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 shortcuts 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 localized region of small |?| values. In the case of modular networks, we can identify contributions of 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 localized 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 diagonalization results.

Modular construction of dynamic nucleodendrimers

Isoguanosine-containing dendritic small molecules self-assemble into decameric nucleodendrimers as observed by 1D NMR spectroscopy, 2D DOSY, and mass spectrometry. In particular, apolar building blocks readily form pentameric structures in acetonitrile while the presence of alkali metals promotes the formation of stable decameric assemblies with a preference for cesium ions. Remarkably, co-incubation of guanosine and isoguanosine-containing nucleodendrons results in the formation of decameric structures in absence of added salts. Further analysis of the mixture indicated that guanosine derivatives facilitate the formation, but are not involved in decameric structures; a process reminiscent of molecular crowding. This molecular system provides a powerful canvas for the rapid and modular assembly of polyfunctional dendritic macromolecules. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.