Emergent Complex Network Geometry

Wu, Zhihao; Menichetti, Giulia; Rahmede, Christoph; Bianconi, Ginestra

doi:10.1038/srep10073

Cited by 124 publications

(145 citation statements)

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“…Scale bar in (g) 10 µm. i A schematic of non-equivalent paths, running in the interface space between CDW puddles, connecting point A to point B in the emergent complex hyperbolic geometry 29,30 , for the superconducting current.The broad phase transition appears to be arrested, as indicated by the size of the CDW puddles ξ a =1/Δh CDW , which does not diverge below T CDW . This behavior is typical of low dimensional systems with quenched disorder.…”

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confidence: 99%

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Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor

Campi

Bianconi

Poccia

et al. 2015

Nature

301

309

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It has recently been established that the high temperature (high-Tc(between atomic and macroscopic scale). Here we report micro X-ray diffraction imaging of the spatial distribution of both the charge-density-wave 'puddles' (domains with only a few wavelengths) and quenched disorder in HgBa 2 CuO 4+y , the single layer cuprate with the highest T c , 95 kelvin [26][27][28] . We found that the charge-density-wave puddles, like the steam bubbles in boiling water, have a fat-tailed size distribution that is typical of selforganization near a critical point 19 . However, the quenched disorder, which arises from oxygen interstitials, has a distribution that is contrary to the usual assumed random, uncorrelated distribution 12, 13 . The interstitials-oxygen-rich domains are spatially anticorrelated with the charge-density-wave domains, leading to a complex emergent geometry of the spatial landscape for superconductivity. 2Although it is known that the incommensurate charge-density-wave (CDW) order in cuprates (copper oxides) is made of ordered, stripy, nanoscale puddles with an average of only 3-4 oscillations, information about the size distribution and spatial organization of these puddles has so far not been available. We present experiments that demonstrate that CDW puddles, have a complex spatial distribution and coexist with, but are spatially anticorrelated to, quenched disorder in HgBa 2 CuO 4+y (Hg1201). The sample we studied is a layered perovskite at optimum doping with oxygen interstitials y=0.12, tetragonal symmetry P4/mmm and a low misfit strain [14][15][16] . The X-ray diffraction (XRD) measurements (see Methods) show diffuse CDW satellites (secondary peaks surrounding a main peak) at q CDW =(0.23a*, 0.16c*), in the b*=0 plane and q CDW =(0.23b*, 0.16c*) in the a*=0 plane (where a*, b*, and c* are the reciprocal lattice units) around specific Bragg peaks, such as (1 0 8), below the onset temperature T CDW =240 K (see Fig. 1a). The component of the momentum transfer q CDW in the CuO 2 plane (0.23a*) in this case is smaller than it is in the underdoped case (0.28a*) 5 . The temperature evolution of CDW-peak profile along a* (in the h direction; Fig. 1b) shows a smeared, glassy-like evolution below T CDW .The CDW-peak intensity reaches a maximum at T=100 K, followed by a drop associated with the onset of superconductivity at T=T c . We investigated the isotropic character of the CDW, in the a-b plane using azimuthal scans, as shown in Fig. 1c. We observed an equal probability of vertical and horizontally striped CDW puddles.Our main result is the discovery of the statistical spatial distribution of the CDW-puddle size and density throughout the sample, which shows an emergent complex network geometry for the superconducting phase. We performed scanning micro X-ray diffraction (SµXRD) measurements (see Methods) to extend the imaging of spatial inhomogeneity previously obtained by scanning tunneling microscopy [7][8][9] , from the surface to the bulk of the sample and from nanoscale to mesoscale spatial inh...

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confidence: 99%

mentioning

confidence: 99%

Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor

Campi

Bianconi

Poccia

et al. 2015

Nature

301

309

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“…Remarkably, widely different complex systems, across diverse natural and social domains, tend to display a common set of structural similarities [124,125,126,127,128]. These include a high clustering coefficient (the formation of communities within a network), short path lengths (most nodes can be reached by a path that travels though only a small number of other nodes), and power-law or scale-free degree distributions (degree is the number of connections at a node).…”

Section: Criticality Indicatorsmentioning

confidence: 99%

Optimality of Social Choice Systems: Complexity, Wisdom, and Wellbeing Centrality

Boik¹

2016

Preprint

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Since circa 1900, civilization has experienced radical changes including changes in the size and distribution of populations, the power of technologies, the magnitude of energy and materials use, and the depth of scientific knowledge. With these have come increasingly complex challenges and elevated risks, and thus a heightened need for wise decision making. Accordingly, the need has grown for efficient and functional decision-making systems, also called social choice systems. I use these terms to refer to economic, governance, and legal systems. The seeming inability of societies, both individually and collectively, to effectively mitigate excessive climate change, poverty, income inequality, pollution, habitat loss, and other major problems suggests that underlying social choice systems are sub-optimal relative to need. I raise two overarching questions: (1) What characteristics would more optimal social choice systems exhibit? (2) How could research and development of more optimal systems best proceed?The answers I explore in this paper are based on the premise that the relative optimality of a social choice system is a measure of its relative capacity to help communities solve problems and organize activities such that collective wellbeing is elevated. The characteristics of complex adaptive systems, successful problemsolving systems found in nature, are explored in order to suggest useful design motifs and monitoring indicators. I emphasize the need for research and development of new social choice system designs, and argue that field testing of these can best occur at the local (e.g., community, city, or county) level. Efforts in this direction by the science and technology sectors and academic community are still nascent. The work described here suggests a new multidisciplinary program that I term wellbeing centrality: the design, testing, promotion, and operation of social choice systems that place wellbeing measurement, evaluation, forecasting, and deliberation at the center of decision-making activities.

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“…More recent "geometric" approaches consider discrete definitions of curvature as in [13,[16][17][18].…”

Section: Introductionmentioning

confidence: 99%

“…While the body of theoretical research on the analysis of networks and related structures has been focused on the properties of the various (discrete) Laplacians (see [10] for an overview on the state of the art), a more recent and related direction concerns the geometrical characterization of real-world and model type networks with discrete curvature, see [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets

Weber

Jost

Saucan

2016

Axioms

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Abstract:We present a viable geometric solution for the detection of dynamic effects in complex networks. Building on Forman's discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. We study the classical Ricci-flow in a network-theoretic setting and introduce an analytic tool for characterizing dynamic effects. The formalism suggests a computational method for change detection and the identification of fast evolving network regions and yields insights into topological properties and the structure of the underlying data.

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Emergent Complex Network Geometry

Cited by 124 publications

References 58 publications

Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor

Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor

Optimality of Social Choice Systems: Complexity, Wisdom, and Wellbeing Centrality

Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets

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