The connectivity hypothesis, central to the increasingly influential symptom network approach to psychopathology, proposes that stronger connectivity among symptoms heightens vulnerability to mental disorders. We provide an analytic derivation of this hypothesis using mean-field Ising models of depression, both in the standard −1/1 formulation and in a 0/1 variant where nodes represent symptoms as absent or present. Applying bifurcation theory, we derive the bifurcation sets and phase transition structure directly from the mean-field equations. This formal characterization elucidates how connectivity shapes system dynamics and, consistent with the network theory of mental disorders, demonstrates that increasing connectivity amplifies the risk of transitions into unhealthy states.