Interacting Vertex-Reinforced Random Walks

• Interacting vertex-reinforced random walks are stochastic processes that use cumulative self- and inter-walker histories to govern transitions, leading to varied phenomena like localization and segregation.
• The model integrates intrinsic weights and interaction parameters on different graph geometries to capture regimes ranging from cooperative behavior to strict competition.
• Methodological advances utilize stochastic approximation, Lyapunov functions, and fixed-point theory for explicit characterizations in complex network settings.

Interacting vertex-reinforced random walks (IVRRWs) constitute a class of stochastic processes in which the transition dynamics of multiple random walkers are governed not only by their own cumulative histories but also by interactions mediated through the collective occupation measures at each vertex. The defining feature is that the probability for a given walker to move to a vertex depends on a reinforcement mechanism that aggregates information from both self-visitation and visits by other interacting walks. IVRRWs generalize classic vertex-reinforced random walks by introducing interaction parameters that encode competition or cooperation between walkers, and provide a framework flexible enough to model a wide variety of dynamics ranging from localization, repulsion, attraction, to spatial segregation and sharing in networks.

1. General Model Architecture and Transition Rule

The canonical model addressed in recent work considers $m$ random walks, each confined to a complete subgraph $G^i = (V_i, E_i)$ of a locally finite undirected graph $G$, with $V = \bigcup_{i=1}^m V_i$. The configuration at time $n$ is described by a collection of empirical occupation measures $X_i(n) = (X^i_v(n))_{v \in V_i}$, giving the normalized frequency of visitation up to time $n$ for walker $i$ at vertex $v$.

For walk $i$ at time $n$, the transition probability to vertex $v \in V_i$, conditionally on the history, is given by

$$\pi^i_v(x) = \frac{x^i_v \, H^i_v(x)}{\sum_{w \in V_i} x^i_w \, H^i_w(x)},$$

where $x$ is the $(x^i_v)_{i, v}$ occupation profile, and the reinforcement function $H^i_v$ encodes both intrinsic attractiveness and interactions: $$H^i_v(x) = \left( \eta^i_v + \sum_{j \in I_v} \theta^{ij}_v x^j_v \right)^\alpha.$$ Here, $I_v = \{ j : v \in V_j \}$ is the set of walks with access to $v$, $\eta^i_v > 0$ is an intrinsic (baseline) weight for vertex $v$ for walk $i$, $\theta^{ij}_v \in \mathbb{R}$ is the interaction parameter encapsulating the influence of walk $j$ on the transition of walk $i$ to $v$, and $\alpha > 0$ is a reinforcement exponent. This structure encompasses a wide range of dynamical regimes by adjusting parameter values: e.g., repulsive ($\theta^{ij}_v < 0$) and cooperative ($\theta^{ij}_v > 0$) interactions.

2. Role of Parameters: Reinforcement, Interaction, and Geometry

Parameter Functions

• Intrinsic parameters $\eta^i_v$: Control baseline preferences of walk $i$ for vertex $v$.
• Interaction strengths $\theta^{ij}_v$: Determine the direction and magnitude of interaction—competition, neutrality, or cooperation—between walkers at vertex $v$. The symmetry $\theta^{ij}_v = \theta^{ji}_v$ is often assumed.
• Exponent $\alpha$: Regulates the "sensitivity" of reinforcement to cumulative visitation; $\alpha = 1$ is linear, $\alpha > 1$ is super-linear, and $\alpha < 1$ is sub-linear.
• Graph geometry: The underlying support of each walk $V_i$ and the patterns of overlap between them (complete graphs, stars, cycles) fundamentally affect the system’s limiting behavior.

The transition dynamics encode memory (self-reinforcement) and inter-dependence between walks (interaction) in a nonlinear, site-dependent fashion.

3. Convergence and Fixed Point Theory

The strong law for the empirical occupation vectors is established via stochastic approximation: $$X(n+1) - X(n) = \pi(X(n)) - X(n) + \text{martingale noise},$$ where $\pi$ is the transition mapping defined by the transition probabilities above. The process is shown to converge almost surely to the set of fixed points of $\pi$, i.e. solutions to $x = \pi(x)$.

A strict Lyapunov function is constructed, e.g.,

$$L(x) = -\sum_{i, v} \eta^i_v x^i_v - \frac{1}{2} \sum_{v} \sum_{i,j \in I_v} \theta^{ij}_v x^v_i x^v_j,$$

which decreases along trajectories except at fixed points. If for each possible support $S$ the linear system obtained from $x = \pi(x)$ is nonsingular, the set of fixed points is finite. This ensures that, for almost all choices of generic parameters, the occupation process converges almost surely to a unique equilibrium.

