Netons: Vibrations of Complex Networks

We consider atoms interacting each other through the topological structure of a complex network and investigate lattice vibrations of the system, the quanta of which we call netons for convenience. The density of neton levels, obtained numerically, reveals that unlike a local regular lattice, the system develops a gap of a finite width, manifesting extreme rigidity of the network structure at low energies. Two different network models, the small-world network and the scale-free network, are compared: The characteristic structure of the former is described by an additional peak in the level density whereas a power-law tail is observed in the latter, indicating excitability of netons at arbitrarily high energies. The gap width is also found to vanish in the small-world network when the connection range r = 1. PACS numbers: 89.75.Fb, 63.20.Dj Netons: Vibrations of Complex Networks 2 A variety of network systems such as computer networks, neuronal networks, biochemical networks and social networks, possesses complex topological structure, which can be described neither by regular networks nor by completely random networks [1]. Among various models for generating such complex networks, the WattsStrogatz (WS) model [2] and the Barabási-Albert (BA) model [3] provide the two most representative ones. Both model networks are characterized by very small network diameters proportional to the logarithm of the network size, which is dubbed “smallworld behavior” and commonly observed in various real networks. Other important properties of many real network systems include high clustering [2] and scale-free degree distributions [3]. While most existing studies of complex networks have focused on the geometrical and topological characterization, there now emerges an increasing number of studies paying attention to the dynamics defined on the networks. Lattice vibrations and associated phonon excitations on local regular networks, e.g., one-dimensional (1D) chains or two-dimensional square lattices, have been a textbook example of the introductory solid-state physics. In the case of such regular networks, atoms located on vertices interact only with nearest neighboring atoms, resulting in the standard linear phonon dispersion. On the other hand, when the interaction between atoms takes the form of the connection topology of a complex network, ingredients of long-range nature enter into the system. For example, the WS network has both local edges and long-range shortcuts [2], which in this work are interpreted as the coexistence of the local and the long-range interactions. Existing studies on phonon excitations have mostly been performed on the local regular networks such as d-dimensional hypercubic lattices or on the fractal geometry [4]. Recently, the spectral properties of lattice vibrations have been studied in a small-world network similar to the WS network [5]. However, effects of the local interaction range and the dependence of the spectral properties on the rewiring probability have not been investigated in detail. The main purpose of this work is to study in detail the vibration spectra of both the WS network and the BA network, in comparison with those of regular lattices. We thus consider atoms located on vertices of a complex network and interacting with each other connected via edges of the network, and investigate lattice vibrations and phonon excitations, with emphasis on the effects of the network topology. The results of such study can have implications in the mechanical property of a system of long chains bent in a complicated way. For example, in a bundle of long flexible polymer chains, some monomers which were separated by a long distance along a chain can make new couplings, building shortcuts. Throughout this work, we refer to phonons in a complex network as netons, to manifest qualitatively different characteristics due to the fact that the underlying geometric structure is not a usual periodic lattice. By means of numerical diagonalization of the dynamic matrix, neton excitation spectra are obtained and the corresponding density of neton levels are computed. It is revealed that the complex network in general has a finite gap in the level density, indicating the absence of low-energy excitations. The dependence of the gap width on the rewiring probability and the local interaction range is investigated in detail. The existence of the gap is Netons: Vibrations of Complex Networks 3 also reflected by the vanishingly small specific heat at low temperatures. In the WS network three singularities are observed in the neton level density: Among them two are shown to be reminiscence of the van Hove singularity present in the local regular network, while the third one originates from the complex network structure. The BA network is also considered and the scale-freeness results in a power-law distribution of the level density, suggesting excitability of netons with arbitrarily high energies. We build the WS network following Ref. [2]: First, a 1D regular network with only local connections of range r is constructed under the periodic boundary conditions. Next, each local edge (or link) is visited once, and with the rewiring probability P removed and reconnected to a randomly chosen vertex. After a whole sweep over the entire network, the average number of shortcuts in the system of size N is given by NPr. In the WS network built as above, an atom is put on every vertex whereas an edge connecting two vertices represents the coupling between the two atoms located at the two vertices. For simplicity, we assume that all atoms are identical, each having mass M and moving only along the direction of the chain. The equation of motion for the lth atom at the position xl then reads Mẍl = C ∑