Tsallis Entropy for Cross-Shareholding Network Configurations

Abstract

In this work, we develop the Tsallis entropy approach for examining the cross-shareholding network of companies traded on the Italian stock market. In such a network, the nodes represent the companies, and the links represent the ownership. Within this context, we introduce the out-degree of the nodes—which represents the diversification—and the in-degree of them—capturing the integration. Diversification and integration allow a clear description of the industrial structure that were formed by the considered companies. The stochastic dependence of diversification and integration is modeled through copulas. We argue that copulas are well suited for modelling the joint distribution. The analysis of the stochastic dependence between integration and diversification by means of the Tsallis entropy gives a crucial information on the reaction of the market structure to the external shocks—on the basis of some relevant cases of dependence between the considered variables. In this respect, the considered entropy framework provides insights on the relationship between in-degree and out-degree dependence structure and market polarisation or fairness. Moreover, the interpretation of the results in the light of the Tsallis entropy parameter gives relevant suggestions for policymakers who aim at shaping the industrial context for having high polarisation or fair joint distribution of diversification and integration. Furthermore, a discussion of possible parametrisations of the in-degree and out-degree marginal distribution—by means of power laws or exponential functions— is also carried out. An empirical experiment on a large dataset of Italian companies validates the theoretical framework.

Approaches to the classification of complex systems: Words, texts, and more

“…Negative values of q, while surprising at first glance, are not uncommon in studies involving the Tsallis q-exponential. [109][110][111] When similar calculations were made for some mammalian mtDNA, 79 the mean value of κ for the Felidae family appeared to be larger than that of the Ursidae ( κ F = 6.3 versus κ B = 5.1) yielding the conclusion that cats generally have longer tails than bears .…”

Complexity in Economic and Social Systems

“…It is applied to analyze the cross-shareholding networks of companies. In this context it offers a measure of market polarisation and a tool for analyzing market self-organization in response to external shocks [5]. Finally, the moving average cluster entropy is proposed to study the long-range dependence in time series and proves useful as a measure capturing endogenous sources of risk over different temporal horizons [6].…”

“…Self-organization of the stock markets and their hierarchical structure can be approached from the angle of information transfer between different sectors in various time intervals [12]. This also refers to the cross-shareholding market structure which self-organizes under the influence of external shocks [5]. Both the external shocks and the internal market events can produce excessive demand for information, which, if properly quantified, may offer a way to monitor oncoming market events that are difficult to predict by using other methods.…”