Stability of P2P Networks Under Greedy Peering (Full Version)

TL;DR

The paper analyzes stability of P2P networks under greedy peering that aims to minimize distance to a miner subset. It develops both a full-information idealized game locating pure Nash equilibria and a practical Single-Exploratory-Greedy protocol with local knowledge, deriving stability conditions and topology properties. Key findings show that stable networks tend to form low-diameter, miner-centered cores, with stability heavily contingent on tie-breaker rules and inbound-capacities; simulations corroborate the emergence of a two-tier structure and fairness trade-offs. These results have practical implications for cryptocurrency overlays, highlighting how greedy peering can drive centralization and disparity in node performance, depending on protocol parameters.

Abstract

Major cryptocurrency networks have relied on random peering choice rules for making connections in their peer-to-peer networks. Generally, these choices have good properties, particularly for open, permissionless networks. Random peering choices however do not take into account that some actors may choose to optimize who they connect to such that they are quicker to hear about information being propagated in the network. In this paper, we explore the dynamics of such greedy strategies. We study a model in which nodes select peers with the objective of minimizing their average distance to a designated subset of nodes in the network, and consider the impact of several factors including the peer selection process, degree constraints, and the size of the designated subset. The latter is particularly interesting in the context of blockchain networks as generally only a subset of nodes are the propagation source for content. We first analyze an idealized version of the game where each node has full knowledge of the current network and aims to select the $d$ best connections, and prove the existence of equilibria under various model assumptions. Since in reality nodes only have local knowledge based on their peers' behavior, we also study a greedy protocol which runs in rounds, with each node replacing its worst-performing edge with a new random edge. We exactly characterize stability properties of networks that evolve with this peering rule and derive regimes where stability is possible and even inevitable. We also run extensive simulations with this peering rule examining both how the network evolves and how different network parameters affect the stability properties of the network. Our findings generally show that the only stable networks that arise from greedy peering choices are low-diameter and result in disparate performance for nodes in the network.

Figures (13)

• Figure 1: For the existence proofs of Lemmas \ref{['lem:nash-capped']} and \ref{['lem:nash_capped_general']} we construct the above network families. Both networks are essentially layered, and the crux of the proof is that the miners are positioned in the "top layer(s)" with the non-miners occupying the remaining layers, such that all nodes except the non-miners in the last layer have full in-capacities. The nodes in the last layer have the highest score and no node higher up in the layers would give up their position to connect to a node in the bottom layer.
• Figure 2: (a) the random graph created from all nodes choosing $d$ random nodes to connect to. (b) the graph we seem to converge to from running Single-Exploratory-Greedy with all nodes being homogeneous miners, size scaled by degree. This structure appears after $\sim$20 rounds.
• Figure 3: Average distance to miners per round for an $n=100$ node network with all miners and $m=10$ miners, for different in-degree caps.
• Figure 4: Distribution of distance to miners(node scores) at round 256 for an $n=100$ node network with all miners and $m=10$ miners, for different in-degree caps.
• Figure 5: Diameter of a 100 node network with all miners and a subset of 10 miners, and the diameter between the 10 miners.

Theorems & Definitions (34)

• Definition 3.1: Tie Breaker Rule
• Definition 3.2: Stability
• Definition 3.3: Special Event
• Lemma 4.1: Every Uncapped Game has a Stable Network
• proof
• Lemma 4.2: Stable Networks in Capped Games with Unit Out-Degree
• proof
• Lemma 4.3: Stable Networks in Capped Games with Arbitrary Out-Degree
• proof
• Proposition 4.4: Clique Miner Stability for Small $m$