases.Fig. 4 compares the first components of xi[n]−x∗ under the explicit and implicit updates at those three k values.Source: §5.1 · p. 11, Figure 4 · p. 11

5.2 PageRank algorithm in an asynchronous setting

  • With misaligned node-update schedules, the proposed PageRank estimates eventually stay close to the scores, whereas the multi-step estimates do not.Fig. 5 plots the proposed estimates as solid lines and PageRank scores as dashed lines.Source: §5.2 · p. 11

6 Conclusion

The paper presents a decentralized framework for designing discrete-time distributed algorithms through emergent blended dynamics. It establishes approximation bounds and identifies an accuracy–iteration-count trade-off, then outlines extensions for future work.

6 Conclusion

  • The framework characterizes collective behavior on a rescaled time axis, establishes approximation bounds, and realizes prescribed blended dynamics through decentralized parameter choices.Source: §6 · p. 11
  • The paper identifies a trade-off between approximation accuracy and iteration count.Source: §6 · p. 12
  • Future work will consider nonautonomous settings, broader stability conditions and coupling structures, and decentralized or adaptive tuning of k.Source: §6 · p. 12

Generative AI disclosure

The authors report using ChatGPT for language editing and manuscript organization, while retaining responsibility for the manuscript. The appendix begins its proof analysis with synchronization error estimates.

Generative AI disclosure

  • ChatGPT assisted with language editing and manuscript organization; the authors reviewed the content and remain responsible for it.Source: Generative AI disclosure · p. 12

A.1 Synchronization error estimates

  • The appendix derives synchronization error estimates used in the proofs of Theorems 1 and 2, beginning with a coordinate decomposition.Source: §A.1 · p. 12
  • The appendix defines V[n] and W[n] as norms measuring the blended-state deviation and transformed disagreement, respectively.Source: §A.1 · p. 12

A.2 Proof of Theorem 1

The proof establishes a finite-horizon bound on the auxiliary state, then uses it to show the error terms vanish as the iteration parameter grows. This yields the approximation claim in Theorem 1.

A.2 Proof of Theorem 1

  • For every fixed T, the auxiliary state s[n,k] stays uniformly bounded through nT,k for sufficiently large k under either update.The bound follows because nT,k/k approaches T/τs and the resulting exponential factor remains bounded.Source: §A.2 · p. 14
  • The proof bounds the forcing term in the error recursion uniformly over the finite horizon using the state bound and (A.5).This supplies a constant BT independent of n and sufficiently large k.Source: §A.2 · p. 14
  • The finite-horizon error component VT is O(1/k), so VT tends to zero as k grows.The bound follows by iteration and taking the maximum over 0 ≤ n ≤ nT,k.Source: §A.2 · p. 14
  • As nT,k grows and the contraction factor remains below one, W[nT,k] also tends to zero, completing the proof of the approximation claim.The argument combines this decay with (A.2) and Lemma 1.Source: §A.2 · p. 14

A.3 Proof of Theorem 2

The proof bounds the synchronization state, establishes stability of the error-recursion matrices, and uses these estimates to obtain approximation bounds. It also derives a uniform-in-time bound as k grows.

A.3 Proof of Theorem 2

  • For each admissible k, the proof bounds the synchronization state Hs[n,k] under either update.The bound is given by Lemma A.2 under Assumption 4.Source: §A.3 · p. 14
  • The error-recursion matrices are shown to be Schur stable in the explicit and implicit cases over their respective parameter ranges.The proof uses bounds on the network contraction and coupling terms to verify stability.Source: §A.3 · p. 15
  • A weighted error measure decays geometrically up to a transient term, yielding the synchronization bound used in Theorem 2.The proof constructs a positive weight from the stable matrix and iterates the resulting inequality.Source: §A.3 · p. 16
  • Combining the synchronization bound with Lemma 2 and the triangle inequality gives the approximation bounds (11a) and (11b).The same estimates yield a finite-iteration error bound for each agent relative to p_i s(tn,k).Source: §A.3 · p. 16
  • For sufficiently large k, the proof obtains constants independent of n and k for the uniform-in-time approximation error.The argument uses the rescaled-time rounding estimate and the O(k^-1) bound on the limiting error.Source: §A.3 · p. 16

A.4 Proof of Theorem 3

The proof shows that the guaranteed iteration count is finite above the accuracy threshold, but diverges both as k grows without bound and as k approaches the threshold from above.

