Interaction-Driven Topologies

• Interaction-driven topologies are network structures defined by complex, multi-component interactions that extend beyond traditional pairwise models.
• They utilize tools such as interaction homology, persistent modules, and the interaction Laplacian to quantify and analyze adaptive, higher-order relationships.
• These methodologies underpin advances in materials science, neural networks, and multi-agent systems by offering actionable insights into dynamic network reconfiguration.

Interaction-driven topologies are network or space structures that are determined, modulated, or fundamentally reconfigured by the underlying pattern, intensity, or heterogeneity of interactions between components, beyond what is captured by classical static or purely pairwise frameworks. This paradigm spans topological data analysis, dynamical collective systems, condensed matter, network science, and statistical learning, addressing domains where the emergence, adaptation, or robustness of structure is a direct consequence of the rules or geometry of interaction.

1. Mathematical Formalisms and Foundational Constructs

Central to interaction-driven topology is the generalization of network and topological invariants to capture the influence of specific interaction patterns, multi-entity or higher-order relationships, and dynamic processes.

Interaction Homology and Generalizations:

Given a family of spaces or subcomplexes $\{X_i\}$, interaction space theory introduces a covering $(X,\{X_i\})$, encoding the "diagram" of mutual overlaps. Homotopy, singular, and simplicial interaction homology are defined by quotienting the tensor product of the constituent chain complexes by the subcomplex generated by simple tensors whose images have empty intersection: $$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$ where $T_*$ is the subcomplex of non-intersecting image tensors. The resulting interaction homology is functorial and invariant under interaction homotopy, with Betti numbers $\beta_p(\{X_i\})$ encoding the multi-space intersection connectivity (Liu et al., 2023).

Interaction Vietoris–Rips Complexes and Persistent Modules:

For a metric space $X$ with distinguished subset $E\subset X$, the persistent interaction Vietoris–Rips complex at scale $\epsilon$ is

$$IR(X,E;\epsilon) = \{ R_X(\epsilon), R_E(\epsilon) \}$$

where $R_X(\epsilon)$ and $(X,\{X_i\})$0 are the classic Vietoris–Rips complexes on $(X,\{X_i\})$1 and $(X,\{X_i\})$2, respectively, and the simplicial structure is restricted to those simplices intersecting $(X,\{X_i\})$3 (Liu et al., 2024). Homology of these complexes generates a persistent module $(X,\{X_i\})$4, stable (via interleaving distance) under perturbations of the filtration parameter.

Interaction Laplacian:

For an interaction chain complex $(X,\{X_i\})$5, the associated Laplacian is

$(X,\{X_i\})$6

whose kernel corresponds to the interaction homology $(X,\{X_i\})$7 and whose spectral gap quantifies the rigidity or robustness of the feature. This provides a spectral invariant sensitive to element-specific or heterogeneously interactive subsystems.

Higher-Order Generalizations:

The IntComplex framework models $(X,\{X_i\})$8-interactions as recursively defined binary trees, supporting GLMY-style homology, $(X,\{X_i\})$9-layer homology (cycles among $$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$0-interactions), and multilayer homology (feedbacks mixing layers), all made persistent via filtration. This enables encoding and stable quantification of intricate multiway relationships not accessible to classical simplicial complexes or hypergraphs (Liu et al., 2024).

2. Dynamics, Adaptation, and Time-Varying Interaction Topologies

Interaction-driven topologies often emerge or evolve through dynamic processes operating on edges, nodes, or higher-order aggregations:

Adaptive and Time-Varying Networks:

In coevolving systems, such as the adaptive contact process, the balance between state dynamics (e.g., infection/recovery) and topology adaptation (e.g., rewiring upon contact state) leads to steady-state degree distributions, variance tuning, and sometimes transitions between high- and low-heterogeneity regimes, all parameterized by the relative interaction rates (Wieland et al., 2015). In networks subjected to bursty edge activation and renewal dynamics, heavy-tailed off-time distributions can collapse degree heterogeneity and fragment connectivity, with direct measurable impact on percolation, transport (random walk mixing times), and evolutionary processes (Zeng et al., 18 May 2025).

Collective Motion and Mixed-Interaction Regimes:

The 3D-KI model interpolates between metric (distance-limited) and topological (fixed-$$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$1-nearest-neighbor) interactions via a continuous parameter $$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$2 controlling the weighting of each scope in self-propelled particle alignment (Kikuchi et al., 13 Jul 2025). Intermediate $$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$3 values maximize global order robustness and enable the formation of resilient, well-aligned sub-flock clusters identified via cluster-level topology (HDBSCAN-based). Purely metric or topological regimes produce either single cohesive or highly fragmented groupings, but only mixed interaction designs exhibit both maximal global order and fluctuation robustness.

Spatially Embedded Higher-Order Interactions:

Triadic percolation, where links are modulated by third-party regulators (e.g., glia modulating synapses), generates time-dependent topologies unattainable by classic percolation. Local triadic interactions in spatial networks induce rich spatiotemporal phases—stripes, octopi, clusters—classified via persistent homology and entropy-complexity metrics (Millán et al., 2023). The dynamical route to chaos, coexistence of distinct topological patterns, and multistability underscore the fundamentally interaction-driven nature of the resultant topology.

3. Topological Transitions, Correlations, and Material Implications

Interaction-driven topology is a central concept in strongly correlated quantum materials and designed nanoscale assemblies.

