Double Spending Analysis of Nakamoto Consensus for Time-Varying Mining Rates with Ruin Theory

TL;DR

This paper develops a ruin-theoretic framework to analyze double spending in Nakamoto consensus under time-varying mining rates. It introduces a matrix-exponential (ME) approach to model honest inter-mining times during block delivery, enabling the incorporation of arbitrary, data-driven delay distributions via $M_{\Theta}(s)$, while assuming a fixed adversary hashrate. By relating the adversary’s lead to a discrete-time ruin problem, the authors derive the $k$-deep confirmation double-spend probability through the Gilmore–Lindley based ruin machinery and first-$k$ masses of the adversary’s block counts, with the key step being $G_{\Phi}(z)=M_{\Theta}(\beta(z-1))$. The method unifies zero-delay, fixed-delay, exponentially-fixed-delay, and variable-delay models as ME-approximations (with CME), and empirical data from Bitcoin networks validate the approach and demonstrate the sensitivity of security probabilities to delay characteristics and $\epsilon$-cutoffs. Overall, the work provides a practical, scalable framework to assess security-latency trade-offs under realistic, time-varying network delays in PoW blockchains.

Abstract

Theoretical guarantees for double spending probabilities for the Nakamoto consensus under the $k$-deep confirmation rule have been extensively studied for zero/bounded network delays and fixed mining rates. In this paper, we introduce a ruin-theoretical model of double spending for Nakamoto consensus under the $k$-deep confirmation rule when the honest mining rate is allowed to be an arbitrary function of time including the block delivery periods, i.e., time periods during which mined blocks are being delivered to all other participants of the network. Time-varying mining rates are considered to capture the intrinsic characteristics of the peer to peer network delays as well as dynamic participation of miners such as the gap game and switching between different cryptocurrencies. Ruin theory is leveraged to obtain the double spend probabilities and numerical examples are presented to validate the effectiveness of the proposed analytical method.

Figures (5)

• Figure 1: The cdf $F_{X}(x)$ obtained with the CME distribution with order $K$ for approximating deterministic variable $X=2$.
• Figure 2: Illustration of the honest chain view.
• Figure 3: The hashrates $\alpha(t)$ of the proposed ME distribution with order $NK+1$ for modeling the variable delays when $\epsilon=0.001$, $K=27$ and $N' \in \{ 8,32,128 \}$.
• Figure 4: The pdf $f_{\Theta}(x)$ obtained with the zero-delay model and additionally the proposed ME distribution with order $NK+1$ for modeling the variable delays when $\epsilon=0.001$, $K=27$ and $N' \in \{ 8,32,128 \}$.
• Figure 5: Double spending probability $q$ as a function of $k$ for various delay models.