Calculator
Example Data Table
| N | rho | Model | Estimated sweeps | Estimated updates |
|---|---|---|---|---|
| 1,000 | 0.50 | Complete graph | ~ 693.147 | ~ 693,147 |
| 5,000 | 0.40 | Complete graph | ~ 3,365.058 | ~ 16,825,290 |
| 10,000 | 0.50 | 2D lattice (scaling) | ~ 10,000*ln(10,000) | ~ 100,000,000+ |
Numbers are illustrative and depend on topology and calibration coefficient.
Formula Used
The classic two-opinion voter dynamics can be summarized by the initial fraction rho of agents holding opinion A in a population of size N. For the mean-field (complete graph) approximation, the expected consensus time has the compact form:
T_sweeps(rho) = C * N_eff * h(rho)
- T_sweeps is measured in Monte Carlo sweeps (about N update attempts).
- N_eff equals N for a complete graph.
- For heterogeneous networks, a common approximation is N_eff = N*(mu1^2 / mu2).
- For 1D and 2D options, the calculator uses tunable scaling forms (N^2 and N ln N) modulated by h(rho).
- The coefficient C lets you match experiments or simulations.
How to Use This Calculator
- Enter the population size N and initial fraction rho.
- Select a topology model that matches your system.
- Choose whether you want time in sweeps or update attempts.
- Provide an update rate if you want seconds, minutes, and hours.
- Optionally enable simulation (complete graph) to sanity-check the estimate.
- Press Calculate to show results above the form.
- Use the download buttons to export CSV or PDF results.
Professional Notes
Consensus time is a finite-size quantity that depends on update rules, topology, and initial imbalance. Mean-field estimates are useful for quick design work, while lattice-like systems often coarsen more slowly.
Treat the scaling options as practical estimates and calibrate the coefficient using measured runs whenever accuracy matters.
Article
1) What the calculator measures
This tool estimates the expected time for a two-opinion voter system to reach full agreement (all A or all B) starting from an initial fraction rho in a finite population N. In physics terms, it tracks an absorbing-state first-passage time for a stochastic, unbiased copying process.
2) Why initial imbalance matters
The slowest typical scenario is a near-even split. When rho approaches 0 or 1, one opinion is already dominant, and consensus is reached quickly. The dependence is captured by the entropy-like factor h(rho) = -rho ln(rho) - (1-rho) ln(1-rho), which peaks at rho=0.5.
3) Mean-field scaling with population size
On a fully mixed population (complete graph), the analytic estimate scales as T_sweeps ≈ C * N * h(rho). For example, at rho=0.5, h(rho)=ln 2 ≈ 0.693, so T_sweeps ≈ 0.693 C N. This implies a linear growth in sweeps with size, while update-attempt time grows as O(N^2).
4) Topology effects via effective size
Real systems are rarely perfectly mixed. A common approximation is to replace N by an effective size N_eff = N*(mu1^2/mu2), where mu1=<k> and mu2=<k^2> summarize degree heterogeneity. Larger mu2 (more hubs) typically reduces N_eff and changes the timescale.
5) Lattice-style coarsening options
In spatial systems, opinions form domains that coarsen over time. The calculator includes practical scaling models: roughly N^2 for a 1D ring and N ln N for a 2D lattice in update attempts. These are engineering estimates that help with planning, not exact closed forms.
6) Converting to wall-clock time
Many projects need seconds rather than abstract steps. Enter an update-attempt rate (updates per second) to obtain wall-clock time. This is useful when mapping Monte Carlo sweeps to experimental sampling, GPU kernels, or agent-based simulations.
7) Calibration and uncertainty
The coefficient C lets you match your specific update rule, sampling schedule, or implementation details. If you have measured consensus times for one size, tune C so the estimate matches, then reuse it to extrapolate across sizes and initial splits.
8) Using simulation as a cross-check
The optional simulation (complete-graph mode) provides empirical mean, median, and percentile times in update attempts. Comparing simulated and analytic values is a fast quality check: large deviations can indicate a mismatch in topology assumptions, an update-rate interpretation issue, or a need to recalibrate C.
FAQs
1) What does “consensus time” mean here?
It is the expected time until all agents share the same opinion, starting from an initial fraction rho, under an unbiased copying update rule.
2) Why is rho=0.5 usually the slowest case?
At rho=0.5 the system is maximally mixed, so it takes longer for random fluctuations to drive the state to either absorbing boundary.
3) Should I report sweeps or update attempts?
Sweeps normalize by population size and are easier to compare across N. Update attempts are closer to raw simulation steps and are used for wall-clock conversion.
4) How do I choose mu1 and mu2?
Use summary statistics of your network: mu1 is the average degree and mu2 is the average of degree squared. They adjust the effective size for heterogeneity.
5) Are the 1D and 2D options exact formulas?
No. They are practical scaling models that reflect slower coarsening in spatial systems. Calibrate the coefficient C if you need closer agreement with data.
6) Why might simulation and analytics disagree?
Differences can come from topology mismatch, finite-size effects, step caps, or implementation details. Try increasing runs, raising the step cap, or adjusting C with a benchmark case.
7) What is a reasonable update rate for wall-clock conversion?
Use the throughput of your actual process: measured updates per second from your code, or an experimental sampling rate mapped to one update attempt definition.
Consensus emerges faster when systems are smaller and mixed.