Friedkin–Johnsen Opinion Dynamics Model

• The Friedkin–Johnsen model is a discrete-time linear system where agents iteratively combine social influence with a persistent pull to their innate opinions.
• It uses matrix formulations to analyze convergence, equilibrium behavior, and phenomena such as polarization and disagreement in social networks.
• Recent extensions and computational algorithms adapt the model to signed networks, hypergraphs, and optimization settings for scalable and precise opinion analysis.

The Friedkin–Johnsen (FJ) opinion dynamics model is a discrete-time linear model in which each agent repeatedly combines interpersonal influence with persistent attachment to an initial or innate opinion. In a common matrix form on a weighted directed graph, the update is $x(t+1)=\Lambda W x(t)+(I-\Lambda)s$, where $W$ is row-stochastic, $\Lambda$ is diagonal, and $s$ is the vector of innate opinions; in a restricted graph-Laplacian form often used on weighted undirected networks, the equilibrium is $z=(I+L)^{-1}s$ (Raineri et al., 9 Apr 2025, Biondi et al., 2022). The model is used to analyze convergence, persistent disagreement, polarization, signed or higher-order influence, external media, and algorithmic intervention, and recent work has also recast it as a boundary-value problem on networks and as a transition system for verification from binary observations (Boudourides, 9 Feb 2026, Xing et al., 6 Apr 2026).

1. Canonical formulations and notational variants

A standard formulation treats the social network as a weighted directed graph with $n$ agents, scalar opinions $x_i(t)\in\mathbb{R}$, a row-stochastic influence matrix $W=(w_{ij})\ge 0$, a diagonal susceptibility matrix $$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$ with $0\le \lambda_{ii}\le 1$, and innate opinions $W$0. The update is

$W$1

or equivalently

$W$2

In this parameterization, $W$3 means totally stubborn and $W$4 means fully open, reducing to DeGroot averaging; the innate opinions act as a persistent external pull toward each agent’s baseline belief (Raineri et al., 9 Apr 2025).

Recent literature explicitly distinguishes several algebraically different FJ variants. The generalized FJ model is

$W$5

with equilibrium

$W$6

A variational form instead averages prejudice and neighbors without explicitly including the node’s own current opinion,

$W$7

and a restricted form sets $W$8,

$W$9

The same paper emphasizes that vFJ is often treated as if it were the same as gFJ, but mathematically it is not; across all three variants, the steady state can be written as $\Lambda$0 for a model-specific linear map $\Lambda$1 (Biondi et al., 2022).

The notation for stubbornness is therefore not uniform. In the generalized formulation above, $\Lambda$2 is susceptibility to social influence; in a separate time-varying formulation, the paper writes

$\Lambda$3

so that $\Lambda$4 weights attachment to the initial condition rather than openness to neighbors. This notational duality is a recurring source of confusion in the contemporary literature (Ballotta et al., 2024).

2. Equilibrium, convergence, and boundary-value structure

For the classical matrix form, a fixed point $\Lambda$5 satisfies

$\Lambda$6

and if $\Lambda$7 is invertible then the equilibrium is unique: $\Lambda$8 Under the usual graph-theoretic condition that there exists at least one non-stubborn agent and the set of non-stubborn nodes is globally reachable in the influence graph, the dynamics are asymptotically stable and converge exponentially to that equilibrium. In that setting, final opinions are convex combinations of innate opinions because the control matrix $\Lambda$9 is stochastic (Raineri et al., 9 Apr 2025).

The single-issue F-J model has also been characterized by exact algebraic and graph-theoretic convergence criteria. With

$s$0

the model is convergent if and only if $s$1 is the only eigenvalue of $s$2 with maximum modulus. A topological restatement is that convergence holds if and only if every independent strongly connected component composed only of non-stubborn agents is aperiodic or primitive. In the undirected case, this simplifies further: convergence is equivalent to each such component having at least one self-loop (Tian et al., 2016).

