Spanning Trees and bootstrap reliability estimation in correlation based networks

We introduce a new technique to associate a spanning tree to the average linkage cluster analysis. We term this tree as the Average Linkage Minimum Spanning Tree. We also introduce a technique to associate a value of reliability to links of correlation based graphs by using bootstrap replicas of data. Both techniques are applied to the portfolio of the 300 most capitalized stocks traded at New York Stock Exchange during the time period 2001-2003. We show that the Average Linkage Minimum Spanning Tree recognizes economic sectors and sub-sectors as communities in the network slightly better than the Minimum Spanning Tree does. We also show that the average reliability of links in the Minimum Spanning Tree is slightly greater than the average reliability of links in the Average Linkage Minimum Spanning Tree.Comment: 17 pages, 3 figure

Dynamic structural and topological phase transitions on the Warsaw Stock Exchange: A phenomenological approach

We study the crash dynamics of the Warsaw Stock Exchange (WSE) by using the Minimal Spanning Tree (MST) networks. We find the transition of the complex network during its evolution from a (hierarchical) power law MST network, representing the stable state of WSE before the recent worldwide financial crash, to a superstar-like (or superhub) MST network of the market decorated by a hierarchy of trees (being, perhaps, an unstable, intermediate market state). Subsequently, we observed a transition from this complex tree to the topology of the (hierarchical) power law MST network decorated by several star-like trees or hubs. This structure and topology represent, perhaps, the WSE after the worldwide financial crash, and could be considered to be an aftershock. Our results can serve as an empirical foundation for a future theory of dynamic structural and topological phase transitions on financial markets

Statistically validated networks in bipartite complex systems

Many complex systems present an intrinsic bipartite nature and are often described and modeled in terms of networks [1-5]. Examples include movies and actors [1, 2, 4], authors and scientific papers [6-9], email accounts and emails [10], plants and animals that pollinate them [11, 12]. Bipartite networks are often very heterogeneous in the number of relationships that the elements of one set establish with the elements of the other set. When one constructs a projected network with nodes from only one set, the system heterogeneity makes it very difficult to identify preferential links between the elements. Here we introduce an unsupervised method to statistically validate each link of the projected network against a null hypothesis taking into account the heterogeneity of the system. We apply our method to three different systems, namely the set of clusters of orthologous genes (COG) in completely sequenced genomes [13, 14], a set of daily returns of 500 US financial stocks, and the set of world movies of the IMDb database [15]. In all these systems, both different in size and level of heterogeneity, we find that our method is able to detect network structures which are informative about the system and are not simply expression of its heterogeneity. Specifically, our method (i) identifies the preferential relationships between the elements, (ii) naturally highlights the clustered structure of investigated systems, and (iii) allows to classify links according to the type of statistically validated relationships between the connected nodes.Comment: Main text: 13 pages, 3 figures, and 1 Table. Supplementary information: 15 pages, 3 figures, and 2 Table

ANALYSIS OF THE TIME EVOLUTION OF NON-LINEAR FINANCIAL NETWORKS

We treat financial markets as complex networks. It is commonplace to create a filtered graph (usually a Minimally Spanning Tree) based on an empirical correlation matrix. In our previous studies we have extended this standard methodology by exchanging Pearson’s correlation coefficient with information—theoretic measures of mutual information and mutual information rate, which allow for the inclusion of non-linear relationships. In this study we investigate the time evolution of financial networks, by applying a running window approach. Since information—theoretic measures are slow to converge, we base our analysis on the Hirschfeld-Gebelein-Rényi Maximum Correlation Coefficient, estimated by the Randomized Dependence Coefficient (RDC). It is defined in terms of canonical correlation analysis of random non-linear copula projections. On this basis we create Minimally Spanning Trees for each window moving along the studied time series, and analyse the time evolution of various network characteristics, and their market significance. We apply this procedure to a dataset describing logarithmic stock returns from Warsaw Stock Exchange for the years between 2006 and 2013, and comment on the findings, their applicability and significance

Dominating Clasp of the Financial Sector Revealed by Partial Correlation Analysis of the Stock Market

What are the dominant stocks which drive the correlations present among stocks traded in a stock market? Can a correlation analysis provide an answer to this question? In the past, correlation based networks have been proposed as a tool to uncover the underlying backbone of the market. Correlation based networks represent the stocks and their relationships, which are then investigated using different network theory methodologies. Here we introduce a new concept to tackle the above question—the partial correlation network. Partial correlation is a measure of how the correlation between two variables, e.g., stock returns, is affected by a third variable. By using it we define a proxy of stock influence, which is then used to construct partial correlation networks. The empirical part of this study is performed on a specific financial system, namely the set of 300 highly capitalized stocks traded at the New York Stock Exchange, in the time period 2001–2003. By constructing the partial correlation network, unlike the case of standard correlation based networks, we find that stocks belonging to the financial sector and, in particular, to the investment services sub-sector, are the most influential stocks affecting the correlation profile of the system. Using a moving window analysis, we find that the strong influence of the financial stocks is conserved across time for the investigated trading period. Our findings shed a new light on the underlying mechanisms and driving forces controlling the correlation profile observed in a financial market

Identifying Metering Hierarchies with Distance Correlation and Dominance Constraints

In this paper, we consider observations from a series of smart meters that are either completely or partially aggregated, and our aim is to estimate the metering hierarchy. We propose to estimate this important metadata through a novel adaptation of the Chow–Liu tree learning procedure. Our approach takes into account prior knowledge from a set of dominance conditions that are easily elicited from the consumption data. In addition to more traditional correlation-based approaches we also introduce a distance-correlation-based method for detecting edges. Synthetic experiments show the benefits of distance correlation and the dominance conditions in recovering tree structure. The paper concludes with a real-world application of the method to infer energy metering hierarchies in a library building

Global and Local Features of Semantic Networks: Evidence from the Hebrew Mental Lexicon

BACKGROUND: Semantic memory has generated much research. As such, the majority of investigations have focused on the English language, and much less on other languages, such as Hebrew. Furthermore, little research has been done on search processes within the semantic network, even though they are abundant within cognitive semantic phenomena. METHODOLOGY/PRINCIPAL FINDINGS: We examine a unique dataset of free association norms to a set of target words and make use of correlation and network theory methodologies to investigate the global and local features of the Hebrew lexicon. The global features of the lexicon are investigated through the use of association correlations--correlations between target words, based on their association responses similarity; the local features of the lexicon are investigated through the use of association dependencies--the influence words have in the network on other words. CONCLUSIONS/SIGNIFICANCE: Our investigation uncovered Small-World Network features of the Hebrew lexicon, specifically a high clustering coefficient and a scale-free distribution, and provides means to examine how words group together into semantically related 'free categories'. Our novel approach enables us to identify how words facilitate or inhibit the spread of activation within the network, and how these words influence each other. We discuss how these properties relate to classical research on spreading activation and suggest that these properties influence cognitive semantic search processes. A semantic search task, the Remote Association Test is discussed in light of our findings