Activation Path Overlaps in Complex Systems

• Activation path overlaps are the simultaneous, intersecting activation channels found in various systems such as networks, neural circuits, and dynamical models that shape connectivity and information flow.
• In network models, overlapping communities boost connectivity through reinforced edge probabilities, while in dynamical systems competing activation routes alter transition kinetics.
• Empirical and mathematical models, including the Community-Affiliation Graph Model and stochastic Markov chains, quantify these overlaps to optimize modular designs and improve memory retrieval.

Activation path overlaps describe the phenomenon wherein multiple structural or dynamical pathways for activation—whether of nodes in a network, states in a dynamical system, or units in a neural architecture—simultaneously interact, intersect, and reinforce one another. This concept arises both in network science, where overlapping communities affect the diffusion of activation, and in dynamical or stochastic systems, where activation may follow more than one concurrent route. The behavior, modeling, and empirical consequences of activation path overlaps have deep implications for the connectivity and dynamics of complex systems, influencing processes such as information flow, memory retrieval, and systemic transitions.

1. Overlapping Structures in Networks and Their Mathematical Modeling

A central observation in real-world networks is that community overlaps—regions where sets of nodes belong to multiple communities—are characterized not by sparse connectivity but rather by high edge density (Yang et al., 2012). Contrary to the prevailing "strength of weak ties" paradigm, empirical investigations across large social and information networks show that nodes sharing multiple community memberships exhibit dramatically increased linkage probabilities. The edge probability between two nodes $u$ and $v$ sharing communities $C_{uv}$ is

$p(u,v) = 1 - \prod_{k \in C_{uv}} (1 - p_k),$

where $p_k$ is the edge probability parameter for community $k$. Each shared community augments the probability of connection, resulting in activation path overlaps—multiple overlapping channels or group affiliations that reinforce the activation.

The Community-Affiliation Graph Model (AGM) formalizes this structure by representing affiliation links as a bipartite graph and stacking per-community chances of edge formation. If $p_c \propto n_c^{-\beta}$ for community size $n_c$ and $0 < \beta < 1$, AGM reproduces empirical densification power laws, with the number of intra-community edges increasing super-linearly with size.

2. Dynamical Systems: Competing and Mixed Activation Pathways

In stochastic dynamical systems, activation trajectories may overlap due to multiple candidate transition routes. For example, in a system of two coupled bistable oscillators (Chen et al., 2014), activation processes exhibit both two-step and one-step pathways, with their nature contingent on coupling strength $\mu$ and force mismatch $\sigma$. For identical subsystems ($\sigma=0$), two parallel routes (each traversing two energy barriers) exist at low coupling, transitioning to a single one-step pathway at high $\mu$. In the presence of mismatch ($\sigma \neq 0$), a region emerges where one path remains two-step, the other becomes one-step, and the transition process is governed by overlapped pathways.

Transition rates, calculated by weighted sums over candidate routes,

$R = p^{(1)} R^{(1)} + p^{(2)} R^{(2)},$

exhibit nonmonotonic dependence on both $\mu$ and $\sigma$, with activation path overlaps modifying the accessibility and kinetics of systemic transitions.

3. Path Overlaps in Excitable Systems Driven by Stochastic Inputs

In excitable units modeled by noisy FitzHugh–Nagumo equations (Franović et al., 2015), the most probable activation paths (MPAPs) cluster stochastic trajectories leading from quiescent to spiking states. The specification of activation events at the spiking branch enforces physically relevant boundary conditions. When two units are coupled (linearly or via nonlinear threshold-like interaction), MPAPs can overlap extensively, producing correlated activation events. With linear coupling, activation is highly synchronous; with nonlinear coupling, variations in MPAP topology arise. The overlap of activation paths dictates collective behavior and can synchronize or decorrelate excitation, with noise intensities $D_1$, $D_2$ shaping the universal statistics and bifurcation transitions between stable and oscillatory regimes.

