Hello everyone,
I'm new to this forum, I hope that I am posting appropriately.
Consider any unweighted and undirected graph, or adjacency matrix, [imath]\bm{G}[/imath], composed of N rows and N columns, i.e., there are N nodes in the graph. The entry [imath]g_{ij}[/imath] of row i and column j of [imath]\bm{G}[/imath] equals 1 if nodes i and j are linked, and 0 otherwise (note [imath]g_{ij}=g_{ji}[/imath]). The Bonacich centralities of nodes are given in vector [math]\bm{B}=[\bm{I}_{N}- \lambda \bm{G}]^{-1}\bm{1},[/math] where row i denotes the Bonacich centrality of node i, where [imath]\bm{I}_{N}[/imath] is the identity matrix of dimension N, [imath]\lambda[/imath] is a parameter such that [imath]0<\lambda<\frac{1}{N-1}[/imath], and [imath]\bm{1}[/imath] 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., [imath]g_{12}=g_{13}=g_{23}=0[/imath]. 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:
[math](\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[/math]
where [imath]m_{ij}[/imath] corresponds to the entry of row i and column j of matrix [imath]\bm{M}=[\bm{I}_{N}- \lambda \bm{G}]^{-1}[/imath], and [imath]b_i[/imath] and [imath]b_j[/imath] correspond to the Bonacich centralities of nodes i and j respectively. It is also worth noting that for any pair i,j, we have [imath]m_{ij}=m_{ji}[/imath].
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!
I'm new to this forum, I hope that I am posting appropriately.
Consider any unweighted and undirected graph, or adjacency matrix, [imath]\bm{G}[/imath], composed of N rows and N columns, i.e., there are N nodes in the graph. The entry [imath]g_{ij}[/imath] of row i and column j of [imath]\bm{G}[/imath] equals 1 if nodes i and j are linked, and 0 otherwise (note [imath]g_{ij}=g_{ji}[/imath]). The Bonacich centralities of nodes are given in vector [math]\bm{B}=[\bm{I}_{N}- \lambda \bm{G}]^{-1}\bm{1},[/math] where row i denotes the Bonacich centrality of node i, where [imath]\bm{I}_{N}[/imath] is the identity matrix of dimension N, [imath]\lambda[/imath] is a parameter such that [imath]0<\lambda<\frac{1}{N-1}[/imath], and [imath]\bm{1}[/imath] 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., [imath]g_{12}=g_{13}=g_{23}=0[/imath]. 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:
[math](\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[/math]
where [imath]m_{ij}[/imath] corresponds to the entry of row i and column j of matrix [imath]\bm{M}=[\bm{I}_{N}- \lambda \bm{G}]^{-1}[/imath], and [imath]b_i[/imath] and [imath]b_j[/imath] correspond to the Bonacich centralities of nodes i and j respectively. It is also worth noting that for any pair i,j, we have [imath]m_{ij}=m_{ji}[/imath].
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!