Statistically validated networks for financial data analysis

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Dipartimento di Fisica Corso di Laurea in Fisica Teorica

Statistically Validated Networks for

financial data analysis

Tesi di laurea magistrale

Relatori:

Prof. Fabrizio Lillo

Prof. Riccardo Mannella

Candidato:

Jacopo Scarpelli

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i

A Zio Franco,

sperando che le mie parole possano arrivare fin lassu.

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List of Figures

2.1 Simplified scheme of portfolio rebalancing . . . 8

3.1 Representation of a graph . . . 18

3.2 Representation of a simple bipartite graph . . . 21

4.1 Number of buy(o) and sell(*) links in function of time. . . 27

4.2 Representation of both buy and sell network of week 56. Yellow dots are MP vertices, blue dots are asset vertices, green and red lines are respective buy and sell links. . . 28

4.3 Number of buy and sell links in function of time for four-monthly resolution. As for fig 2.1, green line labels buy links, red line labels sell link. . . 30

5.1 Time series of the number of links of the original, Bon f erroni and FDR networks for the market’s participants projected networks, panel (a), and for assets networks, panel (b). Green line (o) labels buy links and red line (*) labels sell ones. . . 34

5.2 Bon f erroni, panel (a), and FDR, panel (b), links for the assets networks plotted alone due to the wide range of values, larger respect to the other time series. . . 35

5.3 Fraction of validated links respect to the total number of links in function of time for Buy networks (upper panel) and for Sell networks (bottom panel). Blue line labels MP-networks (o), red line labels A-networks (*). . . 36

5.4 Four-monthly time series of the number of links of the original, Bon f erroniand FDR networks for the market’s participants pro-jected networks, panel (a), and for assets networks, panel (b), with the usual notation for buy (o) and sell (*) links. Panel (c) shows the fraction of validated links of the original networks for both sets of vertices, assets (*) and participants (o). . . 37

5.5 Community detection scheme [21] . . . 39

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LIST OF FIGURES v 5.6 Time series of the number of communities of the three networks

for MP − networks, panel (a), and for A − networks, panel (b). . . 41 5.7 . . . 42 5.8 Four-monthly time series of the number of communities of the

three networks for MP−networks, panel (a), and for A−networks, panel (b). . . 47 6.1 Temporal evolution of observed average ANAD ¯dNA_{( ) versus}

ex-pected hdNA¯ _i(_*_{), first plot, and temporal evolution of observed}

average ANID ¯dNI_{( ) versus expected} _hd¯NI_i(_*_{), second plot, for}

buy networks (panel a) and sell networks (panel b). The time res-olution is weekly. . . 55 6.2 Temporal evolution of observed average ANAD ¯dNA_{( ) versus}

ex-pected hdNA¯ _i(_*_{), first plot, and temporal evolution of observed}

average ANID ¯dNI( ) versus expected hd¯NI_i(_*_{), second plot, for}

buy networks (panel a) and sell networks (panel b). The time res-olution is four-monthly. . . 56 6.3 Time series of Z-scores of the respective motifs for buy networks

set, weekly resolution. . . 58 6.4 Time series of Z-scores of the respective motifs for sell networks

set, weekly resolution. . . 59 6.5 Time series of Z-scores of the respective motifs for four-monthly

buy networks set . . . 61 6.6 Time series of Z-scores of the respective motifs for four-monthly

sell networks set. . . 62 6.7 Ensemble distribution of the number of V-motifs, N(V ) panel (a),

and Λ-motifs, N(Λ) panel(b), in the buy-network of week 93; red vertical dotted lines represent the values of observed motifs abun-dance. The fits are obtained by superimposing a Gaussian distri-bution with the sample mean and variance. . . 64 6.8 Ensemble distribution of N(X ), N(W ) and N(M), panel (a), (b)

and (c), in the buy-network of week 93; red vertical dotted lines represent the values of observed motifs abundance. The fits are obtained by superimposing a Gaussian distribution with the sam-ple mean and variance. . . 65

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Chapter 1 Introduction

The modern scientific research has been characterized by a great and real interest in understanding and modeling complex systems, or rather systems whose be-havior is intrinsically difficult to model due to the dependencies and interactions between their components often in the presence of emerging phenomena. Social systems formed out of people, the brain formed out of neurons, a gas formed out of particles, the weather formed out of air flows are all examples of complex systems. The field of complex systems cuts across all traditional disciplines of science, as well as physics, biology, engineering, economy, finance and medicine. The study of complex systems is about understanding indirect effects. Problems that are difficult to solve are often hard to understand because the causes and ef-fects are not obviously related. Pushing on a complex system "here" often has effects "over there" because the parts are interdependent. A surprising fact is that a large part of the sophisticated mathematical tools needed to investigate this kind of systems have already been performed and improved by the framework of Statis-tical Physics. We consider for instance a gas of interactive particle; it is clear that the behavior of the whole system can’t be exactly determined even if the behavior of each single constituent is known. This problem has been solved by relating the microscopic properties of the individual constituents to the macroscopic proper-ties of the system, which can be easily observed. Again in the example of the gas, we can think about the temperature. The temperature is a macroscopic property of the gas, that it incorporate in a single number many informations about the en-ergy of the gas particles and about the strength of their interactions. This kind of approach is suitable for any other field that is included in the study of complex systems. Now if we restrict our attention to Finance we will see that this is a clear example of a complex system; the financial markets are in fact environments with a large class of heterogeneous individuals who interact with themselves, with dif-ferent modalities but towards a common end. From the previous definition results that the behavior of a financial market can’t be determined by the knowledge of

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CHAPTER 1. INTRODUCTION 2 the behavior of its components; therefore the approach for modeling a financial market can’t be different than the one for modeling a gas of interactive particles and this is the spirit of our work.

1.1 Financial markets

According to the classical definition introduced more than two centuries ago by Adam Smith, markets are places where buyers and sellers meet to each other and the prices of exchanged goods are fixed. This apparently simple process gives life to a surprising thing; the information available to heterogeneous individuals, which may be a very different set for each of them, is incorporated into a single number: the price. Now starting from this definition we could ask ourselves what the connections are between physics and finance, since it is a known fact that in the last few decades a genuine interest was born among physicists for this disci-pline, leading to the development of a new branch of research: the Econophysics. The reasons of this interdisciplinary union must be traced back to the fundamental change in the market structure due first of all to the technological progress. In the past, markets were physical places where people met in person, they were located in the most important and industrialized cities (the New York Stock Exchange or the London Stock Exchange to name a few) and their reachability was limited to a small number of individuals. With the advent of the information age, a new type of market was born: the electronic market. This new structure provides an easily available and transparent background where traders can interact with each other across the world and have real time access to prices of a very large class of differ-ent products. One of the most useful aspect of modern market is the possibility of recording every transaction made by any market participant and this fact implies that a huge amount of data is available. With this great abundance of registered data it becomes possible to change the basis from which economics theories are made. In the past the lack of data meant that a theory must be fully formulated before being tested and consequently a good agreement with data was unlikely, in other words testing a theory in a fully quantitative way was impossible: the corresponding predictions were of qualitative nature. On the contrary instead of deriving everything from a set of first principles, the modern theories try to do exactly the opposite: from the stored data, connections are built to relate known behaviors and ultimately derive models. This type of phenomenological approach should look very familiar to physicists, in fact this is the fundamental principle of modern scientific method [17]. It is clear that modern finance could be a per-fect context where approaches and models specific of physics can be used; so a genuine interest is quite natural not only for physicists but also for other scien-tists (mathematics, engineers ecc). This interdisciplinary approach was

