Tactical Voting in Plurality Elections

How often will elections end in landslides? What is the probability for a head-to-head race? Analyzing ballot results from several large countries rather anomalous and yet unexplained distributions have been observed. We identify tactical voting as the driving ingredient for the anomalies and introduce a model to study its effect on plurality elections, characterized by the relative strength of the feedback from polls and the pairwise interaction between individuals in the society. With this model it becomes possible to explain the polarization of votes between two candidates, understand the small margin of victories frequently observed for different elections, and analyze the polls' impact in American, Canadian, and Brazilian ballots. Moreover, the model reproduces, quantitatively, the distribution of votes obtained in the Brazilian mayor elections with two, three, and four candidates.Comment: 7 pages, 4 figure

Network Geometry and Complexity

(28 pages, 11 figures)Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. More in general, higher-order networks can be cell-complexes formed by gluing convex polytopes along their faces. Interestingly, higher order networks have a natural geometric interpretation and therefore constitute a natural way to explore the discrete network geometry of complex networks. Here we investigate the rich interplay between emergent network geometry of higher order networks and their complexity in the framework of a non-equilibrium model called Network Geometry with Flavor. This model, originally proposed for capturing the evolution of simplicial complexes, is here extended to cell-complexes formed by subsequently gluing different copies of an arbitrary regular polytope. We reveal the interplay between complexity and geometry of the higher order networks generated by the model by studying the emergent community structure and the degree distribution as a function of the regular polytope forming its building blocks. Additionally, we discuss the underlying hyperbolic nature of the emergent geometry and we relate the spectral dimension of the higher-order network to the dimension and nature of its building blocks

Complex Network Geometry and Frustrated Synchronization

12 pages, 5 figuresWe are grateful for financial support from Spanish MINECO (under excelence project FIS2017-84256-P; FEDER funds) and from “Obra Social La Caixa”

Design and characterization of electrons in a fractal geometry

The dimensionality of a quantum system plays a decisive role in its electronic spectral and transport properties. In 3D, electrons behave as a non-interacting Fermi liquid, whereas in 1D interactions are relevant. On the other hand, in 2D exotic phenomena such as charge fractionalization may occur. However, very little is known about electrons in fractional dimensions. Here, we design and characterize an electronic Sierpinski triangle fractal in real and reciprocal space by confining the surface-state electrons of Cu(111) with adsorbed CO molecules. We observe single-electron wave functions in real space with a fractal dimension of 1.58 as well as a subdivision of the wave function in self-similar parts. These results open the path to fractal electronics in a systematic and controlled manner