  Disordered spring networks that are undercoordinated may abruptly rigidify
when sufficient strain is applied. Since the deformation in response to applied
strain does not change the generic quantifiers of network architecture - the
number of nodes and the number of bonds between them - this rigidity transition
must have a geometric origin. Naive, degree-of-freedom based mechanical
analyses such as the Maxwell-Calladine count or the pebble game algorithm
overlook such geometric rigidity transitions and offer no means of predicting
or characterizing them. We apply tools that were developed for the topological
analysis of zero modes and states of self-stress on regular lattices to
two-dimensional random spring networks, and demonstrate that the onset of
rigidity, at a finite simple shear strain $\gamma^\star$, coincides with the
appearance of a single state of self stress, accompanied by a single floppy
mode. The process conserves the topologically invariant difference between the
number of zero modes and the number of states of self stress, but imparts a
finite shear modulus to the spring network. Beyond the critical shear, we
confirm previously reported critical scaling of the modulus. In the
sub-critical regime, a singular value decomposition of the network's
compatibility matrix foreshadows the onset of rigidity by way of a continuously
vanishing singular value corresponding to nascent state of self stress.
