  We consider the Keller-Segel model of chemotaxis on one-dimensional networks.
Using a variational characterization of solutions, positivity preservation,
conservation of mass, and energy estimates, we establish global existence of
weak solutions and uniform bounds. This extends related results of Osaki and
Yagi to the network context. We then analyze the discretization of the system
by finite elements and an implicit time-stepping scheme. Mass lumping and
upwinding are used to guarantee the positivity of the solutions on the discrete
level. This allows us to deduce uniform bounds for the numerical approximations
and to establish order optimal convergence of the discrete approximations to
the continuous solution without artificial smoothness requirements. In
addition, we prove convergence rates under reasonable assumptions. Some
numerical tests are presented to illustrate the theoretical results.
