the network by applying the different procedures for grouping and for filtering the correlation matrix of daily returns. Tola et al. (2008) used clustering algorithms to improve the reliability of portfolios in terms of the ratio of the expected risk to the realised risk. They showed that clustering portfolio optimization methods often outperforms the Markowitz or random matrix theory methods. Khashanah and Miao (2011) applied the MST to the entire financial system, consisting of typical markets (stocks, bonds, derivatives, currencies, and commodities) to study the changes in the structure of the financial system during the economic downturn. With regard to the application of MST techniques on the stock market, a great deal of work has been done in both developed and emerging countries. For example, there has been work on certain emerging markets; Galazka (2011) used the MST to identify the stocks that strongly influenced the price dynamics of other stocks on the same Polish stock market, using a portfolio of 252 stocks in 2007. Majapa and Gossel (2016) applied the MST approach to study the topological evolution before, during and after the 2008-2009 crisis, by constructing a network map of the top 100 companies listed on the Johannesburg Stock Exchange. Sinha and Pan (2007) used data from 201 stocks over the period 1996–2006 to assess the strength of dominance of the largest companies in the Indian economy. Tabak et al. (2010) constructed the MST by using the weekly prices of the 47 stocks on the Brazilian stock exchange from January 7, 2000 to February 29, 2008, and by the correlation matrix for a variety of stocks of different industry indices. Moreover, the study showed that stocks tended to cluster by industry. Situngkir and Surya (2005) found that the Indonesian stock market became stable during the period from 2000 to 2004, just after the economic shock of the currency crisis. They also noted the dominance of several stocks in certain industries. The world has already faced health crises (SARS, MERS and Ebola, among others) that impacted the stock markets and business activities, but less forcefully than the Covid-19 pandemic (Baker et al., 2020). Ashraf (2020) used the daily Covid-19 confirmed cases and death cases, as well as the stock markets returns data from 64 countries and found that stock markets quickly reacted to the Covid-19 pandemic. Liu et al. (2020) found that Asia experienced the most negative abnormal returns among the 21 leading stock market indices in most affected countries. Haroon and Rizvi (2020) found that the overwhelming panic generated by the news outlets has led to the increase in the volatility in the equity markets. Zhang et al. (2020) found that Covid-19 has led to an increase in the global financial market risk Aslam et al. (2020) examined the effects of Covid-19 on 56 global stock indices by using a complex network method and they revealed a structural change in the form of node changes, a reduced connectivity and significant differences in the topological characteristics of the network. As in Yang et al. (2014), the present study aims to apply the MST technique directly to industry indices in order to analyse the structural change of the stock market and to identify the key sectors of the Moroccan economy. The study of the dynamic evolution of the industry indices network will then allow one to see the impact of the crisis on the industrial structure of the stock market. In the second section, of this paper, the MST model is presented and its usefulness shown in terms of simplifying the dependency structure between nodes. The next section presents the details of the study data and their uses for the construction of MSTs of the Moroccan industry indices. In the fourth section, the discussion will turn to the use of some indicators to analyse the dynamics of MSTs. The last section will present the results and draw the relevant conclusions. METHODOLOGY Minimum Spanning Tree Model

DYNAMICS OF THE MOROCCAN INDUSTRY INDICES NET DYNAMICS OF THE MOROCCAN INDUSTRY INDICES NETW DYNAMICS OF THE MOROCCAN INDUSTRY INDICES NETW

OFTHE THE MOROCCAN MOROCCAN INDUSTRY INDUSTRY INDICES INDICES NET NET OF THE MOROCCA The use DYNAMICS ofDYNAMICS the MST to OF translate the financialDYNAMICS taxonomy consists in BEFORE AND DURING THE COVID-19 PANDEMIC BEFORE AND DURING THE COVID-19 PANDEMIC BEFORE AND DURING THE COVID-19 PANDEMIC NDUSTRYstudying INDICES NETWORK BEFORE BEFORE AND AND DURING DURING THE THE COVID-19 COVID-19 PANDEMIC PANDEMIC BEFORE AND DURING the financial assets connections through the evolution of E COVID-19 PANDEMIC their stock prices. In the context of the present study, the focus is on the industrial taxonomy of the Moroccan stock market. In this regard, the daily returns of the industry indices were used to calculate the Minimum Spanning Tree (MST) model Minimum Spanning Tree (MST) model Minimum Spanning Tree (MST) correlation and distance matrix andmodel to trace the MST for the Moroccan Minimum Minimum Spanning Spanning Tree Tree (MST) (MST) model model Minimum Spanning Tree (MST) model stock industries.

Let 𝑁𝑁𝑁𝑁 be the number of industry indices studied. For an industry𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑖𝑖𝑖𝑖(𝑖𝑖𝑖𝑖===1,1,1, , 𝑁𝑁𝑁𝑁), the Let the number industry indices studied. For industry ,. 𝑁𝑁𝑁𝑁), the Let 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 bebe the number ofofof industry indices studied. For anan industry . ..1, .... ,... 𝑁𝑁𝑁𝑁), the rt Let 𝑁𝑁𝑁𝑁be bethe thenumber number ofindustry industry indices indices studied. studied. For For an anindustry industry 𝑖𝑖𝑖𝑖 (𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 (𝑖𝑖𝑖𝑖indices ==1, .studied. ., .𝑁𝑁𝑁𝑁), , 𝑁𝑁𝑁𝑁), the Let 𝑁𝑁𝑁𝑁studied. be the number of industry Let number of industry indices For an industry day 𝑡𝑡𝑡𝑡is: is: day 𝑡𝑡𝑡𝑡 is: day 𝑡𝑡𝑡𝑡 an industry 𝑖𝑖𝑖𝑖 (𝑖𝑖𝑖𝑖 =𝑡𝑡𝑡𝑡 1, . . , 𝑁𝑁𝑁𝑁), the rate of return on day day day 𝑡𝑡𝑡𝑡is:.is: day 𝑡𝑡𝑡𝑡 is: is: 𝑃𝑃𝑃𝑃 (𝑡𝑡𝑡𝑡)

𝑃𝑃𝑃𝑃 is the closing price of 𝑖𝑖𝑖𝑖on on day𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡.𝑡𝑡𝑡𝑡. . on day 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) the closing price 𝑖𝑖𝑖𝑖on day (𝑡𝑡𝑡𝑡) 𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 isisis the closing price ofofof 𝑖𝑖𝑖𝑖 of 𝑖𝑖𝑖𝑖𝑃𝑃𝑃𝑃 (𝑡𝑡𝑡𝑡) isthe theclosing closing price price 𝑖𝑖𝑖𝑖 on 𝑖𝑖𝑖𝑖day onday day .𝑡𝑡𝑡𝑡 . is the closing price of 𝑖𝑖𝑖𝑖 on day 𝑡𝑡𝑡𝑡 . 𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) 𝑖𝑖𝑖𝑖𝑃𝑃𝑃𝑃 𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡)

The Pearson correlation coefficient between two industries 𝑖𝑖𝑖𝑖and and 𝑗𝑗𝑗𝑗: The Pearson correlation coefficient between two industries The Pearson correlation coefficient between two industries 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖and 𝑗𝑗𝑗𝑗:𝑗𝑗𝑗𝑗:𝑗𝑗𝑗𝑗: The The Pearson Pearson correlation correlation coefficient coefficient between between two two industries industries 𝑖𝑖𝑖𝑖 and 𝑖𝑖𝑖𝑖 and 𝑗𝑗𝑗𝑗: The Pearson correlation coefficient between two stries 𝑖𝑖𝑖𝑖 and 𝑗𝑗𝑗𝑗: ������ ��� ��� 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅−𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝚥𝚥𝚥𝚥 ������ ������ 𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝑅𝑅𝑅𝑅��� 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝚥𝚥𝚥𝚥 ��� 𝚤𝚤𝚤𝚤 ��� 𝑅𝑅𝑅𝑅������ 𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥𝚤𝚤𝚤𝚤−𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 ��� 𝚥𝚥𝚥𝚥 𝚥𝚥𝚥𝚥 𝚥𝚥𝚥𝚥 𝜌𝜌𝜌𝜌 = ������ 𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡−1) day 𝑡𝑡𝑡𝑡 is:𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 �𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑃𝑃𝑃𝑃(𝑡𝑡𝑡𝑡−1) 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖The (𝑡𝑡𝑡𝑡)� Pearson correlation coefficient between two industries 𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡) The Pearson between twoofindustries 𝑗𝑗𝑗𝑗: 𝑅𝑅𝑅𝑅 ��𝑃𝑃𝑃𝑃 coefficient = �𝑃𝑃𝑃𝑃�𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡−1) (𝑡𝑡𝑡𝑡) 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡) = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙correlation The Pearson correlation bb (𝑡𝑡𝑡𝑡) is The Pearson correlation coefficient is the the closing closing price 𝑖𝑖𝑖𝑖 on on day 𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡and . coefficient (𝑡𝑡𝑡𝑡−1) 𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡−1) 𝑃𝑃𝑃𝑃 𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡) price of 𝑖𝑖𝑖𝑖 day . (𝑡𝑡𝑡𝑡) 𝑃𝑃𝑃𝑃 is the closing price of 𝑖𝑖𝑖𝑖 on day 𝑡𝑡𝑡𝑡 . (𝑡𝑡𝑡𝑡) 𝑖𝑖𝑖𝑖 𝑃𝑃𝑃𝑃 is the closing price of 𝑖𝑖𝑖𝑖 on day 𝑡𝑡𝑡𝑡 . (𝑡𝑡𝑡𝑡) ������ ��� ��� 𝑃𝑃𝑃𝑃 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑖𝑖𝑖𝑖 of 𝑖𝑖𝑖𝑖 on day 𝑡𝑡𝑡𝑡 . ���� ���� 2 −𝑅𝑅𝑅𝑅 2 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 (𝑡𝑡𝑡𝑡) 𝑃𝑃𝑃𝑃 is the closing price ��� ��� ��𝑅𝑅𝑅𝑅 � �𝑅𝑅𝑅𝑅 𝑖𝑖𝑖𝑖 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 price �of𝑃𝑃𝑃𝑃Banking Journal Finance, 31–50 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖The 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 (𝑡𝑡𝑡𝑡)International is 𝑅𝑅𝑅𝑅 the closing of on��� day . 