# Contagion in Bitcoin networks

## TL;DR

This paper models contagion in Bitcoin networks using Google matrices, revealing a phase transition in user bankruptcy risk related to trade balance thresholds and highlighting the vulnerability of top-ranked users due to their interconnectedness.

## Contribution

It introduces a novel application of Google matrix analysis to Bitcoin transaction networks, modeling contagion and bankruptcy thresholds in a large-scale real-world financial network.

## Key findings

- Phase transition at trade balance threshold κ_c≈0.1
- Top PageRank users are highly interconnected and vulnerable
- Most users remain safe when κ>0.55

## Abstract

We construct the Google matrices of bitcoin transactions for all year quarters during the period of January 11, 2009 till April 10, 2013. During the last quarters the network size contains about 6 million users (nodes) with about 150 million transactions. From PageRank and CheiRank probabilities, analogous to trade import and export, we determine the dimensionless trade balance of each user and model the contagion propagation on the network assuming that a user goes bankrupt if its balance exceeds a certain dimensionless threshold $\kappa$. We find that the phase transition takes place for $\kappa<\kappa_c\approx0.1$ with almost all users going bankrupt. For $\kappa>0.55$ almost all users remain safe. We find that even on a distance from the critical threshold $\kappa_c$ the top PageRank and CheiRank users, as a house of cards, rapidly drop to the bankruptcy. We attribute this effect to strong interconnections between these top users which we determine with the reduced Google matrix algorithm. This algorithm allows to establish efficiently the direct and indirect interactions between top PageRank users. We argue that this study models the contagion on real financial networks.