DeGroot Model: Linear Consensus in Networks

• DeGroot model is a discrete-time opinion dynamical system where agents update their views using convex combinations derived from a row-stochastic influence matrix.
• The model guarantees convergence to consensus under conditions like strong connectivity and aperiodicity, with convergence rates linked to spectral properties.
• Extensions introduce stubborn agents, bias, and alternative update schemes, illustrating the model’s versatility as a baseline for studying advanced consensus phenomena.

The DeGroot model is a discrete-time model of opinion dynamics in which agents update by weighted averaging over a network. In its classical form, for $n$ agents with opinion vector $x(t)\in\mathbb{R}^n$ and a row-stochastic influence matrix $W=[w_{ij}]$, the update rule is

$$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$

with $w_{ij}\ge 0$ and $\sum_{j=1}^n w_{ij}=1$ for every $i$. The model describes linear opinion averaging over a network and has become the baseline against which a large class of nonlinear, stochastic, constrained, and heterogeneous opinion models are defined (Xia et al., 2019, Gadducci et al., 24 Apr 2025).

1. Formal structure and baseline interpretation

In the classical formulation, agents are nodes of a directed influence graph, and an edge $j\to i$ is present when $w_{ij}>0$. The matrix $W$ is row-stochastic, self-weights $x(t)\in\mathbb{R}^n$0 may be present, and the update is synchronous. In graph-theoretic language, strong connectivity means every agent can reach every other through directed paths; irreducibility of $x(t)\in\mathbb{R}^n$1 is the matrix-theoretic equivalent. Aperiodicity is satisfied if the greatest common divisor of cycle lengths is $x(t)\in\mathbb{R}^n$2; a sufficient condition is $x(t)\in\mathbb{R}^n$3 for some $x(t)\in\mathbb{R}^n$4 (Gadducci et al., 24 Apr 2025).

The model is usually presented for scalar opinions, but the same averaging template appears in several equivalent notational forms. Some papers write $x(t)\in\mathbb{R}^n$5 with $x(t)\in\mathbb{R}^n$6 row-stochastic; others use directed graphs with incoming weights normalized per agent. A continuous-time analog also appears in the literature as

$x(t)\in\mathbb{R}^n$7

with solution $x(t)\in\mathbb{R}^n$8 (Wang et al., 2024).

What distinguishes the DeGroot model is not merely linearity, but the fact that each update is a convex combination of current opinions. This places the model at the core of linear consensus theory, Markov-chain-based social learning, and many subsequent generalizations that preserve the same network scaffold while changing the state space, the effective weights, or the local response rule.

2. Consensus, spectral characterization, and finite-time constructions

Under standard assumptions—most commonly, row-stochasticity, strong connectivity, and aperiodicity—the powers of $x(t)\in\mathbb{R}^n$9 converge to a rank-one limit. Equivalently,

$W=[w_{ij}]$0

where $W=[w_{ij}]$1 is the unique stationary left eigenvector satisfying $W=[w_{ij}]$2, $W=[w_{ij}]$3, and $W=[w_{ij}]$4. Hence all agents converge to the same value,

$W=[w_{ij}]$5

so the consensus is the stationary-weighted average of the initial opinions (Xia et al., 2019, Gadducci et al., 24 Apr 2025).

When $W=[w_{ij}]$6 is doubly stochastic, the stationary distribution is uniform and the consensus reduces to the arithmetic mean. In symmetric undirected settings, the nontrivial convergence rate is governed by the second-largest eigenvalue modulus; in regular undirected graphs, the mean-centered component contracts at rate $W=[w_{ij}]$7. By contrast, periodic networks can fail to converge: for undirected bipartite graphs, $W=[w_{ij}]$8 can induce persistent oscillations under the standard update (Bhaskar et al., 2021).

Although DeGroot is usually analyzed asymptotically, exact finite-time consensus can be obtained under special algebraic constructions. The $W=[w_{ij}]$9-method gives sufficient conditions for homogeneous and nonhomogeneous DeGroot products to become stable in finite time, including partial consensus on prescribed subsets. In distributed settings, if a connected graph on $$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$0 vertices contains a spanning subgraph isomorphic to the $$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$1-cube, distributed averaging can be performed in exactly $$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$2 steps (Păun, 20 Oct 2025).

These results show two distinct regimes of analysis. The generic primitive-matrix case yields asymptotic consensus through Perron–Frobenius theory; special partitioned or product-graph constructions yield exact finite-time consensus through algebraic lumping.

3. Stubborn agents, self-appraisal, and endogenous influence

A major extension of the DeGroot model fixes the opinions of some agents and lets the others average around them. In the single-stubborn-agent case, if agent $$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$3 is stubborn and the ordinary-agent block $$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$4 is irreducible, then the weaker condition that at least one ordinary agent places nonzero trust in the stubborn agent is sufficient for convergence of all ordinary agents to the stubborn opinion. With

$$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$5

one has

$$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$6

and row-stochasticity implies $$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$7, so every ordinary agent converges to the stubborn agent’s value (Abrahamsson et al., 2019).

