Topological structures in the equities market network

We present a new method for articulating scale-dependent topological descriptions of the network structure inherent in many complex systems. The technique is based on "Partition Decoupled Null Models,'' a new class of null models that incorporate the interaction of clustered partitions into a random model and generalize the Gaussian ensemble. As an application we analyze a correlation matrix derived from four years of close prices of equities in the NYSE and NASDAQ. In this example we expose (1) a natural structure composed of two interacting partitions of the market that both agrees with and generalizes standard notions of scale (eg., sector and industry) and (2) structure in the first partition that is a topological manifestation of a well-known pattern of capital flow called "sector rotation.'' Our approach gives rise to a natural form of multiresolution analysis of the underlying time series that naturally decomposes the basic data in terms of the effects of the different scales at which it clusters. The equities market is a prototypical complex system and we expect that our approach will be of use in understanding a broad class of complex systems in which correlation structures are resident.Comment: 17 pages, 4 figures, 3 table

A discrete Laplace-Beltrami operator for simplicial surfaces

We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace-Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay triangulations of piecewise flat surfaces revised and expanded. References added. Some minor changes, typos corrected. v3: fixed inaccuracies in discussion of flip algorithm, corrected attributions, added references, some minor revision to improve expositio

The Intrafirm Complexity of Systemically Important Financial Institutions

In November 2011, the Financial Stability Board, in collaboration with the International Monetary Fund, published a list of 29 “systemically important financial institutions” (SIFIs, now referred to as “globally systemically important banks” or G-SIBs), institutions whose failure, by virtue of “their size, complexity, and systemic interconnectedness”, could have dramatic negative consequences for the global financial system. While “size” and “interconnectedness” have been the subject of much quantitative analysis, less attention has been paid to measuring “complexity.” Yet without a consistent way to measure complexity, there is little guarantee that the designated SIFIs capture the complexity that the FSB is concerned about, and little hope of mitigating the consequences that the FSB warns of. In this paper we propose the structure of an individual firm’s majority-control hierarchy as a proxy for institutional complexity. We demonstrate as a proof-of-concept how this method might be used by bank supervisors, particularly the Federal Reserve under its authority as consolidated supervisor, using a data set containing information on the majority-control hierarchies of many of the designated SIFIs. Our mathematical intrafirm network representation (and various associated metrics we propose) provides a uniform way to compare firms with often very disparate organizational structures -- one that is distinct from a simple size comparison

Multiple dynamical time-scales in networks with hierarchically nested modular organization

Many natural and engineered complex networks have intricate mesoscopic organization, e.g., the clustering of the constituent nodes into several communities or modules. Often, such modularity is manifested at several different hierarchical levels, where the clusters defined at one level appear as elementary entities at the next higher level. Using a simple model of a hierarchical modular network, we show that such a topological structure gives rise to characteristic time-scale separation between dynamics occurring at different levels of the hierarchy. This generalizes our earlier result for simple modular networks, where fast intra-modular and slow inter-modular processes were clearly distinguished. Investigating the process of synchronization of oscillators in a hierarchical modular network, we show the existence of as many distinct time-scales as there are hierarchical levels in the system. This suggests a possible functional role of such mesoscopic organization principle in natural systems, viz., in the dynamical separation of events occurring at different spatial scales.Comment: 10 pages, 4 figure

Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \mathbb {R}^{3}, as hyperelliptic curves, and as \mathbb {CP}^{1} modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller

Cryptic Distant Relatives Are Common in Both Isolated and Cosmopolitan Genetic Samples

Although a few hundred single nucleotide polymorphisms (SNPs) suffice to infer close familial relationships, high density genome-wide SNP data make possible the inference of more distant relationships such as 2nd to 9th cousinships. In order to characterize the relationship between genetic similarity and degree of kinship given a timeframe of 100–300 years, we analyzed the sharing of DNA inferred to be identical by descent (IBD) in a subset of individuals from the 23andMe customer database (n = 22,757) and from the Human Genome Diversity Panel (HGDP-CEPH, n = 952). With data from 121 populations, we show that the average amount of DNA shared IBD in most ethnolinguistically-defined populations, for example Native American groups, Finns and Ashkenazi Jews, differs from continentally-defined populations by several orders of magnitude. Via extensive pedigree-based simulations, we determined bounds for predicted degrees of relationship given the amount of genomic IBD sharing in both endogamous and ‘unrelated’ population samples. Using these bounds as a guide, we detected tens of thousands of 2nd to 9th degree cousin pairs within a heterogenous set of 5,000 Europeans. The ubiquity of distant relatives, detected via IBD segments, in both ethnolinguistic populations and in large ‘unrelated’ populations samples has important implications for genetic genealogy, forensics and genotype/phenotype mapping studies

