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Estimating the center of force (mathcal {X}_{0}) in a far range of interaction is not possible since there is not conclusive information about velocity decrements.
Similarly, for a far range of interaction, it is possible to write (hat {vec G}_{text {far}}^{s}(r) = hat {lambda }_{s}).
From this viewpoint, the model of a far range of interaction works until agents present a continuous deceleration through time, which indicates they start approximating to an attractive area.
Based on that, the set of all velocities' magnitudes due to an attractive area in a far range of interaction can be written as ||V_{text{far set)}}||_{2} = {||V_{mathcal{X}_{text{far,1}}}||_{2}, ||V_{mathcal{X}_{text{far,2}}}||_{2},ldots,||V_{mathcal{X}_{text{far,J}}}||_{2}}, where J is the total number of measurements taken in a far range of interaction.
In order to characterize the velocity field in a far range of interaction associated to a segment s, the magnitudes (||U_{k text {far})}^{s}||_{2}) are used to estimate the random variable λ s, such as proposed in Eq. (5).
In this work, it is proposed to represent the velocity's magnitude of an agent localized in r>r switch as the L 2 norm among the velocities experienced in each spatial axis, such that (||V_{mathcal {X}_{text {far}}}||_{2}), where (V_{mathcal {X}_{text {far}}}) represents the agent's velocity in a single location measured in a far range of interaction.
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