In noisy versions with stubborn agents, regular nodes satisfy

$$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$8

with $$x(t+1)=W\,x(t),\qquad x_i(t+1)=\sum_{j=1}^n w_{ij}x_j(t),$$9, $w_{ij}\ge 0$0, stubborn opinions $w_{ij}\ge 0$1, and zero-mean noise on regular agents. When the stubborn set is globally reachable, $w_{ij}\ge 0$2 is Schur stable, the equilibrium mean is

$w_{ij}\ge 0$3

and the stationary covariance solves the discrete Lyapunov equation $w_{ij}\ge 0$4. In the undirected setting studied in the paper, this yields the closed form $w_{ij}\ge 0$5 (Raineri et al., 11 Apr 2025).

A different line of work endogenizes the influence matrix itself. In the DeGroot–Friedkin model, agents discuss a sequence of issues; within each issue, opinions evolve by DeGroot averaging, while across issues agents update their self-confidence levels by reflected appraisal. With $w_{ij}\ge 0$6 the self-confidence vector and $w_{ij}\ge 0$7 the relative interaction matrix,

$w_{ij}\ge 0$8

and the next self-confidence vector is the dominant normalized left eigenvector $w_{ij}\ge 0$9 of $\sum_{j=1}^n w_{ij}=1$0. In the modified DeGroot–Friedkin model of Xu, Liu, and Başar, self-confidence can instead be updated locally in finite time, and when $\sum_{j=1}^n w_{ij}=1$1 is doubly stochastic the unique nontrivial equilibrium is $\sum_{j=1}^n w_{ij}=1$2, so the system converges to a democratic state (Xu et al., 2015). When the relative interaction matrices vary across issues, periodic switching can produce periodic self-appraisal, while arbitrary switching under doubly stochastic irreducible matrices again leads to convergence to $\sum_{j=1}^n w_{ij}=1$3 (Ye et al., 2017).

These variants preserve the DeGroot averaging core but alter the status of the weights: in one case some opinions become fixed boundary conditions, and in another the interpersonal influence structure itself becomes a dynamical variable.

4. Bias, extremization, and polarization

One of the most studied departures from DeGroot replaces linear averaging by state-dependent assimilation. In the nonlinear biased-assimilation model, each agent $\sum_{j=1}^n w_{ij}=1$4 has a bias parameter $\sum_{j=1}^n w_{ij}=1$5, opinions lie in $\sum_{j=1}^n w_{ij}=1$6, and the update is

$\sum_{j=1}^n w_{ij}=1$7

where $\sum_{j=1}^n w_{ij}=1$8 is weighted neighbor support for opinion $\sum_{j=1}^n w_{ij}=1$9 and $i$0 is total incoming weight. Setting $i$1 recovers DeGroot exactly. The regimes $i$2, $i$3, and $i$4 are termed weak, intermediate, and strong bias, respectively. Under balanced neighbor influence, positive bias moves opinions closer to the extremes than DeGroot; for strongly connected networks, the equilibria $i$5 and $i$6 are locally exponentially stable, the neutral consensus $i$7 is unstable for $i$8, and polarization can become locally exponentially stable on complete and two-island networks when bias is strong (Xia et al., 2019).

Other heterogeneous extensions alter the local rule more radically. In the rebels model, conformists average neighbors while rebels update toward the opposite of the local mean and then blend that with their prior through a common self-confidence parameter $i$9. If there is at least one rebel in every closed strongly connected component, then under very weak conditions the entire network converges to $j\to i$0; the paper describes this as the “doctrine of the mean” (Cao et al., 2012).

Confirmation-bias models can either preserve or destroy consensus depending on the response class. In one formal model based on Esteban–Ray polarization, agents discount dissimilar opinions through state-dependent factors $j\to i$1, yet if the influence graph is strongly connected then polarization eventually vanishes; in a regular symmetric circulation, the consensus value is the initial average (Alvim et al., 2021). A related multi-agent model classifies edge-level bias functions into four regions—malleability, receptive-resistant, backfire, and insular—and proves that if every bias is continuous and lies in the receptive-resistant region $j\to i$2, then a strongly connected society converges to consensus (Alvim et al., 2024).

The DeGroot model is also central to the debate on how polarization should be measured. Variance-based disagreement decays under averaging, but group-based measures need not. Extending a limit-analysis tool associated with DeMarzo et al., one paper shows that after centering and normalization the DeGroot trajectory aligns with the graph’s second eigenvector, so continuous group-based polarization measures converge to graph-dependent limits $j\to i$3. In particular, average local agreement in regular graphs satisfies

$j\to i$4

and such measures can increase over time even though opinion variance decreases (Musco et al., 2021). This suggests that the apparent mismatch between DeGroot and rising polarization is partly a question of metric choice rather than only of dynamics.