18, Number 1 (January) 2023, pp:������ model model 𝜌𝜌𝜌𝜌𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖𝚥𝚥𝚥𝚥𝑖𝑖𝑖𝑖 =𝑡𝑡𝑡𝑡Vol. 𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) ������ 𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤� 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖and 𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 ��� ��� ������ ��� 𝑅𝑅𝑅𝑅between 𝑅𝑅𝑅𝑅��� 𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡−1) 2INDICES 𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤 𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥two indu 𝚥𝚥𝚥𝚥NETWORK 𝚤𝚤𝚤𝚤 𝚤𝚤𝚤𝚤��� ���� ���� 2correlation 2 −𝑅𝑅𝑅𝑅 DYNAMICS OF THE MOROCCAN INDUSTRY INDICES 𝚤𝚤𝚤𝚤 𝑅𝑅𝑅𝑅 𝚥𝚥𝚥𝚥−𝑅𝑅𝑅𝑅 DYNAMICS OF MOROCCAN INDUSTRY NETWORK The Pearson ���𝚤𝚤𝚤𝚤 2 ��𝑅𝑅𝑅𝑅 ���𝚥𝚥𝚥𝚥coefficient (𝑡𝑡𝑡𝑡) =price 𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖THE 𝑃𝑃𝑃𝑃 the closing of 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖The on day 𝑡𝑡𝑡𝑡 . (𝑡𝑡𝑡𝑡) is 𝑃𝑃𝑃𝑃 ��𝑅𝑅𝑅𝑅 is the closing price of on day 𝑡𝑡𝑡𝑡 . −𝑅𝑅𝑅𝑅 � 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 𝑖𝑖𝑖𝑖 = 𝜌𝜌𝜌𝜌 = 𝜌𝜌𝜌𝜌 Pearson correlation coefficient between two indu 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 2 ���� The Pearson correlation coefficient between two industries 𝑖𝑖𝑖𝑖 and 𝑗𝑗𝑗𝑗: 𝑖𝑖𝑖𝑖 The Pearson correlation coefficient between two industries 𝑖𝑖𝑖𝑖 and 𝑗𝑗𝑗𝑗: ���� ��� ��� ��𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 � ���� ���� The Pearson correlation coefficient between two industries 𝑖𝑖𝑖𝑖 and 𝑗𝑗𝑗𝑗: ���� ���� 2 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 BEFORE AND DURING THE COVID-19 PANDEMIC 𝚤𝚤𝚤𝚤DURING 𝚥𝚥𝚥𝚥 ��� ��� BEFORE AND THE COVID-19 PANDEMIC ��� ��� ��𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤 𝚤𝚤𝚤𝚤𝑗𝑗𝑗𝑗: −𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 �𝚤𝚤𝚤𝚤 represents 𝚤𝚤𝚤𝚤 𝚤𝚤𝚤𝚤 ��𝑅𝑅𝑅𝑅 𝚥𝚥𝚥𝚥 𝚥𝚥𝚥𝚥 �� 𝚥𝚥𝚥𝚥 𝚥𝚥𝚥𝚥−𝑅𝑅𝑅𝑅 The Pearson correlation coefficient between two industries 𝑖𝑖𝑖𝑖 and (𝑡𝑡𝑡𝑡) yes𝑖𝑖𝑖𝑖 ,𝑖𝑖𝑖𝑖 𝑅𝑅𝑅𝑅and the average of the log-returns 𝑅𝑅𝑅𝑅 over the period studied. For 𝑇𝑇𝑇𝑇 (𝑡𝑡𝑡𝑡) 𝑃𝑃𝑃𝑃 is the closing price of 𝑖𝑖𝑖𝑖 on day 𝑡𝑡𝑡𝑡 . 𝑖𝑖𝑖𝑖 ������ ���𝚤𝚤𝚤𝚤 ��� 𝑖𝑖𝑖𝑖 of the log-returns 𝑅𝑅 (𝑡𝑡) over the period studied. 𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤industries 𝑅𝑅𝑅𝑅𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑇𝑇 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 and sentsstudied. the𝑗𝑗𝑗𝑗:average The Pearson coefficient between ������ ���For The Pearson𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖correlation correlation coefficient between two and 𝑗𝑗𝑗𝑗: 𝑗𝑗𝑗𝑗: : of the log-re 𝑅𝑅𝑅𝑅on 𝑅𝑅𝑅𝑅𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥 the ������ ��� ces ices studied. For an an𝑅𝑅𝑅𝑅 industry industry (𝑖𝑖𝑖𝑖(𝑖𝑖𝑖𝑖 = 1, .���.𝑖𝑖𝚤𝚤𝚤𝚤��� .For ,𝑅𝑅𝑅𝑅𝑁𝑁𝑁𝑁), 𝑁𝑁𝑁𝑁), the the rate rate return on 𝚤𝚤𝚤𝚤 𝑅𝑅𝑅𝑅industries 𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 ��� 𝑅𝑅𝑅𝑅= −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 = 𝜌𝜌𝜌𝜌of 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤1, −𝑅𝑅𝑅𝑅 industry 𝑖𝑖𝑖𝑖return , 𝑅𝑅𝑅𝑅�𝚤𝚤𝚤𝚤two represents average �𝚤𝚤𝚤𝚤 = 1For 𝚤𝚤𝚤𝚤������ 𝚥𝚥𝚥𝚥𝑅𝑅𝑅𝑅 𝚥𝚥𝚥𝚥��� 𝚤𝚤𝚤𝚤., 𝚥𝚥𝚥𝚥��� 𝚥𝚥𝚥𝚥 an 𝑖𝑖𝑖𝑖of 𝑖𝑖𝑖𝑖 = ∑𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡=1 (𝑡𝑡𝑡𝑡) . udied, ������ ��� 1 𝑇𝑇 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 𝑖𝑖𝑖𝑖 𝚤𝚤𝚤𝚤 𝑅𝑅𝑅𝑅𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 ��� 𝚥𝚥𝚥𝚥� = 𝜌𝜌𝜌𝜌 ���� ���� 2 −𝑅𝑅𝑅𝑅 2the 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 = 𝜌𝜌𝜌𝜌 ������ ��� 𝑇𝑇𝑇𝑇 ��� ��� 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 For an industry 𝑖𝑖𝑖𝑖 , 𝑅𝑅𝑅𝑅 represents the average of log-returns 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) over ��𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 � ∑ 𝑅𝑅 (𝑡𝑡) . The Pearson 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥2 2 ���� 𝚤𝚤𝚤𝚤 2 𝚥𝚥𝚥𝚥 𝚤𝚤𝚤𝚤 2 2 𝚤𝚤𝚤𝚤2an industry 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝚥𝚥𝚥𝚥𝑇𝑇𝑇𝑇−𝑅𝑅𝑅𝑅 ���� ���� 2𝚤𝚤𝚤𝚤 industries For 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖,𝑗𝑗𝑗𝑗: the industry ,.𝑅𝑅𝑅𝑅�𝑅𝑅𝑅𝑅�𝚤𝚤𝚤𝚤 𝚤𝚤𝚤𝚤represents represents the ���𝚤𝚤𝚤𝚤an ���𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝑖𝑖𝑖𝑖𝑅𝑅𝑅𝑅 ���� 2 −𝑅𝑅𝑅𝑅 �For ���� ���� 2 −𝑅𝑅𝑅𝑅 2 −𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 coefficient between two and ��𝑅𝑅𝑅𝑅 �𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) ∑ ��� ���𝚥𝚥𝚥𝚥��� = trading studied, 𝑅𝑅𝑅𝑅 𝑇𝑇 𝑡𝑡=1 𝑖𝑖 ��𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤−𝑅𝑅𝑅𝑅 𝚥𝚥𝚥𝚥𝑡𝑡𝑡𝑡=1 ��𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 2𝚥𝚥𝚥𝚥� �days ������ ��� ��� ������ ��� ��� 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 =correlation 𝚥𝚥𝚥𝚥𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤−𝑅𝑅𝑅𝑅 ���� ���� 2𝚤𝚤𝚤𝚤 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 ��� 𝚥𝚥𝚥𝚥 𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤��� −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 ��� 𝚥𝚥𝚥𝚥 𝚥𝚥𝚥𝚥 𝑇𝑇𝑇𝑇 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 𝚥𝚥𝚥𝚥 𝑇𝑇𝑇𝑇 ��𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 � 11 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 2 ���� 2 = � 𝚥𝚥𝚥𝚥 ���� 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 ∑ (𝑡𝑡𝑡𝑡) . 𝑅𝑅𝑅𝑅 trading days studied, 𝑅𝑅𝑅𝑅 ��� ��� 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = =��𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤 −𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤 ��𝑅𝑅𝑅𝑅 𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅𝚥𝚥𝚥𝚥 𝚤𝚤𝚤𝚤 � tradingdays daysstudied, studied,𝑅𝑅𝑅𝑅�𝑅𝑅𝑅𝑅�𝚤𝚤𝚤𝚤 𝚤𝚤𝚤𝚤 ==𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇∑∑𝑡𝑡𝑡𝑡=1 𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡=1 𝑖𝑖𝑖𝑖 trading 𝑡𝑡𝑡𝑡=1𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 ���� ���� 22 ��� 22−𝑅𝑅𝑅𝑅 ���� ���� ��� ���𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤22��� ��� ��𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 Minimum Spanning Treeof (MST) model ������ ��𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 � Tree (MST) model 𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤−𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅indices 𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥 �� an For industry 𝑖𝑖𝑖𝑖 , 𝑅𝑅𝑅𝑅 represents the average of the ll s 𝜌𝜌𝑖𝑖𝑖𝑖Minimum calculatedSpanning for each pair industry 𝑖𝑖 and 𝑗𝑗 form a symmetric 𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 ��� 𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤 an industry 𝑖𝑖𝑖𝑖of , the 𝑅𝑅𝑅𝑅� the𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖average the 𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 Forthe 𝜌𝜌𝜌𝜌industry (𝑡𝑡𝑡𝑡) For ananindustry industry average of log-returns over the per Foran represents averageof the log-returns 𝚤𝚤𝚤𝚤 represents ,𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝚤𝚤𝚤𝚤�2represents For represents theaverage of the log-returns overof the pe 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖,, 𝑅𝑅𝑅𝑅 1 𝑇𝑇𝑇𝑇 𝚤𝚤𝚤𝚤represents 𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡)over ��� (𝑡𝑡𝑡𝑡) the the log-returns 𝑅𝑅𝑅𝑅 the ���� ���� � 𝚤𝚤𝚤𝚤 𝑖𝑖𝑖𝑖 mension 𝑁𝑁 × 𝑁𝑁For and whose diagonal elements equal to unity. In total, the 𝜌𝜌𝜌𝜌 ∑𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡=1 𝑅𝑅𝑅𝑅 (𝑡𝑡𝑡𝑡) . over the per = ∑ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡) days studied, 𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 1��� ��𝑅𝑅𝑅𝑅𝚥𝚥𝚥𝚥 −𝑅𝑅𝑅𝑅 �𝑇𝑇𝑇𝑇 the average 1𝚥𝚥𝚥𝚥 trading For an industry 𝑖𝑖𝑖𝑖studied. 𝑅𝑅𝑅𝑅���𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤�represents of the per 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖period 𝜌𝜌𝜌𝜌 𝚤𝚤𝚤𝚤log-returns 𝚤𝚤𝚤𝚤,−𝑅𝑅𝑅𝑅 𝑇𝑇𝑇𝑇trading 𝜌𝜌𝜌𝜌 � 𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) over the days studied . 