A more structural reformulation casts FJ as a discrete boundary-value problem. If the network is partitioned into interior agents $s$3 and boundary agents $s$4, with susceptibility matrix

$s$5

then after block decomposition the interior dynamics become

$s$6

If

$s$7

the steady state is

$s$8

The resolvent

$s$9

acts as a Green’s operator, with Neumann-series expansion $z=(I+L)^{-1}s$0. A key criterion proved in this framework is

$z=(I+L)^{-1}s$1

which formalizes the idea that all recurrent interior feedback must be damped (Boudourides, 9 Feb 2026).

This convergence theory sharply distinguishes FJ from pure consensus. In DeGroot dynamics, repeated averaging may yield $z=(I+L)^{-1}s$2; in FJ, the generic limit is a prejudice-dependent fixed point, and consensus appears only under additional structural restrictions or special parameter limits (Raineri et al., 9 Apr 2025, Tian et al., 2016).

3. Polarization, disagreement, and signed influence

Because the equilibrium is linear in the innate-opinion vector, much of the theory reduces to studying the operator $z=(I+L)^{-1}s$3 in $z=(I+L)^{-1}s$4. A major survey of polarization under FJ analyzes six indices,

$z=(I+L)^{-1}s$5

with equivalences

$z=(I+L)^{-1}s$6

and invariant relation

$z=(I+L)^{-1}s$7

In the generalized FJ model, local weighted disagreement is always nonincreasing: $z=(I+L)^{-1}s$8 whereas global polarization with respect to $z=(I+L)^{-1}s$9 occurs if and only if the equilibrium operator $n$0 is not doubly stochastic. The same work also shows that on undirected graphs the restricted FJ model is never polarizing in any metric and exhibits no choice shift, $n$1 (Biondi et al., 2022).

On signed graphs, negative edges fundamentally alter both interpretation and computation. With a connected undirected signed graph $n$2, signed adjacency matrix $n$3, signed Laplacian $n$4, internal opinions $n$5, and expressed opinions $n$6, the update is

$n$7

and the equilibrium is

$n$8

In contrast with the unsigned case, $n$9 is generally not nonnegative, so final opinions need not be convex combinations of internal opinions. The equilibrium of node $x_i(t)\in\mathbb{R}$0 admits a random-walk representation

$x_i(t)\in\mathbb{R}$1

where the coefficient on $x_i(t)\in\mathbb{R}$2 is the difference of positive and negative absorbing probabilities on an augmented signed graph (Zhou et al., 2024).

The signed setting also supports a refined decomposition of social tension. For equilibrium $x_i(t)\in\mathbb{R}$3, the paper defines internal conflict

$x_i(t)\in\mathbb{R}$4

disagreement

$x_i(t)\in\mathbb{R}$5

disagreement among friends

$x_i(t)\in\mathbb{R}$6

agreement with opponents

$x_i(t)\in\mathbb{R}$7

and polarization

$x_i(t)\in\mathbb{R}$8

These satisfy

$x_i(t)\in\mathbb{R}$9

which provides a signed-graph analogue of energy decompositions used in unsigned FJ analysis (Zhou et al., 2024).

4. Computational formulations and scalable algorithms

In the restricted model on weighted graphs, equilibrium computation is the solution of a linear system: $W=(w_{ij})\ge 0$0 This representation underlies a sequence of algorithmic results. A paper develops a deterministic local iteration algorithm, BoundLocalIter, which maintains an estimate $W=(w_{ij})\ge 0$1 and residual $W=(w_{ij})\ge 0$2, initialized as $W=(w_{ij})\ge 0$3, $W=(w_{ij})\ge 0$4, and uses local push operations subject to the threshold $W=(w_{ij})\ge 0$5. Its core invariant is

$W=(w_{ij})\ge 0$6

from which the one-sided relative error guarantee follows: $W=(w_{ij})\ge 0$7 The same paper further proves that the SOR-accelerated version BLISOR yields the symmetric bound

$W=(w_{ij})\ge 0$8

and emphasizes that these local methods apply to both directed and undirected graphs, unlike Laplacian-solver approaches restricted to the undirected case (Wang et al., 20 Jul 2025).

