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arxiv: 2604.16069 · v1 · submitted 2026-04-17 · 📡 eess.SP · cs.SY· eess.SY

Convergence Time Distributions for Max-Consensus over Unreliable Networks

Katharina Stich , Bastian Perner , Friedemann Laue , Torsten Reissland , Norman Franchi This is my paper

Pith reviewed 2026-05-10 07:38 UTC · model grok-4.3

classification 📡 eess.SP cs.SYeess.SY
keywords max-consensusconvergence timeunreliable networkslink failuresprobability distributionmulti-agent systemsdeadline-aware design
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The pith

LiFE-CD computes the full probability distribution of convergence times for max-consensus over networks with random link failures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents LiFE-CD as a deterministic method that derives the complete probability distribution of how long max-consensus takes when links fail independently with known probabilities. Earlier work either assumed flawless communication or supplied only expected values and asymptotic bounds. By starting from the network topology and failure rates, the algorithm produces exact distributions for tree-structured networks and tight upper bounds for networks with cycles. This output lets protocol designers select deadlines that achieve explicit reliability targets rather than relying on averages or repeated random trials. The same construction applies to min-consensus through structural symmetry.

Core claim

LiFE-CD iteratively reduces the given network topology while accounting for both unicast and broadcast transmissions under geometrically distributed link delays. For acyclic networks the reduction yields the exact convergence-time distribution; for cyclic networks it produces tight upper bounds by replacing the graph with a shortest-path spanning tree. The resulting distribution supports deadline-aware protocol design that meets specified reliability guarantees, and the approach eliminates the variability and cost of Monte Carlo simulation.

What carries the argument

The LiFE-CD algorithm, which reduces network topology step by step using link failure probabilities and transmission modes to obtain the convergence-time distribution.

If this is right

  • Designers can compute the probability that consensus finishes by any chosen deadline directly from topology and failure rates.
  • Computational effort is deterministic and lower than Monte Carlo simulation while removing run-to-run variability.
  • The same reduction rules and bounds apply without change to min-consensus algorithms.
  • Numerical validation confirms exactness on acyclic graphs and tightness on cyclic graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction technique could be adapted to predict convergence statistics for other distributed coordination tasks that rely on repeated local exchanges.
  • In wireless sensor deployments the ability to set precise timeouts with known success probability may reduce unnecessary retransmissions and energy use.
  • For cyclic networks, comparing the LiFE-CD bound against full-state simulation on modest-sized graphs would quantify how often the bound is achieved.

Load-bearing premise

The method assumes geometrically distributed link delays and that shortest-path spanning trees produce tight upper bounds on convergence time for networks containing cycles.

What would settle it

Compare the exact distribution produced by LiFE-CD on a small acyclic network against the empirical distribution obtained from repeated direct simulations of max-consensus with the same failure probabilities; any systematic mismatch would falsify the exactness claim.

Figures

Figures reproduced from arXiv: 2604.16069 by Bastian Perner, Friedemann Laue, Katharina Stich, Norman Franchi, Torsten Reissland.

Figure 1
Figure 1. Figure 1: Transmission scenarios with node 1 as the source node. (a) Unicast. (b) Broadcast. how the composed delay distributions can be characterized using shortest-path spanning trees. The following subsections address each aspect, forming the building blocks of LiFE-CD. 1) Unicast vs. Broadcast Transmission: The bound in (16) assumes propagation as a sequential transmission chain of length D, assigning each hop t… view at source ↗
Figure 2
Figure 2. Figure 2: Network topologies for LiFE-CD algorithm analysis. (a) Cyclic network with two possible shortest-path trees rooted at node [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: PMF of the consensus convergence time Z: Simulation vs. LiFE-CD for the cyclic network in Fig. 2a. 0 5 10 15 20 0 0.5 1 k FZ (k) Simulation Proposed LiFE-CD [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: CDF of the consensus convergence time Z: Simulation vs. LiFE-CD for the acyclic network in Fig. 2b. in [37] overestimates E[Z] by a factor of almost 8 at p35 = 90% (30 vs. 3.76), with the gap growing without bound as p35 → 1. These results demonstrate that the proposed LiFE-CD significantly improves the upper bound in [37]. The constant E[Z] in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: PMF of the consensus convergence time Z: Simulation vs. LiFE-CD for the cyclic network in Fig. 2a. 0 5 10 15 20 0 0.5 1 k FZ (k) Simulation Proposed LiFE-CD [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: CDF of the consensus convergence time Z: Simulation vs. LiFE-CD for the cyclic network in Fig. 2a. follows. For p35 < 30 %, LiFE-CD computes E[Z] along the shortest-path tree indicated by the green path in Fig. 2a. At p35 = p45 = 30 %, the expected cost of both green and red paths becomes equal, and for p35 ≥ 30 % the dominant propagation path transitions to the red path in Fig. 2a, which is independent of… view at source ↗
Figure 9
Figure 9. Figure 9: Expected convergence time E[Z] as a function of the number of Monte Carlo runs R for the acyclic network in Fig. 2b. Proposed LiFE-CD (red) compared to Monte Carlo simulation (blue) with error bars indicate one standard deviation σˆ. Consequently, achieving a confidence interval below ϵ requires R = O(ϵ −2 ) simulation runs [40], [41], whereas LiFE-CD computes E[Z] analytically in a single run. Unlike Mont… view at source ↗
read the original abstract