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 . = trading days studied, 𝑅𝑅𝑅𝑅 𝑖𝑖𝑖𝑖 -returns 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) over the period studied. For 𝑇𝑇𝑇𝑇 � 𝑇𝑇𝑇𝑇 ∑ (𝑡𝑡𝑡𝑡) . = 𝑅𝑅𝑅𝑅 trading days studied, 𝑅𝑅𝑅𝑅 ∑ (𝑡𝑡𝑡𝑡) 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡) trading days studied, 𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡=1 ��𝚤𝚤𝚤𝚤𝑅𝑅𝑅𝑅 𝚤𝚤𝚤𝚤==𝑇𝑇𝑇𝑇1𝑇𝑇𝑇𝑇 𝑖𝑖𝑖𝑖 average For industry 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ,, 𝑅𝑅𝑅𝑅 represents the (𝑡𝑡𝑡𝑡) over For an andays industry 𝑅𝑅𝑅𝑅 represents the average of≤the the log-returns 𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡) over the the pe p ∑ − 1)/2 correlation coefficients the 1of 𝜌𝜌𝑖𝑖𝑖𝑖𝚤𝚤𝚤𝚤 log-returns ≤ 𝑇𝑇𝑇𝑇1. 𝑡𝑡𝑡𝑡=1 𝑖𝑖𝑖𝑖𝑅𝑅𝑅𝑅 .. . 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 trading studied, 𝑇𝑇𝑇𝑇𝑡𝑡𝑡𝑡=1 𝑖𝑖𝑖𝑖 condition 𝑡𝑡𝑡𝑡=1 �𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤fulfilling 𝑡𝑡𝑡𝑡𝑁𝑁(𝑁𝑁 𝑡𝑡𝑡𝑡. .𝑖𝑖Let ∑ =average studied, 𝑅𝑅𝑅𝑅 𝑇𝑇𝑇𝑇 1 ̅of 𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) 𝑡𝑡𝑡𝑡=1 Let beindustry thetrading number of industry indices studied. For an industry 𝑖𝑖𝑖𝑖 (𝑖𝑖𝑖𝑖 = 1, . . . , 𝑁𝑁𝑁𝑁), the rate of re re (𝑡𝑡) (𝑡𝑡) be the number industry indices studied. For an industry 𝑖𝑖𝑖𝑖 (𝑖𝑖𝑖𝑖 = 1, . . . , 𝑁𝑁𝑁𝑁), the rate of try For , 𝑅𝑅̅𝑁𝑁𝑁𝑁 an the 𝑖𝑖 ,days 𝑅𝑅 average represents of the the log-returns of 𝑅𝑅 the log-returns over the period 𝑅𝑅 studied. over the For period 𝑇𝑇 studie 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑖𝑖𝑁𝑁𝑁𝑁represents 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑇𝑇𝑇𝑇 � � 𝑡𝑡𝑡𝑡 ∑ (𝑡𝑡𝑡𝑡) . = 𝑅𝑅𝑅𝑅 trading days studied, 𝑅𝑅𝑅𝑅 ∑ (𝑡𝑡𝑡𝑡) . = 𝑅𝑅𝑅𝑅 trading days studied, 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡=1 𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤 𝐶𝐶𝐶𝐶 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇of 𝑖𝑖𝑖𝑖average 𝑁𝑁𝑁𝑁 × 𝑁𝑁𝑁𝑁 The correlation calculated forof each pair of industry 𝑖𝑖𝑖𝑖average 𝑡𝑡𝑡𝑡=1 1For an industry 𝑖𝑖𝑖𝑖1𝑖𝑖,coefficients 𝑅𝑅𝑅𝑅�𝑅𝑅 represents the the log-returns 𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖 (𝑡𝑡𝑡𝑡) over the per ̅ 𝑇𝑇 𝑇𝑇 𝚤𝚤𝚤𝚤 day 𝑡𝑡𝑡𝑡 is: (𝑡𝑡) day 𝑡𝑡𝑡𝑡 is: For an industry , represents the of the log-returns over the pe 𝜌𝜌𝜌𝜌 ̅ ̅ 𝑡𝑡𝑡𝑡 𝜌𝜌𝜌𝜌∑ 𝑖𝑖 𝑅𝑅 𝑖𝑖 ∑𝑁𝑁𝑁𝑁𝑡𝑡=1 𝜌𝜌𝜌𝜌 studied, trading 𝑅𝑅of = days 𝑅𝑅𝑖𝑖 (𝑡𝑡) 𝑅𝑅of .𝑖𝑖 𝜌𝜌= . 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 as the distance between 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖studied, 𝑖𝑖𝑖𝑖 indices 𝑖𝑖 industry 𝑖𝑖 (𝑡𝑡) 𝑡𝑡=1 𝑁𝑁𝑁𝑁 𝑇𝑇× 𝐶𝐶𝐶𝐶and of a pair cannot be the 1used 𝑇𝑇𝜌𝜌𝜌𝜌 between between two industries industries 𝑖𝑖𝑖𝑖days 𝑖𝑖𝑖𝑖𝑅𝑅 and 𝑗𝑗𝑗𝑗:𝑗𝑗𝑗𝑗:𝑖𝑖𝑖𝑖 of ×× 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁dimension 𝐶𝐶𝐶𝐶 and form symmetric correlation matrix 1∑𝑇𝑇𝑇𝑇studied. industry 𝑅𝑅 represents the average of log-returns 𝑅𝑅𝑖𝑖 (𝑡𝑡) over of𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡the 𝑖𝑖𝑖𝑖indices 𝑖𝑖𝑖𝑖For an �𝑅𝑅̅𝑖𝑖𝚤𝚤𝚤𝚤the 𝑇𝑇 age of𝑖𝑖 the over𝑖𝑖 a,the period 𝑇𝑇over dustry ,two 𝑅𝑅̅𝑖𝑖 log-returns represents the average of 𝑅𝑅For the period studied. For 𝑇𝑇 the p (𝑡𝑡𝑡𝑡) 𝜌𝜌𝜌𝜌trading . 𝑖𝑖.(𝑡𝑡) ==log-returns 𝑅𝑅𝑅𝑅𝑅𝑅 trading studied, 𝑅𝑅𝑅𝑅̅ 𝑖𝑖 (𝑡𝑡) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡=1 ∑ (𝑡𝑡) days studied, 𝑇𝑇𝑇𝑇 𝑖𝑖 that 𝑖𝑖 a equal 𝑡𝑡=1 use it does not fulfill the three axioms define metric (Mantegna, 𝑇𝑇 𝑇𝑇 and in which its diagonal elements to unity. In total, the 𝜌𝜌𝜌𝜌 (𝑡𝑡𝑡𝑡) 𝜌𝜌𝜌𝜌trading og-returns 𝑅𝑅𝑅𝑅𝑖𝑖 (𝑡𝑡) over studied. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃period 𝑇𝑇 the (𝑡𝑡) log-returns period studied. For 𝑇𝑇 𝑅𝑅𝑖𝑖 (𝑡𝑡) . 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖∑ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡) ∑𝑇𝑇𝑡𝑡=1 𝑖𝑖𝑖𝑖over days studied, 𝑅𝑅̅𝑖𝑖 =For 𝑖𝑖(𝑡𝑡𝑡𝑡) ys studied, 𝑅𝑅̅ = = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙�𝑅𝑅 �the 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡) ��our = 𝑙𝑙𝑙𝑙coefficients 𝑖𝑖 (𝑡𝑡)of.for 𝑡𝑡=1 𝑇𝑇indices 𝑡𝑡𝑡𝑡 1)/2 on coefficients The correlation 𝜌𝜌correlation calculated 𝜌𝜌contains each calculated pair of industry for indices pair of𝑖𝑖we industry and 𝑗𝑗 form indices symmetric 𝑖𝑖 and 𝑗𝑗 form a sy the topological need a afulfilling 𝑇𝑇taxonomy matrix coefficients 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 industry 𝑃𝑃𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(𝑡𝑡𝑡𝑡−1) (𝑡𝑡𝑡𝑡−1) 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 − 𝑃𝑃𝑃𝑃 ofeach 𝑁𝑁𝑁𝑁network, ×correlation 𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡of 𝑁𝑁𝑁𝑁 × of 𝑁𝑁𝑁𝑁 × 𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶 calculated 𝜌𝜌𝜌𝜌The 𝑡𝑡×𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁− 𝐶𝐶𝐶𝐶 of 𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 1)/2 correlation coefficients 𝜌𝜌 for each pair of industry indices 𝑖𝑖 and s distance, the three axioms of Euclidean distance: matrix correlation 𝐶𝐶 𝑡𝑡 ofsatisfying dimension matrix 𝑁𝑁 𝐶𝐶 × of 𝑁𝑁 dimension and whose 𝑁𝑁 diagonal × 𝑁𝑁 and elements whose diagonal equal to elements unity. In equal total, totheunity. In t condition 1 ≤ 𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ≤ 1 . 𝑖𝑖𝑖𝑖 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶the 𝑡𝑡𝑡𝑡 of 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁−−1)/2 1)/2 of 𝑁𝑁𝑁𝑁 × ×𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁indices 𝐶𝐶𝐶𝐶The correlation coefficients 𝜌𝜌 calculated for each pair of industry indices 𝑖𝑖 and 𝑡𝑡 𝑖𝑖𝑖𝑖 dation for each pair of industry 𝑖𝑖 and 𝑗𝑗 form a symmetric coefficients 𝜌𝜌 calculated for each pair of industry indices 𝑖𝑖 and 𝑗𝑗 form a symmetric 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖− correlation matrix of dimension 𝑁𝑁fulfilling × 𝑁𝑁coefficients andthe whose diagonal (𝑡𝑡𝑡𝑡) isis the atrix matrix 1)/2 contains correlation − coefficients 1)/2 condition fulfilling 1 ≤ the 𝜌𝜌elements ≤11.≤ 𝜌𝜌𝜌𝜌equal the𝑁𝑁(𝑁𝑁 closing price of 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑁𝑁(𝑁𝑁 on𝐶𝐶day day (𝑡𝑡𝑡𝑡) 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃correlation closing price of on 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 .. correlation ≤≤1to𝜌𝜌 𝑖𝑖𝑖𝑖 condition of 𝑁𝑁𝑁𝑁 × 𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖contains of 𝑁𝑁𝑁𝑁 × 𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 1 𝑡𝑡 air industry indices 𝑖𝑖 𝑖𝑖and 𝑗𝑗 𝑗𝑗𝑁𝑁 form a awhose symmetric correlation 𝐶𝐶 of dimension × 𝑁𝑁 andequal whose elements equal indices and form symmetric 1coefficients ≤ 𝜌𝜌𝜌𝜌unity. ≤ 1Infulfilling nly ifofof 𝑖𝑖 whose =industry definiteness) and elements equal to unity. In total, the n𝑁𝑁pair matrix 𝐶𝐶𝑗𝑗𝑡𝑡 (positive ofdiagonal dimension 𝑁𝑁 ×matrix and diagonal todiagonal total, 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 cannot correlation matrix contains 𝑁𝑁(𝑁𝑁 − 1)/2 correlation cont The correlation coefficient of a𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 pair ofelements industry indices be thethe −𝑁𝑁 1)/2 𝑡𝑡𝑡𝑡 (𝑡𝑡𝑡𝑡) (𝑡𝑡𝑡𝑡) 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 − 1)/2 e average average of of the the log-returns log-returns 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 over over the the period period studied. studied. For For 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 − 1)/2 of 𝑁𝑁𝑁𝑁 × 𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 − 1)/2 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 correlation matrix contains 𝑁𝑁(𝑁𝑁 − 1)/2 correlation coefficients fulfilling the co e diagonal elements equal to unity. In total, the diagonal elements equal to unity. Inbetween total, the correlation coefficients fulfilling the condition 1two ≤ 𝜌𝜌indices ≤used 1.𝜌𝜌𝑖𝑖𝑖𝑖as𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖cannot nse matrix contains − 1)/2 correlation the 1itas ≤ 𝜌𝜌𝑖𝑖𝑖𝑖 distance ≤ 1. betw distance between two does The Pearson correlation coefficient between industries andcondition 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 − 1)/2 correlation coefficient industries and 𝑗𝑗𝑗𝑗:𝑗𝑗𝑗𝑗:because (positive etry) ifThe andPearson only if used 𝑖𝑖𝑖𝑖𝑁𝑁(𝑁𝑁 =coefficient 𝑗𝑗𝑗𝑗as 𝑖𝑖𝑖𝑖 on coefficient The correlation of a𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 pair ofthe industry ofdefiniteness) aindices pair of𝜌𝜌coefficients industry cannot befulfilling the distance be used between the the 𝑖𝑖𝑖𝑖the − 1)/2 (𝑡𝑡𝑡𝑡) (𝑡𝑡𝑡𝑡) coefficients fulfilling the condition 11axioms ≤≤𝜌𝜌𝜌𝜌 ≤≤ .. nes fulfilling the condition 𝑖𝑖𝑖𝑖three 𝑖𝑖𝑖𝑖 coefficients not fulfill the three that a ifmetric (Mantegna, 1999). 