DeGroot model

While looking through the recent opinion dynamics literature I have started noticing papers exploring various extensions of the DeGroot model [1, 2]. Prior to those papers I haven't even heard or paid much attention to it. So I felt a bit curious.

At a first glance DeGroot model appears to be similar to the trust and suspicion models we have discussed few years ago. Also, in the broadest strokes it likely inspired bounded confidence models.

But let us see what it is actually about!

The model

An agent in this model represents a person holding opinion towards some not necessarily political question. The opinion is encoded by a real number between \( 0 \) and \( 1 \) (actually opinion can be any real number, but lets keep it this way for this post). Opinion held by each agent changes due to interactions between the agents.

The time in this model is discrete. And during each time step an agent interacts with all agents they trust. The trust in another agent is encoded by values between \( 0 \) and \( 1 \). These values are further normalized, so that their sum is exactly \( 1 \). This normalization is meant to conserve the opinion.

After each time step opinion of agent \( i \) changes according to:

\begin{equation} o_i(t+1) = \sum_j T_{i,j} o_j(t) . \end{equation}

In the above \( o_i(t) \) stands for the \( i \)-th agent's opinion at time \( t \). \( T_{i,j} \) is a value encoding how much agent \( i \) trusts agent \( j \).

Model properties

Here we examine just the case with \( 3 \) agents. This is the smallest non-trivial system which still exhibits interesting behaviors (such as divergence).

Observe that the update rule is effectively weighted averaging procedure. So, naturally one would expect that the opinions would converge towards a single value (at least as long as the trust network is strongly connected). Usually this is would be true. For example, with trust matrix

\begin{equation} \mathbf{T} = \begin{pmatrix} 0 & 0.5 & 0.5 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \end{equation}

the system does converge to a fixed shared opinion (consensus) state. Here \( T_{ij} \) stands for how much trust \( i \)-th agent puts into \( j \)-th agent. So in this example the zero-th agent splits their trust equally between the first and second agent.

The system will no longer converge to a consensus state if we instead consider trust matrix

\begin{equation} \mathbf{T} = \begin{pmatrix} 0 & 0.5 & 0.5 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} . \end{equation}

This trust matrix differs from the first one in one minor detail. The second agent instead of trusting the first agent, puts all of their trust into the zero-th agent. Both of these examples were discussed in [3].

To check whether given trust matrix \( \mathbf{T} \) will lead to a consensus state, one needs to observe the behavior of \( \lim_{n\rightarrow\infty} \mathbf{T}^n \). If this limit converges, then the system will also converge to a consensus state.

As raising a matrix to power involves its eigenvalues, for me a convenient shortcut appears to be looking at the eigenvalues of the trust matrix. If a single largest (in absolute value) eigenvalue is equal to \( 1 \), then the system will converge towards consensus state. Otherwise, the system will be non-convergent.

Interactive app

Use the interactive app below to explore the provided examples. Observe that in the second case divergence can be avoided by adding interactions within agents themselves (i.e., self-trust). Feel free to experiment.

In the app above three nodes represent three agents. Hue of each node represents its opinion. Arrows are directed from the agent sharing their opinion to the agent trusting it. The darker the arrow the stronger the influence. Some times nodes will have a border - this border indicates self-influence.

References

• M. H. DeGroot. Reaching a Consensus. Journal of the American Statistical Association 69: 118-121 (1974). doi: 10.2307/2285509.
• H. Noorazar. Recent advances in opinion propagation dynamics: a 2020 survey. The European Physical Journal Plus 135: 521 (2020). doi: 10.1140/epjp/s13360-020-00541-2.
• M. O. Jackson. Social and Economic Networks. Princeton University Press, 2008.