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Decentralized network congestion control for DAG-based distributed ledger system

Mayank Pandey, Rachit Agarwal, Sandeep Kumar Shukla, Nishchal Kumar Verma

arXiv:2609.09961v1cs.CRcs.DCcs.NI

TL;DR

Decentralized DLT congestion control is underexplored, and DAG networks’ fee-free, high-throughput design leaves them vulnerable to transaction spam that nominal PoW does not target. The paper proposes reputation-based variable PoW difficulty with prescribed transaction limits; its game analysis supports prescribed behavior at equilibrium.

  • Problem

    DAG-based DLT combines high throughput and no transaction fees with vulnerability to spamming, while decentralized DLT congestion control remains underexplored.

    Source: §1 · pp. 1–2

  • Method

    The model sets variable PoW difficulty from reputation built on prior-period transaction frequency and token spending, prescribing a transaction limit and disproportionately increasing difficulty for violations.

    Source: §6 · p. 15

  • Results

    The non-cooperative game analysis establishes prescribed behavior at Nash equilibrium, with Kcross transactions per node as the fixed-membership resource-efficient case and violators penalized in the subsequent period.

    Source: §1 · p. 2, §4.1 · p. 10

  • Takeaways & Limitations

    The authors report effective DAG congestion control without compromising overall scalability while giving users equal opportunities to issue transactions.

    Source: §1 · p. 2, §6 · p. 15

    The analysis initially assumes an unchanged number of users and uniformly synchronized time across nodes.

    Source: §3 · p. 6

Abstract

from arXiv · show

We propose a variable and behavior-based node-specific proof-of-work (PoW) model for a directed acyclic graph (DAG)-based distributed ledger technology (DLT) network to mitigate decentralized network congestion control. Network congestion control for centralized communication systems is an established field of study, with detailed and continuous research being done on the subject. However, attention to congestion control in decentralized networks is relatively recent and underexplored, especially with DLT, such as blockchain and DAG-based networks. For the DLT networks, the network congestion is caused by factors such as transaction spamming, an increase in the user base, and the launch of new tokens. We focus on the congestion caused by the spamming of transactions within the blockchain and DAG-based DLT network. Based on the network throughput of transactions per second and consensus procedure, the DAG-based DLT needs to control network spamming more than the blockchain network. The PoW model within the DLT consensus framework is a limited deterrent against spamming. Our model provides equal opportunities for all stakeholders regardless of their computational resources. It prevents and penalizes any node that attempts to spam or dominate the network with more than the prescribed number of transactions. Since the system nodes compete to issue transactions with finite network resources, we display the system behavior through a non-cooperative game. Further, we show that our model enforces prescribed behavior amongst the nodes through the proof of the existence of Nash equilibrium in the game.

1 Introduction

Decentralized DLT congestion control remains underexplored, and DAG networks’ high throughput and lack of transaction fees leave them vulnerable to transaction spamming. The paper proposes a behavior-based variable PoW model and uses Nash equilibrium to support prescribed node behavior.

1 Introduction

  • The paper addresses an underexplored need for decentralized DLT congestion control, focusing on transaction spamming as a source of congestion.The introduction distinguishes malicious spam from congestion associated with legitimate user growth or new-token launches.Source: §1 · p. 1
  • DAG networks’ fee-free, cooperative transaction validation supports higher throughput but also makes them vulnerable to spam-driven congestion.The paper contrasts this with blockchain’s PoW and transaction fees, which deter spamming but can limit throughput and scalability and disadvantage low-resource users.Source: §1 · p. 2
  • Nominal DAG PoW verifies and attaches transactions but was not designed to prevent spam, allowing high-resource nodes to crowd out others.Ledger synchronization limits the number of transactions the network can handle; exceeding that limit can delay broadcasts or invalidate transactions.Source: §1 · p. 2
  • The introduction presents a user-reputation-based, variable PoW difficulty model as a solution to DAG transaction spamming.It reports effective congestion control without compromising overall scalability.Source: §1 · p. 2

Our contributions

  • The proposed model prescribes node behavior and provides equal participation opportunities while penalizing departures from the prescribed transaction limit.The contribution summary also states that the model establishes prescribed behavior through Nash equilibrium in a non-cooperative game and addresses changes in node composition through a uniform change in equilibrium.Source: Our contributions · pp. 2–3

2 Background and Existing shortcomings

The section contrasts blockchain and DAG congestion controls and describes how PoW operates across DAG designs. It motivates a congestion-control difficulty model intended to deter spam without variable fees or reduced throughput capacity.

