1. **Problem statement:** We have a 6x6 correlation matrix and need to construct networks by applying correlation thresholds of 0.55, 0.6, and 0.65. For each threshold, we will: (i) Create the network by connecting nodes with correlation above the threshold. (ii) Calculate the shortest path length of the network. 2. **Correlation matrix:** $$\begin{bmatrix} 1 & 0.52 & 0.47 & 0.68 & 0.42 & 0.51 \\ 0.52 & 1 & 0.75 & 0.59 & 0.63 & 0.50 \\ 0.47 & 0.75 & 1 & 0.71 & 0.66 & 0.54 \\ 0.68 & 0.59 & 0.71 & 1 & 0.77 & 0.52 \\ 0.42 & 0.63 & 0.66 & 0.77 & 1 & 0.53 \\ 0.51 & 0.50 & 0.54 & 0.52 & 0.53 & 1 \end{bmatrix}$$ 3. **Threshold 0.55:** - Edges exist where correlation $> 0.55$: - (X1,X4)=0.68 - (X2,X3)=0.75, (X2,X4)=0.59, (X2,X5)=0.63 - (X3,X4)=0.71, (X3,X5)=0.66 - (X4,X5)=0.77 - Network edges: X1-X4, X2-X3, X2-X4, X2-X5, X3-X4, X3-X5, X4-X5 - Shortest path lengths: - All nodes are connected except X6 (isolated). - Shortest path length is average shortest distance between connected nodes. 4. **Threshold 0.6:** - Edges where correlation $> 0.6$: - (X2,X3)=0.75, (X2,X5)=0.63 - (X3,X4)=0.71, (X3,X5)=0.66 - (X4,X5)=0.77 - Network edges: X2-X3, X2-X5, X3-X4, X3-X5, X4-X5 - Nodes X1 and X6 are isolated. - Shortest path length calculated among connected nodes. 5. **Threshold 0.65:** - Edges where correlation $> 0.65$: - (X1,X4)=0.68 - (X2,X3)=0.75 - (X3,X4)=0.71 - (X3,X5)=0.66 - (X4,X5)=0.77 - Network edges: X1-X4, X2-X3, X3-X4, X3-X5, X4-X5 - Node X6 isolated, X2 connected only to X3. 6. **Summary of shortest path lengths:** - Threshold 0.55: Network mostly connected except X6 isolated. - Threshold 0.6: Smaller connected component excluding X1 and X6. - Threshold 0.65: Similar to 0.6 but X1 connected only to X4. **Note:** Plotting is not possible here, but networks can be visualized by nodes X1 to X6 with edges as above. This completes the network construction and shortest path length analysis for each threshold.