Commutative law for products of infinitely large isotropic random matrices

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size $N \rightarrow \infty$. The notion of spectral commutativity means that the eigenvalue density of a product ABC... of such matrices is independent of the order of matrix multiplication, for example the matrix ABCD has the same eigenvalue density as ADCB. In turn, the notion of self-averaging means that the product of n independent but identically distributed random matrices, which we symbolically denote by AAA..., has the same eigenvalue density as the corresponding power A^n of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of ABCCABC is the same as of A^2B^2C^3. We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues $\lambda \rightarrow 0$. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit $N \rightarrow \infty$: the eigenvalue density of an isotropic random matrix has a power law singularity at the origin $\sim |\lambda|^{-s}$ with a power $s \in (0,2)$ when and only when the density of its singular values has a power law singularity $\sim \lambda^{-\sigma}$ with a power $\sigma = s/(4-s)$. These results are obtained analytically in the limit $N \rightarrow \infty$. We supplement these results with numerical simulations for large but finite N and discuss finite size effects for the most common ensembles of isotropic random matrices.Comment: 15 pages, 4 figure

The fine structure of spectral properties for random correlation matrices: an application to financial markets

We study some properties of eigenvalue spectra of financial correlation matrices. In particular, we investigate the nature of the large eigenvalue bulks which are observed empirically, and which have often been regarded as a consequence of the supposedly large amount of noise contained in financial data. We challenge this common knowledge by acting on the empirical correlation matrices of two data sets with a filtering procedure which highlights some of the cluster structure they contain, and we analyze the consequences of such filtering on eigenvalue spectra. We show that empirically observed eigenvalue bulks emerge as superpositions of smaller structures, which in turn emerge as a consequence of cross-correlations between stocks. We interpret and corroborate these findings in terms of factor models, and and we compare empirical spectra to those predicted by Random Matrix Theory for such models.Comment: 21 pages, 10 figure

Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so called M-transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.Comment: 50 pages, 8 figures, to appear in the Proceedings of the 23rd Marian Smoluchowski Symposium on Statistical Physics: "Random Matrices, Statistical Physics and Information Theory," September 26-30, 2010, Krakow, Polan

Macroeconomic forecasting with statistically validated knowledge graphs

This study leverages narrative from global newspapers to construct theme-based knowledge graphs about world events, demonstrating that features extracted from such graphs improve forecasts of industrial production in three large economies compared to a number of benchmarks. Our analysis relies on a filtering methodology that extracts “backbones” of statistically significant edges from large graph data sets. We find that changes in the eigenvector centrality of nodes in such backbones capture shifts in relative importance between different themes significantly better than graph similarity measures. We supplement our results with an interpretability analysis, showing that the theme categories “disease” and “economic” have the strongest predictive power during the time period that we consider. Our work serves as a blueprint for the construction of parsimonious – yet informative – theme-based knowledge graphs to monitor in real time the evolution of relevant phenomena in socio-economic systems

Scalability and egalitarianism in peer-to-peer networks

Many information-technology innovations are driven, in their early stages, by an egalitarian ethos that empowers individuals through dis-intermediation. Bitcoin and peer to peer financial systems were inspired by these egalitarian ambitions. However, in bitcoin we have recently witnessed a strong centralization around a few large mining pools, which puts control of most of the system in the hands of a few. In this chapter we investigate the physical limits of distributed consensus mechanisms over networks, and discuss whether there are scalability and efficiency reasons that incentivize centralization. We compute the time to reach majority consensus in a variety of settings, comparing egalitarian networks with centralized networks, and quantifying the effect of network topology on the propagation of information

A network perspective on intermedia agenda-setting

In Communication Theory, intermedia agenda-setting refers to the influence that different news sources may have on each other, and how this subsequently affects the breadth of information that is presented to the public. Several studies have attempted to quantify the impact of intermedia agenda-setting in specific countries or contexts, but a large-scale, data-driven investigation is still lacking. Here, we operationalise intermedia agenda-setting by putting forward a methodology to infer networks of influence between different news sources on a given topic, and apply it on a large dataset of news articles published by globally and locally prominent news organisations in 2016. We find influence to be significantly topic-dependent, with the same news sources acting as agenda-setters (i.e., central nodes) with respect to certain topics and as followers (i.e., peripheral nodes) with respect to others. At the same time, we find that the influence networks associated to most topics exhibit small world properties, which we find to play a significant role towards the overall diversity of sentiment expressed about the topic by the news sources in the network. In particular, we find clustering and density of influence networks to act as competing forces in this respect, with the former increasing and the latter reducing diversit

Asymmetric correlation matrices: an analysis of financial data

We analyze the spectral properties of correlation matrices between distinct statistical systems. Such matrices are intrinsically non symmetric, and lend themselves to extend the spectral analyses usually performed on standard Pearson correlation matrices to the realm of complex eigenvalues. We employ some recent random matrix theory results on the average eigenvalue density of this type of matrices to distinguish between noise and non trivial correlation structures, and we focus on financial data as a case study. Namely, we employ daily prices of stocks belonging to the American and British stock exchanges, and look for the emergence of correlations between two such markets in the eigenvalue spectrum of their non symmetric correlation matrix. We find several non trivial results, also when considering time-lagged correlations over short lags, and we corroborate our findings by additionally studying the asymmetric correlation matrix of the principal components of our datasets.Comment: Revised version; 11 pages, 13 figure

Facilitating the Decentralised Exchange of Cryptocurrencies in an Order-Driven Market

This article discusses a protocol to facilitate decentralised exchanges on an order-driven market through a consortium of market services operators. We discuss whether this hybrid protocol combining a centralised initiation phase with a decentralised execution phase outperforms fully centralised exchanges with regards to efficiency and security. Here, a fully efficient and fully secure protocol is defined as one where traders incur no trading costs or opportunity costs and counterparty risk is absent. We devise a protocol addressing the main downsides in the decentralised exchange process that uses a facilitating distributed ledger, maintains an order book and monitors the order status in real-time to provide accurate exchange rate information and performance scoring of participants. We show how performance ratings can lower opportunity costs and how a rolling benchmark rate of verifiable trades can be used to establish a trustworthy exchange rate between cryptocurrencies. The formal validation of the proposed technical mechanisms is the subject of future work

Judgments in the Sharing Economy: The Effect of User-Generated Trust and Reputation Information on Decision-Making Accuracy and Bias

The growing ecosystem of peer-to-peer enterprise – the Sharing Economy (SE) – has brought with it a substantial change in how we access and provide goods and services. Within the SE, individuals make decisions based mainly on user-generated trust and reputation information (TRI). Recent research indicates that the use of such information tends to produce a positivity bias in the perceived trustworthiness of fellow users. Across two experimental studies performed on an artificial SE accommodation platform, we test whether users’ judgments can be accurate when presented with diagnostic information relating to the quality of the profiles they see or if these overly positive perceptions persist. In study 1, we find that users are quite accurate overall (70%) at determining the quality of a profile, both when presented with full profiles or with profiles where they selected three TRI elements they considered useful for their decisionmaking. However, users tended to exhibit an “upward quality bias” when making errors. In study 2, we leveraged patterns of frequently vs. infrequently selected TRI elements to understand whether users have insights into which are more diagnostic and find that presenting frequently selected TRI elements improved users’ accuracy. Overall, our studies demonstrate that – positivity bias notwithstanding – users can be remarkably accurate in their online SE judgments

Interdisciplinary researchers attain better long-term funding performance

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