5. Generalized DeGroot frameworks

A broad contemporary literature retains the DeGroot averaging architecture while enlarging the opinion representation. In the constraint opinion model, opinions and influences are soft constraints over a semiring rather than single real numbers. The canonical update is the semiring-lifted matrix product

$j\to i$5

Choosing the semiring $j\to i$6, one binary topic, probability constraints, and constant row-stochastic entries $j\to i$7 recovers the classical DeGroot update exactly (Gadducci et al., 24 Apr 2025).

Another line adds exogenous stochastic inputs. In the Message-Enhanced DeGroot model, messages evolve as bounded Brownian motions with absorbing boundaries at $j\to i$8 and $j\to i$9, and opinions satisfy

$w_{ij}>0$0

The mean opinion converges to $w_{ij}>0$1, where $w_{ij}>0$2 is the mean of the initial message distribution, but the asymptotic variance is nonzero and depends on $w_{ij}>0$3, $w_{ij}>0$4, and the resolvent of $w_{ij}>0$5. Once the messages absorb at $w_{ij}>0$6, opinions converge to a random fixed point rather than a deterministic consensus (Wang et al., 2024).

Continuous-time noisy DeGroot-type models with feedback produce an even stronger break from consensus. With

$w_{ij}>0$7

the long-run descriptor is a stationary Gaussian law with covariance $w_{ij}>0$8 solving

$w_{ij}>0$9

The paper emphasizes that noisy information destroys consensus formation and can generate a non-equilibrium steady state with a non-zero probabilistic current loop, so the invariant measure is a NESS rather than a reversible equilibrium (Vaidya et al., 2019).

Several works modify DeGroot for robustness or acceleration while keeping linear local communication. The $W$0-DeGroot, or granular DeGroot, rounds opinions to the nearest rational with denominator $W$1; it remains Markovian and stationary, but becomes robust to stubborn agents and to monitoring distortions smaller than $W$2 (Amir et al., 2021). In undirected networks, “memory of local averages” replaces the standard update by

$W$3

which can converge even on periodic networks and, for suitable parameter choices, achieve faster convergence than both standard DeGroot and earlier memory-augmented schemes (Bhaskar et al., 2021).

These constructions show that “DeGroot model” now names not only a single linear recursion, but also a design pattern: row-stochastic local aggregation, possibly lifted to richer state spaces, combined with explicit modifications of memory, noise, message channels, or admissible state representations.

6. Observation, intervention, and empirical deployment

Because DeGroot with stubborn agents has a closed-form equilibrium, it supports explicit optimization problems. In one noisy setting, the objective of observing a subset $W$4 of regular agents is variance reduction in the estimate of the societal average

$W$5

The variance reduction objective

$W$6

is submodular under the paper’s assumptions, so a greedy algorithm achieves the classical $W$7 approximation guarantee (Raineri et al., 11 Apr 2025).

A related intervention problem chooses where to attach stubborn agents in order to shift equilibrium opinions. With the regular-agent equilibrium written as

$W$8

or equivalently $W$9 in the paper’s notation, the mean-opinion objective is monotone and submodular. This permits greedy optimization of stubborn-agent placement under a cardinality budget, and the paper reports nontrivial influence on Twitter networks with tens of thousands of users (Hunter et al., 2018).

The DeGroot framework has also been coupled to a global steering mechanism fitted directly to event streams. In GSM-DeGroot, stochastic agent states $x(t)\in\mathbb{R}^n$00 are generated from current opinions and aggregated into a global signal $x(t)\in\mathbb{R}^n$01, which then feeds back additively: $x(t)\in\mathbb{R}^n$02 Unlike standard DeGroot, the paper proves that if $x(t)\in\mathbb{R}^n$03, consensus is impossible, and it identifies self-cooling and self-exciting regimes according to the share of agents with positive $x(t)\in\mathbb{R}^n$04 (Conjeaud et al., 2022).

Finally, DeGroot has served as the baseline for models in which expression itself becomes endogenous. In the Spiral-of-Silence generalization, silent agents are removed from or filtered through the update. For the memoryless model $x(t)\in\mathbb{R}^n$05, convergence to consensus is guaranteed for cliques but not for general strongly connected aperiodic graphs; for the memory-based model $x(t)\in\mathbb{R}^n$06, convergence is not guaranteed even for clique graphs (Aranda et al., 2024).

Across these applications, the DeGroot model remains the reference linear mechanism: simple enough to admit spectral, probabilistic, and optimization analyses, yet flexible enough to anchor models of stubbornness, cognitive bias, message exposure, memory, silence, uncertainty, and network intervention. The open problems that recur across this literature include global convergence under heterogeneous nonlinearities, topology-dependent attraction basins, finite-time exactness outside special constructions, and extensions to time-varying, sparse, or multi-issue networks (Xia et al., 2019, Gadducci et al., 24 Apr 2025).