Complexity

Volume 2017 (2017), Article ID 9586064, 13 pages

https://doi.org/10.1155/2017/9586064

## The Multiplex Dependency Structure of Financial Markets

^{1}Department of Mathematics, King’s College London, The Strand, London WC2R 2LS, UK^{2}School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK^{3}Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK^{4}Systemic Risk Centre, London School of Economics and Political Sciences, London WC2A 2AE, UK^{5}Dipartimento di Fisica ed Astronomia, Università di Catania and INFN, 95123 Catania, Italy

Correspondence should be addressed to Vito Latora

Received 25 May 2017; Accepted 16 July 2017; Published 20 September 2017

Academic Editor: Tommaso Gili

Copyright © 2017 Nicolò Musmeci et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose here a multiplex network approach to investigate simultaneously different types of dependency in complex datasets. In particular, we consider multiplex networks made of four layers corresponding, respectively, to linear, nonlinear, tail, and partial correlations among a set of financial time series. We construct the sparse graph on each layer using a standard network filtering procedure, and we then analyse the structural properties of the obtained multiplex networks. The study of the time evolution of the multiplex constructed from financial data uncovers important changes in intrinsically multiplex properties of the network, and such changes are associated with periods of financial stress. We observe that some features are unique to the multiplex structure and would not be visible otherwise by the separate analysis of the single-layer networks corresponding to each dependency measure.

#### 1. Introduction

In the last decade, network theory has been extensively applied to the analysis of financial markets. Financial markets and complex systems in general are comprised of many interacting elements, and understanding their dependency structure and its evolution with time is essential to capture the collective behaviour of these systems, to identify the emergence of critical states, and to mitigate systemic risk arising from the simultaneous movement of several factors. Network filtering is a powerful instrument to associate a sparse network to a high-dimensional dependency measure and the analysis of the structure of such a network can uncover important insights on the collective properties of the underlying system. Following the line first traced by the preliminary work of Mantegna [1], a set of time series associated with financial asset values is mapped into a sparse complex network whose nodes are the assets and whose weighted links represent the dependencies between the corresponding time series. Filtering correlation matrices has been proven to be very useful for the study and characterisation of the underlying interdependency structure of complex datasets [1–5]. Indeed, sparsity allows filtering out noise, and sparse networks can then be analysed by using standard tools and indicators proposed in complex networks theory to investigate the multivariate properties of the dataset [6, 7]. Further, the filtered network can be used as a sparse inference structure to construct meaningful and computationally efficient predictive models [7, 8].

Complex systems are often characterised by nonlinear forms of dependency between the variables, which are hard to capture with a single measure and are hard to map into a single filtered network. A multiplex network approach, which considers the multilayer structure of a system in a consistent way, is thus a natural and powerful way to take into account simultaneously several distinct kinds of dependency. Dependencies among financial time series can be described by means of different measures, each one having its own advantages and drawbacks, and this has led to the study of different types of networks, namely, correlation networks, causality networks, and so on. The most common approach uses Pearson correlation coefficient to define the weight of a link, because this is a quantity that can be easily and quickly computed. However, the Pearson coefficient measures the linear correlation between two time series [9], and this is quite a severe limitation, since nonlinearity has been shown to be an important feature of financial markets [10]. Other measures can provide equally informative pictures on assets relationships. For instance, the Kendall correlation coefficient takes into account monotonic nonlinearity [11, 12], while other measures, such as the Tail dependence, quantify dependence in extreme events. It is therefore important to describe quantitatively how these alternative descriptions are related but also differ from the Pearson correlation coefficient and also to monitor how these differences change in time, if at all.

In this work we exploit the power of a multiplex approach to analyse simultaneously different kinds of dependencies among financial time series. The theory of multiplex network is a recently introduced framework that allows describing real-world complex systems consisting of units connected by relationships of different kinds as networks with many layers, where the links at each layer represent a different type of interaction between the same set of nodes [13, 14]. A multiplex network approach, combined with network filtering, is the ideal framework to investigate the interplay between linear, nonlinear, and Tail dependencies, as it is specifically designed to take into account the peculiarity of the patterns of connections at each of the layers but also to describe the intricate relations between the different layers [15].

The idea of analysing multiple layers of interaction was introduced initially in the context of social networks, within the theory of frame analysis [16]. The importance of considering multiple types of human interactions has been more recently demonstrated in different social networks, from terrorist organizations [14] to online communities; in all these cases, multilayer analyses unveil a rich topological structure [17], outperforming single-layer analyses in terms of network modeling and prediction as well [18, 19]. In particular, multilayer community detection in social networks has been shown to be more effective than single-layer approaches [20]; similar results have been reported for community detection on the World Wide Web [21, 22] and citation networks [23]. For instance, in the context of electrical power grids, multilayer analyses have provided important insight into the role of synchronization in triggering cascading failures [24, 25]. Similarly, the analyses on transport networks have highlighted the importance of a multilayer approach to optimize the system against nodes failures, such as flights cancellation [26]. In the context of economic networks, multiplex analyses have been applied to study the World Trade Web [27]. Moreover, they have been extensively used in the context of systemic risk, where graphs are used to model interbank and credit networks [28, 29].

