The Anatomy of Leadership in Collective Behaviour.

Understanding the mechanics behind the coordinated movement of mobile animal groups provides key insights into their biology and ecology while also yielding algorithms for bio-inspired technologies and autonomous systems. It is becoming increasingly clear that many mobile animal groups are composed of heterogeneous individuals with differential levels and types of influence over group behaviors--often considered as "leaders". The ability to infer this differential influence, or leadership, is critical to understanding group functioning in these collective animal systems. Due to the broad interpretation of leadership, many different measures and mathematical tools are used to describe and infer "leadership", e.g., position, causality, influence, information flow. But a key question remains: which, if any, of these concepts actually describes leadership? We argue that instead of asserting a single definition or notion of leadership, the complex interaction rules and dynamics typical of a group implies that leadership itself is not merely a scalar quantity, but rather, a complex combination of many different components. In this manuscript we develop an anatomy of leadership, identify several principle components and provide a general mathematical framework for discussing leadership. With real and synthetic examples we then illustrate how to use this framework. We believe this multifaceted approach to leadership definition will allow for a broader understanding the role of leadership and its inference from data in mobile animal groups and beyond.