where m is an overall rate scalar, S~fs jk g is a K|K symmetric matrix and P~diag(p 1 , . . . , p K ). Infinitesimal rate matrices of this form generate reversible Markov chains, such that p j l jk~pk l kj and p j p jk (t)~p k p kj (t), ð3Þ placing many restrictions on the underlying geographic process. We discuss these limitations and modeling extensions that allow for irreversible chains in the Discussion. In its most general timereversibile (GTR) form, L contains (Kz2)(K{1)=2 free parameters, with P donating K{1 together with mS's K(K{1)=2 off-diagonal entries. Following standard practice, we normalize entries in S such that m measures the expected (with respect to P) number of transitions per unit time t. One illuminating perspective from which to view a CTMC is that of a random walk on a graph G. From this perspective, the possible realizations of the chain fS k g correspond to the vertex set of G. Between the vertices lie edges that record the infinitesimal transition rates. For example, between S j and S k sits l jk . As a continuous-time random walk, a particle, starting at vertex S j at time 0, first waits an Exponential amount of time with rate {l jj and then randomly decides to which neighboring vertix S k to move with probability {l jk =l jj . Now on S k , the process repeats. Neighboring vertices are those for which a single edge connects them. For character evolution, ''complete'' graphs find almost exclusive use, such that edges exist between all pairs of vertices. At a minimum, however, the graph must remain ''connected'', such that it remains possible to walk between any two vertices on G.