4. Activation Path Overlaps in Percolation and Particle Dynamics

Interacting particle systems subject to critical bond percolation exhibit path overlaps through the activation of spatial clusters (Junge, 2019). Dormant particles (e.g., sleeping frogs) at network vertices are clustered by percolation geometry, and an active particle activates all particles within a cluster upon contact. The coupling to critical first passage percolation (FPP) controls the timing and extent of activations:

<table> <tr><th>Structure</th><th>Activation Overlap Mechanism</th><th>Consequence</th></tr> <tr><td>Critical Percolation Clusters</td><td>Multiple activation events latch onto the same cluster</td><td>Rapid, explosive spread</td></tr> <tr><td>Regular Trees</td><td>Independence between subtrees</td><td>Guaranteed explosion</td></tr> <tr><td>Euclidean Lattices</td><td>Spatial correlation via percolation</td><td>Superlinear expansion</td></tr> </table>

Activation overlaps arise as clusters can be repeatedly reached by different active particles, and activation jumps exhibit heavy-tailed distributions. For specific graph topologies and particle multiplicities, the process can "explode," activating infinitely many vertices in finite time, with the superlinear scaling ($M_t/t \to \infty$) highlighting the impact of overlapping activation routes.

5. Activation Path Overlaps in Neural Complexity and Network Modularization

Neural network architectures manifest activation path overlaps through compositional and path-based complexity measures (Li et al., 2020, Ngo et al., 1 Nov 2024). Path-based norms,

$$\|\theta\|_p = \sum_{k=1}^m |a_k|(\|b_k\|_1 + |c_k|),$$

and their extensions to general activations and deep residual networks encapsulate the total "weight" traversed by every input–output path. When general activations are approximated by ReLU subnetworks, the resulting modified norm aggregates contributions from both full-length paths and "short" or partial paths starting at intermediate layers, capturing all overlap effects.

Activation-driven modular training directly regulates layer activations so that class-specific modules require minimal neuron overlap: intra-class affinity, inter-class dispersion, and compactness objectives force distinct activation pathways for different classes, reducing weight overlap by up to 3.5× compared to mask-based modularization (Ngo et al., 1 Nov 2024). This decrease in activation path overlap enables efficient module extraction, replacement, and reusability in large neural models, optimizing both resource and adaptation efficiency.

6. Biological Systems: Activation Pathway Competition and Specificity

In immunological models of T cell activation, stochastic binding produces overlapping Markov chains corresponding to productive (cognate APC) and nonproductive (non-cognate APC) engagement (Wong et al., 2 Oct 2025). The kinetic scheme features shared initial binding states but diverges into distinct terminal outcomes, with the backward transitions enabling resets (kinetic proofreading). The overlap of these activation paths is central to both protection from degradation (via nonproductive interactions) and the specificity of activation. The key measure—the mean first-passage time to activation and the overall activation probability—depends on the interplay between pathway overlaps and kinetic parameters (forward bias, reset rates).

7. Spurious Activation Path Overlaps in Associative Memory Networks

In Hopfield-type attractor networks, spurious overlaps between the network state and nonretrieved memories are conventionally viewed as detrimental noise. However, with learning rules inferred from neurobiological data, the cumulative effect of spurious overlaps induces a mean-reducing shift in synaptic currents (Benedetti et al., 20 Oct 2025). This shift sparsifies neural activity, effectively gating activation paths to favor retrieval of the target memory and suppress collateral activity:

$$S = \sum_{\alpha \neq \mu} \exp\left[-\frac{a\alpha}{Nc}\right] b \langle f(\tilde{r}) \rangle m^\alpha,$$

where $S$ is the spurious current and $f(\tilde{r})$ is negative on average, promoting sparsity. Paradoxically, this overlap augments memory capacity by increasing attractor separability and reducing interference.

Summary

Activation path overlaps, occurring in a diverse range of systems from networked communities and dynamical models to artificial and biological neural circuits, are a fundamental organizing principle of connectivity and dynamics. Mathematically grounded models—such as AGM, stochastic Markov chains, percolation processes, and path-norm complexity metrics—quantify the degree and consequences of these overlaps. Empirical studies consistently demonstrate that dense overlaps, rather than diffuse weak boundaries, underlie rapid activation propagation, increased memory capacity, and robust modularization. Understanding and controlling activation path overlaps is crucial for optimizing network interventions, designing scalable neural systems, and interpreting biological activation processes.