This paper proposes the LiFE-CD algorithm for convergence time analysis of the max-consensus algorithm in multi-agent systems under Bernoulli-distributed link failures. Unlike existing approaches, which either assume ideal communication or provide asymptotic upper bounds on the expected convergence time, LiFE-CD deterministically computes the full probability distribution of the convergence time from network topology and individual link failure probabilities, without simulation. The full probability distribution enables deadline-aware protocol design with specified reliability guarantees. Based on geometrically distributed link delays, the proposed algorithm iteratively reduces the given network topology considering both unicast and broadcast transmissions. LiFE-CD yields exact results for acyclic networks and, for cyclic networks, tight upper bounds on the convergence time via shortest-path spanning tree construction. Numerical results confirm analytical exactness for acyclic networks, validate tightness for cyclic networks, and demonstrate improvement over existing approaches. Our complexity analysis shows reduced computational cost compared to Monte Carlo simulations, while eliminating stochastic variability and enhancing reproducibility. All results extend directly to min-consensus by structural equivalence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. This manuscript presents the LiFE-CD algorithm, which computes the full probability distribution of the convergence time for the max-consensus algorithm in multi-agent systems subject to Bernoulli-distributed link failures. The algorithm is based on iterative graph reduction considering unicast and broadcast transmissions under geometrically distributed link delays. It provides exact distributions for acyclic networks and tight upper bounds for cyclic networks by constructing a shortest-path spanning tree. Numerical results validate the approach, and complexity analysis shows advantages over Monte Carlo simulations. The results extend to min-consensus.

Significance. The provision of the complete probability distribution, rather than just expected values or asymptotic bounds, is a notable advance that supports deadline-aware protocol design with explicit reliability guarantees. The deterministic computation without simulation improves reproducibility and eliminates variability. If the tightness of the bounds for cyclic networks holds generally, this would be a valuable tool for network protocol analysis in unreliable environments.

major comments (2)
  1. [Abstract] Abstract: The assertion that shortest-path spanning tree construction yields tight upper bounds on the convergence time distribution for cyclic networks is load-bearing for the central claim but insufficiently justified. In cyclic graphs, max-value propagation occurs over the union of all paths (i.e., the minimum hitting time across parallel routes), whereas the spanning-tree reduction replaces this with a single path per branch and therefore produces a stochastically larger random variable. Tightness is topology- and failure-probability-dependent; numerical confirmation on selected instances does not establish that the bound remains sufficiently tight for the claimed deadline-aware reliability guarantees across arbitrary cases.
  2. [Section describing the cyclic-network reduction] Section describing the cyclic-network reduction (and associated numerical validation): The iterative graph-reduction procedure and its mapping to a shortest-path spanning tree must be accompanied by an explicit argument showing that the resulting distribution is a valid upper bound and under what conditions it is tight. The geometric-delay assumption is used throughout, yet its interaction with redundant paths in cycles is not analyzed; a counter-example or sensitivity study with varying redundancy levels would be required to support generality.
minor comments (2)
  1. The pseudocode or step-by-step description of the LiFE-CD algorithm would benefit from an explicit listing of input/output variables and termination conditions to improve reproducibility.
  2. In the numerical-results figures, ensure that exact versus bound distributions are overlaid with clear legends and that all plotted probabilities sum to one for each parameter set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the justification of the cyclic-network bounds in our LiFE-CD algorithm. We address each major comment below and will revise the manuscript to strengthen the presentation with additional formal arguments and analysis.

