Selective Neighborhood Contraction in Networks

• Selective neighborhood contraction is a framework for selectively reducing local connectivity in networked systems and nonlinear control to achieve consensus and safety.
• Models like asynchronous averaging and input-selective contraction in excitable systems illustrate its effectiveness, even with local edge removals or dynamic neighbor updates.
• Theoretical guarantees leverage martingale methods, blockwise contraction metrics, and average dwell-time analysis to secure global stability amidst nonuniform dynamics.

Selective neighborhood contraction refers to a class of mechanisms and analytical frameworks in networked dynamical systems and nonlinear control theory characterized by the selective reduction of “neighborhoods” or coupling sets, typically in a state- or agent-dependent, nonuniform, and potentially endogenous fashion. This paradigm appears in models ranging from asynchronous averaging on dynamic graphs to input-dependent contraction in excitable systems and componentwise reachability analyses of nonlinear ODEs. The central theme is the selective restriction or contraction of interaction domains—whether discrete neighborhoods in a time-varying network or metric contraction regions in state space—addressed with the goal of consensus, reliability, or scalable safety analysis.

1. Asynchronous Averaging Models and Selective Neighborhood Contraction

In discrete-time consensus models on dynamic graphs, selective neighborhood contraction is instantiated as an endogenous reduction in the updated agent’s local connectivity. At each time $t$, a state vector $x(t) \in \mathbb{R}^n$ evolves according to an asynchronous averaging dynamic: an agent $i$ is selected with probability $p_i$, its state is replaced by the average of its current neighbors $N_i(t)$, and its neighborhood may then contract—i.e., $N_i(t+1) \subseteq N_i(t)$ according to a possibly random rule $f_i$ that deletes edges incident to $i$. Non-selected agents may add or remove edges independently without monotonicity constraints. This structure yields a time-varying communication graph $G_t$ with endogenous and exogenous updates, generalizing classic models with fixed or globally changing topology (Li, 25 Dec 2025).

The key property is that local contraction (neighborhoods shrinking under specified rules only for the updated node) does not preclude network consensus, provided the evolving graph satisfies recurrent global connectivity constraints. No a priori lower bound on minimum neighborhood size is required, and contraction events at individual nodes are permitted without global monotonicity in edge sets.

2. Infinitely-Often Connectivity: Structural Assumptions and Dynamics

The central structural assumption enabling convergence in the face of selective contraction is “infinitely-often connectivity.” Formally, the sequence $G_0, G_1, \ldots$ is assumed to satisfy: there exists a sequence of times $t_1, t_2, \ldots$ and a fixed window $T$ such that the union of the graphs over $[t_k, t_k + T]$ is connected for each $k$. This allows extensive local neighborhood reduction (so long as non-selected nodes’ rewiring or exogenous mechanisms maintain union connectivity on a suitable timescale).

The model thus supports a highly nonuniform and asynchronous contraction process at the node level, robust to arbitrary exogenous addition, removal, and rewiring of other neighborhoods. This result extends previous consensus theories that required stronger uniformity or monotonicity in connectivity evolution (Li, 25 Dec 2025).

3. Theoretical Guarantees: Martingale Arguments and Consensus

The almost-sure consensus result established in (Li, 25 Dec 2025) pivots on the edge-disagreement supermartingale: $Z_t = \sum_{(i,j) \in E_t} (x_i(t) - x_j(t))^2$ and the observation that after each asynchronous update,

$\mathbb{E}[Z_{t+1} \mid \mathcal{F}_t] \leq Z_t$

while the decrease $$Z_t - Z_{t+1} \geq 2 \sum_{k=1}^n (x_k(t)-x_k(t+1))^2$$ guarantees non-increase of edge disagreement and controls the total motion via telescoping sums. Each agent’s trajectory $x_i(t)$ is almost surely Cauchy, and recurrent union connectivity forces all state differences $|x_i(t) - x_j(t)|$ to converge to zero as $t \to \infty$. The contraction operates locally and selectively, but the global structure is maintained by recurring intervals of global connectivity, ensuring diffusion of information and joint agreement.

No explicit exponential rate is given, but convergence is almost sure; selective neighborhood contraction is compatible with robust global agreement mechanisms under the stated connectivity regime.

