Spectra of Modular and Small-World Matrices

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate spectra of a certain class of small-world matrices generated from random graphs by introducing short-cuts via additional random connectivity components. Both adjacency matrices and the associated graph Laplacians are investigated. For the Laplacians, we find Lifshitz type singular behaviour of the spectral density in a localised region of small $|\lambda|$ values. In the case of modular networks, we can identify contributions local densities of state from individual modules. For small-world networks, we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states, exhibiting only localised states in the band-gaps. Results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances, and with direct diagonalisation results.Comment: 18 pages, 5 figure

Effects of membrane alterations on bacterial retention

The study shows the respective roles of skin and support of an ultrafiltration membrane in the retention mechanisms of bacteria (Escherichia coli). For this, pinholes defects of 5–200 μm in diameter were performed through ultrafiltration polymeric membranes and their impact was assessed on bacterialretention in a stirred cell when the transmembrane pressure is set at 0.5 bars. Various techniques have been used to make the defects such as a microhardness tester or femtosecond lasers. As long as the selective skin is not altered through its whole thickness, the membrane keeps a retention efficiency equivalent to the one of an uncompromised membrane. The retention by the macroporous support is also investigated. In case of membrane with defects of cylindrical geometry, experimental results are compared to calculated data obtained with a pore flow model, and the validity of this model is discussed