Bonacich centrality (graph theory)

[DSR75]

Hello everyone,

I'm new to this forum, I hope that I am posting appropriately.

Consider any unweighted and undirected graph, or adjacency matrix, $\bm{G}$, composed of N rows and N columns, i.e., there are N nodes in the graph. The entry $g_{ij}$ of row i and column j of $\bm{G}$ equals 1 if nodes i and j are linked, and 0 otherwise (note $g_{ij}=g_{ji}$). The Bonacich centralities of nodes are given in vector $$\bm{B}=[\bm{I}_{N}- \lambda \bm{G}]^{-1}\bm{1},$$ where row i denotes the Bonacich centrality of node i, where $\bm{I}_{N}$ is the identity matrix of dimension N, $\lambda$ is a parameter such that $0<\lambda<\frac{1}{N-1}$, and $\bm{1}$ is vector of one's, composed of N rows.

Suppose that some nodes 1, 2 and 3 are not linked between each other, i.e., $g_{12}=g_{13}=g_{23}=0$. Is it possible that node 1 gains more Bonacich centrality by linking with node 2 than with node 3, that node 2 gains more Bonacich centrality by linking with node 3 than with node 1, and that node 3 gains more Bonacich centrality by linking with node 1 than with node 2?

It is worth noting that the Bonacich centrality that any agent i gains by linking with some agent j equals:

$$(\frac{1-\lambda m_{ij}}{(1-\lambda m_{ij})^2-\lambda^2m_{ii}m_{jj}}-1)b_i+(\frac{1}{(1-\lambda m_{ij})^2-\lambda^2m_{ii}m_{jj}})\lambda m_{ii}b_j$$
where $m_{ij}$ corresponds to the entry of row i and column j of matrix $\bm{M}=[\bm{I}_{N}- \lambda \bm{G}]^{-1}$, and $b_i$ and $b_j$ correspond to the Bonacich centralities of nodes i and j respectively. It is also worth noting that for any pair i,j, we have $m_{ij}=m_{ji}$.

Even if you do not know or find the answer to the question above, any hint on how I could arrive to the answer is appreciated. Thanks a lot!