A complementary line of work studies sublinear-time estimation under query access, without reading the whole graph. For undirected weighted graphs, the equilibrium remains

$W=(w_{ij})\ge 0$9

but the computation is related to personalized PageRank. In $$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$0-regular graphs, the paper shows that the FJ equilibrium can be expressed by a PageRank-type vector with teleport parameter

$$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$1

and derives a deterministic local approximation whose runtime depends only on $$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$2 and $$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$3, not on the graph size. This yields the structural conclusion that in constant-degree regular graphs, each node’s equilibrium opinion can be approximated by looking only at a constant-size neighborhood, independently of network size (Neumann et al., 2024).

For signed networks, direct inversion of $$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$4 is replaced by a reduction to an unsigned symmetric diagonally dominant system. The signed Laplacian solver is built from the $$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$5 SDD matrix

$$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$6

which is the Laplacian of an associated unsigned graph. The resulting routine

$$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$7

returns $$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$8 satisfying

$$\Lambda=\mathrm{diag}(\lambda_{11},\dots,\lambda_{nn})$$9

with expected running time $0\le \lambda_{ii}\le 1$0. The same framework yields nearly-linear-time approximation of $0\le \lambda_{ii}\le 1$1, $0\le \lambda_{ii}\le 1$2, $0\le \lambda_{ii}\le 1$3, $0\le \lambda_{ii}\le 1$4, and $0\le \lambda_{ii}\le 1$5, as well as an optimal $0\le \lambda_{ii}\le 1$6 algorithm and a $0\le \lambda_{ii}\le 1$7 approximation algorithm for the signed opinion-optimization problem (Zhou et al., 2024).

These algorithmic developments reinforce a central computational point: in many important FJ variants, equilibrium analysis is less a simulation problem than a linear-algebra problem, and the main distinctions are the structure of the operator—unsigned Laplacian, signed Laplacian, local residual push, or PageRank-like propagation.

5. Temporal, higher-order, and media-driven extensions

A recent temporal generalization studies diminishing competition or stubbornness through

$0\le \lambda_{ii}\le 1$8

If competition is uniform across agents and $0\le \lambda_{ii}\le 1$9, then

$W$00

which coincides with the nominal DeGroot consensus. The same work proves that diminishing competition slows convergence according to its own rate of decay and provides upper and lower bounds on the convergence-rate function $W$01. In the illustrative case $W$02, the rate becomes polynomial,

$W$03

rather than exponential. The paper also shows that vanishing stubbornness is not sufficient in the non-uniform case: $W$04 Thus the final consensus is preserved by uniform decay, but not by heterogeneous decay (Ballotta et al., 2024).

The FJ-MM model extends FJ to include memory and higher-order neighbors: $W$05 For the one-step memory case,

$W$06

the paper proves that exponential stability is equivalent to the stability of the comparison FJ system

$W$07

and that both systems share the same equilibrium

$W$08

The convergence rate, however, is not preserved: under $W$09,

$W$10

so memory slows convergence. The numerical section further reports that memory and multi-hop influence can reduce polarization and pull final opinions toward intermediate values (Raineri et al., 9 Apr 2025).

Higher-order interactions have also been incorporated through hypergraphs. In a weighted hypergraph $W$11 with node-specific contribution proportions $W$12, each hyperedge is converted into effective pairwise interactions by either an undirected clique projection or a directed projection

$W$13

After projection, the equilibrium remains

$W$14

but the aggregate behavior changes: for undirected projections, the fundamental matrix is doubly stochastic and total expressed opinion is conserved, whereas for directed projections it is generally only row-stochastic, so $W$15. The same work gives a spanning converging forest interpretation

$W$16

and reports that directed higher-order effects yield larger polarization than undirected clique projections (Xu et al., 2023).

External media sources lead to another family of FJ generalizations. With one or two media nodes attached to the social graph, the generalized equilibrium under two stubborn media sources is

$W$17

On $W$18-regular graphs, the total opinion admits an exact formula

$W$19

showing how the sign and magnitude of the bias depend on media strength and coverage. The same paper proves that a single non-stubborn media source is much weaker: $W$20 whereas stubborn media can shift the system by a factor $W$21 (Out et al., 11 Apr 2025).