The limit set of the process is described by a finite union of connected sets, each characterized as the solution set of a linear system specified uniquely by sub-supports for each walk.

4. Limiting Regimes and Phase Structure: Geometry-Parameter Interplay

The structure of limiting points is determined by both interaction geometry and reinforcement parameters. For the competitive regime ($\theta^{ij}_v < 0$ for $i \neq j$), the following behaviors arise:

• Complete graphs: For two interacting walks and $\epsilon = 0$ (i.e., no self-penalization), each walk’s limiting support is disjoint from the other’s (zero overlap in the limit), while for $\epsilon > 0$ a unique solution with explicit limiting occupation proportions exists and includes partial overlap, driven by $\epsilon$ and the sizes of respective supports.
• Star graphs (bipartite $K_{1,m}$): When $\epsilon = 0$, only one walk occupies the center in the limit; for $0 < \epsilon < 1/2$, a subset of walks can share the central vertex, with explicit formulas connecting $\epsilon$ to the share; for $\epsilon > 1/2$, all walks share the center symmetrically.
• Cyclic graphs ($C_m$): The set of fixed points divides into four distinct families ($C_1$–$C_4$), classified by the "mixing" or segregation status of edge occupation. For $m$ not divisible by $4$, almost sure convergence falls into the families $C_3 \cup C_4$.

Table: Examples of Limiting Behavior

Geometry	Competitive Regime ($\epsilon = 0$)	With Strong Self-Interaction ($\epsilon > 1/2$)
Complete Graph	Disjoint supports (zero overlap)	Partial overlap, unique vector, explicit formula
Star Graph	Only one walk occupies center (others isolated)	All walks symmetrically share center
Cycle Graph	Segregated or periodic patterns in occupation	Categorical families ($C_1$–$C_4$), geometry-dependent

The complexity of the fixed-point set increases with the richness of overlaps and the choice of system geometry but remains generically discrete under nonsingularity conditions.

5. Comparison with Related Interaction Models

The IVRRW framework generalizes both classical VRRW and prior studies on multi-particle reinforced interacting walks (Chen, 2012), exponential interaction models (Pires et al., 2020), and competitive walk models on the integer lattice (Prado et al., 2020). Unlike models where the transition probability depends only on a walk's own visitation history or simple sum-based interactions, the present framework allows for intricate forms of reinforcement and competition determined by the interaction parameter matrices and vertex access sets.

Notably, when interaction parameters are positive (cooperation), walkers may asymptotically synchronize, resulting in occupation measures with maximal overlap. In the strong negative interaction (competition) regime, full spatial segregation is observed (cf. small joint support in (Chen, 2012)).

6. Methodological Innovations

The analysis makes essential use of stochastic approximation methods, Lyapunov function construction, and a fixed-point theory of the nonlinear mapping $\pi$. Asymptotic results leverage spectral analysis of associated Jacobians to determine stability and uniqueness of equilibria. The model is highly adaptable to different reinforcement exponents ($\alpha$ not restricted to unity), an advance over many earlier reinforcement schemes.

Furthermore, explicit formulae are provided for finite complete graphs and star/cycle configurations, allowing for a detailed phase description in applications.

7. Applications and Future Directions

Applications include but are not limited to:

• Modeling competitive resource partitioning in ecological contexts (each walk as a species, vertices as resource patches);
• Load balancing and segmentation in multicomponent distributed systems and network clustering (walks as agents or processes);
• Understanding segregation and coexistence phenomena in social or economic networks.

Potential future directions include analysis of non-generic parameter regimes yielding continuum sets of limiting measures, paper of critical transitions in the nature and number of equilibria as a function of geometry and interaction, and extension to time-dependent interaction parameters or reinforcement laws with different functional forms.

A plausible implication is that this general framework can inform algorithmic heuristics for fair division, competitive routing, or cooperative reinforcement learning in multi-agent systems, although direct numerical results for such applications would require further model specification.

The results discussed are synthesized from (Prado et al., 21 Aug 2025), which provides rigorous law of large numbers for the limiting occupation profile, explicit classification of fixed points in various geometric settings, and the analytical tools for spectral and dynamical paper necessary for modern work on IVRRWs. The developed theory incorporates and extends foundational results from prior works on VRRW and interacting reinforcement processes (Chen, 2012, Pires et al., 2020, Prado et al., 2020), firmly placing interacting vertex-reinforced random walks at the intersection of stochastic process theory, dynamical systems, and statistical mechanics.