**5. Conclusions**

The present work analyses co-jump structures in high frequency markets. We investigate the distribution of co-jump sizes for 300 stocks on 1 min returns. We highlight features of this distribution, such as the finite size effect in the tail and the divergence of small crash frequencies from the distribution. We show how the ranking and structure of crash frequency throughout stocks changes drastically through a phase transition between small and large crash sizes at size 5. We quantify this with the Spearman correlation between crash frequency ranks at different crash sizes. We then apply a null model of crash frequency at each crash size to test the hypothesis of a phase transition. Finally, we highlight how larger crashes are dominated not by the less liquid stocks present in small crashes, but rather by highly liquid stocks which are present in most flash crashes as the crash size grows. Preliminary results, which we leave for future work, find these stocks to be systemic in communities and core-periphery like structures of co-crashes. We sugges<sup>t</sup> that these systemic events can be viewed as communities centered around these most influential stocks.

We know from the literature that these structures can be indeed vulnerable and highly unstable, as well as fragmented if characterized by multiple cores. One of the possible reasons for this can be inferred from the interviews with different market players following the crash of May 6th [3]. Many HFTs highlight the centralized risk constraints for volatility and P&L, which cause them to withdraw from the market in the case of extreme conditions or losses. As they constitute much of the liquidity in the market in particular for smaller stocks, withdrawing from those causes liquidity droughts. These are often systemic, as players have central risk constraints and withdraw from the entire market as those are triggered. Further, as systemic stocks crash, arbitrageurs come into play to level prices across the market, thus making the isolated event a systemic one. In this view, well-known stocks are not systemic per se, but rather as a result of non-siloed trading by HFTs and ETFs.

In light of the present results, future works shall investigate the asynchronous price changes of securities and model spreading dynamics of flash crashes and their directed structure. Lead–lag investigations of causality of these larger crashes are also suggested for future work. Already from our results, one can monitor, in particular, the most systemic stocks from larger flash crashes for co-jumps of size 5 and higher and induce trading halts or limitations to avoid further spreading of these systemic events. This is crucial, as our results combined with those of [10] sugges<sup>t</sup> a systemic self-excited process in both frequency and magnitude of those crashes.

We leave the investigation of this structure for future work and highlight that this is of high importance for practitioners and regulators when dealing with market efficiency and stability, particularly as trading frequencies rise and electronic trading becomes widespread across securities.

We conclude by observing that volatility and P&L-based trading breaks used by market players may worsen these events and their systemic characteristics since they cause liquidity withdrawals throughout stocks and market players. This introduces systemic synchronization throughout the market and makes individual assets more susceptible to small trading volumes. Further, we sugges<sup>t</sup> to monitor the stocks we find to be systemic throughout larger crashes to model the contagion of liquidity crises and halt trading before these spread and distort a larger number of assets. This should also be topic of future work aimed at smart and efficient regulation in high frequency markets.

**Author Contributions:** J.D.T. conducted the computational analysis and drafted the manuscript. T.A. guided the work and interpretation of results and reviewed and edited the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** J.D.T. acknowledges support from EPSRC (EP/L015129/1). TA acknowledges support from ESRC (ES/K002309/1), EPSRC (EP/P031730/1) and EC (H2020-ICT-2018-2 825215).

**Data Availability Statement:** The data used in this work was obtained from the LOBSTER dataset (https://lobsterdata.com (accessed on 2 January 2022)) under academic license to the Financial Computing and Analytics group at University College London. We are therefore unable to publish the raw data.

**Acknowledgments:** J.D.T. acknowledges Riccardo Marcaccioli for useful discussions and support with the jump detection method. J.D.T. acknowledges Charles-Albert Lehalle for useful feedback which motivated us to investigate the role of liquid stocks in co-crash structures and prompted further works on the topic.

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