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success-CHAPTER 1. INTRODUCTION 3 fully used in financial modeling: application include arbitrage theory, portfolio optimization theory, statics of extreme events, etc (for a complete introduction to these topics see [16], [31], [3], [6] and [29]) acknowledged unanimously by the scientific community (we may just think of the Nobel Prize in Economy won by Merton and Scholes in 1997 for their work on option pricing [8]).

The aim of this thesis is to provide a different approach for analyzing and better understanding the financial markets. At variance with the large part of previous approach, we focus directly on the trading activity of market participants, i.e. the purchase and sell of an arbitrary quantity of financial assets that are present in the considered market. In this approach we focus on some general properties of the system and from these we extract informations on the behaviour of the system’s elements. Our attitude is different from the previous frameworks, where the study of single constituents had a central role. For this purpose we have used a special database, maintained by Euroclear Finland, that reports the activity, in terms of purchased and sold shares, of all the financial institutions present in the Finnish market. To describe dynamically this system and concentrate the large amount of information in an analytical tractable structure, we have used a class of mathematical objects: the complex networks. The study of the properties of these objects, in particular of those which detect some kind of hierarchical structure in the system, it is the central part of this thesis and has provided several results. The idea behind our approach is simple, we first observe a set of topological properties of the networks and we reduce them to some fundamental features of the financial markets. Consequently we introduce some methods to assess if these properties are statistically explainable or they contain a dynamical contribution. Finally we investigate if more complex properties could be explained by the fundamental ones and we introduce a method to evaluate the accuracy of our predictions.

The thesis is structured as follows:

In chapter 1 we introduce the main topics.

In chapter 2 we present some research fields where we aim to contribute, by presenting an excursus of the more significant literature concerning these topics. More precisely we focus on the vulnerability of financial systems with respect to endogenous factors (such as synchronized behaviors of the individuals that are present in the system). In financial language this problem is known as systemic risk.

In chapter 3 we present a formal description of complex networks, we in-troduce some useful properties and we provide the mathematical basis for their understanding.

Chapters 4,5 and 6 are the original parts of this thesis. In chapter 4 we present our database and we construct the set of networks. We separate market partic-ipants and assets in two distinct set of vertices, and we describe an interaction between two generic elements of this sets by linking together the

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correspond-CHAPTER 1. INTRODUCTION 4 ing vertices. We study the time evolution of the number of interactions (i.e. the number of links) and we observe a mean reverting trend in the considered time interval.

In chapter 5 we introduce a method to statistically validate the links of these networks in order to detect which connections are statistically significant; we also present a community detection algorithm that allows us to study the hierarchical structure of the validated networks in terms of the presence of investors and as-sets clusters. Our results show the accuracy of this method to identify significant links and false positive links, i.e. links due to statistical fluctuations. The com-munity detection algorithm shows the presence of highly connected groups in all the considered networks. Therefore we show that the combination of a statistical validation test and a community detection algorithm reveals and well describes preferential relationships among the heterogeneous elements of the considered bi-partite complex system.

Finally in chapter 6 we investigate which properties of our observed networks are reproducible by a statistical ensemble of randomized networks, that preserve only a part of the attributes of our starting graphs. We will evaluate the good-ness of the predictions of our model through an appropriate statistical test. The results of this section show that the accuracy of the predictions of our model are very good with respect to the different topological properties of the considered networks. Furthermore we show how complex features (such as high order cor-relations between vertices) can be observed by considering structures defined in terms of fundamental properties.

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Chapter 2 Fire sales spillovers and systemic

risk

The full analysis of financial markets and the interactions between its components are important not only to study the process of price formation and perform more profitable strategies, but also to prevent global crises. The latter topic has become extremely interesting after the two great crises (the 2007 crisis of subprime and the 2010 sovereign debt crisis) that put the global economy in serious risk. The researchers focus their attention mainly on the role of financial institutions in the creation and in the ensuing spreading of systemic risk; indeed the literature on evaluation and anticipation of systemic events is wide (see [20], [5], [22], [28], [43] and [30] among many contributions).

Although the financial distress could propagate between institutions through many channels, one of the main drivers of systemic risk is the so called fire sales spillovers, i.e. the synchronized heavy sale of one or more assets by a set of finan-cial institutions. The more relevant factors driving fire sales spillovers are asset’s illiquidity and common portfolio holding; the latter factor (discussed in detail in many academic works such as [10], [11] and [9]) means that shared investments create a significant overlap of portfolios between financial institutions; this fact represents an important source of contagion since partial liquidation of assets by a single market player is expected to affect all other market participants who share with it a large fraction of their own investments. The former is a more general con-cept concerning the structure of financial markets and means that fire sales move prices due to the finite liquidity of assets and to market impact; in a perfectly liquid market there will be no fire sale contagion at all. Finally the feedback of fire sales spillover are amplified by the leverage and the constrains imposed to financial institutions [38]. We will present in this section a review of the litera-ture concerning the topic of fire sales spillover, we will focus on the detection, the evaluation and the prevention of these events and eventually we will show the

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CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 6 innovations that our work could provide in the study of this problem.

2.1 Financial institutions’ structure

A deep understanding of fire sales spillover’s effects is impossible if we do not consider the financial institutions’ structure and the role that leverage has in their market’s activity. In the modern financial system the financial institutions actively manage their balance sheets in response to price changes and to changes in mea-sured risk, that it is usually estimated by the "value at risk" (VaR). Since leverage and value at risk occupy a central role in the problem of fire sales, they need a detailed definition. Suppose that a certain financial institution (F) is financed with a mix of equity e and debt d and holds a total number of assets that is a, then a= e + d. Following the notation of [37] the leverage l is defined as the ratio between the total assets and the equity, i.e. l = a_e. The VaR instead is a common estimate for the measured risk and defines the risk of a level of loss (V ) corre-sponding to a given probability of loss (P) over the time interval τ [16]. This definition means, for example, that (in the case of P = 1%) a loss greater than V over a time interval of τ = 1 day happens once every 100 days on average. We denote by V the value at risk per dollar of assets held by the financial institution, then the total VaR will be VaR = V × a. According to the definition of VaR, F has to maintains capital e to meet the total value at risk then e = VaR = V × a. Now following the definition of leverage we deduce that l = a_e =_V1.

