I have a large sparse nonnegative asymmetric matrix $A$. Since the matrix $A$ is nonnegative, its spectral radius $\rho(A)$ is an eigenvalue of it. But $A$ may have other eigenvalues being the same modulus as $\rho(A)$.
Can the power method give $\rho(A)$?
No. This has nothing to do with the moduli of other eigenvalues. For instance, when $A$ is not primitive, even if the moduli of other eigenvalues of $A$ are strictly smaller than $\rho(A)$, the power method still doesn't always converge. For an illustrative example, consider $$ A=\pmatrix{0&2\\ 1&0},\ x_0=\pmatrix{1\\ 1}. $$ Using the power method, the iterates will oscillate between $x_0$ and $x_1=(\frac23,\frac13)^T$ and none of them is an eigenvector of $A$.