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arxiv: 1906.09073 · v1 · pith:SZ4CU3ROnew · submitted 2019-06-21 · 💻 cs.DC

MinMax Algorithms for Stabilizing Consensus

Bernadette Charron-Bost , Shlomo Moran This is my paper

Pith reviewed 2026-05-25 18:51 UTC · model grok-4.3

classification 💻 cs.DC
keywords stabilizing consensusMinMax algorithmtime-varying topologydynamic networksroot agentanonymous agentssynchronous distributed systemsinput stabilization
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The pith

A generic MinMax algorithm solves stabilizing consensus in time-varying networks with periodic roots.

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

The paper shows that stabilizing consensus can be achieved in synchronous networks of identical anonymous agents connected by changing links. Agents start with input values and must eventually all output the same one of those inputs. The algorithm succeeds whenever every bounded time interval contains a root agent that can reach every other agent, possibly through a chain of messages and with the root changing from interval to interval. No central coordinator or knowledge of network size is required.

Core claim

Our main result is a generic MinMax algorithm that solves the stabilizing consensus problem in this model when, in each sufficiently long but bounded period of time, there is an agent, called a root, that can send messages, possibly indirectly, to all the agents. Such topologies are highly dynamic (in particular, roots may change arbitrarily over time) and enforce no strong connectivity property (an agent may be never a root). Our distributed MinMax algorithms require neither central control (e.g., synchronous starts) nor any global information (e.g., on the size of the network), and are quite efficient in terms of message size and storage requirements.

What carries the argument

The generic MinMax algorithm, which propagates and selects among input values using min and max operations whenever a root reaches the network.

If this is right

  • Stabilizing consensus is possible without any agent knowing the network size.
  • The same algorithm works when the identity of the root changes arbitrarily between periods.
  • No permanent strong connectivity or fixed spanning tree is needed.
  • Message size and local storage remain small even as the network grows.
  • The solution applies to fully anonymous agents that start without synchronized clocks or identifiers.

Where Pith is reading between the lines

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

  • The periodic-root condition may be close to the weakest topology assumption that still permits consensus without global knowledge.
  • Similar MinMax rules could be tested in simulations of mobile ad-hoc or sensor networks whose connectivity fluctuates.
  • The approach suggests that other agreement tasks in dynamic settings may also reduce to local min-max selections under comparable reachability conditions.

Load-bearing premise

In every sufficiently long but bounded time period there exists at least one root agent that can reach all others through the current links.

What would settle it

An execution of the algorithm on a network in which some bounded period contains no root that reaches all agents, yet all outputs still stabilize to the same input value.

Figures

Figures reproduced from arXiv: 1906.09073 by Bernadette Charron-Bost, Shlomo Moran.

Figure 1
Figure 1. Figure 1: Distributed implementation of a MinMax algorithm [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
read the original abstract

In the stabilizing consensus problem, each agent of a networked system has an input value and is repeatedly writing an output value; it is required that eventually all the output values stabilize to the same value which, moreover, must be one of the input values. We study this problem for a synchronous model with identical and anonymous agents that are connected by a time-varying topology. Our main result is a generic MinMax algorithm that solves the stabilizing consensus problem in this model when, in each sufficiently long but bounded period of time, there is an agent, called a root, that can send messages, possibly indirectly, to all the agents. Such topologies are highly dynamic (in particular, roots may change arbitrarily over time) and enforce no strong connectivity property (an agent may be never a root). Our distributed MinMax algorithms require neither central control (e.g., synchronous starts) nor any global information (eg.,on the size of the network), and are quite efficient in terms of message size and storage requirements.

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

1 major / 0 minor

Summary. The paper claims to introduce a generic MinMax algorithm that solves the stabilizing consensus problem for identical anonymous synchronous agents connected by a time-varying directed topology. The key enabling assumption is that in every sufficiently long but bounded time interval there exists (possibly changing) a root agent that can reach all others, possibly indirectly; the algorithm requires neither synchronous initialization nor global parameters such as network size and is claimed to be efficient in message size and local storage.

Significance. If the correctness claim holds, the result would be significant for distributed computing in dynamic networks: it supplies an explicit, parameter-free algorithm for stabilizing consensus under a weak and time-varying connectivity condition that is strictly weaker than perpetual strong connectivity, while avoiding any central control or knowledge of n.

major comments (1)
  1. [Abstract / Main result statement] The abstract asserts existence and correctness of the MinMax algorithm under the periodic-root condition, yet the provided text contains no proof sketch, invariant, or inductive argument establishing that outputs eventually stabilize to a common input value. Without the detailed analysis the central claim cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for explicit verification of the central correctness claim. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / Main result statement] The abstract asserts existence and correctness of the MinMax algorithm under the periodic-root condition, yet the provided text contains no proof sketch, invariant, or inductive argument establishing that outputs eventually stabilize to a common input value. Without the detailed analysis the central claim cannot be verified.

    Authors: The full manuscript contains a complete correctness proof (Section 4) that proceeds by exhibiting a non-increasing potential function on the multiset of output values, proving an invariant that every output remains within the convex hull of the inputs, and showing that the periodic-root condition forces all outputs to converge to the same input value in finite time. A concise proof sketch and the key invariant are also stated in the introduction. If the version seen by the referee omitted these sections, we are happy to ensure they are clearly visible; otherwise we can expand the sketch in the introduction for clarity. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under external topology assumption

full rationale

The paper states its main result as the existence of a generic MinMax algorithm that solves stabilizing consensus precisely when the network satisfies an externally given condition: in each sufficiently long bounded interval there exists a root agent that can reach all others (possibly indirectly). This topology precondition is presented as an input assumption rather than derived from or redefined in terms of the algorithm's outputs. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the provided description. The algorithm is claimed to require neither synchronous starts nor global parameters, and correctness is asserted relative to the stated model, making the derivation independent of its own results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard synchronous distributed-computing assumptions plus the domain-specific periodic-root topology condition; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Synchronous communication rounds with identical anonymous agents
    Standard model assumption invoked to define the setting in which the algorithm operates.
  • domain assumption Time-varying directed graph topology contains a root in every sufficiently long bounded period
    This is the load-bearing condition stated in the abstract that enables the MinMax algorithm to guarantee stabilization.

pith-pipeline@v0.9.0 · 5696 in / 1261 out tokens · 22754 ms · 2026-05-25T18:51:06.512394+00:00 · methodology

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Reference graph

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