Financial intermediaries (i.e. the commercial and investment banks, the mu-tual/pension funds and the broker dealers) leveraged so their net worths are ex-tremely sensitive to price and measured risk changes. This type of institutions en-sure that changes in leverage and changes in balance sheets are positively related so the leverage is procyclical for them; i.e. financial intermediaries adjust their balance sheets in such a way that leverage is high during economics booms and low during busts. Procyclical leverage translates directly to the counter-cyclical nature of VaR (the measured risk is high during busts and low during booms). This type of behavior is readily understandable from the point of view of financial intermediaries but it has serious consequences for the financial system. First of all since market-wide events are felt simultaneously by all market participants, the subsequent reactions to such events are synchronized. If such synchronized reac-tions lead to declines in asset prices and consequently a higher levels of measured risk, there is the potential for a spiral of synchronized reactions. This mechanism is easily explained by the following example. We consider a commercial bank that hold 10 worth of assets and it has funded this holding with debt worth 9 and equity worth 1 and we suppose that price of debt is approximately constant for small changes in total assets. We also suppose that the commercial bank manages

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CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 7 its balance to maintain a constant leverage ratio of 10. Now we suppose that the price of the assets increases by 1%, so the new assets’ value is 10.1, consequently the equity’s value rises to 1.1 and the leverage’s value falls to l = a_e = 9.18. In order to maintain the fixed leverage ratio, the bank must take on additional debit of d∗in to purchase d∗worth of assets. The right value of d∗solves the equation l= a+d_e ∗ = 10.1+d_1.1 ∗ = 10 and it is d∗= 0.9. Thus, an increase in the assets’ price of 0.1 implies an increased holding worth 0.9 and we can assert that the demand curve is upward sloping. We notice that after the rebalancing the leverage comes back to 10 and the assets are worth. Now if we suppose a shock of -0.9% to the assets’ price, that drives down the assets’ value to approximately 10.9, we can see that the mechanism operates also in the reverse case. In fact when the assets’ value falls to 10.9, the equity’s value is reduced to 1 but the debt is still equal to 9.9, so the leverage increases to 10.9. For this reason the bank must sell assets worth 0.9 and pay debt worth 0.9 in order to adjust again the leverage to its initial level of 10, the supply curve is downward sloping and the balance-sheet comes back to the starting state (figure 2.1). This perverse nature of demand and supply curves is stronger when the leverage of financial intermediaries is pro cyclical, due to the fact that when the assets price goes up, the upward adjustment of leverage implies purchases of assets that are larger than in the case of constant leverage.

Starting from this simple but very interesting model, we can provide a deeper and quantitative description of the problem. In particular we are interested primar-ily in two fundamental aspects: the first aspect is providing a model that allows us to estimate the vulnerability of the financial system with respect to fire sales, the second is quantifying the impact of fire sales on the volatility and correlations of asset returns. In the literature the main works concerning these two topics are respectively "Vulnerable Banks" by Greenwood et al. [32]. and "Fire sales foren-sics: Measuring endogenous risk" by R.Cont and L.Wagalath [4] (for many other important contributions see [1], [35], [41] and [7]). In the following section we will present these works and the corresponding results.

2.2 Vulnerable Banks and systemic risk

We focus now on the problem that emerges when financial stress experienced by bank contaminates other banks and spirals into a shock that distresses the broader financial system. Measuring systemic risk has become the financial regulator’s priority since the 2008 collapse of Lehman Brothers triggered financial distress among the collectivity of US financial institutions; this need for accurate mea-sure has been also strengthened by the 2010 sovereign debt crisis. Following the framework of Greenwood et al. we present a model that is able to quantitatively measure the systemic risk and it is also able to distinguish between bank’s

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con-CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 8 Assets • 10 Liabilities • Equity 1 • Debt 9 Leverage • 10 Assets • 10.1 Liabilities • Equity 1.1 • Debt 9 Leverage • 9.18 Assets • 11 Liabilities • Equity 1,1 • Debt 9,9 Leverage • 10 Assets • 10.9 Liabilities • Equity 1 • Debt 9.9 Leverage • 10.9 + 1% rebalancing - 0.9% rebalancing

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CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 9 tribution to financial sector fragility (that is the definition of systemicness) and a bank’s vulnerability to the deleveraging of other banks. The set up of the model is the following: We consider two periods t=1,2 and N banks. Each bank is fi-nanced with a mix of debt dnt and equity ent; At is the N × N diagonal matrix of

bank’s assets, so each diagonal term is equal to ant= ent+ dnt, and L is the N × N

diagonal matrix of leverage such that each diagonal term is ln=d_e_ntnt. There is a set

of assets K and each bank n holds an amount of mnk (that is the weight in terms

of dollars’ amount) of the asset k and M is the N × K matrix of weights. In each period, the vector of banks’ unlevered returns is:

R_t = MF_t

when Ft is a K × 1 vector that denotes assets net returns. With this simple model

we can describe realistically a distressed financial system by introducing three assumptions (that are a good approximation of the financial institutions’ and mar-ket’s behavior):

• Asset trading in response to bank return shock

We assume [37] that banks scale up or down their total assets in order to maintain a fixed leverage level in response to positive/negative shocks. • Target exposures remain fixed in percentage terms

We assume that banks sell assets in response to negative shock in such a way as to hold the matrix M constant between dates 1 and 2. If we indicate with φ the K × 1 vector of net asset purchases by all banks in period 2, according to the previous assumption we have φ = M0A1LR1. As an illustrative

exam-ple we can consider a bank holding 10 percent cash, 20 percent in stocks and 70 percent in bonds. If the bank scales down its portfolio by ten units, it will sell 2 units of stocks, 7 units of bonds and it reduces its cash by 1 unit. This assumption is admittedly strong because in practice banks may first sell their most liquid assets.

• Fire sales generate price impact

We assume that asset sales φ in the second period generate a price impact according to the linear model: F2= Iφ , where I is the matrix of price impact

ratios; for simplicity it is assumed to be diagonal (i.e. fire sales of one asset do not directly affect prices of other assets). This assumption allows us to compute the banks’ returns in period 2: R2= MF2= MIφ = (MIM0LA1)R1.

