  Centrosymmetry often mediates Perfect State Transfer (PST) in various complex
systems ranging from quantum wires to photosynthetic networks. We introduce the
Deformed Centrosymmetric Ensemble (DCE) of random matrices, $H(\lambda) \equiv
H_+ + \lambda H_-$, where $H_+$ is centrosymmetric while $H_-$ is
skew-centrosymmetric. The relative strength of the $H_\pm$ prompts the system
size scaling of the control parameter as $\lambda = N^{-\frac{\gamma}{2}}$. We
propose two quantities, $\mathcal{P}$ and $\mathcal{C}$, quantifying centro-
and skewcentro-symmetry, respectively, exhibiting second order phase
transitions at $\gamma_\text{P}\equiv 1$ and $\gamma_\text{C}\equiv -1$. In
addition, DCE posses an ergodic transition at $\gamma_\text{E} \equiv 0$. Thus
equipped with a precise control of the extent of centrosymmetry in DCE, we
study the manifestation of $\gamma$ on the transport properties of complex
networks. We propose that such random networks can be constructed using the
eigenvectors of $H(\lambda)$ and establish that the maximum transfer fidelity,
$F_T$, is equivalent to the degree of centrosymmetry, $\mathcal{P}$.
