The friendship quotient: A measure of local neighborhood persistence in reconstructed phase space

Complex signals are commonly characterized using measures of irregularity, recurrence, and predictability. Although recurrence-based methods quantify the occurrence, duration and timing of returns to previously occupied regions of phase space, they do not directly track whether the same members of an initially defined local neighborhood remain associated under synchronized forward temporal evolution. We introduce the Friendship Quotient (FQ), a measure of this local relational persistence in reconstructed phase space. Following time-delay embedding of a scalar time series, local neighborhoods are tracked through consecutive forward images, and FQ estimates the proportion of neighbors that maintain coherent trajectories, weighted by the relative spatial extent of the surviving neighborhood. High FQ values indicate stable local organization and temporally coherent dynamics; low values reflect rapid neighborhood disruption, characteristic of chaotic or stochastic behavior. Applied to the logistic map, FQ closely tracks the Lyapunov exponent profile across the period-doubling route to chaos and resolves periodic windows embedded within chaotic regimes. In a controlled mixing model spanning the continuum from periodic to stochastic dynamics, FQ exhibits a characteristic non-monotonic response profile that is absent from both recurrence determinism and sample entropy, suggesting sensitivity to an aspect of local dynamical organization not captured by established measures. A reliability analysis confirms stable estimation under strongly chaotic conditions. Applications to EEG signals demonstrate sensitivity to cognitive-state changes and to the progressive reorganization of neural dynamics during epileptic seizure evolution. FQ is easy to compute, assumption-light, and may be useful for characterizing local dynamical organization in short recordings across biological, physical, and ecological systems.