  When randomly displacing the nodes of a crystalline and unstressed spring
network, we find that the Possion's ratio decreases with the increase of
structural disorder and even becomes negative. Employing our finding that
longer springs tend to contribute more to the shear modulus but less to the
bulk modulus, we are able to achieve negative Poisson's ratio with lower
structural disorder by attributing each spring a length dependent stiffness.
Even with perfect crystalline structure, the network can have negative
Possion's ratio, if the stiffness of each spring is set by its virtual length
after a virtual network distortion. We also reveal that the nonaffine
contribution arising from the structural or spring constant disorder produced
in some cooperative way by network distortion is essential to the emergence of
negative Poisson's ratio.
