Transport in deformed centrosymmetric networks

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, $\mathcal{P}$.