Syntropy in complex systems: A complement to Shannon's Entropy

TL;DR

This work introduces syntropy as a complement to Shannon entropy, built on an expectancy framework E_N using a convex φ, with a primary choice φ(α)=e^{α}-1 leading to E_N(α) = [1/(N(e^{1/N}-1))] ∑ (e^{α_i}-1) and S_N(A) = β_N ∑ α_i (e^{α_i}-1). The theory develops a suite of properties (sub/supramultiplicativity, sub/supraadditivity, symmetry, stationarity, differentiability, robustness, and dimensional generalization) and analyzes the equilibrium between entropy and syntropy, including a distance-to-equilibrium measure DAE. The paper then applies these constructs to time-series and spectral data through time-series syntropy, spectral syntropy (SS), and vectorial transformations that yield correlation matrices, followed by a dynamic-syntropy framework for evolving systems via Φ_{i,k} and state counts. Across these layers, the authors illustrate how syntropy captures ordered, coherent structure alongside the probabilistic variability described by entropy, and they discuss potential applications in biological signals and dynamic systems, with a call for further mathematical and computational exploration. Overall, the work provides a mathematical scaffold for integrating order, coherence, and predictability into complexity metrics, offering new tools for analyzing real-world signals and dynamical processes.

Abstract

This study introduces the syntropy function ($S_N$) and expectancy function ($E_N$), derived from the novel function $φ$, to provide a refined perspective on complexity, extending beyond conventional entropy analysis. $S_N$ is designed to detect localized coherent events, whereas $E_N$ encapsulates expected system behaviors, offering a comprehensive framework for understanding system dynamics. The manuscript explores essential theorems and properties, underscoring their theoretical and practical implications. Future research will further elucidate their roles, particularly in biological signals and dynamic systems, suggesting a deep interplay between order and chaos.

Figures (13)

• Figure 1: Simplex with minimum in $\frac{1}{N}$ and maximum in $\delta(0)$ and $\delta(1)$ for a probability '$p$' and '$q = 1-p$'.
• Figure 2: The plot displays the behavior of the $g(N)$ function for positive values of $N$, where it approaches zero as $N$ approaches zero and asymptotically approaches 1 as $N$ increases.
• Figure 3: This 3D visualization illustrates the supramultiplicative inequality comparison for the syntropy function $S_N(\textbf{A})$ within the domain of probabilities. The X-axis represents the likelihood of event p, the Y-axis the likelihood of event q, and the Z-axis the magnitude of syntropy. The light green surface corresponds to the left side of the inequality $e^{pq} - 1$, while the purple surface represents the right side $(e^p - 1)(e^q - 1)$. The overlapping of the purple surface over the light green highlights the regions where the supramultiplicativity holds true.
• Figure 4: This 3D visualization graphically represents the supraadditive inequality $e^{p+q} \geq e^{p} + e^{q} - 1$. The X and Y axes represent the likelihoods of events p and q, respectively, within the domain $[0, 1]$. The green surface illustrates the left side of the inequality, $e^{p+q}$, while the purple surface depicts the right side, $e^{p} + e^{q} - 1$. The overlay of these surfaces visually demonstrates the regions where the inequality holds, emphasizing the supraadditive nature of this exponential function in the context of probabilistic events.
• Figure 5: Complementary Symmetry of the Syntropy Function $S_N(x)$. The plot illustrates $S_N(x)$ as a solid line and $S_N(1-x)$ as a dashed line. The vertical gray dashed line at $x = 0.5$ serves as a reference to demonstrate the symmetry. The overlap and mirroring of the two curves around $x = 0.5$ highlight the complementary symmetry property, indicating that the values of $S_N(x)$ at $x$ and $1-x$ are equal for any probability mass function (pmf) $X$.

Theorems & Definitions (9)

• Lemma 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Proposition 1
• Proposition 2