**1. Introduction**

### *1.1. Literature Review*

The crucial and most obvious features of economic systems are cooperation and interaction, e.g., supply chains, ownership dependencies, cooperation networks, and financial networks. The companies and their activities form complex networks of relationships and dependencies [1–3]. The idea that the economy can be considered as a complex system can be found at the end of the XX century in economic literature, e.g., [4]. The features of the network depend on various parameters, such as technology, law, regulations, culture, climate, weather, resources, and many other parameters. Among various aspects of economic activity, the financial market plays a special role. Besides fundraising for the business, they allow monitoring the condition of enterprises and even whole economic sectors. The stock market indices are regularly published and are considered to be not only a measure of the change of companies' group values, but also a test of the economy's status [5]. Therefore, within this study, the cross-correlations among stock time series of S&P 500 index components are analysed. This index was chosen because it is based on quotations of 500 shares of the largest stocks that trade on the New York Stock Exchange and NASDAQ. Considering the importance of those companies, as well as their variety and number, this set can be considered to be representative of the USA's economy and, therefore, can be the basis of the analysis of cross-correlations among them. Particularly, these stocks can be analysed from the point of view of cross-correlation network structures formed by those companies. The cross-correlation analysis consists of two steps: the distance (usually based on linear correlation [6]) matrix calculation and its analysis by constructing a network of random matrix approaches [7–9]. The alternative cross-correlation analysis should be here mentioned. Recently, detrended cross-correlation analysis based on detrended fluctuation analysis (DFA) [10] and their modifications have been very popular; see the recent review [11], or the power law classification scheme [12]. However, the most popular strategy is based

on the network construction and its structure analysis [13,14]. The most popular choice is the minimum spanning tree (MST) analysis [3,9,15–21]. MST application results primarily from the portfolio optimisation problem [22], but is also due to the proper recovering of the industrial sectors [3,6]. Besides MST analysis, the second most commonly used group of methods are those which construct networks based on assumed threshold [23]. This approach is used also in this paper—the distance matrix is filtered assuming that nodes are connected when the distance fulfills a given condition. Within this analysis, the properties of the most typical, weak, strong or significant correlations are investigated. The network generation procedures are described in detail in Section 3. The main aim of the paper is the description of the structural changes observed during the COVID-19 pandemic and their comparison to the changes in cross-correlation network properties during other, recently observed crises.

Another important aspect is the influence of the external parameters on the network structure. It is a truism to say that the state of the market depends on the macroeconomic situation. Particularly, the reaction of the market to crashes is also widely discussed from the point of view of network structure, e.g., [9,24,25], or in the analysis of globalization processes [26–28]. However, the present pandemic situation should not be considered as the typical shock but rather a change of the "economic environment". The most important fact is that the pandemic was expected and induced serious changes in the economy. The government, due to the pandemic situation, introduced special rules lasting a relatively long time. The restrictions form special conditions which are expected to change the structure of cross-correlations among companies. The network structure of shares' cross-correlations is the subject of interest of various studies [9,21,29]. A natural extension of time series analysis through the network methods is evolving network analysis, since the economy time series are non-stationary, and thus, should be described by an evolving network rather than a static network. Evolving network theory was initially applied to systems naturally described by network structures, such as social networks [30], scientific collaboration networks [31] and economy time series [8,25,32,33], to mention a few examples.

#### *1.2. COVID-19 History*

Although the COVID-19 epidemic is a contemporary event, for the convenience of future readers, a short description of the epidemic key points in the USA is presented below.

**December 2019** The first known cases have been identified in Wuhan, China.

**January 2020** The epidemic spreads to other provinces of China.

**February 2020** Italy is affected with a rapidly growing number of infected and fatal cases. **March 2020** The USA overtakes China and Italy with the highest number of confirmed cases in the world.

The present situation is the subject of various studies. More detailed history and discussion on the influence of pandemic on stock markets from the standard time series analysis point of view can be found at [34–38].

#### *1.3. Paper Structure*

The present paper is organised as follows. Section 2 describes the data analysed. The Section 3 defines the methods used: the statistical distance, the network construction algorithms and the network parameters (node entropy, cycle entropy, averaged clustering coefficient and transitivity) calculated and analysed in the study. Section 4 presents the obtained results, including the evolution of the node and cycle entropy and the averaged clustering coefficient and transitivity. It is worth stressing that the parameters introduced here (the node and the cycle entropy) are sensitive to economic crises. Moreover, the performed analysis shows that the structure of the defined networks changes significantly in the crisis. The main outcome of the paper is the analysis of a representative group of USA companies and the observation of their reaction to changing economic situations. A very promising outcome of the study is that four periods during the COVID-19 pandemic are distinguished, which shows that the reactions to various factors are different and that this analysis is capable of observing this.
