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(I already know that this answer will disappoint you - as it did disappoint me when I found it out) The current algorithms around the tangle depend on a central node run by the Foundation, known as the Coordinator (or Coo). This node will publish a signed milestone transaction approximately every minute. The exact rules which tips will be selected for this ...

4

Variable z is just the index over which the sum in equation (13) iterates. This is similar to when you have \sum_{i=1}^n ... where i is the standard index (so no need to "define" i separately). Now, notation z:z⇝x simply means All z such that z approves x or All z such that z⇝x. So we sum over all transactions z that directly approve our transaction x. ...

4

A random walker on it's way to tip to approve must randomly select a path at each step. The random walker have the choice between all transactions (i.e. site) directly approving it's current site. Here is an example, let's assume that the random walker is currently on a site (i.e. a transaction) directly approved by 3 transactions. Let's call those 3 ...

3

This important question should be answered clearly & with detail. There are three parts to your question: How does IOTA confirm transactions now? How will IOTA confirm transaction after coordicide? How is this different from a blockchain confirmation? I will answer these in order. How does IOTA confirm transactions now? tl;dr The coordinator, run by ...

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r/(r+λh) is a mean value ( < 1 ) r is tips before time [t-h] r+λh is a slightly higher number adding all tips from time interval [t-h,t] since the node does not know λh tips are no tips any more at time index t (assuming λh is always stationary value), we have a total probability of choosing a tip of r/(r+λh) ( < 1 ) Total mean number of chosen tips ...

2

1/ln2 from equation (3) in page 8. (in version 1.2 of the whitepaper)

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The k from the footnote of section 1 is the required number of transactions to be referenced by new transactions (as is noted by you in your question). The k from section 3 is just an index/placeholder in the context of number of tips overall (not the number of tips to be referenced), and one could have used any letter here. For example, The limit of P[L(...

2

In both cases, k is a "variable" used to make the English sentence easier to understand, the variable is both defined and used within the sentence, but nowhere else. The meaning of these two k is therefore different. So the second example could be rewritten as One may also study similar systems where each transaction must approve a constant number of ...

2

H(t) represent the expected cumulative weight at time t. ∂ is a way to note a very small delta time. The first equation explicit the value the H function relatively to the value of the H function a very small time before. The second equation is the differential of the first one. In other words, dH(t)/dt is the slope of the function H(t)

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It's Taylor series of the exp function to the 1st order: https://en.wikipedia.org/wiki/Taylor_series#Exponential_function exp(-x) = 1 + (-x)^1/1! + (-x)^2/2! + .... Since white paper assumes x to be small, you forget about higher order terms and only take 1st order. What remains is an approximation for exp(-x), since x^2 is really tiny, once x is already ...

2

Transactions are conflicting if they try to spend the same money from the same address (and the address does not include enough money to satisfy both at the same time - but this should not happen anyway, as compliant clients will move the remaining balance to a change address - but can of course happen if an adversary tries this). Either this can be because ...

2

Currently we have the Coo node running and placing milestones at regular intervals and these milestones are used by the nodes to determine the 'direction' that consensus is forming among all the transactions. When a wallet requests some tips to verify, the node makes two random walks from within the tangle to find those tips. The first random walk begins ...

1

Unfortunately, this doesn't look correct. Regardless of the actual numbers, you are not treating the sum as a sum but rather as a single summand. On top of that, you seem to be missing the exp function in a few places during your P_xy calculation. Below I am pasting a sample calculation for a transaction x that is referenced by three other transactions y_1,...

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Selection of start points of random walkers is quite random in the Tangle. First, it considers a subset of the Tangle with a cumulative weight between the range L and 2L where L is reasonably large enough. (The idea to select a subset between L and 2L and keeping L somewhat large enough is to place the particle deep into the tangle network so that the walk ...

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Here W^{(n)} denotes the time it takes to complete a Proof-Of-Work of difficulty n, where n is a positive integer (of acceptable size, cf. first paragraph of Section 2).

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Parameter N was defined in an earlier version of the whitepaper as "total number of transactions". However, the dependency of h on L and N (as seen in your quote) has since been removed, and all the references to N should have been updated. Bottom line, it seems like a typo and you can think of h(L,N) as just h.

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Expected increment of # of tips at time t is: 1 - 2*r/(r+λh) .....eqn(1) where "1" refers to new tip created by the tx and "2*r/(r+λh)" refers to expected # of "erased" tips. If the two "erased" tips were already approved (usually for small L(t)), then "2*r/(r+λh)" = 0, thus eqn(1) becomes positive or increasing. If the two "erased" tips were not ...

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