ppsim 0.0.4

A package for simulating population protocols.

ppsim Python package

The ppsim package is used for simulating population protcols. The core of the simulator uses a batching algorithm which gives significant asymptotic gains for protocols with relatively small reachable state sets. The package is designed to be run in a Python notebook, to concisely describe complex protocols, efficiently simulate their dynamics, and provide helpful visualization of the simulation.

Installation

The package can be installed with pip via

pip install ppsim

The most important part of the package is the Simulation class, which is responsible for parsing a protocol, performing the simulation, and giving data about the simulation.

from ppsim import Simulation

Example protcol

A state can be any hashable Python object. The simplest way to describe a protocol is a dictionary mapping pairs of input states to pairs of output states. For example, here is a description of the classic 3-state approximate majority protocol. There are two initial states A and B, and the protocol converges with high probability to the majority state with the help of a third "undecided" state U.

a, b, u = 'A', 'B', 'U'
approximate_majority = {
    (a,b): (u,u),
    (a,u): (a,a),
    (b,u): (b,b)
}

Example Simulation

To instantiate a Simulation, we must specify a protocol along with an initial condition, which is a dictionary mapping states to counts. Let's simulate approximate majority with in a population of one billion agents with a slight majority of A agents.

n = 10 ** 9
init_config = {a: 0.501 * n, b: 0.499 * n}
sim = Simulation(init_config, approximate_majority)

Now let's run this simulation for 10 units of parallel time (10 * n interactions). We will record the configuration every 0.1 units of time.

sim.run(10, 0.1)

The Simulation class can display all these configurations in a pandas dataframe in the attribute history.

sim.history

	A	B	U
time
0.000000	501000000	499000000	0
0.100010	478285585	476280966	45433449
0.200039	459449200	457428961	83121839
0.300056	443650105	441607805	114742090
0.400098	430264769	428196457	141538774
...	...	...	...
9.601324	352755117	314377928	332866955
9.701327	353418258	313744457	332837285
9.801359	354106871	313093847	332799282
9.901364	354824761	312423976	332751263
10.001375	355556526	311728627	332714847

101 rows × 3 columns

sim.history.plot()

Without specifying an end time, run will run the simulation until the configuration is silent (all interactions are null). In this case, that will be when the protcol reaches a silent majority consensus configuration.

sim.run()
sim.history.plot()

As currently described, this protocol is one-way, where these interactions only take place if the two states meet in the specified order. We can see this by checking print_reactions.

sim.print_reactions()

A, B  -->  U, U      with probability 0.5
A, U  -->  A, A      with probability 0.5
B, U  -->  B, B      with probability 0.5

Here we have unorder pairs of reactants, and the probability 0.5 is because these interactions as written depended on the order of the agents. If we wanted to consider the more sensible symmetric variant of the protocol, one approach would explicitly give all non-null interactions:

approximate_majority_symmetric = {
    (a,b): (u,u), (b,a): (u,u),
    (a,u): (a,a), (u,a): (a,a),
    (b,u): (b,b), (u,b): (b,b)
}
sim = Simulation(init_config, approximate_majority_symmetric)

But a quicker equivalent approach is to tell Simulation that all interactions should be interpreted as symmetric, so if we specify interaction (a,b) but leave (b,a) as null, then (b,a) will be interpreted as having the same output pair.

sim = Simulation(init_config, approximate_majority, transition_order='symmetric')
sim.print_reactions()
sim.run()
sim.history.plot()

A, B  -->  U, U
A, U  -->  A, A
B, U  -->  B, B

A key result about this protocol is it converges in expected O(log n) time, which surprisingly is very nontrivial to prove. We can use this package to very quickly gather some convincing data that the convergence really is O(log n) time, with the function time_trials.

from ppsim import time_trials
import numpy as np

ns = [int(n) for n in np.geomspace(10, 10 ** 8, 20)]
def initial_condition(n):
    return {'A': n // 2, 'B': n // 2}
df = time_trials(approximate_majority, ns, initial_condition, num_trials=100, max_wallclock_time = 30, transition_order='symmetric')
df

	n	time
0	10	14.100000
1	10	9.000000
2	10	7.700000
3	10	14.900000
4	10	7.200000
...	...	...
1359	42813323	48.937840
1360	42813323	50.391446
1361	42813323	56.390054
1362	100000000	50.265050
1363	100000000	53.851442

1364 rows x 2 columns

This dataframe collected time from up to 100 trials for each population size n across a many orders of magnitude, limited by the budget of 30 seconds of wallclock time that we gave it. We can now use the seaborn library to get a convincing plot of the data.

import seaborn as sns
lp = sns.lineplot(x='n', y='time', data=df)
lp.set_xscale('log')