DeGroot Model on Distributed Systems

• The DeGroot model is a foundational framework for distributed consensus, where agents iteratively update opinions using weighted averages from neighbors.
• Extensions addressing asynchronous updates, network heterogeneity, and adversarial influences enhance its practical reliability in real-world distributed systems.
• Innovations such as finite-time convergence and global feedback mechanisms offer scalable solutions for robust and efficient multi-agent consensus.

The DeGroot model is a foundational framework for understanding distributed consensus and opinion dynamics, where agents iteratively update their opinions according to the weighted average of their neighbors' states. In distributed systems, variants and extensions of the DeGroot model have been developed to address practical, structural, and behavioral factors such as asynchronous updates, network heterogeneity, stubborn agents, random actions, finite-time convergence, global feedback mechanisms, message influence, homophily, and algorithmic observability. The following sections present detailed technical perspectives and advances on the DeGroot model in distributed systems, based exclusively on peer-reviewed and preprint research.

1. Heterogeneous Dynamics: Conformists, Rebels, and Spectral Moderation

The classical DeGroot model prescribes each agent $j$ to update its opinion $x_j(t)$ using the rule: $$x_j(t+1) = \lambda x_j(t) + (1-\lambda)\sum_k A_{jk}x_k(t)$$ where $A$ is stochastic ($A_{jj}=0$, row sums $=1$), and $\lambda\in [0,1]$ is the agent's confidence. In a heterogeneous extension, agents may be "rebels," defined by $u_j=0$, who update: $$x_j(t+1) = \lambda x_j(t) + (1-\lambda)\left[1 - \sum_k A_{jk}x_k(t)\right]$$ Collectively, with $U=\mathrm{diag}(u_1,\ldots,u_n)$, the coupled update is: $$x(t+1) = \lambda x(t) + (1-\lambda)\left[ UA x(t) + (I - U)(\mathbf{1} - Ax(t)) \right]$$ A central finding is the moderating effect of rebels: even a small proportion "steers" the system to the "doctrine of the mean." Provided at least one rebel exists per strongly connected component and mild spectral conditions (e.g., $-1\notin \sigma(A)$ for all-rebel, $1\notin\sigma((2U-I)A)$ for mixed types), all opinions converge to $0.5$ regardless of initial conditions. The stabilizing effect is robust, with exponential convergence rate determined by $|2\lambda-1|$ and the spectral properties of the corresponding update matrix. The result is significant for designing distributed protocols with contrarian or error-correcting agents, ensuring system-wide moderation (Cao et al., 2012).

2. Asynchronous Dynamics and Fairness in Update Scheduling

Asynchronous variants of DeGroot dynamics account for random or otherwise non-simultaneous state updates. In the Poisson clock-driven model, each node $v$ updates at time $t$ to the average $f_t(v)=\sum_u P_{vu}f_{t-}(u)$ upon its clock's ring, where $P$ encodes network structure (typically a Markov chain or graph Laplacian). The expected consensus time can scale polynomially or polylogarithmically in the number of nodes, depending on the structure (e.g., undirected graphs or Eulerian digraphs with i.i.d. initial conditions reach $\varepsilon$-consensus in polylog$(n)$ time). The limiting consensus is a random weighted sum of initial opinions, with variance tightly estimated by graph parameters and initial distribution (Elboim et al., 2022).

A formal treatment of update sequencing introduces the framework of Opinion Transition Systems (OTS), where fairness conditions are critical. Standard strong fairness (every edge scheduled infinitely often) does not guarantee consensus; "m-bounded fairness"—requiring every run segment of $m$ consecutive complete windows—does. Consensus is achieved if the influence graph is strongly connected, influence weights per edge in $(0,1)$ (puppet-free), and the run is m-bounded fair with $m\geq n-1$. For time-varying (dynamic) influence weights, consensus holds provided all influences remain within fixed bounds $[L,U]$ with $0Aranda et al., 2023).

3. Robustness to Adversaries, Noise, and Discrete Updates

The DeGroot model's vulnerability to stubborn or adversarial agents has prompted robust variants. The $1/m$-DeGroot model restricts opinion values to the discrete grid $\{k/m:k\in \mathbb{Z}\}$, where opinions are rounded to the nearest such value post-update, with ties favoring the current opinion. This granularity confines the adversarial impact to a finite radius—outside this radius, consensus approximates the true state robustly, even in infinite or irregular networks. The model is Markovian, stationary, and Lyapunov-contractive, ensuring energy decrease and stability (Amir et al., 2021).

In the presence of stubborn agents and noise, optimal selection of observation nodes is essential for estimation problems. Given opinions $X_i$ evolving per noisy DeGroot dynamics with some agents fixed,

$X(t+1) = A X(t) + B u + V(t+1)$

the variance reduction achieved by observing a subset $K$ for estimating the mean opinion is proven submodular: $$F(K) = (\mathbf{C}1)_K^T (\mathbf{C}_{KK})^{-1} (\mathbf{C}1)_K$$ where $\mathbf{C}$ is the equilibrium covariance. This enables a Greedy Algorithm providing a $(1-1/e)$ approximation to the optimal set, validated on synthetic and empirical networks (Raineri et al., 11 Apr 2025).

