An introduction to spectral distances in networks (extended version)

Many functions have been recently defined to assess the similarity among networks as tools for quantitative comparison. They stem from very different frameworks - and they are tuned for dealing with different situations. Here we show an overview of the spectral distances, highlighting their behavior in some basic cases of static and dynamic synthetic and real networks

Dispersion relations and wave operators in self-similar quasicontinuous linear chains

We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of nonlocal particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasicontinuous linear chain of infinite length with a spatially self-similar distribution of nonlocal interparticle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function that exhibits self-similar and fractal features. We also derive a continuum approximation, which relates the self-similar Laplacian to fractional integrals, and yields in the low-frequency regime a power-law frequency-dependence of the oscillator density

Bose-Einstein Condensation on inhomogeneous complex networks

The thermodynamic properties of non interacting bosons on a complex network can be strongly affected by topological inhomogeneities. The latter give rise to anomalies in the density of states that can induce Bose-Einstein condensation in low dimensional systems also in absence of external confining potentials. The anomalies consist in energy regions composed of an infinite number of states with vanishing weight in the thermodynamic limit. We present a rigorous result providing the general conditions for the occurrence of Bose-Einstein condensation on complex networks in presence of anomalous spectral regions in the density of states. We present results on spectral properties for a wide class of graphs where the theorem applies. We study in detail an explicit geometrical realization, the comb lattice, which embodies all the relevant features of this effect and which can be experimentally implemented as an array of Josephson Junctions.Comment: 11 pages, 9 figure

Graphs and networks theory

This chapter discusses graphs and networks theory

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

Maximal accretive extensions of Schr\"odinger operators on vector bundles over infinite graphs

Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential self-adjointness of a perturbation of this Laplacian by an operator-valued potential. Additionally, we give a sufficient condition for the resulting Schr\"odinger operator to serve as the generator of a strongly continuous contraction semigroup in the corresponding l^{p}-space.Comment: We have made significant revisions of the previous version. In particular, this version has a new title: "Maximal Accretive Extensions of Schr\"odinger Operators on Vector Bundles over Infinite Graphs." The final version will appear in Integral Equations and Operator Theory and will be availableat Springer via http://dx.doi.org/10.1007/s00020-014-2196-

Analysis of heat kernel highlights the strongly modular and heat-preserving structure of proteins

In this paper, we study the structure and dynamical properties of protein contact networks with respect to other biological networks, together with simulated archetypal models acting as probes. We consider both classical topological descriptors, such as the modularity and statistics of the shortest paths, and different interpretations in terms of diffusion provided by the discrete heat kernel, which is elaborated from the normalized graph Laplacians. A principal component analysis shows high discrimination among the network types, either by considering the topological and heat kernel based vector characterizations. Furthermore, a canonical correlation analysis demonstrates the strong agreement among those two characterizations, providing thus an important justification in terms of interpretability for the heat kernel. Finally, and most importantly, the focused analysis of the heat kernel provides a way to yield insights on the fact that proteins have to satisfy specific structural design constraints that the other considered networks do not need to obey. Notably, the heat trace decay of an ensemble of varying-size proteins denotes subdiffusion, a peculiar property of proteins