2 Background and Existing shortcomings

  • Blockchain PoW and PoS discourage spamming, but their resource-consuming or high-stakes consensus mechanisms favor users with greater resources.The blockchain transaction pool, block rewards, and transaction fees provide incentives for transaction inclusion and confirmation.Source: §2 · p. 3
  • Block-based DAGs use PoW for block attachment and retain blockchain-like transaction fees and pool-to-block processing, while blockless DAGs add transactions directly or after accumulation.Block-based DAGs are described as comparable to blockchain in their measures against transaction spamming; low-resource users are disadvantaged in block-based systems.Source: §2 · p. 3
  • Blockless DAG designs include IOTA and its variants, fee-based Graphchain, and Hashgraph’s parallel chains, which can favor high-resource nodes.IOTA’s variants differ in transaction verification and attachment limits, while Graphchain leaves transaction fees for users who attach and verify transactions.Source: §2 · p. 3
  • DAG networks use nominal or adaptive PoW against transaction spamming, with adaptive difficulty responding more to transaction frequency than methods in other networks.The section identifies IOTA’s difficulty as adaptive and gives its difficulty parameters as a base level, transaction count over a prior time window, and reputation-related mana.Source: §2 · pp. 3–4
  • PoW difficulty determines the average computational work required for a transaction or block hash to meet a specified output target.The section states that increasing difficulty by one doubles the average work, while decreasing it by one halves the work.Source: §2 · p. 4
  • The proposed objective is to control spamming through difficulty without variable transaction fees or compromising network throughput capacity.The section also states an aim of viability across PoW-based DLT networks, whether nominal or competitive.Source: §2 · p. 4

3 Proposed congestion control method

The method assigns node-specific PoW difficulty from recent transaction behavior and reputation to discourage spamming while respecting finite DAG-network capacity. It updates capacity and reputation parameters as nodes join or leave, then evaluates prescribed behavior through a non-cooperative game.

3 Proposed congestion control method

  • The model targets network capacity Λ over period T, with nodes attaching transactions to selected tips after verification.The passage specifies two tips as the required number in the described setting.Source: §3 · p. 6
  • The proposed reputation-based difficulty model penalizes nodes that exceed prescribed transaction limits while aiming to minimize computational and communication costs.Difficulty is assigned for the next period using the node's prior-period transaction frequency.Source: §3 · pp. 6–7
  • Node reputation combines token-spending and transaction-frequency components, with transaction-frequency reputation diminishing beyond a capacity-based threshold.The frequency-based component is designed to encourage participation while discouraging excessive transaction issuance.Source: §3 · pp. 6–7
  • The authors identify Kcross as the local minimum of difficulty multiplied by transaction frequency, making it the target transaction rate for each node.The model's prescribed behavior keeps each node at or below Kcross.Source: §3 · p. 8
  • Exceeding Kcross by even one transaction imposes a considerably heavier computational burden in the following period.A one-unit change in difficulty doubles or halves average resource requirements.Source: §3 · p. 8

3.1 Joining and leaving of nodes

  • When node membership or network capacity changes, the model updates average capacity and reputation parameters while retaining the fixed-set difficulty rule.A joining node can begin issuing transactions in the period after it joins.Source: §3.1 · pp. 8–9
  • The authors frame prescribed behavior as a non-cooperative game and seek maximum transaction capacity with minimum total computational resources for fixed and variable node sets.The following section establishes Nash equilibrium for both settings.Source: §3.1 · p. 9

4 Optimal Resource consumption via non-cooperative games

The authors model transaction issuance as a non-cooperative game under finite network capacity. They show that the prescribed per-node transaction limit is a Nash equilibrium for fixed membership and shifts uniformly when membership changes.

4 Optimal Resource consumption via non-cooperative games

  • The game models nodes choosing transaction counts without communication or binding agreements, under a shared physical limit on transactions per period.Its objective combines full use of current network capacity with optimal resource consumption in the next period.Source: §4 · p. 9, §4.1 · p. 9

4.1 Nash equilibrium for a fixed set of nodes

  • For a fixed node set, issuing Kcross transactions per node is a Nash equilibrium when ζ1/(2ζ2) = Kcross.The equilibrium holds for every node when the game has a bounded strategy set.Source: §4.1 · p. 10
  • At that fixed-set equilibrium, every node issuing at most Kcross supports maximum network use with minimum total computational resources.A node exceeding the limit is penalized in the next period with a disproportionately higher difficulty level.Source: §4.1 · p. 10