read point-by-point responses

  1. Referee: The assertion that shortest-path spanning tree construction yields tight upper bounds on the convergence time distribution for cyclic networks is load-bearing for the central claim but insufficiently justified. In cyclic graphs, max-value propagation occurs over the union of all paths (i.e., the minimum hitting time across parallel routes), whereas the spanning-tree reduction replaces this with a single path per branch and therefore produces a stochastically larger random variable. Tightness is topology- and failure-probability-dependent; numerical confirmation on selected instances does not establish that the bound remains sufficiently tight for the claimed deadline-aware reliability guarantees across arbitrary cases.

    Authors: We agree that the abstract claim requires more explicit support. The shortest-path spanning tree yields a valid upper bound because convergence time in the original graph is determined by the minimum hitting time over all paths, while the tree uses only a subset of paths; additional paths can only decrease (or leave unchanged) the hitting time, so the tree's convergence time stochastically dominates the true one. We will add a formal lemma and proof of this stochastic dominance in the revised cyclic-network section. We also acknowledge that tightness depends on topology and failure probability (tight when redundant paths offer no improvement over the tree paths). We will expand the numerical results with a sensitivity study over varying redundancy levels and failure probabilities to characterize the bound's tightness and support its use for conservative deadline-aware guarantees. revision: yes

  2. Referee: The iterative graph-reduction procedure and its mapping to a shortest-path spanning tree must be accompanied by an explicit argument showing that the resulting distribution is a valid upper bound and under what conditions it is tight. The geometric-delay assumption is used throughout, yet its interaction with redundant paths in cycles is not analyzed; a counter-example or sensitivity study with varying redundancy levels would be required to support generality.

    Authors: We will revise the cyclic-network reduction section to include an explicit argument and proof that the shortest-path spanning tree distribution is a valid upper bound. Under independent geometric delays, path delays are negative binomial and the node hitting time is the minimum over paths; the tree omits some redundant paths, which can only increase the min hitting time. We will analyze the interaction with cycles by showing that redundant paths strictly reduce the hitting time distribution (or leave it unchanged) and state conditions for tightness (e.g., when the tree captures all minimum-delay paths or when high failure probabilities make extra paths unlikely to help). We will also add a counter-example showing looseness (such as a cycle with low failure probability and high redundancy) and a sensitivity study with varying redundancy levels to demonstrate generality. revision: yes

Circularity Check

0 steps flagged

No circularity: LiFE-CD is an input-driven iterative graph reduction algorithm

full rationale

The derivation chain consists of an explicit iterative graph-reduction procedure that ingests the network topology and per-link Bernoulli failure probabilities as independent external inputs and produces the convergence-time distribution (exact on trees, upper-bounded on cycles via a shortest-path spanning tree). No equation or step equates the output distribution to a fitted parameter, a self-defined quantity, or a result imported solely via self-citation. Numerical validation on separate instances is cited to support exactness and tightness claims, but the algorithm itself does not presuppose those outcomes. The method therefore remains self-contained against its stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The central claim rests on modeling assumptions for link failures and delays, plus graph-theoretic reductions; no fitted parameters or invented physical entities are described.

axioms (3)
  • domain assumption Link failures follow independent Bernoulli distributions
    Stated as the model for unreliable networks in the abstract.
  • domain assumption Link delays are geometrically distributed
    Used as basis for iterative network reduction in the proposed algorithm.
  • domain assumption Shortest-path spanning tree yields tight upper bounds for cyclic networks
    Invoked for handling cycles where exact computation is not claimed.
invented entities (1)
  • LiFE-CD algorithm no independent evidence
    purpose: Deterministic computation of convergence time probability distribution
    Newly proposed iterative reduction method for the analysis.

pith-pipeline@v0.9.0 · 5488 in / 1482 out tokens · 64118 ms · 2026-05-10T07:38:36.069258+00:00 · methodology

discussion (0)

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