4. Input-Selective and Componentwise Contraction Perspectives

A related but non-network instance appears in excitable system theory, where contraction is not global but exists only in selected (input-dependent or “neighborhood”) regions of the state space. In (Bin et al., 4 Nov 2025), input-selective contraction is defined for nonautonomous ODEs: for a metric $d(x_a, x_b) = \sqrt{(x_a - x_b)^T P (x_a - x_b)}$, contraction with rate $\lambda > 0$ holds whenever two states remain in a convex, possibly input-dependent region $C(u)$: $$J(x,u)^T P + P J(x,u) \leq -2\lambda P, \quad \forall x \in C(u)$$ This selective contraction property supports strong convergence only when trajectories enter contraction regions, while expansion is possible elsewhere. An average dwell-time argument quantifies overall contractive behavior based on the time spent in such neighborhoods, yielding explicit finite-time trajectory convergence data for excitable models such as the FitzHugh–Nagumo system, and forming the basis of reliable neuronal computation and robust regulation in small excitable networks (Bin et al., 4 Nov 2025).

5. Blockwise and Simulation-Based Selective Contraction in Nonlinear Reachability

In high-dimensional nonlinear systems, selective neighborhood contraction is operationalized via componentwise contraction properties. Given a system $\dot{x}(t) = f(t, x(t))$, the state is partitioned into $k$ components, and the Jacobian $J$ is represented in block form. Matrix measures $\mu$ associated with (possibly distinct) component norms yield blockwise contraction rates $\mu_{p_i}(J_{ii})$ and mixed-norm coupling bounds $\|J_{ij}\|_{p_i,p_j}$. A comparison matrix $C$ summarizes contraction/expansion along and between blocks: $$C_{ii} \geq \mu_{p_i}(J_{ii}), \quad C_{ij} \geq \|J_{ij}\|_{p_i,p_j}, \ i \neq j$$ Differential inequalities, propagated via a parameterized sensitivity equation, enable simulation-based reachability bounds: $|x_i(T) - z_i(T)| \leq [e^{C T} \epsilon]_i$ Automatic optimization over weighted norms (using LMIs and SDPs) enables tight bounding of contraction rates within each component, supporting scalable conservative analysis in the presence of parameter uncertainty (Arcak et al., 2017). The selective aspect lies in exploiting contraction properties that are valid only in certain directions or blocks, and dynamically updating the neighborhood of the state (i.e., blockwise balls or tubes) in accordance with these properties.

6. Illustrative Simulations and Practical Implications

Numerical experiments in (Li, 25 Dec 2025) highlight the practical impact of selective neighborhood contraction in asynchronous averaging on dynamic graphs. For $n=100$ agents on an Erdős–Rényi $G(n,p)$ graph seeded for connectivity, both the nodewise contraction parameter $q_\mathrm{shrink}$ and the exogenous rewiring parameter $q_\mathrm{flip}$ are varied. Increased contraction probability $q_\mathrm{shrink}$ delays consensus, especially when network density $p$ is low, but moderate exogenous rewiring $q_\mathrm{flip}$ restores connectivity and accelerates convergence. These results confirm that local selective contraction delays but does not prevent agreement, as long as global recurrent connectivity is maintained. A plausible implication is the design of consensus protocols robust to aggressive local edge deletion, so long as auxiliary mechanisms ensure intermittent global connectivity.

Model/Setting	Selectivity Mechanism	Key Result/Guarantee
Asynchronous Averaging (Li, 25 Dec 2025)	Node-selective neighborhood contraction after update	A.s. consensus under infinitely-often connectivity
Excitable Networks (Bin et al., 4 Nov 2025)	Input-/state-selective contraction regions	Reliability and robust regulation from average contraction
Nonlinear Reachability (Arcak et al., 2017)	Component-/blockwise contraction metrics	Scalable, componentwise reachability bounds

7. Connections and Distinctions Across Formulations

Across these domains, selective neighborhood contraction abstracts the notion of nonuniform, state-/agent-/input-dependent contraction mechanisms. In networked systems, contraction acts on discrete neighborhoods and is governed by communication rules; in excitable ODEs, contraction holds only for states traversing specific regions parametrized by input; in reachability analysis, contraction is measured per state component, shaped by the system’s modularity or uncertainty structure.

A salient distinction is between endogenous and exogenous selectivity: in (Li, 25 Dec 2025), only the updated node contracts its neighborhood, while others evolve exogenously; in (Bin et al., 4 Nov 2025), the contraction region is determined by both the trajectory and the input; in (Arcak et al., 2017), block-partitioning is selected for analytical tightness and scalability. Selective contraction thus supports flexible design and analysis of systems with locally or directionally varying stability properties, extending classical theory centered on global contraction and uniform network structures.

Selective neighborhood contraction, in all these settings, yields new analytical tools and guarantees for agreement, reliability, and safety in systems that are modular, time-varying, uncertain, or excitable, provided that adequate global or averaged structural conditions are enforced.