A tool for filtering information in complex systems

Abstract

We introduce a technique to filter out complex data sets by extracting a subgraph of representative links. Such a filtering can be tuned up to any desired level by controlling the genus of the resulting graph. We show that this technique is especially suitable for correlation-based graphs, giving filtered graphs that preserve the hierarchical organization of the minimum spanning tree but containing a larger amount of information in their internal structure. In particular in the case of planar filtered graphs (genus equal to 0), triangular loops and four-element cliques are formed. The application of this filtering procedure to 100 stocks in the U.S. equity markets shows that such loops and cliques have important and significant relationships with the market structure and properties.cluster analysis ͉ complex networks ͉ correlation analysis S everal complex systems have been investigated recently from the perspective of the (weighted) networks that are linking the different elements comprising them (1-4). Indeed, complex systems are in general made of several interacting elements, and it is rather natural to associate to each element a node and to each interaction a link yielding to a graph. Examples include food webs (5), scientific citations (6), social networks (7, 8), communication networks (9), sexual contacts among individuals (10), company links in a stock portfolio (11), the Internet (12), and the World Wide Web (13). The properties of such graphs have been studied with the aim of catching basic features of the investigated systems (14-16). However, the complexity of the system is generally reflected in the associated graph, which results in an intricate interweaved and densely connected structure. There is therefore a general need to find methods that are able to single out the key information by filtering such a complex graph into a simpler relevant subgraph. Such a filtering is especially essential for correlation-based graphs where, in the absence of any filtering procedure, all links among elements are present.In this work, we introduce a filtering procedure that extracts a representative subgraph with a controlled complexity and maximal information content out of the graph describing the system. To illustrate the method, we present a concrete example dealing with 100 stocks belonging to a U.S. equity portfolio. In the modeling of equity portfolios, a natural starting point is the investigation of cross-correlation among time series of returns of stock pairs. The correlation provides a similarity measure among the behavior of different elements in the system. It was shown by one of us that a powerful method to investigate financial systems consists in the extraction of a minimal set of relevant interactions associated with the strongest correlations belonging to the minimum spanning tree (MST) (11). However, the reduction to a minimal skeleton of links is necessarily very drastic in filtering correlation-based networks, losing therefore valuable information. The necessity of a less drastic filtering ...

Cluster analysis for portfolio optimization

“…However, the single linkage is just one possible correlation based filtering methods. Other methods have also been applied to financial portfolios [20,22,24,25,26,27,28,30,34,35]. Each method puts a specific emphasis on some aspects of the original matrix and is usually able to point out a series of aspects that might not be elucidated by different filtering procedure.…”

“…In recent years, other filtering procedures of the correlation coefficient matrix performed using correlation based clustering procedures has also been proposed in the econophysics literature [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. These methods also select information of the correlation coefficient matrix which is representative of the entire matrix and it is often less affected by the statistical uncertainty and therefore more stable than the entire matrix during the time evolution of the system.…”

A firm’s perspective on econophysics-based currency risk analysis

“…After all the appropriate links are added, such a graph is always reduced into a tree [Tumminello et al 2005;Tumminello, Lillo, Mantegna 2010].…”

Untangling Hairballs

“…Sparse connected subgraphs of edges not likely to be between cohesive groups have been proposed, e.g., by van Ham and Wattenberg [28] (planar graphs) and Tumminello et al [27] (graph of bounded genus). A minimally connected subgraph of edges with high weights is a maximum spanning tree (MST), and Mantegna [14] proposed these as a backbone.…”