4.2 Nash equilibrium for variable node set

  • The variable-set analysis updates effective players and average capacity across periods, with the changed equilibrium applying after membership changes take effect.The membership change occurs in one period and affects the subsequent period's capacity and equilibrium.Source: §4.2 · p. 10
  • When the user set changes, the Nash equilibrium shifts uniformly across the system, so no node gains a preferred transaction limit from the membership change.The authors conclude that prescribed behavior remains continuously enforced for a dynamic node set.Source: §4.2 · pp. 10–11

5 Numerical Examples

The examples compare IOTA’s behavior-based difficulty with the proposed reputation model, then illustrate payoffs and Nash equilibria under fixed and changing node sets. The proposed model’s equilibrium shifts when a node joins.

Example 1 (Analysis of IOTA congestion control model)

Example 2

  • Difficulty di(5T) is d0 when wi(5T) ≤ 40 and d0 + 1 when 40 < wi(5T) ≤ 100; during [4T, 5T], 90% spending keeps it at d0 + 2, while 10% gives d0 + 3.Source: Example 2 · pp. 11–12

Example 3

  • For two users, the Nash equilibrium is 100, equal to Kcross, where the best-response functions intersect.The payoff surface peaks at 5000.Source: Example 3 · p. 13

Example 4

  • With three nodes, the unique equilibrium is 80 per node, a shift of 20 from the prior equilibrium of 100.The updated payoff distribution rises toward the 70–90 range.Source: Example 4 · p. 14

6 Conclusion

The paper proposes a variable, behavior-based PoW model for DAG-based DLT congestion control. Its difficulty rule is designed to limit transaction spamming, and a non-cooperative game analysis establishes a unique pure Nash equilibrium.

6 Conclusion

  • The proposed model addresses transaction-spamming congestion in DAG-based DLT by serving as proof of attachment for transactions.The paper identifies spamming by an unscrupulous individual or group as the main cause of DLT network congestion.Source: §6 · p. 15
  • The model sets PoW difficulty from users’ previous-period transaction frequency and token spending, with quadratic penalties for exceeding prescribed transaction limits.It is intended to give users equal opportunities regardless of computational resources.Source: §6 · p. 15
  • The non-cooperative game has a unique pure Nash equilibrium where each node uses the full transaction limit with minimum possible average computational resource.The model also applies to a dynamic node set while retaining the prescribed behavior.Source: §6 · p. 15
  • The authors report that their difficulty model is better for DAG-based ledgers than static or linearly increasing difficulty.Source: §6 · p. 15

A Values of di(CT) for different values of wi(CT)

The appendix describes how difficulty varies with the node’s transaction behavior weight, including the baseline and threshold cases.

A Values of di(CT) for different values of wi(CT)

B Proof of Lemma 4.1

The proof establishes a unique Nash equilibrium for the fixed-user game at the prescribed threshold Kcross. It uses standard equilibrium existence conditions and derives the equilibrium through best-response functions.

B Proof of Lemma 4.1

  • For a bounded strategy set and fixed user count, wi(CT) = Kcross is the Nash equilibrium when ζ1/2ζ2 = Kcross.Source: B Proof of Lemma 4.1 · p. 18
  • Existence follows from a compact, convex strategy set and payoff functions that are strictly concave and continuous.The proof checks negative definiteness of the Hessian to establish strict concavity.Source: B Proof of Lemma 4.1 · p. 18
  • The proof obtains uniqueness by deriving best responses and induction from two users to a generalized user set.Source: B Proof of Lemma 4.1 · pp. 18–20
  • At equilibrium, Kcross is independent of the payoff-share ratio κR, provided κ2 ≠ 0.Source: B Proof of Lemma 4.1 · p. 20

C Proof of Lemma 4.2

The proof considers how the equilibrium changes when the node set changes between periods. It finds that the equilibrium tracks the current period’s Kcross value, with the shift determined by the change in the corresponding parameter ratio.

C Proof of Lemma 4.2

  • The proof derives the updated game parameters and best responses for the revised node set before solving for its equilibrium.Source: C Proof of Lemma 4.2 · pp. 21–22
  • The shift between periods’ equilibrium values is the absolute difference between the current and previous parameter ratios ζ1/(2ζ2).Source: C Proof of Lemma 4.2 · p. 22
  • When nodes join or leave, the proof gives the new Nash equilibrium as the current period’s Kcross for the updated user set.For a three-node set changing to two nodes, it likewise identifies the new equilibrium at Kcross.Source: C Proof of Lemma 4.2 · p. 22
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