Data for several river networks show algebraically decaying distribution functions for the mean annual discharge and the dissipation rate (Rodr¿guez-Iturbe et al., 1992). We derive a relation between the exponent &agr; in the integrated mean annual discharge distribution P(Q≥q)~q-&agr; and the topological dimension Dt of the network, &agr;=1-1/Dt. Using the experimentally determined value Dt≈1.8 (Tarboton et al., 1988) we find &agr;≈0.45, in good agreement with the data of (Rodr¿guez-Iturbe et al., 1992). For the random model (Shreve, 1967, 1969; Smart and Werner, 1974) we find an exponent of 1/2. ¿ American Geophysical Union 1994