𝑖𝑖𝑖𝑖of (positive The coefficient a1.1. pair indices 𝜌𝜌𝑖𝑖𝑖𝑖 be used and only 𝑖𝑖𝑖𝑖 that =cannot 𝑗𝑗𝑗𝑗(Mantegna, definitene 𝑑𝑑𝑑𝑑define ≤the 𝜌𝜌𝜌𝜌(M 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 − 𝑖𝑖 and two𝑗𝑗inequality) industries because it𝑖𝑖correlation does and not because fulfill itthe does not axioms fulfill the that three define axioms a if metric define a metric 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 −𝑗𝑗1)/2 1)/2 (symmetry) triangle 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖of=industry 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖d 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 11asas ≤ 𝜌𝜌𝜌𝜌 ≤ 𝜌𝜌𝜌𝜌 ≤ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ≤ 𝜌𝜌𝜌𝜌 ≤ ������ ��� ��� ������ ��� ��� (positive The correlation coefficient of a pair of industry indices 𝜌𝜌 cannot be used the 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 = if and only if 𝑖𝑖𝑖𝑖 = 𝑗𝑗𝑗𝑗 definiteness) 𝑑𝑑𝑑𝑑 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 𝚤𝚤𝚤𝚤 𝚥𝚥𝚥𝚥 Indeed, to analyse the topological taxonomy of the industry indices 𝑖𝑖𝑖𝑖= 𝚤𝚤𝚤𝚤 analyse 𝚥𝚥𝚥𝚥be 𝚤𝚤𝚤𝚤used 𝚥𝚥𝚥𝚥𝑖𝑖taxonomy dustry indices 𝜌𝜌 cannot distance between the 𝑖𝑖𝑖𝑖as 𝑖𝑖𝑖𝑖 the ation coefficient of a pair of industry indices 𝜌𝜌 cannot be used as the distance between the ≤ 𝜌𝜌𝜌𝜌 ≤ if and only if 𝑖𝑖𝑖𝑖 = 𝑑𝑑𝑑𝑑 = if and only if 𝑖𝑖𝑖𝑖 = 𝑑𝑑𝑑𝑑 d, to 1999). analyse Indeed, the topological to the topological of our taxonomy industry indices of our network, industry indices we need network, a we two industries and 𝑗𝑗 because it does not fulfill the three axioms that define a 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 (symmetry) =𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 � 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 1 ≤ 𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ≤ 1 𝑖𝑖𝑖𝑖 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖be (triangle inequality) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = − 1)/2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝜌𝜌+𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2 variable 2 = s𝑑𝑑𝑑𝑑𝜌𝜌 cannot used the distance between the 2and two 𝑖𝑖 𝑗𝑗 because it does not fulfill the three axioms that define network, a is needed that can function as distance, satisfying ���� ���� 2industries ces cannot be used as the distance between the ���� ���� 2as 22−𝑅𝑅𝑅𝑅 (symmetry) 𝑖𝑖𝑖𝑖function 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 ��� ��� ot fulfill the three axioms that define a metric (Mantegna, tries 𝑖𝑖 and 𝑗𝑗 because it does not fulfill the three axioms that define a metric (Mantegna, ��� ��� � 𝑖𝑖𝑖𝑖 ��𝑅𝑅𝑅𝑅 can variable as that distance, can function satisfying as distance, the three satisfying axioms of the Euclidean three axioms distance: of Euclidean distance: −𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 � 1999). Indeed, to analyse the topological taxonomy of our industry indices n ≤ 𝜌𝜌𝜌𝜌 ≤ ��𝑅𝑅𝑅𝑅 −𝑅𝑅𝑅𝑅 ��𝑅𝑅𝑅𝑅 � 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ≤ 𝜌𝜌𝜌𝜌 ≤ 𝚥𝚥𝚥𝚥 (symmetry) 𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤 duced by Mantegna 𝚤𝚤𝚤𝚤𝚤𝚤𝚤𝚤(1999), from𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖the distance =𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(symmetry) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ��𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝚥𝚥𝚥𝚥𝚥𝚥𝚥𝚥is 𝚥𝚥𝚥𝚥determined ≤ Euclidean 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 +𝑑𝑑𝑑𝑑taxonomy inequality) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (triangle 1999). Indeed, to analyse the topological of our industry indices the three axioms of Euclidean axioms define adistance metric (Mantegna, ethree three axioms that define acan metric (Mantegna, al taxonomy ofthat our industry indices network, we𝑖𝑖𝑖𝑖 industry need a 0 indices deed, analyse the topological taxonomy ofdistance: our network, we need a defin (positive = if and only if 𝑖𝑖𝑖𝑖 = 𝑗𝑗𝑗𝑗 𝑑𝑑𝑑𝑑 variable that function as variables distance, satisfying the three axioms of Euclidean distin (triangle 𝑑𝑑𝑑𝑑 ≤ 𝑑𝑑𝑑𝑑 +𝑑𝑑𝑑𝑑 inequality) 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 akes ittopossible to relate the of two (i.e. industry indices) 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (triangle (positive 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖axioms ≤≤𝑖𝑖𝑖𝑖𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖= 0 ≤definiteness) ifdefiniteness) only if 𝑗𝑗𝑗𝑗+𝑑𝑑𝑑𝑑 defin 𝑑𝑑𝑑𝑑satisfying (triangle (positive 𝜌𝜌𝜌𝜌and ≤ = 00ififfunction and only if 𝑖𝑖𝑖𝑖 = 𝑗𝑗𝑗𝑗 𝑑𝑑𝑑𝑑(positive 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (positive 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 1 = if and only if 𝑖𝑖𝑖𝑖 = 𝑗𝑗𝑗𝑗 𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖+𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 variable that can as distance, the three of Euclidean disi 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 y of our industry indices network, we need a my of our industry indices network, we need a (positive = and only if 𝑖𝑖𝑖𝑖 = 𝑗𝑗𝑗𝑗 definiteness) 𝑑𝑑𝑑𝑑 sfying the three axioms of Euclidean distance: at can function as distance, satisfying the three axioms of Euclidean distance: = 0For if and only 𝑑𝑑𝑖𝑖𝑖𝑖if=𝑖𝑖𝑖𝑖𝑖𝑖0 = if� 𝑗𝑗�𝑑𝑑𝑑𝑑and ifdefiniteness) 𝑖𝑖the =only 𝑗𝑗 (positive definiteness) 𝑖𝑖𝑖𝑖represents 𝑖𝑖𝑖𝑖 =only ents: (symmetry) 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 (positive (𝑡𝑡𝑡𝑡) � if and if 𝑖𝑖𝑖𝑖 = 𝑗𝑗𝑗𝑗 definiteness) For an industry 𝑖𝑖𝑖𝑖 , 𝑅𝑅𝑅𝑅 represents average of the log-returns 𝑅𝑅𝑅𝑅 over the period studied (𝑡𝑡𝑡𝑡) an industry , 𝑅𝑅𝑅𝑅 the average of the log-returns 𝑅𝑅𝑅𝑅 over the period studied 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝚤𝚤𝚤𝚤 𝑖𝑖𝑖𝑖 𝚤𝚤𝚤𝚤 𝑖𝑖𝑖𝑖 (symmetry) 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 definiteness) ness) (symmetry) == hree axioms Euclidean 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 �(symmetry) �𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑distance: three axioms Euclidean 𝑖𝑖𝑖𝑖distance: 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖= 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖if 𝑑𝑑� and only ififif 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖= (positive �2�1 (1) 𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 � of = and only = 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗(positive definiteness) 𝑑𝑑𝑑𝑑 (symmetry) =𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 0∑ if(symmetry) and only =𝑗𝑗� definiteness) = 𝑑𝑑trading = 𝑑𝑑𝑗𝑗𝑗𝑗� 𝑇𝑇𝑇𝑇 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖= 𝑇𝑇𝑇𝑇if {of𝑑𝑑𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗−(symmetry) (triangle inequality) � 𝑑𝑑𝑑𝑑(positive ≤ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖+𝑑𝑑𝑑𝑑 +𝑑𝑑𝑑𝑑 inequality) (symmetry) (𝑡𝑡𝑡𝑡) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 . 𝑅𝑅𝑅𝑅 trading days studied, 𝑅𝑅𝑅𝑅 ∑ (𝑡𝑡𝑡𝑡) . = 𝑅𝑅𝑅𝑅 days studied, 𝑅𝑅𝑅𝑅 � 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ≤ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (triangle (positive 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑑𝑑 = if and only if 𝑖𝑖 = 𝑗𝑗 definiteness) 𝚤𝚤𝚤𝚤 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡=1 𝚤𝚤𝚤𝚤 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡=1 𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖 (positive ositive 𝑑𝑑 if and only if 𝑖𝑖 = 𝑗𝑗 definiteness) (triangle 𝑑𝑑𝑑𝑑 ≤ 𝑑𝑑𝑑𝑑 +𝑑𝑑𝑑𝑑 inequality) 𝑇𝑇𝑇𝑇 (triangle 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑 ≤ 𝑑𝑑𝑑𝑑 +𝑑𝑑𝑑𝑑 inequality) 𝑇𝑇𝑇𝑇 �2�1 𝑖𝑖𝑖𝑖 = 0 definiteness) 𝑑𝑑𝑑𝑑 = − 𝜌𝜌𝜌𝜌 � 𝑖𝑖𝑖𝑖𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑗𝑗𝑗𝑗 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖inequality) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖≤ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 (symmetry) 𝑑𝑑𝑖𝑖𝑖𝑖𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = (symmetry) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖inequality) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 {𝑑𝑑𝑑𝑑 = (symmetry) = 𝑑𝑑𝑑𝑑 (triangle 𝑑𝑑𝑑𝑑0𝑑𝑑𝑑𝑑𝑑𝑑 +𝑑𝑑𝑑𝑑 � (triangle � ≤ 𝑑𝑑𝑖𝑖𝑖𝑖 +𝑑𝑑𝑘𝑘𝑘𝑘 (triangle 𝑑𝑑𝑖𝑖𝑖𝑖 ≤ 𝑑𝑑𝑖𝑖𝑖𝑖inequality) +𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖= 𝑖𝑖𝑖𝑖≤ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 only if and 𝑗𝑗𝑗𝑗 (positive definiteness) 𝑑𝑑𝑑𝑑 𝑘𝑘𝑘𝑘 𝑑𝑑𝑑𝑑{𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑 +𝑑𝑑𝑑𝑑 iteness) �2�1 = −if𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �=inequality) =𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (triangle 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖(1) 𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑 𝑗𝑗𝑗𝑗 (symmetry) 𝑑𝑑niteness) �2�1−−𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖�� =�2�1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑖𝑖𝑖𝑖 = 𝑑𝑑𝑗𝑗𝑗𝑗 (symmetry) (triangle 𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖distance, ≤ inequality) ≤ +𝑑𝑑𝑑𝑑 inequality) (triangle ≤𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖+𝑑𝑑 +𝑑𝑑𝑑𝑑introduced inequality) 𝑘𝑘𝑘𝑘 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 The metric by Mantegna (1999), is determined 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (triangle 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 � 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑑𝑑 𝑖𝑖𝑖𝑖 =≤ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(symmetry) (triangle +𝑑𝑑by inequality) 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖(1999), 𝑘𝑘𝑘𝑘 Mantegna ality) 𝑑𝑑distance, 𝑑𝑑𝑖𝑖𝑖𝑖 +𝑑𝑑 inequality) introduced metric distance, by introduced is(Coelho determined (1999), is determined the fromdistance the Euclidean 𝑖𝑖𝑖𝑖 ≤The 𝑘𝑘𝑘𝑘 (triangle from the𝑑𝑑𝑑𝑑Mantegna Euclidean distance et al., from 2007), andEuclidean it is permitted (triangle ≤ 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖the +𝑑𝑑𝑑𝑑 inequality) �2�1 𝑑𝑑𝑑𝑑construct =variables − 𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌of 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 s., are therefore used to calculate distances and to a matrix of 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 the 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖��two 2007) (Coelho and makes et al., it 2007) possible and makes to relate it the possible distance to relate of two distance (i.e. industry variables indices) industry The metric distance, introduced by Mantegna (1999), is determined the �2�1 𝑑𝑑𝑑𝑑 = − . 𝑑𝑑𝑑𝑑 distance to their (i.e.from �2�1 ==�2�1 −−𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌of �2�1− 𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖the 𝜌𝜌𝜌𝜌 𝑖𝑖𝑖𝑖�𝑖𝑖𝑖𝑖 variables𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖(i.e. industry 𝑖𝑖𝑖𝑖indices) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖�� 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖= 𝑖𝑖𝑖𝑖two 11𝐷𝐷≤≤ ≤ ≤relate 11 metric 𝑡𝑡 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖to 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 The distance, introduced by Mantegna (1999), is determined from the 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 ylation indices of the same dimension as the 𝑁𝑁 × 𝑁𝑁 correlation matrix. Small . 𝑑𝑑𝑑𝑑 (1999), is determined from the Euclidean distance cntegna distance, introduced by Mantegna (1999), is determined from the Euclidean distance �2�1 = − 𝜌𝜌𝜌𝜌 � to coefficients: their correlation coefficients: (Coelho et al., 2007) and makes it possible to relate the distance of two variables (i 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 of 𝑁𝑁𝑁𝑁 × 𝑁𝑁𝑁𝑁 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 coefficients,𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖as(1) correlation in Equation (1): 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖. . 𝐶𝐶𝐶𝐶 of 𝑁𝑁𝑁𝑁 × 𝑁𝑁𝑁𝑁 𝐶𝐶𝐶𝐶 (Coelho et al., 2007) and makes it possible to relate the distance of two variables ( �2�1 𝑑𝑑𝑑𝑑 = − 𝜌𝜌𝜌𝜌 � is from the Euclidean distance �2�1 𝑑𝑑𝑑𝑑 = − 𝜌𝜌𝜌𝜌 ly high correlations between industry indices. � 99), isdetermined determined from the Euclidean distance 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 e9), to relate the of two variables (i.e. industry indices) al., 2007) anddistance makes it possible to relate the distance of two variables (i.e. industry indices) 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 to their correlation coefficients: ∑(𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 tovariables their∑correlation coefficients: e distance ofoftwo (i.e. industry indices) he distance two variables (i.e. industry indices) rrelation coefficients: 𝑑𝑑𝑑𝑑 (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 � ∑∑(𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST (1) 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖− =𝜌𝜌 �2�1 − (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (positive definiteness) definiteness) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖.. graph 𝐺𝐺 in which = − 𝜌𝜌draw (1) )𝑖𝑖𝑖𝑖𝑑𝑑𝑑𝑑a= . 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖directionless 𝐷𝐷𝑗𝑗𝑗𝑗𝑡𝑡𝑗𝑗𝑗𝑗√2(1 , (positive we can the 𝑖𝑖𝑖𝑖𝑑𝑑 𝑖𝑖𝑖𝑖 )and fully connected 𝑖𝑖𝑖𝑖..𝑖𝑖𝑖𝑖√2(1 𝑑𝑑𝑑𝑑 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 −between 1)/2 𝑁𝑁𝑁𝑁(𝑁𝑁𝑁𝑁 − 1)/2 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 . them 𝑑𝑑𝑖𝑖𝑖𝑖 are = √2(1 − 𝜌𝜌𝑖𝑖𝑖𝑖by ) the distances 𝑑𝑑𝑖𝑖𝑖𝑖 . and the links measured The coefficients therefore, used to calculate the (1) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖.. correlation ∑are 𝑑𝑑𝑑𝑑(1) 𝑑𝑑𝑖𝑖𝑖𝑖 = √2(1 − The 𝜌𝜌𝑖𝑖𝑖𝑖 ) ∑ (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = √2(1 − 𝜌𝜌𝑖𝑖𝑖𝑖 )∑ 𝑑𝑑𝑑𝑑 𝑖𝑖nequality) 𝑑𝑑𝑑𝑑 (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST ∑ 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑 inequality) (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST The 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 on coefficients The correlation are therefore coefficients to therefore calculate used the11 distances and the to construct distances aand matrix to construct of am distances and to are construct a (1) matrix the distances of the industry ∑ 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖used The The (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST ≤toof ≤ 11 ≤ 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖calculate (1) ∑ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ≤ . correlation 𝑑𝑑𝑑𝑑 (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST 𝑡𝑡 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 between 𝑡𝑡 are (𝑁𝑁 − 1) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖direct ed using links the 𝑁𝑁 nodes of 𝐺𝐺 such that the sum The coefficients therefore used to calculate the distances and to co of the theindustry distances indices of the 𝐷𝐷 industry of the indices same dimension 𝐷𝐷 of the as same the 𝑁𝑁 dimension × 𝑁𝑁 correlation as the 𝑁𝑁 matrix. × 𝑁𝑁 correlation Small matri indices of the same dimension as the correlation matrix .The ∑ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∑(𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST 𝑡𝑡 The correlation coefficients areimply therefore used tothis calculate theasadistances and to c ∑(𝑖𝑖,𝑗𝑗)∈MST these links 𝑑𝑑𝑖𝑖𝑖𝑖 isimply minimal. MST is constructed in way used to calculate the distances and to construct a matrix of ation coefficients are therefore toThe calculate the distances and to construct matrix of he matrix distances imply inthe high the matrix correlations between high correlations industry between indices. industry indices. distances of used the industry indices 𝐷𝐷 of the same dimension the 𝑁𝑁 × 𝑁𝑁 correl small distances in the matrix the high correlations between The 𝑡𝑡 The ∑ the distances of thecorrelation industry indices of𝑁𝑁 the×between same as theSmall 𝑁𝑁 × 𝑁𝑁 corr 𝑑𝑑𝑑𝑑construct culate the distances and adistances; matrix of The The theindustry distances and to𝐷𝐷 construct aimply matrix of (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST 𝑖𝑖𝑖𝑖𝑡𝑡𝑁𝑁 𝑖𝑖𝑖𝑖smallest flculate the same dimension as to the ×matrix 𝑁𝑁 matrix. ces ofnodes the indices of the same dimension as𝐷𝐷Small the 𝑁𝑁 correlation matrix. the together with the starting by linking thedimension two industry indices. distances in the high correlations industry indices. The (positive = if and only if 𝑖𝑖𝑖𝑖 = 𝑗𝑗𝑗𝑗 definiteness) 𝑑𝑑𝑑𝑑 (positive = if and only if 𝑖𝑖𝑖𝑖 = 𝑗𝑗𝑗𝑗 definiteness) 𝑑𝑑𝑑𝑑 The 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 (1) (1) 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑡𝑡 𝑡𝑡 distances in the matrix imply high correlations between industry indices. imension as 𝑁𝑁 correlation matrix. Small asthe the 𝑁𝑁× 𝑁𝑁matrix correlation matrix. Small odes, the next is×can added to a𝐷𝐷 the provided that the addedand distance isin which graph tions between industry indices. ndimension theFrom imply high correlations between industry indices. ance matrix the 𝐷𝐷node distance ,𝑁𝑁 we draw directionless , MST we can draw and a fully directionless connected graph fully𝐺𝐺connected the 𝐺𝐺 in whic The The (symmetry) 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖From =does 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖the (symmetry) 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 � 𝑡𝑡 it can � en industry indices. een industry indices. dition of this node not close the loop with the nodes already added to From the distance matrix, be seen that there is a directionless distance 𝐷𝐷 between ,𝑡𝑡measured we can them draw a directionless connected ustrynodes indices