Spatially quantifying the leadership effectiveness in collective behavior

Among natural biological flocks/swarms or mass social activities, when the collective behavior of the followers has been dominated by the direction or opinion of one leader group, it seems difficult for later-coming leaders to reverse the orientation of the mass followers, especially when they are in quantitative minority. This paper, however, reports a counter-intuitive phenomenon, i.e. Following the Later-coming Minority, provided that the later-comers obey a favorable distribution pattern that enables them to spread their influence to as many followers as possible within a given time and to be dense enough to govern these local followers they can influence directly from the beginning. We introduce a discriminant index to quantify the whole group's orientation under competing leaderships, with which the eventual orientation of the mass followers can be predicted before launching the real dynamical procedure. From the application point of view, this leadership effectiveness index also helps us to design an economical way for the minority later-coming leaders to defeat the dominating majority leaders solely by optimizing their spatial distribution pattern provided that the premeditated goal is available. Our investigation provides insights into effective leadership in biological systems with meaningful implications for social and industrial applications.