A search for "algebraic connectivity" of a graph may be helpful, as well as the extensive literature on rapid mixing.
The problems mainly occur when the graph is nearly disconnected because different component-like sets have too few edges between them, or edges with weights too close to zero (which is the weight of a non-edge). If you make certain edges exponentially small, the cover time also becomes exponential.
Your adjacency matrix has Perron root and spectral radius $W$, and the usual bounds on mixing or covering time are in terms of the eigenvalue of second-highest magnitude, as a fraction of $W$. (Bipartite graphs are a special case, where $-W$ is also an eigenvalue.) The limiting distribution, in this case uniform, is the $W$ eigenvector, and is the only all-positive eigenvector. Convergence to uniform is exponential, but with base the ratio
of $W$ to other eigenvalues, by the spectral decomposition theorem. Sometimes you can actually get a clue to the worst component-like pieces from the positive and negative parts of large eigenvectors.