We can iterate multiple rounds of deleveraging following an initial shock, by further multiplying for the transition matrix MIM0LA1, but for simplicity

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CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 10 We suppose, fixing the notation, multiple scenarios of financial shock and we analyze the reaction of the whole aggregate banks’ system. The first aspect that we consider is measuring the aggregate exposure to deleveraging (so called Aggregate Vulnerability). The scenario is the following: we suppose a negative shock −F1= (− f1, · · · , − fk) to assets returns, it translates into shocks to banks’

asset given by A1MF1. The aggregate direct effect on all bank assets is simply

10A₁MF₁ where 1 is a N × 1 vector of ones and this effect doesn’t involve any contagion, it is simply the change in asset value. Now if we follow assumption 3, we can compute the aggregate effect of shock F1 on bank assets to fire sales

by simply pre-multiply MIM0LA₁MF₁by 10A₁. If we normalize this quantity with the total bank equity pre-deleveraging E1, we define the aggregate vulnerability:

AV = 1

0_A

1MIM0LA1MF1

E₁

AV is a measure of the percentage of aggregate bank equity that would be wiped-out by bank deleveraging as a result of a negative shock −F1to asset’s returns. We

can extract very useful information by writing −R1= −MF1= (−r1t, · · · , −rnt)0

and rearranging the definition of AV in the following way: AV× E1=

∑

n

γnlnan1rn1

where γn= ∑k(∑mammnk)ikmnk measures the connectedness of bank n and ik is

a measure of assets’ liquidity. From the last equation we can notice that the four terms (connectedness, leverage, size and exposure) enter multiplicatively in the definition of AV and have the same relevance. Therefore the systemic risk in-creases if we have large banks (large an1), in the presence of high leverage (large

l_n1), highly exposed to the considered shock (rn1) or highly connected (large γn).

Using the same notation we can introduce another important measure that is the contribution of each bank to the aggregate vulnerability, i.e. the Systemicness. The situation is identical to the previous one but with the difference that the shock F1affects only the bank n. The Systemicness is defined as:

S(n) = 1 0_A 1MIM0LA1δnδ 0 nMF1 E1

where δn is a N × 1 vector of all zeros except the nth component. Since we can

interpret the Systemicness as the contribution of bank n to the aggregate vulnera-bility it implies that AV = ∑nS(n). Again if we expand the definition of S(n) the

previous equation will reads:

S(n) = γn×

a_n E₁

× ln× rn1

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CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 11 • A bank is more systemic if it is more connected (larger γn), i.e the bank

owns assets that are both illiquid and strongly held by other banks. • A bank is more systemic if it is big (bigger an

E1).

• A bank is more systemic if it is more levered (larger l_n), a shock to a more levered bank is going to induce to sell more, which generates more price impact.

• Obviously a bank is more systemic if it receives a bigger shock rn1.

This model has been tested by Greenwood et al. on a database provided by European Bank Authority, which is composed by the 90 largest banks in the EU27 countries. The authors have figured an hypothetic scenery where the financial system is affected by a 50% write-down (a very large shock) on all sovereign debit of GIIPS countries (Greece, Italy, Ireland, Portugal and Spain). The results are very interesting, the authors have found that a 50% write-down on all GIIPS debt wiped out 111% of the equity of the average bank through direct impact, while the indirect impact from deleveraging would create a surprising additional loss of 302% of equity.

2.3 Fire sales analysis and endogenous risk measure

The second approach to the problem of fire sales that we will consider is the one introduced by R.Cont [4] and it concerns the effect of fire sales on the volatil-ity and correlations of assets. In particular, unexpected spikes in the correlations across asset returns have been frequently observed during market downturns and they lead to a loss of diversification benefits for the investors. On a large scale ex-ample, we can just think the great deleveraging of financial institutions’ portfolios that is followed to the default of Lehman Brothers in 2008, it led to an unprece-dented peak in correlations across assets returns. The authors of the framework propose a method for modeling and estimating the impact of fire sales in multi-ple funds, they provide a structural explanation for the variability observed in the measures of cross sectional dependence in asset returns, by linking such increases in cross-sectional correlation to the deleveraging of large portfolios. The authors also show how these parameters could be estimated from empirical observation of price series. All these results provide a quantitative framework for the analysis of the impact of fire sales and distressed selling, that are explained with two em-pirical examples: the August 2007 hedge fund losses and the great deleveraging of bank portfolios following the default of Lehman Brothers in September 2008. Since the mathematics behind this framework is very complex, we will show only

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CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 12 the final results needed for understanding the results of the empirical examples. A complete description can be found in the paper [4]. The central result concerns the definition of the realized covariance matrix of asset returns in the presence of fire sales and it is captured by the following corollary: The realized covariance be-tween the time 0 and T ( no fire sales) and bebe-tween T and T + τ_liq(where T + τ_liq is the end of the liquidation) are respectively equal to

C_{[0,T ]}= 1 T Z T 0 c_tdt= Ξ and C_{[T,T +τ}_liq_]= 1 τliq Z T+τ_liq T c_tdt = Ξ + LM0ΠΞ + ΞΠM0L+ O(||Λ2||)

with the following assumptions and definitions:

• the authors consider valid the following assumption: there are no fire sales between 0 and T and each fund j liquidates between T and T + τliq at a

constant rate γj

• ct is the instantaneous covariance matrix of returns

• Ξ is the ’fundamental’ covariance matrix of returns, it describes the corre-lation structure of returns in the absence of fire sales.

• M0= ∑1≤ j≤J γj V₀j× α j_(αj₎t

is a matrix that characterizes the excess re-alized covariance and it reflects the magnitude of the fire sales. The dif-ferent funds are labeled by j ,V₀j is the starting value of the portfolio j,

αj=    α1 .. . α_k  

is the vector of positions of fund j and α

j_(αj₎t _{is a k × k}

sym-metric matrix that represents the projection of fund j’s positions. Hence M0

is a sum of projectors and it captures the direction and intensity of liquida-tions in the j funds.

• L is a diagonal matrix with the i-th diagonal term equal to 1

λi where λiis the

market depth for asset i and is interpreted as the number of shares that an investors has to buy/ sell in order to increase/decrease the price of asset i by 1%.

• Π is a diagonal matrix with i-th term equal to _τ1

liq

RT+τliq

T Ptidtwhere Ptiis the

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CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 13

• Λ = (Λ1, · · · , ΛJ) is a vector where each Λj=

   α1 λ1 .. . αn λn  

represents the position of fund j in each market as a fraction of the respective market depth. Clearly the real matrix M0is unknown, so we need to estimate its from a set of

ob-served data. If we denote by bM(τ) the estimator of the matrix M0and we consider

its asymptotic distribution, we are able to test whether M0 6= 0 i.e. if

signifi-cant fire sales occurred between T and T + τ_liq. In fact under the null hypothesis H₀: M0= 0 that there are no fire sales between T and T + τliq, the estimator bM(τ)

will verify a proper central limit theorem. This result allows to test whether the variability in the realized covariance of asset return during [T, T + τliq] may be

ex-plained by the super position of homoscedastic fundamental covariance structure and feedback effects from fire sales. In practice, this test translates in estimating the matrix M0and testing the nullity of the liquidation volumes of fire sales.

