I am studying a paper called "Effect of the media on the opinion dynamics in online social networks" and I cannot understand a passage.
Let's first define a matrix A (n x n) which elements are $a_{ij}=\frac{1-v_i}{(1-v_i)d_i+v_i}$ where $v_i$ can assume values from 0 to 1 (continuous values) and $d_i$ can assume values from 1 to n-1 (discrete values).
I don't understand this passage:
"Matrix A meets the conditions (1) $a_{ij} \ge 0 \forall i,j$ and (2) $\sum_j{a_{ij}} \le 1 \forall i$, so is sub-stochastic matrix which can’t be reduced. Define $\rho (A)=max_i|\lambda_i(A)|$ as the spectral radius of A. According to the Perron-Frobenius theorem, the largest eigenvalue of A should be positive, where $ 0<\lambda_1<1$ and $\rho(A)=\lambda_1$. Hence, $lim_{T\rightarrow \infty}A^T=0$."
My problems are:
Why does $ 0<\lambda_1<1$ holds and how can I prove it? Is it a consequence of the fact that A is sub-stochastic? or is it a consequence of the Perron-Freobenius theorem?
Why does the limit hold and how can I prove it?
Someone can help? Thank you in advance! :)
Yes, $0 < \lambda_1 < 1$ is stated to be a consequence of the Perron-Frobenius theorem (cf. , e.g., point 11 in the list under "Statement" in the linked Wikipedia page).
The limit of $A^T$ for $T \rightarrow \infty$ is dominated by a contribution from the largest eigenvalue (the eigenvalue with the largest absolute value), which behaves as $\lambda_1^T$. Which goes to zero because $0 < \lambda_1 < 1$. This is also mentioned in the Wikipedia article: