# Computing Persistent Homology of Directed Flag Complexes
## 2 Results
### Sample Computations
For \(\beta_{3}\) the theoretical bound is clearly not as good, but here as in all other cases where precise computation was not carried out, the trajectory of the approximated Betti numbers as the approximation parameter grows suggests that the number computed is actually much closer to the real Betti numbers than the theoretical error suggests (Figure 1). \begin{table}
\begin{tabular}{l r r r r r r} \hline \hline
**Approx.** & \(\delta_{1}\) & \(\delta_{2}\) & \(\delta_{3}\) & \(\delta_{4}\) & \(\delta_{5}\) \\ \hline
1 & 7,724,934 & 56,746,602 & 19,092,109 & 664,481 & 4162 \\
10 & 6,714,811 & 15,944,792 & 796,559 & 372 & 0 \\
100 & 179,794 & 4,107,019 & 6414 & 0 & 0 \\
1000 & 14,209 & 1,902,229 & 42 & 0 & 0 \\
10,000 & 1002 & 1,054,397 & 0 & 0 & 0 \\
100,000 & 27 & 664,857 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Number of skipped columns of the coboundary matrix for the Blue Brain Project graph with respect to different values of the Approximate parameter. The approximation accuracy of \(\beta_{i}\) depends theoretically on the number of columns skipped in the coboundary matrices for \(\delta_{i-1}\) and \(\delta_{i}\). \begin{table}
\begin{tabular}{l r r r r r r} \hline \hline
**Approx.** & **Time** & **Memory** & \(\beta_{1}\) & \(\beta_{2}\) & \(\beta_{3}\) & \(\beta_{4}\) & \(\beta_{5}\) \\ \hline
1 & 40 min 44 s & 11.77 GB & 14,640 & 9,831,130 & 30,219,098 & 513,675 & 785 \\
10 & 38 min 44 s & 11.96 GB & 15,148 & 14,831,585 & 12,563,309 & 13,278 & 40 \\
100 & 40 min 29 s & 12.69 GB & 15,188 & 13,956,297 & 7,171,621 & 9898 & 37 \\
1000 & 55 min 40 s & 15.42 GB & 15,283 & 14,598,249 & 5,780,569 & 9822 & 37 \\
10,000 & 2 h 24 min 17 s & 23.78 GB & 15,410 & 14,872,057 & 5,219,666 & 9821 & 37 \\
100,000 & 12 h 15 min 3 s & 52.96 GB & 15,438 & 14,992,658 & 4,951,674 & 9821 & 37 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of Flagser on the Blue Brain Project graph with respect to different values of the Approximate parameter. The Betti numbers computed are an approximation to the true Betti numbers. See Table 4 for accuracy. We performed a similar computation with an Erdos-Renyi random directed graph with the same number of vertices as the Blue Brain Project column and the same average probability of connection (Tables 5 and 6). Notice that there was a very substantial difference in performance between the Blue Brain Project graph and a random graph with the same size and density parameters. The difference can be explained by the much higher complexity of the Blue Brain Project graph, as witnessed by the Betti numbers. A remarkable feature of Flagser is its parallelisable construction algorithm of the directed flag complex. To demonstrate the capability of the software, we computed some large scale examples. In particular, we computed the cell counts of two sample circuits produced by the Blue Brain Project. The first is a reconstruction of the somatosensory cortex of a rat (Figure 2), and the second is the so called PL region [8] of the reconstruction of the full neocortex of a mouse (Figure 3). We also computed the simplex counts on some large non-biological networks (Figure 4). \begin{table}
\begin{tabular}{l c c c} \hline
**Approx.** & \(\delta_{1}\) & \(\delta_{2}\) & \(\delta_{3}\) \\ \hline
1 & 7,692,010 & 719,041 & 124 \\
10 & 1,501,909 & 3 & 0 \\
100 & 371,920 & 0 & 0 \\
1000 & 219,501 & 0 & 0 \\
10,000 & 174,628 & 0 & 0 \\
100,000 & 155,685 & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 6: Number of skipped columns of the coboundary matrix for an Erdős-Renyi graph with respect to different values of the Approximate parameter in dimensions from 1 to 3. Figure 1: Trajectories of \(\beta_{2}\) and \(\beta_{3}\) as a function of the approximation parameter. A change in order of magnitude in the approximation parameter results in a small change in the Betti number computed. Figure 4: _Cont_  
Figure 3: The simplex count for the PL-Left region of a mouse neocortex (local connections only).