Stakeholder Influence Network: Opinion Dynamics Boardroom

Setting up the lab bench...

Presets

Live readouts

Graphs

Opinion of each stakeholder over discussion rounds (dotted lines are equilibria)

Variance of opinions (disagreement) on a log scale

Opinion now, at the start and at equilibrium per stakeholder

Long-run influence (stationary vector x8) and opinion shift

Mean opinion and share of supporters

Numerical checks

running...

How to use

Pick a preset and press play (Space pauses, R resets). Click a stakeholder (or use the Selected menu) to read and change their anchor and susceptibility. Try: (1) Open forum: everybody converges to the weighted mean; (2) Sponsor dominates: raise the extra attention to the authority figure and watch the consensus move toward 0.85; (3) Two camps, thin bridge: the two groups stay apart for many rounds, then raise the bridge weight; (4) Anchored views: lower the susceptibility and see that agreement never completes (dotted lines and green ticks mark the equilibrium); (5) Regulator holds out: one stubborn person drags the whole group; (6) use Persuade +/- to give the selected stakeholder a one-off nudge and see it fade when they are anchored. Orange means opposed, grey neutral, blue supportive; the coloured ring on each station is the stakeholder identity colour.

The science

What the model is. Eight stakeholders each hold an opinion x_i between 0 (oppose) and 1 (support) about a proposal. Every discussion round each person revises their view by averaging what the people they listen to think. Who listens to whom is a weighted directed graph; this is the DeGroot model of opinion dynamics, extended to the Friedkin-Johnsen model in which people stay partly anchored to their first opinion. It is a model of social influence on a network, not of management advice.

Trust matrix. Stakeholder i gives attention A_ij ≥ 0 to j (set by the network type, the extra attention paid to the authority figure and the bridge weight between camps) and self-trust s. The row-stochastic trust matrix is

W_ij = A_ij / (s + Σ_k A_ik), W_ii = s / (s + Σ_k A_ik), Σ_j W_ij = 1

DeGroot / Friedkin-Johnsen update. With susceptibility a_i (1 = fully open to others) and prejudice x⁰_i (the starting opinion):

x(t+1) = Λ W x(t) + (I - Λ) x⁰, Λ = diag(a₁, ..., a₈)

If the matrix I - ΛW is invertible (every closed group of stakeholders contains someone with a_i < 1) the opinions converge to the unique fixed point

x* = (I - ΛW)^-1 (I - Λ) x⁰

and disagreement can persist forever: people who partly keep their first view never fully agree. If all a_i = 1 and the graph is strongly connected with self-trust (so the chain is aperiodic), opinions converge to a consensus equal to the weighted mean

x_∞ = π · x⁰, π W = π, Σ π_i = 1

The stationary vector π is the long-run influence of each stakeholder: a person who is heavily listened to (the sponsor with a large authority weight, or the hub of the star network) gets a large π_i and so drags the consensus toward their own view. The quantity π·x(t) is conserved by every round when a = 1, which the page checks. Because each new opinion is an average of old ones, no opinion ever leaves the range [min x⁰, max x⁰] (maximum principle).

Speed of agreement. The rate of convergence is set by the second-largest eigenvalue modulus of W. A thin bridge between two camps makes it close to 1, so agreement is very slow ("Two camps, thin bridge"); shortcuts in a ring or a dense forum make it much smaller. The page reports the number of rounds until every opinion is within 0.01 of the equilibrium.

Checks in the page. (1) the simulated all-susceptible limit against π·x⁰ with π from a linear solve; (2) the simulated fixed point against the direct matrix solution (I - ΛW)^-1(I - Λ)x⁰ by Gaussian elimination; (3) the invariant π·x(t) after 60 rounds; (4) row sums of W; (5) the maximum principle. The 3D page also compares the drawn column heights with the model.

Honest simplifications. Eight abstract actors, one scalar opinion on one issue, fixed trust weights, no learning of who to trust, no noise, no bounded confidence, no strategic behaviour or bargaining. Real organisations are far messier; the model shows only how network structure, authority and anchoring shape agreement. Names are generic roles, not real people or companies.