As we mentioned early, the authors test this model first in a simulated discrete-time market and after the great deleveraging of fall (2008) and in the hedge fund losses of August 2007. They estimate all these parameters using the observed time series of S&P500 and Eurostoxx50 index and they find a good agreement between theoretical predictions and empirical observations. More precisely in the considered cases, we can consider reliable the hypothesis where the increases of correlations and volatility of assets are driven by the excess liquidation due to fire sales.

2.4 A new approach to the problem

The two approaches previously presented have led to very important results, they are very useful for regulators in view of systematically investigating all systemic risk events in financial systems. In spite of the flexibility of such methods, they have some limitations. First of all, none of the two approaches allows to directly observe fire sales spillovers, they propose methods to indirectly observe those through the study of phenomena strictly dependent on them or methods to predict the effects of their presence. Secondarily the two models use a large number of parameters and need specific databases, that often are hardly achievable for their estimation (we can just think at market depth, banks’ leverage, bank’s portfolios ecc to name a few). In this present framework we propose a new type of approach that allows us to directly observe the fire sales. The two fundamental features of our approach are the specific database considered and the particular object used to investigate the financial market: the bipartite network of investors-assets. The database, described in details in the chapter 4, records the daily investors’ activity,

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CHAPTER 2. FIRE SALES SPILLOVERS AND SYSTEMIC RISK 14 in terms of the shares’ purchases and sells, that appear in the Finnish financial market. Therefore we are able to see all the transactions made by investors in any available asset during the considered time interval. The bipartite network of investors-assets allows us to collapse the large amount of provided information in an essential, flexible and analytical tractable object. The time dependence of many relevant characteristics of the considered financial system, can be studied by analyzing the time dependence of some topological properties of this network. From the perspective of fire sales spillovers, since they are essentially strong and synchronized sells, they are directly observable with our approach and we can introduce many network properties that are able to capture the magnitude and synchronization of such events. Thus the scope of the present framework is not only to provide a different point of view for the problem of fire sales spillover, but also to introduce a different analysis of the financial market that could be useful for studying and understanding a large class of other phenomena.

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Chapter 3 Complex networks

The interacting components of a complex system form a network, which is a col-lection of discrete objects and relationships between them and it is usually repre-sented by a graph of vertices connected by links, called edges. In the real world there is a large number of systems that match this description coming from very different fields; biological networks such as the neural connections or protein in-teractions, technological ones such as the Internet, psychological and social net-works, infrastructural systems such as roads and highways and financial networks. Research involving networks started up as early as the beginning of 20th century, later on we will introduce a brief overview of the past contributions in network research.

This section is inspired by the work of G.Ghoshal, for a deeper discussion see [15].

3.1 Historical overview

Despite the fact that research involving networks have mainly grown up in the past 100 years, we can found a primitive introduction of this topic in Leonhard Eu-ler’s work Solutio problematis situs pertinentis, in which a solution for the ’Seven bridge of Konigsberg problem’ is presented. The problem was concerned with a city (Konigsberg) that was set on both side of Pregel River and it included two islands connected to each other and to the mainland by seven bridges. The ques-tion was to find a walk through the city that could cross each bridge once and only once. The Euler’s solution was formulated in abstract terms by considering only the islands and the bridges, in modern terms he replaced the landmass with an ab-stract vertex and each bridge with an abab-stract connection (an edge). The resulting mathematical object was a graph and the Euler’s formulation of the problem laid the foundations of the modern mathematical graph theory. In this theory has been

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CHAPTER 3. COMPLEX NETWORKS 16 improved only from the 19th century by the contributions of Thomas Kirkman, William Hamilton and Gustav Kirchoff which firstly employed graph theoreti-cal ideas in practitheoreti-cal purposes (the theoreti-calculation of electritheoreti-cal currents in circuits). The developments that emerged from their works concerned mainly theoretical mathematical objects rather than systems that can describe the real world (the lat-ter definition is what we mean by network theory in modern context). The first appearance of real-world networks’ studies goes back to 1930 in the context of social sciences. The main contribution came from the psychiatrist and sociolo-gist Jacob Moreno [18] who pioneered the systematic recording and analysis of social interaction in small groups. The general method that social scientists used to gather data was by directly querying participants ([26],[36]), this obviously involved a labor-intensive effort and a limited size of the constructed networks (the maximum size is of the order of few hundreds of vertices). The most impor-tant change in real-networks studies has developed in 1990s with the proliferation of widespread computing and technological resources. The birth of large scale communication systems (such as Internet) has inevitably changed the size of the analyzed networks (from a few hundreds of vertices to millions or even billions) and consequently also the methods for studying them have changed. Rather than focusing on the properties of a single vertex or edge (that is clearly impossible when the number of available vertices is of the order of thousands or greater), it is more useful to look at statistical properties of networks (such as vertices’ degree distribution or vertices’clusters that affect network connectivity). This change of approach naturally has led to physicists interest, since the study of large and com-plex systems is one of the main topics of physical research and the techniques from statistical physics are well suited to be employed in such studies (see [23], [27] and [19] among many others contributions). In the last recent years physicists extended the empirical studies of real-word networks to different research fields including social, biological, technological and financial systems. The latter, that is the topic of this present work, showed to be breeding ground for many applica-tions of graphs theory. Many innovaapplica-tions have been introduced in different areas, such as : identification of clusters of investors [24], assessment of systemic risk due to fire sale spillovers [14] or study country exportation through World Trade Web analysis ([13] and [12]) to mention only a few.

3.2 Types of graphs

In the previous section we have defined a graph as a collection of vertices linked together by edges. In mathematical terms a more formal definition is the follow-ing:

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CHAPTER 3. COMPLEX NETWORKS 17 empty set of vertices, V (G), a finite set of edges E(G) and a incidence function ψGthat associates an unordered pair of distinct vertices with each edges. An edge

is incident with a vertex v if v is one of its ends and two vertices joined by an edge are adjacent [2].

Starting from this simple definition we can define several types of graphs and their respective topological properties. These properties allow us to perform a quantitative description of these objects, very useful in practical applications for describing ad understanding the system that the network represents. The different types of graphs are defined by adding particular attributes to the set of vertices and edges. The simplest case is the unipartite network: a clear example of this type could be an e-mail network, where vertices represent individuals and an edge represents the action of sending an e-mail. The first aspect that we can consider is the direction of an edge. Considering the previous example again, direction means that individual A sends an e-mail to individual B but not vice versa so the edge can point only in one direction. A graph with this type of edge, is called directed graph. Another important property for the edge is the weight: individual Acan send an arbitrary number (greater than one) of e-mails to individual B so the respective link will contain an addictive information and will become a weighted link, the corresponding graph is called a weighted graph. Attributes can be added not only to edges but also to vertices, in fact they can take on characteristics that represent different membership categories (for the previous example gender, na-tionality ecc). If we introduce different types of vertices, it is natural to divide a network with respect to these various types. The simplest network’s partition can be performed by introducing only two types of vertices, with edges running only between the two types and not within them. This type of partition will identify a so called bipartite network. In general, multiple types of vertices, can be de-fined with edges running only between unlike types and the respective types of networks are called multipartite networks.