A.4 Proof of Theorem 3

  • For k above kε, the limiting error is below ε, so the guaranteed iteration count Nε(k) is finite.The finite-iteration error converges to its limiting bound as n grows.Source: §A.4 · p. 16
  • Under Theorem 3's first condition, Nε(k) tends to infinity as k grows without bound.For each fixed n, the error eventually exceeds ε, so the guarantee requires more than n iterations.Source: §A.4 · p. 17
  • Under the second condition of Theorem 3, Nε(k) also tends to infinity as k decreases to kε from above.The transient error remains positive for each fixed n near kε.Source: §A.4 · p. 17

B.1 Proof of Proposition 3

The proof represents each component of the state through a nonnegative linear recursion and establishes convergence to a positive equilibrium.

B.1 Proof of Proposition 3

  • Each component χ(ℓ)[n] follows a linear recursion with a nonnegative, irreducible state matrix whose row sums are at most one and whose ℓ-th row sum is smaller than one.These properties imply ρ(Aℓ) < 1.Source: §B.1 · p. 17
  • Because Aℓ is nonnegative and irreducible, with row sums at most one and the ℓ-th strictly smaller, ρ(Aℓ) < 1; hence χ(ℓ)[n] converges to χ(ℓ)* = (Eℓ + kL)^-1 1N.Source: §B.1 · p. 17
  • Taking ℓ = i establishes convergence for each component, and the nonnegative state matrix and positive input ensure positivity.Source: §B.1 · p. 17

B.2 Derivation for Example 1

Example 1’s derivation shows how the iteration count controls when rounding recovers the target value; the accompanying proof establishes approximation of the continuous-time trajectory for both update forms.

B.2 Derivation for Example 1

  • For N = 2 and δ_1 = δ_2 = 0.5, the iteration matrix has eigenvalues 0 and (4k+1)/(4k+2).For k > 2, the stated initial bounds apply to both agents’ deviations from the equilibrium.Source: §B.2 · p. 17
  • For n ≥ N_0.5(k), nearest-integer rounding returns 2 in Example 1.The passage states that the error bounds are both below 0.5 when n > Γ_0.5(k).Source: §B.2 · p. 17

C.1 Proof of Lemma 1

  • For either explicit or implicit updates, the agent state approaches the continuous-time trajectory as k grows on the rescaled time axis.The proof uses a finite-time argument and continuity of the trajectory.Source: §C.1 · pp. 17–18

C.2 Proof of Lemma 2

The proof material establishes approximation bounds and analyzes how the spectral radius of W governs the long-run behavior of the updates. Below one, both updates converge to a shared equilibrium; above one, trajectories can be unbounded.

C.2 Proof of Lemma 2

  • Under Assumption 4, Lemma C.1 considers discrete and continuous trajectories from arbitrary initial values and establishes a finite-iteration bound.The proof derives the bound using the error terms and constants defined in the preceding argument.Source: §C.2 · p. 18
  • The Lemma 2 argument takes the finite-iteration bounds to the long-time regime, with the discrete-time sampling offset bounded by τ_s/k.The initial-condition terms and G_n terms vanish as T tends to infinity.Source: §C.2 · p. 18
  • When λ = ρ(W) < 1, both updates converge globally exponentially to the same unique equilibrium for sufficiently large k, and that equilibrium tends to zero as k grows.The contraction argument gives this result for both explicit and implicit updates.Source: D The Cases ρ(W)̸ = 1 · pp. 18–19
  • For λ > 1, for every sufficiently large k, each update has an unbounded trajectory; when F(0) = 0, such trajectories can start arbitrarily near the origin.The region C is forward invariant, and every trajectory starting there with u[0] ≠ 0 has ∥u[n]∥ growing geometrically.Source: D The Cases ρ(W)̸ = 1 · pp. 18–19
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