Hierarchical Computable Aggregations

The concept of hierarchical structure is central when considering complex systems. On one hand, many complex systems exhibit a hierarchical structure, on the other hand, the idea of defining a hierarchical structure to cope with the complexity of the system is widely present in literature related to many different fields. Hierarchies are also present in the field of aggregation with the definition of hierarchical aggregation processes. Broadly speaking, a hierarchical system is a system formed by several components (subsystems) structured at different levels, implying a sort of ranking or ordering relation among them. Positioning in the levels could be related to many different aspects or properties of the components: priority, abstraction, granularity, specificity, precision, etc. Computable aggregations have recently been introduced as a natural approach to classical aggregation functions, in which the emphasis is placed on the program (the implementation) that makes possible the aggregation instead of simply considering the function or the algorithm that permits the aggregation process. Considered as a program that implements an aggregation process, a computable aggregation is also suitable for being interpreted in terms of a hierarchical process. In this paper, the idea of hierarchical computable aggregation is considered, exploring those situations where the aggregation process involves some intrinsic structure that can be interpreted in hierarchical terms (like priorities and veto), as well as those other situations where the hierarchical approach is mostly related to computational considerations (like recursion and parallelization). These and other types of hierarchical computable aggregation will be presented and analyzed.