More links, faster agreement?
Both presets start with the same six default values. Compare the steps and exchanges needed to agree.
01 / Coordination · Beginner
Distributed average
consensus.
Can a group agree without a leader?
Six agents. Six different numbers.
Each one listens to its neighbours and adjusts its own value.
Before they exchange
A10
A22
A34
A48
A510
A612
After one step
A13
A24
A35
A47
A58
A69
An interactive browser simulation of six abstract agents. No physical motion or live mission.
Before you begin
Where will the numbers end up?
Make a prediction. Then change the connections.
Choose an answer to think about, then test it below.
Method & assumptionsLinear rule · leaderless decisions
Synchronous steps · bidirectional links · no delay
- Algorithm variant
- Linear · constant edge gain
Each neighbor difference contributes with the same gain, α = 1/12.
- Decision architecture
- Decentralized · leaderless
Every agent decides from its own value and its neighbors’ values.
- Update timing
- Synchronous · discrete time
All agents read step k, then update together to step k + 1.
- Communication model
- Undirected · unweighted
Links carry values both ways, with equal gain and no delay or random loss.
Execution: six agents in one browser simulation. Decentralized describes their decisions; the simulator computes the global mean and disagreement for you to inspect.
01 / Try it
Paused · ready to exploreA conversation in numbers.
Abstract values · no physical unitsSame run in 2D and 3D
Step0of 1,000
Disagreement12.0000max − min
Mean6.0000initial 6.0000
Connected groups1all agents connected
Mean, disagreement and connected groups are measurements for you. Each agent receives only its neighbours’ values.
Watch the difference
max − minAre the values getting closer?
Look inside
What each agent knows
| Agent | Value | Neighbors | Group |
|---|
Group IDs describe the network for the observer; agents do not receive them.
30 directed scalar exchanges / next step0 exchanges so far · Model counts, not network bytes
02 / Follow a question
Same rule.
Different questions.
Change one condition. Look for what stays the same.
Two groups. Two answers.
Two disconnected groups can agree internally and still disagree with each other. Run to step 1,000 to see it.
Bring the conversation back.
Load a chain with A3–A4 removed at step 0. Advance to step 100, restore A3–A4, then run to 1,000. Does agreement return?
Agreed. But is it correct?
Shift every starting value by 100. The group can still agree, but this model has no external truth to check.
Every exercise starts paused. Take a moment to predict before pressing Play.
Compare measured reference runs 5 runs · 1,000 steps each
Computed locally using the same model, with starting values [0, 2, 4, 8, 10, 12]. Recovery restores A3–A4 at step 100. Shared bias adds 100. These comparisons do not change your experiment.
| Experiment | First agreement | Exchanges to agreement | Final range |
|---|
03 / Go deeper
A simple rule.
A precise set of assumptions.
Each agent moves a little toward its neighbours’ values. All six read the same step before updating together.
One agent, one step
A1 starts at 0. On the complete graph, its neighbours sum to 36.
0 + 36 / 12 = 3
The gain is fixed. A1 is not given the group average.
Read the rule, assumptions & source
xᵢ[k + 1] = xᵢ[k] + ¹⁄₁₂ ∑ⱼ∈Nᵢ (xⱼ[k] − xᵢ[k])
- xᵢ[k]
- Agent i’s value at step k.
- Nᵢ
- The neighbors currently linked to agent i.
- ∑
- Add one difference for each neighbor.
- α = 1/12
- The fixed gain applied to each neighbor difference.
With six agents, α = 1 / (2N) = 1/12. This is a linear weighted sum of the previous values. Each neighbor has weight α; the agent’s own value has weight 1 − α × its number of neighbors. Removing a link changes this self-weight while the edge gain stays fixed.
Even with five neighbors, an agent retains a positive share of its own value. Each update is a weighted average, so values stay within their previous range.
Why the average survives
Every undirected exchange adds to one agent exactly what it subtracts from another. The sum stays constant. A fixed connected graph converges to the initial mean.
Where this model stops
Six fixed agents, synchronous steps, equal undirected links, no delays or random message loss. No flight dynamics, GPS, membership changes, or external truth measurement.
Foundational context: Olfati-Saber & Murray (2004), Consensus Problems in Networks of Agents With Switching Topology and Time-Delays ↗. This lesson implements the discrete averaging rule shown above, not every case in the paper.
A fixed connected graph approaches the initial mean. Agreement alone does not establish that the shared value is correct.
Keep the thread
They can agree.
Who does what?
Continue the coordination path with task allocation.