Robust physical methods that enrich genomic regions identical by descent for linkage studies: confirmation of a locus for osteogenesis imperfecta

<p>Abstract</p> <p>Background</p> <p>The monogenic disease osteogenesis imperfecta (OI) is due to single mutations in either of the collagen genes ColA1 or ColA2, but within the same family a given mutation is accompanied by a wide range of disease severity. Although this phenotypic variability implies the existence of modifier gene variants, genome wide scanning of DNA from OI patients has not been reported. Promising genome wide marker-independent physical methods for identifying disease-related loci have lacked robustness for widespread applicability. Therefore we sought to improve these methods and demonstrate their performance to identify known and novel loci relevant to OI.</p> <p>Results</p> <p>We have improved methods for enriching regions of identity-by-descent (IBD) shared between related, afflicted individuals. The extent of enrichment exceeds 10- to 50-fold for some loci. The efficiency of the new process is shown by confirmation of the identification of the Col1A2 locus in osteogenesis imperfecta patients from Amish families. Moreover the analysis revealed additional candidate linkage loci that may harbour modifier genes for OI; a locus on chromosome 1q includes COX-2, a gene implicated in osteogenesis.</p> <p>Conclusion</p> <p>Technology for physical enrichment of IBD loci is now robust and applicable for finding genes for monogenic diseases and genes for complex diseases. The data support the further investigation of genetic loci other than collagen gene loci to identify genes affecting the clinical expression of osteogenesis imperfecta. The discrimination of IBD mapping will be enhanced when the IBD enrichment procedure is coupled with deep resequencing.</p

Integration of sequence data from a consanguineous family with genetic data from an outbred population identifies PLB1 as a candidate rheumatoid arthritis risk gene

Integrating genetic data from families with highly penetrant forms of disease together with genetic data from outbred populations represents a promising strategy to uncover the complete frequency spectrum of risk alleles for complex traits such as rheumatoid arthritis (RA). Here, we demonstrate that rare, low-frequency and common alleles at one gene locus, phospholipase B1 (PLB1), might contribute to risk of RA in a 4-generation consanguineous pedigree (Middle Eastern ancestry) and also in unrelated individuals from the general population (European ancestry). Through identity-by-descent (IBD) mapping and whole-exome sequencing, we identified a non-synonymous c.2263G>C (p.G755R) mutation at the PLB1 gene on 2q23, which significantly co-segregated with RA in family members with a dominant mode of inheritance (P = 0.009). We further evaluated PLB1 variants and risk of RA using a GWAS meta-analysis of 8,875 RA cases and 29,367 controls of European ancestry. We identified significant contributions of two independent non-coding variants near PLB1 with risk of RA (rs116018341 [MAF = 0.042] and rs116541814 [MAF = 0.021], combined P = 3.2×10-6). Finally, we performed deep exon sequencing of PLB1 in 1,088 RA cases and 1,088 controls (European ancestry), and identified suggestive dispersion of rare protein-coding variant frequencies between cases and controls (P = 0.049 for C-alpha test and P = 0.055 for SKAT). Together, these data suggest that PLB1 is a candidate risk gene for RA. Future studies to characterize the full spectrum of genetic risk in the PLB1 genetic locus are warranted. © 2014 Plenge et al

The intrafirm complexity of systemically important financial institutions

In November 2011, the Financial Stability Board, in collaboration with the International Monetary Fund, published a list of 29 “systemically important financial institutions” (SIFIs, now referred to as “globally systemically important banks” or G-SIBs), institutions whose failure, by virtue of “their size, complexity, and systemic interconnectedness”, could have dramatic negative consequences for the global financial system. While “size” and “interconnectedness” have been the subject of much quantitative analysis, less attention has been paid to measuring “complexity.” Yet without a consistent way to measure complexity, there is little guarantee that the designated SIFIs capture the complexity that the FSB is concerned about, and little hope of mitigating the consequences that the FSB warns of. In this paper we propose the structure of an individual firm’s majority-control hierarchy as a proxy for institutional complexity. We demonstrate as a proof-of-concept how this method might be used by bank supervisors, particularly the Federal Reserve under its authority as consolidated supervisor, using a data set containing information on the majority-control hierarchies of many of the designated SIFIs. Our mathematical intrafirm network representation (and various associated metrics we propose) provides a uniform way to compare firms with often very disparate organizational structures -- one that is distinct from a simple size comparison