Fig. 1

1 Behavioral dynamics underlying social characteristics (coordinate system: horizontal axis = \( \widehat{i} \) and vertical axis = \( \widehat{j} \)). Following the simulation, the left plot shows a displacement that separates individuals with a relative position structure controlled by the initial setting. This implies that although the pattern of individual behavior depends on a localized view of the initial conditions, a slight change in individual characteristics [individual’s upward velocity (\( \vec{v}_{i} \)) resulting in a loss of group heading (\( \vec{v}_{avg} \))] underlying its social influence [calculated from the social ties (\( k = St \)) multiplied by the mutation rate (\( u \))] has a remarkably diverging (a) or converging (b) effect on its displacement. The blue dots represent their position in an \( x \), \( y \) coordinate plane, and the red lines denote links (see Additional file 1: Figure S5 for more detail). 2 Approximation of the evolution underlying interconnected interactions (coordinate system: horizontal axis = \( \widehat{i} \) and vertical axis = \( \widehat{j} \)). The plots indicate that the patterns that occur correspond to the relative value. With certain defaults of their relativity, a slight change in the scalar value (social ties = St) produces a dramatic impact at a certain point [a = St(0.55), b = St(0.56), c = St(0.57)]: blue dots = individuals, red lines = links, background = density with symmetrical characteristics between the individual and group headings. Notice that as the social ties increases a–c, symmetrical characteristics are biased to one side c (see Additional file 1: Figure S6 for more detail)