From a visual perspective, Optimism displays an inverse relationship between each couple of variables, while the same doesn’t happen for Arbitrum. A deeper analysis with correlation coefficients confirms that the variables considered have opposite correlations in the two rollups.

Correlation Matrix for Arbitrum, on the left, and Optimism, on the right. Source: https://dune.com/filarm/rollups

In Optimism, a moderate negative correlation exists (-0.25) between the amount of data posted to Ethereum (in Mb) and the cost per Mb, indicating that as the amount posted increases, the cost per Mb decreases. This suggests potential economies of scale or efficiency gains with larger data volumes. Additionally, a negative correlation (-0.36) between the number of transactions processed and the cost per transaction suggests that as transaction volumes increase, the cost per transaction tends to decrease, implying further economies of scale or operational efficiencies with higher transaction volumes. Conversely, in Arbitrum, a moderate positive correlation (0.37) between the amount of data posted to Ethereum and the cost per Mb suggests that as data volume increases, the cost per Mb tends to increase, potentially indicating diseconomies of scale compared to Optimism. Similarly, a positive correlation (0.26) between the number of transactions and the cost per transaction implies that as transaction volumes rise, the cost per transaction tends to increase, reflecting potential differences in cost structures or operational dynamics that may lead to diseconomies of scale.