# Stochastic Modelling of Selfish Mining in Proof-of-Work Protocols +
## 5 Results
### Relative Selfish Revenue
#### 5.1.3 Network Topologies  
Next, we consider the effect of varying network topologies on relative selfish revenue. Figure 4 shows relative selfish revenue as a function of selfish mining power under uniform random, Erdos-Renyi and Barabasi-Albert topologies. Again, we show results for two latency regimes. It is apparent that the dynamics of the system are very robust to changes in topology for both latency regimes.  
Overall, we can consolidate that the profitability of a selfish mining strategy is very robust to changes in the underlying network topology or hashing power distribution. The exception being powerful honest nodes in environments where block diffusion is very slow. In addition, the system in which hashing power follows a power-law distribution is a relatively unfavourable environment for a selfish miner.  
Furthermore, to investigate the importance of centrality within the topology, the selfish miner is assigned to various positions in the network using various measures. In particular, we compare the performance difference between a random assignment, an assignment based on the standard betweenness centrality, and an assignment based on the new concept of _hashing betweenness_ (introduced in Section 3.2).  
Figure 3: Relative selfish revenue as a function of selfish hashing power (_a_) under different distribution: (**a**,**b**) in which the hashing powers of all miners follow the uniform distribution; in (**c**,**d**) miners’ hashing powers follow the power-law distribution; and in (**e**,**f**) miners’ hashing powers follow the exponential distribution. In addition, the subfigures in left column show the results when network has very high latency \(\lambda_{nd}=0.03\), and those in right column show the results with low network latency \(\lambda_{nd}=79.43\).  
In Figure 5, for various centrality measures, we show the relative selfish revenue as a function of both latency (for \(\alpha=0.4\)) and selfish hashing power (for \(\lambda_{nd}=79.43\)) under the Barabasi-Albert topology and exponential hashing power distribution. Finally, we do not observe any notable effect of any centrality measure, despite choosing a heterogeneous network topology. The reason for this is likely to be the relatively small network size (\(N=100\)), as well as the fact that the network is very well connected (nodes have an average degree of 10). Although it seems that the location of the selfish miner in the network has almost no effect on the profitability, the final conclusion still needs more experiments using more realistic network parameters. Therefore, future research could re-investigate the effect of centrality measures on a larger network.  
Figure 4: Relative selfish revenue as a function of selfish hashing power (\(\alpha\)) for different network topologies: in (**a**,**b**) the network is in the uniform random topologies; in (**c**,**d**) the network is in the Erdos-Renyi topologies; and in (**e**,**f**) the network is in the Barabasi-Alber topologies. In addition, the subfigures in left column show the results when network has very high latency \(\lambda_{nd}=0.03\), and those in right column show the results with low network latency \(\lambda_{nd}=79.43\).