Generalized coefficients of clustering in (un)directed and (un)weighted networks: An application to systemic risk quantification for cryptocoin markets

Abstract

A coefficient of clustering is one of the important network parameters to describe clustering characteristics around a node in a network. It accounts for the number of cycles of length 3 (triangles) relative to the number of paths of length 2 in which this node participates as their inner node. In practice, the number of longer paths and cycles may be larger than the number of 2-paths and 3-cycles, thereby requiring its generalized version. In this paper, we aim at proposing the so-called λ-clustering coefficient to characterize the node clustering tendency at level λ≥3. We define it by counting the number of λ-cycles relative to the number of (λ−1)-paths containing a fixed inner node. In addition, we introduce the total clustering coefficient by considering the total number of all cycles relative to the total number of all paths consisting of this inner node. We formulate these generalized clustering coefficients for each of the following cases: undirected and unweighted networks, undirected but weighted networks, directed but unweighted networks, and directed and weighted networks. We then implement them for correlation- and conditional risk measure-based networks that represent conventional cryptocurrency markets and stablecoin markets. They can be viewed as new measures of systemic risk. The larger their values, the larger the tendency of risk transmission from one market to other subsequent markets forming longer cycles, and therefore the higher the systemic risk level. The systemic risk quantification is appropriate when they fulfill certain properties. In summary, the novelty of this study lies in the mathematical formulation of the generalized clustering coefficients and their computation for the cryptocoin networks with the purpose of appropriately managing systemic risk in the represented cryptocoin markets.