areand industry the links indices between andmatrix the them links are by the are distances measured𝑑𝑑by . thefully distances 𝑑𝑑𝑖𝑖𝑖𝑖 . grap 𝑖𝑖𝑖𝑖and 𝑡𝑡 𝑑𝑑𝑑𝑑 From the distance matrix 𝐷𝐷 , the we can a directionless fullyby connected gra (triangle ≤ 𝑑𝑑𝑑𝑑fully +𝑑𝑑𝑑𝑑industry inequality) (triangle 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖The ≤ +𝑑𝑑𝑑𝑑 inequality) thm is used for this purpose. A adetailed approach to draw constructing MSTs w a directionless and connected graph 𝐺𝐺1in which the distance matrix , we can draw directionless and fully connected 𝐺𝐺and inindices which the and nodes aregraph industry 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖are 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝐷𝐷 𝑖𝑖𝑖𝑖fully 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 nodes indices and links between them are measured the distance nodes are industry indices and the links between them are measured by the distanc less and fully connected graph 𝐺𝐺 in which the nless and fully connected graph 𝐺𝐺 in which the setween presented inare (Yang etconstructed al., 2014). links are the them measured by thedirect distances industry indices the links between them measured the distances 𝑑𝑑𝑁𝑁𝑖𝑖𝑖𝑖that . theofsum (𝑁𝑁 (𝑁𝑁𝑑𝑑are then The constructed MST isand using then − 1) using links between − direct thelinks 𝑁𝑁bynodes between of 𝐺𝐺 the such nodes 𝐺𝐺 such that 𝑖𝑖𝑖𝑖 .1) are measured by the distances 𝑑𝑑 . m are measured by the distances 𝑑𝑑 . 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 (𝑁𝑁 (𝑁𝑁 ∑ ∑ The MST is then constructed using − 1) direct links between the 𝑁𝑁 1) distances of the of − these 1) distances links (𝑖𝑖,𝑗𝑗)∈MST of these𝑑𝑑links The 𝑑𝑑𝑖𝑖𝑖𝑖MST is minimal. is constructed The MST in this is constructed waynodes of in 𝐺𝐺 (𝑖𝑖,𝑗𝑗)∈MST 𝑖𝑖𝑖𝑖 is minimal. (𝑁𝑁 The MST is then constructed using − 1) links between the 𝑁𝑁 nodes of en nodes is considerably reduced by using minimum spanning tree (MST). The MST is then constructed by using direct links between the (𝑁𝑁 −vely direct links between the 𝑁𝑁 nodes of 𝐺𝐺 such that the sum is1)then constructed using − 1) direct links between the 𝑁𝑁 nodes of 𝐺𝐺 such that the sum (𝑁𝑁−together ∑the of�2�1 the −𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌1)�all distances oftogether these𝑡𝑡 links 𝑑𝑑𝑖𝑖𝑖𝑖 distances; isbyminimal. Thetwo MST is con linking by progressively the nodes linking the with nodes the smallest with distances; smallest starting linkingstarting the by linking (𝑖𝑖,𝑗𝑗)∈MST �2�1 𝑑𝑑𝑑𝑑distances = �distance 𝑑𝑑𝑑𝑑all = −of 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖the 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝐺𝐺 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖∑ (𝑁𝑁 ∑ of the − 1) distances of these links 𝑑𝑑 is minimal. The MST is co (𝑁𝑁 nks between the 𝑁𝑁 nodes such that the sum ×− (𝑁𝑁 − 1)/2 in the matrix 𝐷𝐷 to only − 1) shortest nodes of ,such that the sum of the distances of these inks between 𝑁𝑁 nodes of 𝐺𝐺 such that the sum (𝑖𝑖,𝑗𝑗)∈MST 𝑖𝑖𝑖𝑖 added 𝑑𝑑𝑖𝑖𝑖𝑖 (most is of minimal. Thenode MST constructed inisthis way 1) closest distances links 𝑑𝑑next minimal. The MST is constructed in thisthe way correlated) nodes, correlated) the next nodes, is isadded the the node MST added provided to the that MST the provided distance that is added di (𝑖𝑖,𝑗𝑗)∈MST (𝑖𝑖,𝑗𝑗)∈MST bythese progressively linking all the nodes together with smallest distances; starting 𝑖𝑖𝑖𝑖tois isisminimal. The MST is inof way by progressively linking all the nodes together smallest distances; startin minimal. The MST isconstructed constructed inthis this way ether with the smallest distances; starting by linking the two linking all the nodes together with the smallest distances; starting by linking thetotwo 𝑖𝑖sively and that the the smallest addition and of that this the node addition does not this close node does loop not with close the nodes the the loop with the added nodes already closest (most correlated) nodes, the next node is with added toalready the MST provided that tha 35to closest (most correlated) nodes, the next node added toapproach thethe MST provided he distances; starting by linking the two thesmallest smallest distances; starting bythat linking the node is added tonodes, the MST provided that the added distance is does ost correlated) the next node is added totwo the MST provided that the added distance isthethat ruskal's the algorithm MST. Kruskal's issmallest used algorithm for this purpose. isthe used for A detailed this purpose. approach Ais detailed tonotconstructing MSTs constructin the and addition of this node close loop with nod the smallest and that the addition of this node does not close the loop with the to no ded to the MST provided that the added distance is . 𝑑𝑑𝑑𝑑 . 𝑑𝑑𝑑𝑑 to the MST provided that the added distance is node does not close the loop with the nodes already added to st and that the addition of this node does not close the loop with the nodes already added to l's algorithm using Kruskal's is presented algorithm inKruskal's (Yang is presented etalgorithm al., 2014). in (Yang et al., the MST. is used for2014). this purpose. A detailed approach 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 the MST. Kruskal's algorithm is used for this purpose. A detailed approach to ot close the loop with the nodes already added to not close the loop with the nodes already added to orKruskal's this purpose. A detailed to constructing MSTs algorithm isKruskal's usedapproach foralgorithm this purpose. A detailed approach constructing MSTs using is presented in (Yang et 2014). 11al.,to

(1)−−𝜌𝜌𝜌𝜌𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)) 𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖==√2(1 (1) (1) and to construct )) The correlation √2(1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 coefficients are therefore used to calculate the distances (1) 𝑡𝑡 to The correlation coefficients are therefore used calculate the and a matr he correlation coefficients therefore used distances to𝑁𝑁to construct a matrix are therefore used toofcalculate the distances and to construct a distances matrixasand ofthe the distances the are industry indices 𝐷𝐷to ofcalculate the samethe dimension × construct 𝑁𝑁 correlation m 𝑡𝑡 𝑡𝑡 𝑡𝑡 the distances of the industry indices 𝐷𝐷 of the dimension as 𝑁𝑁distances ×pp:correlation 𝑁𝑁of correlation matrix. and to construct aused matrix of he distances of the indices 𝐷𝐷 ofand the same dimension theathe 𝑁𝑁 × 𝑁𝑁 SmS The International Journal of Banking Vol. 18, Number 1as (January) 2023, 31–50 yscients indices 𝐷𝐷therefore of the same dimension as the 𝑁𝑁Finance, × 𝑁𝑁same correlation matrix. distances inindustry the matrix imply high correlations between industry indices. ents are are The therefore correlation used to coefficients to calculate calculate the are the distances therefore distances and used and to toto construct construct calculate the aSmall matrix matrix of and The correlation coefficients are therefore used calculate the distances andtomatrix. toconstruct construc .imply 𝑑𝑑𝑑𝑑between 𝑡𝑡matrix 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖the in the imply high correlations industry indices. × 𝑁𝑁 correlation matrix. Small istances inthe the matrix correlations between industry indices. ydistances high correlations indices. ustry indices indices the distances 𝐷𝐷 𝐷𝐷𝑡𝑡to ofof the ofof same same the industry dimension dimension indices asasthe the 𝐷𝐷 𝑁𝑁 of ×of ×𝑁𝑁 the 𝑁𝑁 correlation correlation same dimension matrix. matrix. asSmall the edustry distances and construct aindustry matrix of distances thehigh industry indices 𝐷𝐷𝑡𝑡𝑁𝑁𝑡𝑡between the same dimension asSmall the𝑁𝑁𝑁𝑁××𝑁𝑁𝑁𝑁correlation correlationmm 𝑡𝑡 From the distance matrix 𝐷𝐷 , we can draw a directionless and fully connected graph 𝐺𝐺 in w mply imply high correlations ininthe between matrix between imply industry industry high indices. indices. correlations as thehigh 𝑁𝑁distances ×correlations 𝑁𝑁 correlation matrix. Small distances the matrix imply high correlationsbetween betweenindustry industryindices. indices. 𝑡𝑡 𝐷𝐷 𝑡𝑡𝑑𝑑𝑑𝑑, 𝑖𝑖𝑖𝑖we 𝑡𝑡 From ∑ links is minimal. In this way, the MST is constructed the distance matrix can draw a directionless and fully connected graph 𝐺𝐺 in which (𝑖𝑖𝑖𝑖,𝑖𝑖𝑖𝑖)∈MST 𝑖𝑖𝑖𝑖 rom the distance matrix 𝐷𝐷 , we can draw a directionless and fully connected graph 𝐺𝐺 in which , we can draw a directionless and fully connected graph 𝐺𝐺 in which the nodes are industry indices and the links between them are measured by the distances 𝑑𝑑𝑖𝑖𝑖𝑖the . t y indices. 