In online settings, the cascade-driven FJC model replaces persistent neighborhood averaging by updates conditioned on actual post propagation. If $W$22 is the set of predecessors that reshared the post and exposed node $W$23, the update becomes

$W$24

This layer-by-layer cascade ordering produces different influence pathways from standard FJ. The empirical conclusion reported is that cascades can amplify the influence of central opinion leaders and often yield higher polarization than classical FJ when diffusion is sufficiently active, although real Twitter cascades with very small repost probabilities can produce low or even negative polarization relative to the standard model (Biondi et al., 19 Jun 2025).

6. Multi-issue dynamics, inference, optimization, and control

The FJ model has been extended from single-issue evolution to sequences of issues. With

$W$25

and issue-to-issue path dependence

$W$26

the induced cross-issue map is governed by

$W$27

This framework shows that consensus can arise over issue sequences even when single-issue consensus fails; under the stated assumptions and with no fully stubborn agents, consensus occurs if and only if there exists a partially stubborn agent that has directed paths to all other partially stubborn agents in $W$28 (Tian et al., 2016). A related best-response framework embeds FJ as the special case

$W$29

and then uses the consensus map across repeated issues to update self-appraisals through reflected appraisals, thereby connecting opinion dynamics to the evolution of interpersonal influence structures and social power (Rey et al., 2017).

Parameter identification has likewise become a topic in its own right. One paper studies FJ models observed only through binary thresholded outputs

$W$30

with state equation

$W$31

By augmenting the state, defining a customized $W$32-approximate simulation relation that controls both state and Hamming-distance output errors, and discretizing initial opinions and stubbornness, the paper constructs finite abstract FJ models for consistency verification from binary observations. The practical implication is that verification can be performed on the finite abstraction and transferred back to continuous-parameter FJ models with a controlled error margin (Xing et al., 6 Apr 2026).

Optimization under incomplete or strategic information has developed rapidly. When innate opinions are unknown and only a limited query budget is available, one framework proceeds in three stages: select queried nodes, reconstruct the innate-opinion vector, and optimize an FJ objective using the reconstruction. For six classical objectives—directed and undirected versions of polarization, disagreement, and their combination—the paper proves the error-propagation bound

$W$33

with explicit Lipschitz constants for each objective, thereby quantifying how opinion-reconstruction error degrades the final intervention quality (Cinus et al., 27 Jan 2025). In a different strategic setting, the equilibrium

$W$34

is used to study election sway via targeted changes in resistance parameters. The optimization of the equilibrium median is shown to be NP-hard and inapproximable within any constant factor for $W$35-budget interventions, while projected Huber, sigmoid-based, and lazy greedy algorithms provide practical attack strategies (Ristache et al., 3 Feb 2025).

Control-theoretic and game-theoretic extensions push the model beyond passive opinion formation. In a Stackelberg steering framework, stubborn agents act as leaders with controlled opinions

$W$36

while regular agents act as followers with openness variables $W$37,

$W$38

The regular-agent dynamics become

$W$39

and the coupled problem is solved by forward quadratic programming for the followers and backward dynamic programming for the leaders (Rastgoftar, 8 Sep 2025). A different extension makes leadership endogenous by replacing fixed susceptibility with a dynamic state $W$40,

$W$41

$W$42

so that agents become more leader-like when conviction dominates local misalignment and more follower-like in the opposite regime. The resulting analysis gives sufficient conditions under which specific agents necessarily satisfy $W$43 and emerge as leaders (Alutto et al., 4 Jun 2026).

Across these developments, the Friedkin–Johnsen model remains recognizable by a single structural idea: expressed opinion is neither pure averaging nor pure persistence, but a controlled compromise between social influence and attachment to prior belief. What varies across the literature is the substrate on which that compromise is defined—unsigned graphs, signed graphs, hypergraphs, cascade trees, community-structured random networks, or controlled multi-issue systems—and the analytic lens applied to it, from spectral theory and resolvents to optimization, random walks, and formal verification (Zhou et al., 2024, Boudourides, 9 Feb 2026).