**5. Conclusions**

The presented study analysed the impact of the pandemic on the structure of crosscorrelation networks among the most important companies on S&P500 components. The stock market crashes strongly influence cross-correlation network structure. Four different networks have been introduced and investigated: strongly, weakly, most typically, and significantly connected companies. The first 2 networks are based on 25% of links while the 2 latter networks are constructed on 50% of links. In general, all constructed networks are sensitive to large price fluctuations. Of particular interest was the crisis induced by the COVID-19 pandemic, where four stages of the market reaction were distinguished. It is worth stressing that the observed changes in the network structure can be related to particular features of the 2020 situation. The essential result is that the discussed changes can be quantified by rank entropy and cycle entropy measures, as well as the standard network parameters, such as the averaged clustering coefficient and the transitivity. The networks of strongly connected companies react in a different way to the crisis than the networks of weakly connected companies. Besides the type of the network and its features, the optimal size of the time window to calculate cross-correlation has been investigated. The optimal window size is a month, *T* = 20*d*. In the analysis based on the *T* = 20*d* cross-correlation time window, the fluctuations are suppressed such that important trends can be seen and discussed. On the other hand, in the analysis performed for the shortest time window (*T* = 5 days), the averaged clustering coefficient and the transitivity for the network of the significantly connected companies decreases in the crises, while for the networks of strongly and typically connected companies, these parameters visibly increase. This situation might be the effect of the short window time wherein the Pearson correlation coefficient is calculated on the five data point sets. This observation supports the conclusion that the optimal time window for the analysis of the daily time series returns is a month period.

The presented results show that the proposed network structures are capable of describing and measuring the changes resulting from crises on the stock markets. Moreover, the introduced parameters, the rank network entropy and the cycle entropy, are useful parameters in the analysis of structure changes and crises recognition. Particularly, the rank entropy, which is capable of quantitatively characterising network structure changes and those parameters, might be useful in crash analysis. On the other hand, the introduced network structures, which are composed of strongly, significantly, typically and weakly correlated companies, do not introduce as strong of constraints as the frequently used MST structures. For example, the GE company, which is the centre of the MST in [16], is one of the highly connected companies here, but is not so prominent as in the MST structure. The number of links of GE is comparable to the median level of the number of links for a given network type.

Besides the main results of the paper, it has been observed that the rank entropy is likely to change its value in a step-like function, showing that, according to the market situation, the network will change to some well-established structures. This is a very intriguing hypothesis which deserves further study.

**Author Contributions:** Conceptualization, J.M.; methodology, J.M.; software, J.M.; investigation, J.M.; data verification, J.M.; results discussion, J.M., D.B.-K.; writing—original draft preparation, J.M.; writing—review and editing, D.B.-K.; visualization, J.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data used in this study were downloaded from the web page www.stoog.pl (accessed on 28 March 2021).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Companies List**

The analysis of this paper is based on the following companies quotes (using the standard abbreviations):