Here, we extend the multiplex approach to financial market time series, with the purpose of analysing the role of different measures of dependencies, namely, the Pearson, Kendall, Tail, and Partial correlation. In particular we consider the so-called Planar Maximally Filtered Graph (PMFG) [2–4, 7] as filtering procedure to each of the four layers. For each of the four unfiltered dependence matrices, the PMFG filtering starts from the fully connected graph and uses a greedy procedure to obtain a planar graph that connects all the nodes and has the largest sum of weights [3, 4]. The PMFG is able to retain a higher number of links, and therefore a larger amount of information, than the Minimum Spanning Tree (MST) and can be seen as a generalization of the latter which is always contained as a proper subgraph [2]. The topological structures of MST and PMFG have been shown to provide meaningful economic and financial information [30–34] that can be exploited for risk monitoring [35–37] and asset allocation [38, 39]. The advantage of adopting a filtering procedure is not only in the reduction of noise and dimensionality but more importantly in the possibility of generating sparse networks, as sparsity is a requirement for most of the multiplex network measures that will be used in this paper [14]. Other kinds of filtering procedures, including thresholding based methods [35, 40], could have been considered. However, PMFG has the advantage to produce networks with fixed a priori () number of links that make the comparison between layers and across time windows easier. It is worth mentioning that the filtering of the Partial correlation layer requires an adaptation of the PMFG algorithm to deal with asymmetric relations. We have followed the approach suggested in [41] that rules out double links between nodes. The obtained planar graph corresponding to Partial correlations has been then converted into an undirected graph and included in the multiplex.

#### 2. Results

##### 2.1. Multiplex Network of Financial Stocks

We have constructed a time-varying multiplex network with layers and a varying number of nodes. Nodes represent stocks, selected from a dataset of US stocks which have appeared at least once in S&P500 in the period between 03/01/1993 and 26/02/2015. The period under study has been divided into rolling time windows, each of trading days. The network at time can be described by the adjacency matrix , with and . The network at time window has nodes, representing those stocks which were continuously traded in time window . The links at each of the four layers are constructed by means of the PMFG procedure from Pearson, Kendall, Tail, and Partial dependencies. The reason for this choice is to provide a complete picture of the market dependency structure: Pearson layer accounts for linear dependency, Kendall layer for monotonic nonlinearity, and Tail dependency for correlation in the tails of returns distribution while Partial correlation detects direct asset-asset relationships which are not explained by the market (see Materials and Methods for details).

Figure 1(a) shows how the average link weight of each of the four dependency networks changes over time. We notice that the average edge weight is a meaningful proxy for the overall level of correlation in one of these dependency layers, since the distribution of edge weights within a layer is normally quite peaked around its mean. The curves shown in Figure 1(a) indicate an overall increase of the typical weights in the examined period 1993–2015 and reveals a strongly correlated behaviour of the four curves (with linear correlation coefficients between the curves range in ). In particular they all display a steep increase in correspondence with the 2007-2008 financial crisis, revealing how the market became more synchronized, regardless of the dependence measure used. This strong correlation in the temporal patterns of the four measures of dependence may lead to the wrong conclusion that the four networks carry very similar information about the structure of financial systems. Conversely, we shall see that even basic multiplex measures suggest otherwise. In Figure 1(b) we report the average edge overlap , that is, the average number of layers of the financial multiplex network where a generic pair of nodes is connected by an edge (see Materials and Methods for details). Since our multiplex network consists of four layer, takes values in , and in particular we have when each edge is present only in one layer, while when the four networks are identical. The relatively low values of observed in this case reveal the complementary role played by the different dependency indicators. It is interesting to note that the edge overlap displays a quite dynamic pattern, and its variations seem to be related to the main financial crises highlighted by the vertical lines in Figure 1(b). Overall, what we observe is that periods of financial turbulence are linked to widening differences among the four layers. Namely, the effect of nonlinearity in the cross-dependence increases, as well as correlation on the tails of returns: the dependence structure becomes richer and more complex during financial crisis. This might be related to the highly nonlinear interactions that characterise investors activities in turbulent periods and that make fat-tail and power-law distributions distinctive features of financial returns. Indeed, if returns were completely described by a multivariate normal distribution, the Pearson layer would be sufficient to quantify entirely the cross-dependence and its relation with the other layers would be trivial and would not change with time. Therefore any variation in the overlapping degree is a signature of increasing complexity in the market. In particular, the first event that triggers a sensible decrease in the average edge overlap is the Russian crisis in 1998, which corresponds to the overall global minimum of in the considered interval. Then, starts increasing towards the end of year 2000 and reaches its global maximum at the beginning of 2002, just before the market downturn of the same year. We observe a marked decrease in 2005, in correspondence with the second phase of the housing bubble, which culminates in the dip associated with the credit crunch at the end of 2007. A second, even steeper drop occurs during the Lehman Brothers default of 2008. After that, the signal appears more stable and weakly increasing, especially towards the end of 2014. Since each edge is present, on average, in less than two layers, each of the four layers effectively provides a partial perspective on the dependency structure of the market. This fact is made more evident by the results reported in Figure 1(c), where we show, for each layer , the fraction of edges that exist exclusively in that layer (see Materials and Methods for details). We notice that, at any point in time, from to of the edges of each of the four layers are unique to that layer, meaning that a large fraction of the dependence relations captured by a given measure are not captured by the other measures. For instance, despite the fact that Pearson and Kendall show similar behaviour in Figure 1(c), still between 30% and 40% of the edges on each of those layers exist only on that layer. This indicates that the Pearson and Kendall layers differ for at least 60% to 80% of their edges. In general, each of the four layers is contributing information that cannot be found in the other three layers. It was shown in a recent paper by some of the authors [36] that information filtering networks can be used to forecast volatility outbursts. The present results suggest that a multilayer approach could provide a further forecasting instrument for bear/bull markets. However, this requires further explorations. Interestingly, we observe an increase of for all the layers since 2005, which indicates a build-up of nonlinearity and tail correlation in the years preceding the financial crisis: such dynamics might be related to early-risk warnings.