Heisenberg model in a random field: phase diagram and tricritical behavior

By using the differential operator technique and the effective field theory scheme we study the tricritical behavior of Heisenberg classical model of spin-1/2 in a random field. The phase diagram in the T-h plane on a square and simple cubic lattice for a cluster with two spins is obtained when the random field is bimodal distributed.Comment: 10 pages, 1 figur

Syntonets: Toward A Harmony-Inspired General Model of Complex Networks

We report an approach to obtaining complex networks with diverse topology, here called syntonets, taking into account the consonances and dissonances between notes as defined by scale temperaments. Though the fundamental frequency is usually considered, in real-world sounds several additional frequencies (partials) accompany the respective fundamental, influencing both timber and consonance between simultaneous notes. We use a method based on Helmholtz's consonance approach to quantify the consonances and dissonances between each of the pairs of notes in a given temperament. We adopt two distinct partials structures: (i) harmonic; and (ii) shifted, obtained by taking the harmonic components to a given power $\beta$, which is henceforth called the anharmonicity index. The latter type of sounds is more realistic in the sense that they reflect non-linearities implied by real-world instruments. When these consonances/dissonances are estimated along several octaves, respective syntonets can be obtained, in which nodes and weighted edge represent notes, and consonance/dissonance, respectively. The obtained results are organized into two main groups, those related to network science and musical theory. Regarding the former group, we have that the syntonets can provide, for varying values of $\beta$, a wide range of topologies spanning the space comprised between traditional models. Indeed, it is suggested here that syntony may provide a kind of universal complex network model. The musical interpretations of the results include the confirmation of the more regular consonance pattern of the equal temperament, obtained at the expense of a wider range of consonances such as that in the meantone temperament. We also have that scales derived for shifted partials tend to have a wider range of consonances/dissonances, depending on the temperament and anharmonicity strength

On degree-degree correlations in multilayer networks

We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterogeneity between layers on the spreading of diseases. Our results highlight the need of studying degree-degree correlations on multilayer systems, instead of on aggregated networks.Comment: 8 pages, 3 figure

A polynomial eigenvalue approach for multiplex networks

We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive interesting results based on an interpretation of the traditional eigenvalue problem. More specifically, we reduce the dimensionality of our matrices but increase the power of the characteristic polynomial, i.e, a polynomial eigenvalue problem. Such an approach may sound counterintuitive at first glance, but it allows us to relate the quadratic problem for a 2-Layer multiplex system with the spectra of the aggregated network and to derive bounds for the spectra, among many other interesting analytical insights. Furthermore, it also permits to directly obtain analytical and numerical insights on the eigenvalue behavior as a function of the coupling between layers. Our study includes the supra-adjacency, supra-Laplacian, and the probability transition matrices, which enable us to put our results under the perspective of structural phases in multiplex networks. We believe that this formalism and the results reported will make it possible to derive new results for multiplex networks in the future.Comment: 15 pages including figures. Submitted for publicatio