Triple of Binary Relations

• Triple of binary relations is a configuration of three distinct binary relationships that capture higher-order interactions in systems like NLP and ecosystem modeling.
• The approach utilizes cascade binary tagging to extract (s, r, o) triples, effectively addressing entity overlap and enhancing inference accuracy.
• It also models dynamic triple-helix ecosystems using coupled Lotka–Volterra equations, revealing oscillatory behaviors and structural redundancies.

A triple of binary relations refers to a structured configuration in which three binary relations define or mediate the interactions among three distinct types of entities or elements. This mathematical arrangement is fundamental in diverse fields such as natural language processing (for relational information extraction) and the modeling of complex social or informational ecosystems (such as University-Industry-Government structures). The triple configuration imposes specific structural and dynamic properties on the set of relations, with repercussions for both inference and redundancy in their respective domains.

1. Formal Characterization of Triples of Binary Relations

A triple of binary relations $\mathcal{T} = (R_1, R_2, R_3)$ consists of three binary relations among elements drawn from possibly overlapping sets. For example, in the context of knowledge extraction from natural language, a triple is an ordered tuple $(s, r, o)$ in which $r$ acts as a binary relation between subject $s$ and object $o$—a basic construct of knowledge graphs and information extraction frameworks (Wei et al., 2019). In the context of ecosystem modeling (triple-helix models), the triple of relations captures the bilateral channels among three agents—universities (U), industries (I), and governments (G)—with the set of binary relations being $(\text{U–I}, \text{U–G}, \text{I–G})$ (Ivanova et al., 2013).

The defining feature of such triples is the entailed mutual contingency and potential for overlapping: entities can participate in multiple distinct relations, and the structure reflects more than just pairwise association—it encodes configurational properties and higher-order dependencies.

2. Binary Relational Extraction: Cascade Tagging and Factorization

In information extraction, a fundamental problem is the extraction of all $(s, r, o)$ triples from unstructured text, especially in the presence of overlap and ambiguity, where multiple triples can share the same entities. The CasRel framework recasts the triple extraction task by treating each relation $r$ as a function $f_r$ mapping subject spans $s$ to their corresponding object spans $o$, rather than treating relations as discrete class labels over ordered pairs. This decomposition is formalized as:

$$P((s,r,o)\mid x) = P(s \mid x) \cdot P_r(o \mid s, x) \cdot \prod_{r' \neq r} P_{r'}(o_\emptyset \mid s, x)$$

Here, $P(s \mid x)$ is the probability of a subject span in sentence $x$, $P_r(o \mid s, x)$ is the probability that object $o$ is associated with subject $s$ under relation $r$, and for all other relations $r'$, $P_{r'}(o_\emptyset \mid s, x)$ corresponds to the "null" (i.e., no object) event (Wei et al., 2019).

The inference proceeds in a cascade: first extract subjects using binary sequence taggers over contextual embeddings, then, conditioned on each subject, extract objects for each relation with relation-specific binary taggers. This approach allows natural support for overlapping and entangled relational triples and achieves state-of-the-art extraction F1 scores, including robust performance on datasets with heavy overlap (Wei et al., 2019).

3. Modeling Triple-Helix Ecosystems: Lotka–Volterra Formalism

The Triple Helix (TH) approach conceptualizes the structure of university–industry–government relations as three intertwined binary channels (U–I, U–G, I–G). These are modeled via two time-dependent three-dimensional vectors $P(t)$ (sending) and $Q(t)$ (receiving), where the components index the three binary relations.

The state evolution of $P$ and $Q$ is governed by a pair of coupled vector Lotka–Volterra equations:

$$\dot{P} = -2g (P \times Q),\qquad \dot{Q} = +2g (P \times Q)$$

with $g$ the coupling strength and $(P \times Q)$ the vector cross product. The dynamics drive the three binary relationships in oscillatory, coupled trajectories, corresponding to periodic reconfigurations of relational dominance and interaction (Ivanova et al., 2013).

Conserved quantities under this formulation include the squared norms $\|P(t)\|^2$ and $\|Q(t)\|^2$, and the difference in squared norms governs the system's configurational redundancy.

4. Redundancy, Decomposition, and Frequency Analysis

Within the triple-helix framework, instantaneous redundancy for each channel $i = 1,2,3$ is defined as:

$R_i(t) = P_i(t)^2 - Q_i(t)^2$

Marginalizing over the channels yields the total redundancy $R(t)$, which in the noise-free model remains constant, while the partial redundancies $R_i(t)$ oscillate with frequencies determined by initial conditions and the interaction strength $g$. The sum $R(t)$ is proportional to the scalar mutual redundancy expressed informationally as:

$$R_{UIG} = H_U + H_I + H_G - H_{UI} - H_{UG} - H_{IG} + H_{UIG}$$

where $H_\cdot$ are the Shannon-type entropies for marginals and joint distributions (Ivanova et al., 2013). Fourier decomposition of the time series $R_i(t)$ enables analysis of the dominant frequencies in each subsystem, elucidating the tempo and qualitative nature of restructuring or synergy generation in the tripartite system.

5. Empirical Applications and Inference Protocols

Information Extraction from Text

In relational extraction, the application of a cascade of binary taggers (as realized in CasRel) enables robust mining of all valid relational triples $(s, r, o)$ from unstructured corpora—even under entity and relation overlaps. The framework is implemented atop deep contextual encoders (BERT), with binary classifiers for start/end positions of subjects and, conditioned on subject spans, objects. This allows direct, thresholded extraction of triples without explicit search over all possible subject-object pairs, leading to scalable, high-fidelity knowledge base construction (Wei et al., 2019).

Analysis of Triple-Helix Dynamics

For empirical evaluation of the triple of binary relations among U, I, and G, time-series bibliometric data are encoded in quadruple address tags (U, I, G, F), with Shannon-type entropies used to compute mutual redundancies within all triads. Fourier analysis reveals dominant cycles (e.g., high-frequency oscillations in U–I–G triads, slower ones in I–G–F), reflecting underlying systemic temporalities. Oscillation frequencies and amplitudes extracted from $R_i(t)$ illuminate the independence and interplay of interaction strength and frequency, supporting nuanced examination of system symmetries, asymmetries, and response dynamics under external or internal perturbations (Ivanova et al., 2013).

6. Structural, Dynamical, and Inferential Properties

Triples of binary relations, across domains, impose higher-order structure and interactions not present in pairwise-only configurations. In information extraction, factorization of triple-level probabilities into cascaded binary decisions allows for overlapping and non-mutually exclusive relationship structures to be inferred efficiently. In TH modeling, the explicit vectorial representation and coupled evolutionary equations endow the system with oscillatory dynamics, conserved redundancy, and compositional decomposition—facilitating both analytic solution and empirical spectral analysis.

A plausible implication is that systems or algorithms explicitly designed around the triple structure are more robust to overlaps, capable of decomposing synergistic or redundant interaction patterns, and better suited for empirical investigation of multifaceted relational phenomena.