3.3 Standard network metrics

In this section we present some standard statistical and topological measures to quantify some important properties of the network structure.

3.3.1 Adjacency and incidence matrix

The simplest and most common way to represent a network is by introducing the so-called adjacency or incidence matrix. The full set of structural properties of a undirected graph is uniquely determined by these two matrices. Let G be a graph with vertex set V (G) = v1, · · · , vpand edge set E(G) = e1, · · · , eq. The incidence

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CHAPTER 3. COMPLEX NETWORKS 18

Figure 3.1: Representation of a graph

matrix of G is the p × q matrix of the form:

M(G) = mi j with

(

m_{i j} = 1 if the edge ej is incident to the vertex vi

m_{i j} = 0 otherwise

Similarly the ad jacency matrix of G is the p x p matrix of the form:

A(G) = ai j with

(

a_{i j}= 1 if the vertices i and j are connected a_{i j}= 0 otherwise

A(G) is a symmetric matrix (in the case of directed networks these matrix is still not symmetric) with non negative entries, in the case of a weighted graph these entries take the values of the respective assigned weights. If we consider, for example, the simple network in figure 3.1, the incidence (M) and adjacency (A) matrix are: M=       E1 E2 E3 E4 E5 E6 V1 1 1 1 0 0 1 V2 1 0 1 1 0 0 V3 0 0 1 0 1 1 V4 0 0 0 1 1 0 V5 1 0 0 0 0 0       A=       V1 V2 V3 V4 V5 V1 0 1 1 0 1 V2 1 0 1 1 0 V3 1 1 0 1 0 V4 0 1 1 0 0 V5 1 0 0 0 0      

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CHAPTER 3. COMPLEX NETWORKS 19 The matrix representation is a powerful tool for analyzing networks, in fact a lot of informations can be extracted by using common matrix properties. One of the most intuitive property that we can consider is the number of edges incident on a particular node, called the degree of the node, and this is easily computed by this relation: d_i= p

∑

j=1 A_{i j}

where diis the degree of vertex i and the set of diis usually called degree sequence

and is indicated with {d}_ip

3.3.2 Degree distribution

Despite its simplicity, the degree sequence and consequently the degree distribu-tion, it is one of the most important and studied property of network structure. The degree distribution pd is defined as the fraction of vertices that have degree

d and it is an actual probability distribution; p_d represents the usual probability measure i.e. the probability of a random chosen vertex to have exactly d edges connected to it. Given its nature of probability distribution, pd satisfies the usual

normalization condition:

∞

∑

d=0

p_d= 1

and all the various moments can easily be calculated. For example the first mo-ment (i.e. the average degree for the network) is:

hdi =

∞

∑

d=0

d p_d

In real world networks the degree distribution comes with very different types, the most common is when vertices have degrees that are highly right-skewed that implies a long right tail of the corresponding distribution. Typically these take on the form of power laws:

p_d≈ d−β

where β in the most cases is between 2 and 3. The degree distribution is useful for several reasons. First of all most of the mathematical models are based on it; the statistical validation and ensemble sampling algorithms shown in this paper are only a few examples. Secondarily, in terms of real-world network, vertices’s degree is the simplest measurable quantity and a large class of more complex topological properties are strictly dependent on it.

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3.3.3 Transitivity and Clustering

Transitivity is a property observed in a large class of real networks and is strictly connected with the concept of clustering. Roughly speaking, transitivity is the measure of the presence of triangles in the network (we mean by triangle a triplet of vertices that are all connected to each other) and mathematically is quantified by the so called clustering coefficient [42]:

Ci=

ni

∆

n_tripletsi

where Ciis the clustering coefficient, ni_∆is the number of triangle and ni_triplets(we

mean by triplets a single vertex with edges running to a pair of other vertices) having i as a vertex. The total clustering coefficient for the network is:

C=∑iCi p

with p is the total number of vertices and C is included in the range [0,1]. It is clear from the previous definition that C measures the fraction of the closed triplets by a third edge, or in other words is the mean probability of two neighbors to share a common third neighbor.

3.3.4 Assortativity

Very often real world networks involve vertices of different types. For example we may consider a network of academics in a university setting, vertices here represent professors or researchers and edges might represent some manner of acquaintance. An obvious way to distinguish between vertices in this case, is if there is some scalar attribute which represents academic discipline (physics, mathematics, political science etc). We can ask if vertices of a certain type connect only to vertices of similar type (in our example two physicists are more likely to know each other than someone who is a political scientist). This type of selective linking is conventionally referred to assortativity or homophily. A special and very common case of assortativity mixing is based on the degrees of vertices and is often called degree correlation. In this case we are interested in detecting if high-degrees vertices are mainly connected with other high-degree vertices or else they are connected with low-degrees ones. These correlations can be quantified by the average nearest neighbour degree (ANND) considering again the adjacency matrix A:

d_inn(A) =∑j6=i∑k6= jai jajk ∑j6=iai j

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CHAPTER 3. COMPLEX NETWORKS 21 where d_inn(A) is the ANND of vertex i and a_{i j}is a generic element of the adjacency matrix A. Usually the d_inn are plottered against the degree sequence {d}i in order

to assess if a high-degree vertex shows high value of ANND or not.

3.4 Bipartite networks

Figure 3.2: Representation of a simple bipartite graph

A bipartite network is a special case of the aforementioned class of multipar-titenetworks and it is able to represent a large class of systems including the case investigated in this thesis.

3.4.1 Biadjacency matrix

The particular property featuring bipartite networks ( i.e. the presence of two dis-tinct set of vertices) induces a very intuitive matrix representation of this object. Let V1= {v1, v2, . . . , vP} denotes the first set of vertices and V2= {vP+1, vP+2, . . . , vP+Q

the second; the adjacency matrix A(G) is a (P + Q) × (P + Q) square block matrix that takes the form:

A(G) =

0 B(G)

B(G)T 0

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CHAPTER 3. COMPLEX NETWORKS 22 B(G) is a P × Q rectangular matrix called biadjacency matrix and takes the form B(G) = bpq with

(

bpq= 1 if the vertices vp ∈ V1and vq ∈ V2are connected

bpq= 0 otherwise

If we consider, for example, the simple bipartite graph in figure 3.2, its biadaja-cency matrix will be:

B(G) =   AS1 AS2 AS3 AS4 AS5 MP1 1 1 0 0 0 MP2 0 1 1 1 0 MP3 0 0 0 0 1  

The biadjacency matrix is useful even if we want to study the connection between nodes of the same set. As specified above in a bipartite graph vertices belonging to the same set can’t directly link to each other but introducing connections be-tween them is still possible. The simplest way of inferring the presence of links between any two nodes of the same set is connecting each other with a weighted edge depending on the number of common neighbours shared by the considered vertices. The mathematical object that comes out is again a square matrix, called monopartite projectionon the considered set, and is defined by the following re-lation:

O(G) = opp0 with

(

o_pp0 = w where w is the number of common neighbours

shared by vertex p and p0

We show as illustrative example again the two monopartite projections on both sets of graph in figure 3.2:

O(G)MP=   MP1 MP2 MP3 MP1 0 1 0 MP2 1 0 0 MP3 0 0 0   O(G)AS=       AS1 AS2 AS3 AS4 AS5 AS1 0 1 0 0 0 AS2 1 0 1 1 0 AS3 0 1 0 1 0 AS4 0 1 1 0 0 AS5 0 0 0 0 0      

Given the important amount of information encoded in the monopartite projection, in the present work we will recursively use this object especially in the sections of statistically validated links (section 5.1) and communities detection (section 5.2).

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3.4.2 Degree sequence and other metrics

The entire set of metrics introduced for standard graphs can be extended to the bi-partite case with specific arrangements, again all the considered topological prop-erties can be extracted from the biadjacency matrix. We now briefly introduce such extensions, used in section 6.

Degree sequence

The degree sequence is the fundamental property also for bipartite networks. For-mally it can be defined exactly as the monopartite case with the substantial differ-ence that now we have two distinct degree sequdiffer-ences, one for each set, and they can be calculated by simply summing over rows and columns of the biadjacency matrix: (dset1)p= Q

∑

q=1 b_pq (dset2)q= P

∑

p=1 b_pq Assortativity

The correlation between the degrees of adjacent vertices can be measured with respect to the two distinct sets of nodes by extending the concept of ANND. Let d1_p to be the degree sequence of set 1 and d_q2the one of set 2, the ANND of set 1 is defined as: dNN1_p (B) =∑ Q q=1bpqdq2 d1 p

and the ANND of set 2 is:

d_qNN2(B) = ∑

P

p=1bpqd1p

d2 q

From the above definition it is clear that the correlation of a generic node belong-ing to a generic set is defined in terms of the degrees of all the nodes linked to it (that obviously belong to the other set) since direct connections of vertices of the same layer are forbidden.

Transitivity

Introducing the concept of transitivity in the bipartite case is more complex and less intuitive with respect to the other metrics. The main reason is that triangles can not be observed in bipartite network due to the impossibility of linking ver-tices of the same set, so no clustering coefficient can be calculated. Therefore we have to consider a new class of objects, commonly known as motifs in order to

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V− moti f V3− moti f W− moti f

Λ − moti f Λ3− moti f M− moti f

X− moti f

Table 3.1: Visual example of different motifs. Green dots label MP vertices and orange dots label assets vertices.

detect the hierarchical structure of the network. The motifs are a class of sub-networks, with a structure that can be more or less complex depending on the order of correlation between nodes that we wish to study.

The first class of moti f s that we will consider are the V -motifs and Λ-motifs [33]. The former is a couple of investors trading the same assets, quantifying the synchronization of the investors trading strategies; the latter is a couple of assets are traded by the same investor, quantifying the diversification of investors’ strategies. Mathematically we can define the number of respective motifs in the following way: N_V(A) = N

∑

n=1 N

∑

n0_=n+1 M

∑

m=1 a_nma_n0_m= M

∑

m=1 dA m 2 N_Λ(A) = M

∑

m=1 M

∑

m0=m+1 N

∑

n=1 a_nma_nm0= N

∑

n=1 dI n 2

We can generalize the two previous measures by rising the number of investors ac-tive on the same asset, as well as the number of assets traded by the same investor,

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CHAPTER 3. COMPLEX NETWORKS 25 leading to the definition of V (Λ) motifs of n-th order:

NVn(A) = M

∑

m=1 dA m n N_Λ_n(A) = N

∑

n=1 dI n n

We can capture high-order correlations by extending the concepts of motifs to a more general class of sub-networks. We can increase the number of both sets’ element simultaneously leading to the definition of more complex motifs: these are the X , M and W − moti f s. The X − moti f is composed by a couple of investors which trades the same two assets (or vice versa) and its formal definition is: N_X(A) =

_∑

n<n0_m<m

∑

0 a_nma_nm0a_n0_ma_n0_m0 =

_∑

n<n0 V_nn0 2 =

_∑

m<m0 Λmm0 2

Similarly we define the M and W − moti f s by enlarging the set of investors or assets: N_M(A) =

_∑

n<n0 V_nn0 3 N_W(A) =

_∑

m<m0 Λ_mm0 3

Table 3.1 shows the introduced motifs. The accuracy of BiCM model in repro-ducing the number of motifs measured in the real networks, can be quantifies by computing the z-score:

z_x=Nx(A) − hNxi σx

where σx=phNx2i − hNxi2 and x indicating a particular motif considered. If the

distribution of Nx is Gaussian, we can consider the z-score as the usual

standard-ized variable. In the following, we will present the results of motifs research as usual for both the available databases [13].

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Chapter 4 Database and Networks construction

In this chapter we will introduce the specific database used for our analysis. After we will describe the set of constructed networks, we will observe some of the fun-damental properties of these objects and we will study their temporal evolutions. Finally we will provide a quantitative description of the investigated financial sys-tem and of the interaction between its constituents.

4.1 Data

We investigate the trading activity of the financial institutions registered in a spe-cial database maintained by Euroclear Finland. The database is a register that re-ports the activity, in terms of bought and sold shares, of all the institutions present in the Finnish market. The total number of register’s members is 1483 and for each individual member we study their trading actions on a set of 78 different stocks. For our database there are two resolutions available: A daily resolution, where the considered time interval starts from the 02/01/2007 until the 31/12/2008 for a total of 515 registered days, and a monthly resolution, with a larger interval start-ing from the 01/1995 until the 12/2016 for a total of 216 registered months. We will perform our analysis on both datasets and the respective results will presented separately.

4.2 Networks construction

Daily resolution

Starting by the 515 registered days observed that the density of the respective net-works, i.e. the fraction of links, is too small and this reason could be a problem

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CHAPTER 4. DATABASE AND NETWORKS CONSTRUCTION 27 0 10 20 30 40 50 60 70 80 90 100 110 Week 100 150 200 250 300 350 400 450 500 550 600 N of links

Figure 4.1: Number of buy(o) and sell(*) links in function of time.

when we will apply our statistical links validation and communities detection al-gorithms to the entire set of networks. Taking this into account we have decided to aggregate the database every 5 days and this longer time interval involves a larger density of the respective graphs. After the aggregation has been made, we split the buy transactions from the sell ones and for each week we have two bi-partite networks; one with only buy links (that we call buy network) and another with only sell links (sell network). Formally the object is a multipartite network where three distinct set of vertices are present:

• V (B) the set of buyer id • V (S) the set of seller id • V (A) the set of assets

We recall from the definition of multipartite network that a link can exists only between V (B) and V (A) or V (S) and V (A). The time dependence of the number of buy and sell links is shown in the figure 4.1.