4. Finite-Time Consensus: The $G$ Method and Grouped Matrix Products

Standard DeGroot consensus is typically asymptotic, but the "G method" provides finite-time convergence by organizing update matrices into partitions and factors, producing a grouped matrix (using "−+" superscripts) whose stability implies consensus. For products $P_1P_2\dots P_t$ arranged via appropriate chains of partitions $(\Delta_1,\ldots,\Delta_{t+1})$, if the final partition is the trivial (one-block) partition $\Delta_{t+1}=(\langle r\rangle)$, grouped products yield: $$(P_1P_2\dots P_t)^K = e^T(P_1^{-+}P_2^{-+}\dots P_t^{-+})$$ guaranteeing consensus on $K$. For distributed averaging in graphs with $2^m$ vertices and a spanning $m$-cube subgraph, averaging occurs in $m$ steps: $$W_m W_{m-1} \dots W_1 = e^T \left(\frac{1}{2^m},\dots,\frac{1}{2^m}\right)$$ This result generalizes to graphs with $n_1 n_2 \dots n_t$ vertices under analogous structural conditions. Subgroup properties allow hierarchical consensus in modular networks (Păun, 20 Oct 2025).

5. Dynamic Network Structure and Homophily

In dynamic networks where each time-step's interaction matrix $X_t$ is stochastic and ergodic, opinions evolve as: $$p^{(t)} = X^{(t)}p^{(0)}, \qquad X^{(t)} = X_t X_{t-1} \dots X_1$$ Consensus (or weak ergodicity) holds if products $X^{(t)}$ eventually become strictly positive, yielding asymptotic beliefs: $p_i^{(\infty)} = \sum_{j=1}^n \pi_j p_j^{(0)}$ Collective intelligence requires that maximal influence vanishes as $n$ grows, $\max_{i\leq n}\mathbb{E}[\pi_i]\to 0$. In sparse networks, randomness bridges isolated nodes and accelerates mixing; in well-connected networks, added randomness may slow learning.

The initial topology ("skeleton") and overlap ("homophily") of agents' models and information sets define mutual influence and rate of belief convergence. Agents $i$ allocate self-weight $T_{ii}$, with off-diagonal weights $T_{ij}$ proportional to model overlap, and degree of homophily is computed by: $$h_{ij} = \frac{|(M_i^*\setminus I_i)\cap I_j|}{\sum_{k\neq i}|(M_i^*\setminus I_i)\cap I_k|}$$ These design choices significantly affect consensus robustness and speed (Mudekereza, 18 Feb 2025).

6. Global Feedback, Messages, and Extended Models

Extensions incorporating global feedback (e.g., GSM-DeGroot) and external message processes (e.g., Message-Enhanced DeGroot) model real-world factors such as mass media or trending topics. In GSM-DeGroot, opinions are driven by both local neighbor averaging and a global steering term based on the aggregate of agents’ event-driven states: $X_{i,t+1} = \beta_i g(S_t) + \sum_j w_{ji}X_{j,t}$ where $g(S_t)$ reflects the network-wide state, and $\beta_i$ modulates reaction (positive or negative).

Message-Enhanced DeGroot models incorporate stochastic messages, evolving as bounded Brownian motion subject to absorbing bounds. The opinion dynamics are: $\dot{o}_t = (\alpha W - I) o_t + (1-\alpha) U s_t$ with $W$ (influence among agents), $U$ (valuation of sources), and $\alpha$ (relative weight). Long-term mean opinion converges to the mean of message distribution, with asymptotic variance: $$\lim_{t\to \infty} \mathrm{D}(o_t) = \mu(1-\mu)(1-\alpha)^2 \, \mathrm{diag}\left[(\alpha W - I)^{-1}U U^T(\alpha W^T - I)^{-1}\right]$$ These models are empirically validated in social media scenarios and suggest that accounting for global signals and exogenous stochastic processes is vital in distributed consensus and decision-making (Conjeaud et al., 2022, Wang et al., 29 Feb 2024).

7. Modified Self-Confidence and Reflected Appraisal in Distributed Updates

The Modified DeGroot–Friedkin model updates agents’ self-confidence (“social power”) via local neighborhood interactions, permitting finite-step updates without global consensus per issue. With a doubly stochastic interaction matrix $C$, self-confidence $x_i(s)$ evolves as: $$x_i(s+1) = x_i^2(s) + \sum_j (1 - x_j(s)) c_{ji} x_j(s)$$ or in matrix form,

$x(s+1) = C^T x(s) + X(s)x(s) - C^T X(s)x(s)$

The unique nontrivial equilibrium (for $n\geq 3$) is the democratic state $x^*=(1/n)\mathbf{1}$, achieved under mild conditions without centralized computation. This distributed update mechanism is robust and scalable but currently lacks full theoretical treatment for general non-doubly-stochastic cases (Xu et al., 2015, Ye et al., 2017).

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Each aspect above demonstrates technical innovation in extending, stabilizing, and leveraging the DeGroot model for robust, scalable consensus in distributed systems, addressing heterogeneity, stochasticity, topology, fairness, generalization, and practical observability as central concerns in modern networked applications.