Interaction-Driven Topological Phases in Quantum Materials:

Monolayer TaIrTe$$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$4 displays an array of phases—quantum spin Hall insulator (QSHI), trivial insulator, higher-order (HOTI), and metallic—realized by tuning filling near van Hove singularities, dielectric screening, and strain (Li et al., 23 Jun 2025). The interplay between electron-electron interactions and the band structure leads to interaction-induced phase boundaries, mapped via Hartree–Fock mean-field calculations. Topological invariants (Fu–Kane Z$$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$5, Z$$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$6, quadrupole moment) are used to diagnose phase character, and experimental transport signatures (quantized conductance, nonlocal voltage, corner states) confirm theoretical predictions. Phase boundaries correspond to interaction-driven parity inversions at specific Brillouin zone points, with prevalence and tunability controlled via external parameters.

Aggregation and Post-Processing in Programmable Materials:

Hierarchical networks assembled from the aggregation of cliques/simplexes with tunable defect probabilities can realize hyperbolic, highly clustered, or modular geometries (Tadic et al., 2019). Attachment rules governed by chemical affinity and defect compatibility generate desired topological features (connectivity, hyperbolicity metric $$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$7, Q-analysis structure vectors). Post-growth removal of defect edges provides an in situ means of reconfiguring for target properties (fragmenting cliques, introducing cycles, modifying connectivity or curvature), yielding "programmable" material topologies engineered via interaction design.

4. Quantification of Interactions in Complex Systems

Topological Detection of Statistical Interactions in Neural Networks:

Interaction-driven topology extends to detecting and quantifying functional interdependencies (statistical interactions) in trained neural networks. Persistent interaction detection (PID) leverages 0-dimensional persistent homology of network-weight graphs to score feature sets by the persistence of their connected subgraphs to network outputs across weight thresholds (Liu et al., 2020). The resulting measure, $$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$8, is stable under small weight perturbations, efficiently computable, and empirically outperforms classical or previous neural methods for interaction identification. This bridges topological data analysis and machine learning, providing a disciplined approach to extracting data-driven interaction structure from the inner topology of learned models.

Exclusive Topology Classification in High-Granularity Experiments:

In liquid argon TPC neutrino experiments, "exclusive topology" analysis defines event categories and cross-sections purely by observed final-state particle multiplicities, independent of model-driven channel definitions (Palamara et al., 2013). The resulting cross-sections are sensitive to underlying nuclear effects and correlations, giving model-agnostic, data-dominated quantification of interaction signatures relevant for precision oscillation and nuclear modeling.

5. Implications for Control, Robustness, and Cooperative Design

Iso-Connectivity and Isospectral Families in Cooperative Systems:

Graphs or interaction networks with identical algebraic connectivity ($$IS_*(\{X_i\}) = \left(\bigotimes_{i=1}^n S_*(X_i)\right)\big/\!T_*(\{X_i\}),$$9 of the Laplacian) but differing edge patterns (iso-connectivity, and, more stringently, isospectrality) provide a space of alternative, functionally equivalent configurations (Dutta et al., 2016). Analytical construction via Laplacian similarity transforms or permutation matrices realizes these families, supporting reconfiguration (agent mobility, obstacle avoidance) without loss of consensus speed or robustness. In dense graphs, a "zone of no connectivity change" exists, where nodes can shift position without altering global connectivity metrics, a feature with direct application to adaptive and resilient multi-agent network design.

Chimera States and Topology-Driven Dynamical Regimes:

The topology of interactions (local, nonlocal, global, modular, temporal, multilayer) shapes the emergence and stability of chimera states, where coherent and incoherent domains coexist in oscillatory media. In each regime, specific interaction structures (nonlocal rings, temporal switching, multiplex couplings) drive distinct symmetry-breaking, pattern-formation, and robustness properties. Theoretical and practical understanding remains sensitive to the detailed topology of interaction (Bera et al., 2017).

6. Generalizations, Open Problems, and Future Directions

Open Mathematical Directions:

Interaction-driven topologies motivate extensions to persistent interaction homology on filtrations, direct relations to intersection and nerve cohomologies, and the development of sheaf-theoretic or multi-graded generalizations (Liu et al., 2023). The stability of such invariants, their algorithmic tractability in high dimensions or layered complex systems, and their sensitivity to various interaction structures remain active topics.

Future Empirical and Engineering Opportunities:

Programmable interaction-driven topologies underpin emerging strategies in metamaterials, photonic devices, catalysis, and network control, where targeted modulation of connections (by type, time, context, or regulation) can dynamically shape system function. The interaction-based paradigm provides a rigorous framework for both quantification and design, with immediate relevance in heterogeneous molecular assemblies, complex adaptive networks, and artificial intelligence systems.

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References:

• (Liu et al., 2023) "Interaction homotopy and interaction homology"
• (Liu et al., 2024) "Persistent interaction topology in data analysis"
• (Liu et al., 2024) "IntComplex for high-order interactions"
• (Liu et al., 2020) "Towards Interaction Detection Using Topological Analysis on Neural Networks"
• (Kikuchi et al., 13 Jul 2025) "Intermediate Interaction Strategies for Collective Behavior"
• (Millán et al., 2023) "Triadic percolation induces dynamical topological patterns in higher-order networks"
• (Tadic et al., 2019) "Topology of nanonetworks grown by aggregation of simplexes with defects"
• (Li et al., 23 Jun 2025) "Interaction-Driven Topological Transitions in Monolayer TaIrTe$T_*$0"
• (Wieland et al., 2015) "Analytic description of adaptive network topologies in steady state"
• (Zeng et al., 18 May 2025) "Bursty Switching Dynamics Promotes the Collapse of Network Topologies"
• (Palamara et al., 2013) "Exclusive Topologies reconstruction in LAr-TPC experiments"
• (Dutta et al., 2016) "Role of Iso-connectivity Topologies in Multi-agent Interactions"
• (Bera et al., 2017) "Chimera states: Effects of different coupling topologies"
• (Agliari et al., 2010) "A Hebbian approach to complex network generation"