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 progressively by linking all the nodes together with the smallest nodes are industry indices and the links between them are measured by the distances 𝑑𝑑 . ected graph 𝐺𝐺 in which the odes industry and the links between them measured by distances 𝑑𝑑𝑖𝑖𝑖𝑖 .graph nd links between are measured by the distances 𝑑𝑑𝑖𝑖𝑖𝑖 . 𝐺𝐺𝐺𝐺ininwhich 𝑖𝑖𝑖𝑖 rix x 𝐷𝐷the 𝐷𝐷are , ,we we From can can draw the draw distance a athem directionless directionless matrix 𝐷𝐷 and and , ,we fully fully can connected connected draw aaare directionless graph graph and which the theconnected From theindices distance matrix 𝐷𝐷 we can draw directionless andfully fully connected graph𝐺𝐺𝐺𝐺ininw distances. can beare started by the twothem closest correlated) he 𝑑𝑑 .are 𝑖𝑖𝑖𝑖are fully connected graph 𝐺𝐺Itinthem which the ices es distances and and the nodes the links links between between industry them indices are measured and measured the links by by− the between the distances 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖 are .𝑖𝑖𝑖𝑖(most . measured 𝑑𝑑𝑑𝑑𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖. .th (𝑁𝑁 The MST isindustry then constructed using 1)distances direct links between theby 𝑁𝑁the nodes of 𝐺𝐺 such nodes indices and thelinking links between them are measured by thedistances distances The nodes, the next node is added to the MST, with the condition that (𝑁𝑁 The MST is then constructed using − 1) direct links between the 𝑁𝑁 nodes of 𝐺𝐺 such that the (𝑁𝑁 he MST is then constructed using − 1) direct links between the 𝑁𝑁 nodes of 𝐺𝐺 such that the su (𝑁𝑁 ed using − 1) direct links between the 𝑁𝑁 nodes of 𝐺𝐺 such that the sum sured byofthe (𝑁𝑁 − 1)𝑑𝑑distances thedistances of these links ∑(𝑖𝑖,𝑗𝑗)∈MST 𝑑𝑑𝑖𝑖𝑖𝑖 is minimal. The MST is constructed 𝑖𝑖𝑖𝑖 . the added distance must be the smallest and also the addition of that (𝑁𝑁 ∑ of the − 1) distances of these links 𝑑𝑑 is minimal. The MST is constructed in this nodes of 𝐺𝐺 such that the sum (𝑁𝑁 ∑ fstructed the − 1) distances of these links 𝑑𝑑 is minimal. The MST is constructed in this w ∑MST these links 𝑑𝑑 isconstructed minimal. MST constructed in this (𝑖𝑖,𝑗𝑗)∈MST (𝑁𝑁 (𝑁𝑁−−1) (𝑁𝑁 ructed using using The isis 1)then direct direct constructed links links between between using the the 𝑁𝑁𝑁𝑁 −is nodes 1) ofof𝐺𝐺𝐺𝐺links such such between that thatway the the sum the sum ofof𝐺𝐺𝐺𝐺 such (𝑖𝑖,𝑗𝑗)∈MST (𝑁𝑁 𝑖𝑖𝑖𝑖 by progressively all theThe nodes together with the smallest distances; starting by linkt The MST then using −nodes 1)𝑖𝑖𝑖𝑖direct direct links between the𝑁𝑁𝑁𝑁nodes nodes such (𝑖𝑖,𝑗𝑗)∈MST 𝑖𝑖𝑖𝑖linking node should not close the loop with the nodes that are already added MST is constructed in this way by progressively linking all the nodes together with the smallest distances; starting by linking the yses progressively linking all the nodes together with the smallest distances; starting by linking the tw (𝑁𝑁 ∑ ∑(𝑖𝑖,𝑗𝑗)∈MST ∑ the nodes together with the smallest distances; starting by linking the two een 𝑁𝑁 nodes of 𝐺𝐺 such that the sum closest (most correlated) nodes, the next node added to the MST provided that the added of ofthe these these ofof links the links − 1) distances 𝑑𝑑 𝑑𝑑 is is minimal. of minimal. these links The The MST MST is is constructed constructed 𝑑𝑑 is minimal. in in this this The way way MST is constructed (𝑁𝑁 ∑ the − 1) distances of these links 𝑑𝑑 is minimal. The MST is constructe (𝑖𝑖,𝑗𝑗)∈MST 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑖𝑖,𝑗𝑗)∈MST (𝑖𝑖,𝑗𝑗)∈MST 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 to the MST. Kruskal’s algorithm is used for this purpose. A detailed closest (most correlated) nodes, the next node is added to the MST provided that the added distan es; starting by linking the two losest (most correlated) nodes, the next node is added to the MST provided that the added distance des, the next node is added to the MST provided that the added distance is the smallest and that the addition of this node does not close the loop with the nodes alread al. The MST is constructed in this way ng g all all theby nodes nodes progressively together togetherwith with linking thesmallest all smallest distances; distances; together starting starting with the by linking smallest linking the the distances; two two starting by progressively linking allthe thenodes nodes together withby the smallest distances; startingby bylink lin approach to constructing MSTs using Kruskal’s algorithm isMST presented the smallest and that the addition of this node does not close the loop with the already add that the added distance isalgorithm he smallest and that the addition of this node does not close the loop with nodes already added dition ofthe this node does not close the loop with the nodes already added tothe MST. Kruskal's isthe used for this detailed tothe construc d) )ded nodes, nodes, closest the the next next (most node node correlated) is is added added to to nodes, the the MST MST provided next provided node that that isispurpose. added the the added added to the distance distance MST provided isapproach is nodes that added st distances; starting by linking the two closest (most correlated) nodes, the next node added toA the provided that the adde in Yang etand al. (2014). MST. Kruskal's algorithm is used this purpose. A detailed approach to constructing M ththe the nodes already added to he MST. Kruskal's algorithm is used for this purpose. detailed approach to constructing MS isprovided used for this purpose. A detailed approach tothe constructing MSTs using Kruskal's algorithm is presented in (Yang etA al., 2014). he ehm addition addition the ofof smallest this this node node and does does that not not the close close addition the thefor loop of loop with with node the does nodes nodes not already already close the added added loop toto with the MST that the added distance is the smallest that the addition ofthis this node does not close the loop with thenodes nodesalrea alre using Kruskal's algorithm is presented in (Yang et al., 2014). proach to constructing MSTs sing Kruskal's algorithm is presented in (Yang etapproach al., 2014). she presented inused (Yang etthis al., 2014). gorithm lgorithm is the is used MST. for for Kruskal's this purpose. purpose. algorithm AAdetailed isisused approach for purpose. to constructing constructing AAdetailed MSTs MSTs approach loop with the nodes already added todetailed the MST. Kruskal's algorithm used forthis thisto purpose. detailed approachtotoconstru constr The number ofal., links between considerably reduced by using The number of links between nodesnodes is considerably reduced by using minimum spanning t hm m isispresented presented using Kruskal's in into (Yang (Yang algorithm etalgorithm et al., 2014). 2014). isispresented ininis (Yang etetal., 2014). etailed approach constructing MSTs using Kruskal's presented (Yang al., 2014). 𝑡𝑡 number of links between nodes is − considerably reduced by using minimum spanning (M the minimum spanning tree (MST). The present study therefore, will heThe number links between nodes considerably reduced using minimum spanning treetree (MST en nodes is of considerably reduced using minimum spanning tree (MST). (𝑁𝑁 We therefore go from 𝑁𝑁 by ×is(𝑁𝑁 1)/2 distances inbythe distance matrix 𝐷𝐷 to only − 𝑡𝑡 𝑡𝑡 𝑡𝑡 (𝑁𝑁 go from the d distances in the distance matrix to only We therefore go from 𝑁𝑁 × (𝑁𝑁 − 1)/2 in the distance matrix 𝐷𝐷 to only − 1) sho spanning tree (MST). (𝑁𝑁 We therefore from 𝑁𝑁 × (𝑁𝑁 − 1)/2 distances in the distance matrix 𝐷𝐷 to only − 1) shorte (𝑁𝑁 ×inimum (𝑁𝑁 − 1)/2 distances in the distance matrix 𝐷𝐷 to only − 1) shortest distances in the MST. tween etweennodes nodes The isnumber isconsiderably considerably ofoflinks between reduced by nodes by using using isisminimum considerably minimum spanning spanning reduced tree tree by (MST). (MST). using Thenumber linksreduced between nodes considerably reduced by usingminimum minimumspanning spanning 𝑡𝑡 in1)/2 the MST. (𝑁𝑁 distances in the MST. 