This article was first published on IOTA - Medium
In today’s blog post, we’ll have a closer look at the journey of random walkers. Similar to one of our previous posts on mixing time in the Tangle, we are interested in understanding how long random walkers have to walk to select a tip and, more importantly, how long they take to reach their goal. For this analysis, we will simulate different Tangles and, for each one, we will release 100,000 random walkers starting from the Genesis and measure their velocity while performing a random walk on the Tangle. This article assumes a basic understanding of how the Tangle is built and probability theory, as we will touch topics such as the parameter α, direct approvers, and probability density functions (PDF). If you are not familiar with these topics, we suggest you read up on them first.
We can start by defining what exactly we mean by velocity. Consider a Tangle with N transactions. To each transaction, we assign an ID according to the time of announcement to the network i.e., the Genesis transaction has ID 1, the first transaction that approves the genesis has ID 2, and so on. We can assume that those transactions are announced to the network at times t_1 < t_2 < … < t_N. We define the following quantities:
- distance(x-y) = ID_y - ID_x
- velocity(x-y) = distance(x-y) / steps
Basically, we define the distance between two transactions x and y as the number of transactions announced to the network within the interval that goes from tx to ty. Similarly, we define the velocity of a random walker transitioning from a transaction x to a transaction y as the distance covered by the walker divided by the total number of steps required to complete that transition.
Now, to give you a graphical representation of ...
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