This simple plot is a first qualitative measure of the synchronization between market’s members. As we can see, there are many time intervals where the num-ber of buy links dominates the sell one and vice versa many intervals where this two numbers are very close to each other. This fact allows us to infer that there are periods where the activity of investors is strongly synchronized in one of the two possible state (buy/sell) and periods where is present a kind of equilibrium between states. Finally the two time series appear to show a decreasing trend in the investigated time interval.

The plot in figure 4.1 allows us to extract general informations about the prop-erties of the consider graph’s set, specifically they are useful to show the time

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CHAPTER 4. DATABASE AND NETWORKS CONSTRUCTION 28

(a) Buy bipartite network of week 56.

(b) Sell bipartite network of week 56.

Figure 4.2: Representation of both buy and sell network of week 56. Yellow dots are MP vertices, blue dots are asset vertices, green and red lines are respective buy and sell links.

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CHAPTER 4. DATABASE AND NETWORKS CONSTRUCTION 29 dependence of these properties, but they aren’t able to describe specific attributes of the elements of the network. For this purpose, the best solution is directly vi-sualize the considered network. For example, in figure 4.2 we show the bipartite buy and sell networks of the week 56 (chosen due to the high density of both net-works). The dotted colors label the different sets of vertices (blue labels market participants (MP) set, yellow asset set) and the sizes of the nodes are proportional to the respective degrees. We labeled also the edges with different weights and col-ors proportionally to the degrees of the adjacent vertices (darker and wider lines denote links between high degrees nodes). We can extract very useful information by simply looking at these figures; indeed we can see directly all the links in the respective networks and we can quantify qualitatively the graphs’ structures and properties. For example, the figure shows that the biggest nodes tend to have links with the same assets, involving a high synchronization of their trading activities (this aspect will be quantitatively analyzed in the chapter 6 with the introduction of motifs). For a deeper structural analysis we refer to the section of communities detection.

4.2.1 Monthly resolution

For the monthly dataset we have found the same problem as the daily one. Due to the small network’s density we had to aggregate the dataset at a four-month resolution, ranging from a time series of 264 months to a series of 66 four-month. After that, the same analysis has been performed, the multipartite networks have been created and they have been divided between buy and sell networks. Figure 4.3 shows the number of links of the respective set of networks in function of time, exactly as figure 4.1. The figure 4.1 and 4.3 have both similarities and differences. First of all the two monthly time series have quite similar behavior, they still show a trend that seems to be mean reverting. Especially for the buy time series the trend seems to be increasing in the first 20 four-months, roughly constant in the following 30 four-months and decreasing at the end of the con-sidered time interval; for the sell one this trend still appears but it is not so well defined. The plot also shows this behavior but in this case the number of buy and sell links are quite close together for all the considered time intervals and there are few and not very marked periods where one state dominates the other; so for monthly resolution the investors seem to persist in an equilibrium between the two possible states. In the next chapters of this framework we will deeply study all the characteristics shown by this plot and we will provide quantitative result.

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CHAPTER 4. DATABASE AND NETWORKS CONSTRUCTION 30 0 10 20 30 40 50 60 70 Four-month 100 200 300 400 500 600 N of links

Figure 4.3: Number of buy and sell links in function of time for four-monthly resolution. As for fig 2.1, green line labels buy links, red line labels sell link.

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Chapter 5 Statistical validated networks

Our set of networks shows many properties, in this chapter we introduce some methods to assess if these properties are statistically explainable or they contain a dynamical contribution. Several approaches have been developed in literature, they are based on the construction of statistical ensembles of completely random or with specified constraints graphs, systemically used as a reference to iden-tify non-random patterns in real networks. In the following framework first we will present two different approaches for the statistical validation of networks, both well know in the literature and successfully used in various research fields. The differences between these methods have specific theoretical bases; in fact in statistical physics terms, the ensembles generated by the two approaches are microcanonical for the former one and grancanonical for the latter one. After we will apply these methods to our specific case and we will show the goodness of our results. More precisely we will focus on the interactions between the ele-ments of our system (i.e. the considered financial market), we will assess if these relations contain dynamical information and if they are useful to investigate the hierarchical structure of the system.

5.1 Hypergeometric null hypothesis validation

The first method proposed is the network links validation against a proper null hypothesis. This type of methodology is one of the fundamental bases of statistic analysis, and its application to networks is logic and useful ([25] and [24]).

5.1.1 Introduction

This method provides a direct validation of each link in the network and it is articulated in the following steps:

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CHAPTER 5. STATISTICAL VALIDATED NETWORKS 32 1. First we introduce a null hypothesis of random connectivity between ele-ments in the starting bipartite network that take into account the differences of both sets of elements.

2. We associate a p-value with each link of the projected network, in order to test the presence of the link against the selected null hypothesis.

3. Finally we introduce an appropriate correction of the statistical significance level in the presence of multiple hypothesis testing.

5.1.2 The mathematics of the model

In section 3.4.1 we have introduced the projected network extracted from the corresponding bipartite network. Once defined the projected network we can statistically validate each link against a null hypothesis of random occurrence of common neighbors that takes into account the degree heterogeneity of elements of both disjoint sets (called now A and B) of the starting bipartite graph. For this purpose we first decompose the bipartite network in subsets ak consisting of all

the N_Bk elements belonging to set B with a given degree k and of all the elements of A linked to them. Now we consider two elements i and j of the set A and we assume that they have N_{i j}k common neighbors in set Bk (the set of elements of

B with a degree equal to k). Under a null hypothesis of randomly connection of elements i and j to the elements of B_k, the probability that elements i and j share X neighbors in set Bk is given by the hypergeometric distribution [25]:

H(X |N_Bk, N_ik, Nk_j) = N_ik X N_Bk−N_ik N_ik−X Nk B Nk_j

where N_ik, Nk_j are respective the degree of element i and j. Consequently we can associate a p-value p(N_{i j}k) with the real number N_{i j}k of neighbors of element i and

j: p(N_{i j}k) = 1 − Nk i j−1

∑

X=0 H(X |N_Bk, N_ik, Nk_j)

Now the next step of the method is to set a level of statistical significance p, which takes into account the fact that we are performing a multiple hypothesis test (more precisely a test for each pair of elements of A for each subsystem ak). Several

methods that provide the correction for multiple hypothesis test are present in literature [25], we will consider two of them that are the Bon f erroni correction and the FDR (False discovery rate) correction. If we suppose that the threshold

Statistically validated networks for financial data analysis