𝐷𝐷 only − 1) ,the shortest in the MST. (𝑁𝑁 (𝑁𝑁− using minimum spanning tree (MST). mxyistances 𝑁𝑁distances 𝑁𝑁 ××to (𝑁𝑁 (𝑁𝑁 We −We −1)/2 therefore distances distances go from infrom inthe the 𝑁𝑁 × distance matrix matrix distances 𝐷𝐷 𝐷𝐷𝑡𝑡 𝑡𝑡totoonly only ininthe distance − 1)1)shortest shortest matrix (𝑁𝑁− therefore goshortest 𝑁𝑁distance ×(𝑁𝑁 (𝑁𝑁−−1)/2 1)/2 distances the distance matrix𝐷𝐷𝐷𝐷𝑡𝑡 𝑡𝑡totoonly only(𝑁𝑁 distances ininthe MST. (𝑁𝑁 nce matrix 𝐷𝐷 𝑡𝑡 to only − 1) shortest distances the MST. Data Data Data Data The we datawill for study the study werethe the prices prices of 𝑁𝑁 = 21 industry The data concern industryindices indicesofofthe the Moroccan stoc Data Data data we will study concern the prices of 𝑁𝑁 = industry indices of the Moroccan exch heThe data we will study concern the prices of 𝑁𝑁 = industry indices of the Moroccan cern the prices of 𝑁𝑁 = industry indices of the Moroccan stock exchange Moroccan stock exchange from January 23, 2013 to November 30, from January 23, 2013 to November 30, 2020, i.e. a total of 1945 trading daysstock 𝑡𝑡 stock = exchan 1,2, ... from January 23, 2013 to November 30, 2020, i.e. a total of 1945 trading days 𝑡𝑡 = 1,2, . . . , 1945 the Moroccan stock exchange om January 23, 2013 to November 30, 2020, i.e. a total of 1945 trading days 𝑡𝑡 = 1,2, . . . , 1945. ovember 30, 2020, i.e. a=total 1945 trading days 1,2, . industry .stock , stock 1945. 1,945 trading days yconcern concernThe the theprices data prices of ofwill 𝑁𝑁will 𝑁𝑁 = study 21of industry industry concern indices indices the of ofthe the of𝑡𝑡ofMoroccan 𝑁𝑁= Moroccan exchange indices exchange The datawe we study concern theprices prices 𝑁𝑁== 21.industry indicesofofthe theMoroccan Moroccanstoc sto days 𝑡𝑡of = . 2020, . .2020, , 1945. As in1,2, Onnela et al. (2003), Tabak ettrading al. (2010) and et (2014) anddays for analysis and oto November November from 30, January 30, 23, i.e. i.e. 2013 a2013 atotal total totoof November of 1945 1945trading 30, 2020, days daysi.e. 𝑡𝑡i.e. 𝑡𝑡 =a=Yang 1,2, 1,2,.of ..of .. ,.al. 1945 1945. ,1945 1945. trading 𝑡𝑡 𝑡𝑡 == 1,2, .. ndices the Moroccan stock exchange from January 23, November 30, 2020, atotal total trading days 1,2, As in et Onnela et𝑡𝑡inal. (2003), et al. (2010) and Yang et al. (2014) and for analysis and smoo As intrading Onnela et (2003), et al. (2010) Yang et al. (2014) and for analysis smoothi Tabak al.days (2010) and Yang et al. (2014) and analysis and smoothing purposes, the data are divided into 𝑤𝑤 and = 1.2 … .89 windows width 𝑊𝑊 = and . The st Asal. Onnela al. (2003), Tabak etfor al. (2010) and Yang etofal. (2014), = 1,2, .Tabak .et. ,Tabak 1945. the data are divided into 𝑤𝑤 = 1.2 … .89 windows of width 𝑊𝑊 = . The step bet nd for analysis and smoothing urposes, the data are divided into 𝑤𝑤 = 1.2 … .89 windows of width 𝑊𝑊 = . The step betwe edpurposes, into 𝑤𝑤 = 1.2 … .89 windows of width 𝑊𝑊 = . The step between two consecutive windows is set at trading days 𝑑𝑑𝑑𝑑 = . The choice of 𝑊𝑊 = i 003), 03), Tabak Tabak As et in et al. Onnela al. (2010) (2010) et and al. and (2003), Yang Yang et et Tabak al. al. (2014) (2014) et al. and (2010) and for for analysis and analysis Yang and and et smoothing al. smoothing (2014) and for analysis and and for analysis and smoothing purposes, the data were divided into As in Onnela et al. (2003), Tabak et al. (2010) and Yang et al. (2014) and for analysis an two consecutive windows is set at trading days 𝑑𝑑𝑑𝑑 = . The choice of 𝑊𝑊 = is motiv 𝑊𝑊 = . The step between wo consecutive windows is set at trading days 𝑑𝑑𝑑𝑑 . The choice of 𝑊𝑊 = is motivat sdivided set at trading days 𝑑𝑑𝑑𝑑 = . The choice of 𝑊𝑊 = is motivated by the fact that the period of the Covid-19 pandemic studied is days (between March ivided purposes, into 𝑤𝑤 𝑤𝑤 =analysis =the 1.2 1.2… data …and .89 .89 are windows divided windows of ofwidth width 𝑤𝑤𝑤𝑤 =𝑊𝑊 1.2 ==…… .89 . The The windows step step between of between 𝑊𝑊𝑊𝑊 == 185 (2014)into and for smoothing windows step between two purposes, the data are dividedinto into =𝑊𝑊 1.2 .89 windows ofwidth width 185. .The Thes the fact that the of the Covid-19 pandemic studied is days (between 01, 20i ice ofis 𝑊𝑊 = isperiod motivated yofby the fact that the ofstudied the Covid-19 pandemic studied is (between March 2020 of Covid-19 pandemic is days (between March 01, 2020 to November 30, 2020); this period corresponds roughly to the time between the official ann ows ws is set set two at at20 consecutive trading trading days days windows 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 == is20 set . .The at The choice choice trading of ofdays 𝑊𝑊 𝑊𝑊 =𝑑𝑑𝑑𝑑 = =185 isdays is motivated motivated .The choice ofMarch == width 𝑊𝑊 = .period The step between consecutive windows was set at20 trading days choice two consecutive windows is185 set at trading days 𝑑𝑑𝑑𝑑 = .The The choice of𝑊𝑊 𝑊𝑊01, November 30, 2020); this period corresponds roughly to the time between the official announce s (between March 01, 2020 to November 30, 2020); this period corresponds roughly to the time between the official announceme period corresponds to the time between the official announcement riod od ofofthe the by Covid-19 Covid-19 the that pandemic pandemic the period studied studied ofofthe isthe is Covid-19 days days (between (between pandemic March studied 01, 01, isis 2020 2020 todays to (between . The choice offact 𝑊𝑊 = is motivated by the factroughly that thewas period Covid-19 pandemic studied 185days (betweenMarch March motivated by the fact that the March period of the Covid-19 een the official announcement his period period November corresponds corresponds 30, roughly 2020); roughly to tothe period the time time corresponds between betweenthe the roughly official official announcement the time between sthis days (between March 01,this 2020 to November 30, 2020); this period corresponds roughly toannouncement theto time betweenthe theofficial officialanan pandemic studied was days (between March 01,to 2020 November ime between 30, the 2020); officialthis announcement period corresponded roughly to the time between the official announcement of the pandemic and the announcement of33the the announcement of the vaccine. The The 3step of 𝑑𝑑𝑑𝑑 = 20 corresponded roughly corresponds vaccine availability. roughly toto of trading days month.number Thus, this distribution of data consists the theper average of trading days per month. Thus, of thisstudying distribution s each corresponding a window 185 stockthe market prices of with89a 20-day of data to consisted ofofstudying evolution MSTs, shift each window to another. The lasttowindow studied corresponds to the entire of corresponding a window of 185 stock market prices withperiod a 20-day mic (March 01, to November 2020).toThis choice arbitrary in shift2020 (overlap) from one30, window another. Theremains last window studied at the choicecorresponded of window width is a compromise toopandemic noisy and too to the entire period of thebetween Covid-19 (March mall and large01, window respectively (Onnela al., 2003). Similarlywith for the 2020 widths to November 30, 2020). Thisetchoice was arbitrary the ice of very small dW may raise concerns about the inability to identify episodic fact that the choice of window width had been a compromise between works as the too network carries large proportion of the noisy and too asmoothed data for small andobservations large windowbetween widths, (De Carvalhorespectively & Gupta, 2018). (Onnela et al., 2003). Similarly, for the overlap step, the ofThe verystep small concerns relatedtoto its inability ement of the choice vaccine. of 𝑑𝑑𝑑𝑑 might = 20cause roughly corresponds ed for time windows are notchanges independent each other (two successive ays perdifferent month.toThus, thisthe distribution of data in consists of studying the the identify episodic the of networks because network ding days over an analyzed MST period of trading days), but form a series sponding to a window of 185 stock market prices with a 20-day shift , this multitude trees isstudied interpreted as a sequence of evolutionary 36 of nother. The last window corresponds to the entire period of steps of a tree (Onnela al., 2003)30, 2020). This choice remains arbitrary in 01, 2020 to et November ce of window width is a compromise between too noisy and too carried a huge portion of the observations between the successive windows (De Carvalho & Gupta, 2018). The constructed MSTs for different time windows were not independent of each other (two successive MSTs share 165 trading days over an analyzed MST period of 185 trading days), but form a series over time. Therefore, this multitude of trees was interpreted as a sequence of evolutionary steps of a single dynamic asset tree (Onnela et al., 2003) The 21 stock market industry indices studied are classified as in Table 1. The MSTs corresponding to the two periods, the pre-pandemic period (January 23, 2013 to February 29, 2020) and the pandemic period (March 1, 2020 to April 30, 2020) are as given in Figure 1.