**4. Discussion**

Spectral clustering techniques can be used to discover the equity index structure. On the one hand, clusters help us to overcome the hardship of heterogeneity and make diversification more efficient. In our paper we shed some light on the relations between spectral, geographical and qualitative clustering. It also turned out that Gaussian-kernel based clusters are more suitable than geographical and qualitative categorizations. In addition, spectral cluster-wise linear regressions give time stationary and significant results.

On the other hand, we stress that correlation does not convey enough information about the network; hence linear dependency-based diversification is not optimal (Sharpe 1964; Maldonado and Anthony 1981). We compared various similarity kernels and spectral clustering methods to demonstrate the inadequacy of a normalized Laplacian approach (Takumasa et al. 2015) and underpin the applicability of the proposed Newman–Girvan cut. Moreover, we highlighted that daily closing prices incorporate the network level information. The results unveiled that tail events have little effect on the dense network structure, in other words, market shocks have no effect on the cluster components; thus, index co-movements are not affected by large price changes.

All of these imply spectral clustering can eliminate non-linear effects, thus regular mean-standard deviation representation gives cluster-wise reliable figures. Instead of qualitative categorization, we sugges<sup>t</sup> that portfolio managers should use Gaussian-based normalized modularity clusters to diversify global non-systematic risk.

An interesting field of further research would be analyzing the evolution of the network to identify patterns that could help us to understand the life cycle of hubs and the vulnerability of the current equity network.

**Author Contributions:** Conceptualization, L.N. and M.O.; Methodology, L.N. and M.O.; Validation, L.N. and M.O.; Formal Analysis, L.N. and M.O.; Investigation, L.N. and M.O.; Writing-Original Draft Preparation, L.N. and M.O.; Writing-Review & Editing, L.N. and M.O.; Visualization, L.N.; Supervision, M.O.

**Funding:** Research fund was provided by Pallas Athéné Domus Educationis.

**Acknowledgments:** The authors would like to gratefully acknowledge the valuable comments and suggestions of three anonymous referees that contributed to a substantially improved paper. Mihály Ormos acknowledges the support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and the support of the Pallas Athéné Domus Educationis Foundation. The views expressed are those of the authors and do not necessarily reflect the official opinion of the Pallas Athéné Domus Educationis Foundation.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