A, AAL, AAP, AAPL, ABBV, ABC, ABT, ACN, ADBE, ADI, ADM, ADP, ADS, ADSK, AEE, AEP, AES, AFL, AIG, AIV, AIZ, AJG, AKAM, ALB, ALGN, ALK, ALL, ALLE, ALXN, AMAT, AMD, AME, AMG, AMGN, AMP, AMT, AMZN, ANSS, ANTM, AON, AOS, APA, APD, APH, APTV, ARE, ARNC, ATVI, AVB, AVGO, AVY, AWK, AXP, AYI, AZO, BA, BAC, BAX, BBY, BDX, BEN, BFb, BHF, BIIB, BK, BKNG, BKR, BLK, BLL, BMY, BPYU, BRKb, BSX, BWA, BXP, C, CAG, CAH, CAT, CB, CBOE, CBRE, CCI, CCL, CDNS, CERN, CF, CFG, CHD, CHRW, CHTR, CI, CINF, CL, CLX, CMA, CMCSA, CME, CMG, CMI, CMS, CNC, CNP, COF, COG, COO, COP, COST, COTY, CPB, CPRI, CRM, CSCO, CSX, CTAS, CTSH, CTXS, CVS, CVX, D, DAL, DD, DE, DFS, DG, DGX, DHI, DHR, DIS, DISCA, DISCK, DISH, DLR, DLTR, DOV, DRE, DRI, DTE, DUK, DVA, DVN, DXC, EA, EBAY, ECL, ED, EFX, EIX, EL, EMN, EMR, EOG, EQIX, EQR, EQT, ES, ESS, ETN, ETR, EW, EXC, EXPD, EXPE, EXR, F, FAST, FB, FBHS, FCX, FDX, FE, FFIV, FIS, FISV, FITB, FL, FLIR, FLR, FLS, FMC, FRT, FTI, FTV, GD, GE, GILD, GIS, GL, GLW, GM, GOOG, GOOGL, GPC, GPN, GPS, GRMN, GS, GT, GWW, HAL, HAS, HBAN, HBI, HCA, HD, HES, HIG, HLT, HOG, HOLX, HON, HP, HPE, HPQ, HRB, HRL, HSIC, HST, HSY, HUM, IBM, ICE, IDXX, IFF, ILMN, INCY, INFO, INTC, INTU, IP, IPG, IPGP, IR, IRM, ISRG, IT, ITW, IVZ, J, JBHT, JCI, JEF, JNJ, JNPR, JPM, JWN, K, KDP, KEY, KHC, KIM, KKR, KLAC, KMB, KMI, KMX, KO, KR, KSS, KSU, L, LB, LEG, LEN, LH, LHX, LKQ, LLY, LMT, LNC, LNT, LOW, LRCX, LUMN, LUV, LYB, M, MA, MAA, MAC, MAR, MAS, MAT, MCD, MCK, MCO, MDLZ, MDT, MET, MGM, MHK, MKC, MMC, MMM, MNST, MO, MOS, MPC, MRK, MRO, MS, MSFT, MSI, MTB, MTD, MU, NAVI, NCLH, NDAQ, NEE, NEM, NFLX, NI, NKE, NLOK, NLSN, NOC, NOV, NRG, NSC, NTAP, NTRS, NVDA, NWL, NWS, NWSA, O, OKE, OMC, ORCL, ORLY, OXY, PAYX, PBCT, PCAR, PCG, PDCO, PEAK, PEG, PEP, PFE, PFG, PG, PGR, PH, PHM, PKG, PKI, PLD, PM, PNC, PNR, PNW, PPG, PPL, PRGO, PRU, PSA, PSX, PVH, PWR, PXD, PYPL, QCOM, QRVO, RCL, RE, REG, REGN, RF, RHI, RL, RMD, ROK, ROP, ROST, RRC, RSG, RTX, SBAC, SBUX, SCHW, SEE, SHW, SIG, SJM, SLB, SLG, SNA, SNPS, SO, SPG, SPGI, SRCL, SRE, STT, STX, STZ, SWK, SWKS, SYF, SYK, SYY, T, TAP, TDG, TEL, TFC, TGT, TJX, TMO, TNL, TRIP, TRV, TSCO, TSN, TXN, TXT, UA, UAA, UAL, UDR, UHS, ULTA, UNH, UNM, UNP, UPS, URI, USB, V, VAR, VFC, VIAC, VLO, VMC, VNO, VRSK, VRSN, VRTX, VTR, VTRS, VZ, WAT, WBA, WDC, WEC, WELL, WFC, WHR, WLTW, WM, WMB, WMT, WRK, WU, WY, WYNN, XEC, XEL, XLNX, XOM, XRAY, XRX, XYL, YUM, ZBH, ZION, ZTS.

#### **Appendix B. Graph Examples**

Here, a few of the network examples generated and analysed in the study are presented (Figures A1–A4). The figures were obtained using Mathematica 11 with the "SpringElectricalEmbeding" algorithm. This algorithm optimises the position of nodes with respect to its rank. However, due to the number of nodes and links, the graphs are a bit unclear, particularly in the case of the network of significantly connected companies in Figure A4. However, even a cursory observation shows that the proposed structures give significantly different results. Particularly, networks representing the state of the stock market during the pandemic vary significantly, even though loss of network connectivity is observed. It goes beyond the scope of this paper, but the detailed analysis of the network's evolution from the point of view of the role of a particular company or the reaction of a group of companies to the pandemic situation might be very interesting; however, this is left for another study.

**Figure A1.** Examples of the graphs obtained for the network of strongly correlated companies. The presented graphs show the network state before and in the first and second stage of the COVID-19 pandemic. The top graphs correspond to the shortest time window *T* = 5*d*, the middle three graphs represent networks for the month time window and the bottom graphs present the examples for the quarter time window.

**Figure A2.** Examples of the graphs obtained for the network of weakly connected companies. The presented graphs show the network before and in the first and second stage of the COVID-19 pandemic. The top graphs correspond to the shortest time window *T* = 5*d*, the middle three graphs represent networks for the month time window and the bottom graphs present the examples for the quarter time window.

**Figure A3.** Examples of the graphs obtained for the network of the typically connected companies. The presented graphs show the network before and in the first and second stage of the COVID-19 pandemic. The top graphs correspond to the shortest time window *T* = 5*d*, the middle three graphs represent networks for the month time window and the bottom graphs present the examples for the quarter time window.

**Figure A4.** Examples of the graphs obtained for the network of significantly connected companies. The presented graphs show the network before and in the first and second stage of the COVID-19 pandemic. The top graphs correspond to the shortest time window *T* = 5*d*, the middle three graphs represent networks for the month time window and the bottom graphs present the examples for the quarter time window.
