  The paper attempts to develop a suitable accessibility index for networks
where each link has a value such that a smaller number is preferred like
distance, cost, or travel time. A measure called distance sum is characterized
by three independent properties: anonymity, an appropriately chosen
independence axiom, and dominance preservation, which requires that a node not
far to any other is at least as accessible.
  We argue for the need of eliminating the independence property in certain
applications. Therefore generalized distance sum, a family of accessibility
indices, will be suggested. It is linear, considers the accessibility of
vertices besides their distances and depends on a parameter in order to control
its deviation from distance sum. Generalized distance sum is anonymous and
satisfies dominance preservation if its parameter meets a sufficient condition.
Two detailed examples demonstrate its ability to reflect the vulnerability of
accessibility to link disruptions.
