# Evaluating Methods for the Identification of Off-Chain Transactions in the Lightning Network
## 5 Results
### Linking
This means if one is closing many channels unilaterally at the same time, consolidation of all timelocked outputs into one timelocked transaction is a natural thing to do; this would cause graph overlap as defined in heuristic three, but also create a high-value output from the timelocked transaction which could be used to fund new channels. Heuristic 3 which used graph overlap or multi-input timelocked transactions provided 12% on mainnet, and 24% on the testnet, of the total connections, found using both heuristic two and three. If we combine our results of using all three heuristics on the channel sets, in the mainnet we get a total of 4436 relationships between the channel graphs. This enabled us to create 2182 links between channels. The reason for this difference is redundancy of connections and key reuse within a single channel graph. This means that 2254 of our relationships, or about 50% where redundant, but only 36 of them being the actual reuse of the same key in the same channel graph. The set of linked channels can be seen visualized in Figure 15, were we have created a channel network as discussed in Section 4.3. The network consists of 2324 smaller components, with each node having an average degree of 1.1. While this graph does not distinguish users, we can reveal additional information about the users participating in the channels. As edges represent at least one common user being present in both nodes, we can create the property: for adjacent nodes, there can be at most three different users and a minimum of two users. This means if the distance between vertices is more than one, we can no \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Mainnet** & **Testnet** \\ \hline Total relationships found & 1586 & 5715 \\ Founding output relationships & 931 & 2824 \\ Closing output relationships & 30 & 2 \\ Timelocked output relationships & 435 & 1499 \\ Graph overlap relationships & 190 & 1390 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relationships found between channel graphs using heuristic one and two. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **Mainnet** & **Testnet** \\ \hline Unique Keys found & 26,824 & 50,050 \\ Unique keys reused & 1370 & 3250 \\ Instances of key reuse & 2850 & 7240 \\ Maximum number of reuses for a single key & 23 & 171 \\ Average reuse of keys & 2 & 2.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Key reuse in channel graphs found in complete runs. longer be sure if there is common user participation, but we cannot rule it out either. More importantly, it has an impact on complete components--i.e., every node is connected to every other node. Because the property we defined must hold for every node in such a complete component, there are only two possibilities with regard to user participation: there can be at most three total users in the component, or that one user participates in all nodes, but with a different second user in each. We can see a few examples of such components in Figure 15, and some components with a similar structure but with additional nodes which are not part of the complete core of the component. We did the same for the set of channel graphs from the testnet. The total number of relationships using all three heuristics was 12,955, while the number of links we could create was 7084. Again, 45% of the connections we found where redundant for actually linking the channels. Here the number of reused keys within a single channel graph was 113, which is relatively small compared to the 5758 other redundant relationships. The channel network is shown in Figure 16 is more connected than the mainnet in Figure 15, with the testnet network having over twice as many nodes, but only having 617 more components. We can see this difference clearly with the large component in the middle of the testnet graph. As the LN is one network, it should consist of one dynamic component, meaning a historic network graph should also ideally have one large component.