Example networks

pyphi.examples collects the substrates, systems, TPMs and transitions used throughout the IIT literature and this documentation, so that a worked example is one function call away. Every example is registered in pyphi.examples.EXAMPLES, a mapping from category (substrate, system, tpm, transition) to the functions that build them; xor_substrate, for instance, is EXAMPLES["substrate"]["xor"]. The cards and connectivity graphs below are drawn from the objects themselves when this page is built. The cards leave out the transition probability matrices; load an example to see its TPM.

Show code cell source

Hide code cell source

import inspect

import matplotlib.pyplot as plt
from IPython.display import Markdown, display

import pyphi
from pyphi.examples import EXAMPLES
from pyphi.visualize import plot_system

pyphi.config.progress_bars = False
# The cards below omit each substrate's TPM grid; the object carries it.
pyphi.config.repr_verbosity = 1


def summary(func):
    """The first paragraph of a function's docstring."""
    doc = inspect.getdoc(func) or ""
    return " ".join(doc.split("\n\n")[0].split())

Substrates

Show code cell source

Hide code cell source

for name, func in EXAMPLES["substrate"].items():
    display(Markdown(f"### {name}\n\n{summary(func)}"))
    try:
        substrate = func()
        display(substrate)
        system = pyphi.System(
            substrate,
            state=(0,) * substrate.size,
            node_indices=tuple(range(substrate.size)),
        )
        plt.figure(figsize=(4, 4))
        plot_system(system)
        plt.show()
        plt.close()
    except Exception as exc:  # noqa: BLE001
        display(Markdown(f"*Could not render:* `{type(exc).__name__}: {exc}`"))

grid3

3-node grid substrate.

Substrate3 units · A, B, C

Units	A, B, C
State space	binary

Connectivity

	A	B	C
A	1	1	·
B	1	1	1
C	·	1	1

basic

A 3-node substrate of logic gates.

Substrate3 units · A, B, C

Units	A, B, C
State space	binary

Connectivity

	A	B	C
A	·	·	1
B	1	·	1
C	1	1	·

basic_noisy_selfloop

Based on the basic_substrate, but with added selfloops and noisy edges.

Substrate3 units · n0, n1, n2

Units	n0, n1, n2
State space	binary

Connectivity

	n0	n1	n2
n0	1	·	1
n1	1	1	1
n2	1	1	1

residue

The substrate for the residue example.

Substrate5 units · A, B, C, D, E

Units	A, B, C, D, E
State space	binary

Connectivity

	A	B	C	D	E
A	·	·	·	·	·
B	·	·	·	·	·
C	1	·	·	·	·
D	1	1	·	·	·
E	·	1	·	·	·

xor

A fully connected system of three XOR gates. In the state (0, 0, 0), none of the elementary mechanisms exist.

Substrate3 units · A, B, C

Units	A, B, C
State space	binary

Connectivity

	A	B	C
A	·	1	1
B	1	·	1
C	1	1	·

propagation_delay

A version of the primary example from the IIT 3.0 paper with deterministic COPY gates on each connection. These copy gates essentially function as propagation delays on the signal between OR, AND and XOR gates from the original system.

Substrate9 units · A, B, C, D, E, F, G, H, I

Units	A, B, C, D, E, F, G, H, I
State space	binary

Connectivity

	A	B	C	D	E	F	G	H	I
A	·	1	·	·	·	·	·	·	1
B	·	·	·	1	·	·	·	·	·
C	1	·	·	·	·	·	·	·	·
D	·	·	1	·	1	·	·	·	·
E	·	·	·	·	·	·	1	·	·
F	·	·	·	1	·	·	·	·	·
G	·	·	·	·	·	1	·	1	·
H	1	·	·	·	·	·	·	·	·
I	·	·	·	·	·	·	1	·	·

macro

A substrate of micro elements which has greater integrated information after coarse graining to a macro scale.

Substrate4 units · A, B, C, D

Units	A, B, C, D
State space	binary

Connectivity

	A	B	C	D
A	1	1	1	1
B	1	1	1	1
C	1	1	1	1
D	1	1	1	1

blackbox

A micro-substrate to demonstrate blackboxing.

Substrate6 units · A, B, C, D, E, F

Units	A, B, C, D, E, F
State space	binary

Connectivity

	A	B	C	D	E	F
A	·	·	1	·	·	·
B	·	·	1	·	·	·
C	·	·	·	1	1	·
D	·	·	·	·	·	1
E	·	·	·	·	·	1
F	1	1	·	·	·	·

rule110

A substrate of three elements which follows the logic of the Rule 110 cellular automaton with current and previous state (0, 0, 0).

Substrate3 units · A, B, C

Units	A, B, C
State space	binary

Connectivity

	A	B	C
A	1	1	1
B	1	1	1
C	1	1	1

rule154

A substrate of five elements which follows the logic of the Rule 154 cellular automaton.

Substrate5 units · A, B, C, D, E

Units	A, B, C, D, E
State space	binary

Connectivity

	A	B	C	D	E
A	1	1	·	·	1
B	1	1	1	·	·
C	·	1	1	1	·
D	·	·	1	1	1
E	1	·	·	1	1

fig1a

The substrate shown in Figure 1A of the 2014 IIT 3.0 paper.

Substrate6 units · A, B, C, D, E, F

Units	A, B, C, D, E, F
State space	binary

Connectivity

	A	B	C	D	E	F
A	·	1	1	·	·	·
B	1	·	1	·	1	·
C	1	1	·	·	·	·
D	1	·	·	·	·	·
E	·	·	·	·	·	·
F	·	·	·	·	·	·

fig3a

The substrate shown in Figure 3A of the 2014 IIT 3.0 paper.

Substrate4 units · A, B, C, D

Units	A, B, C, D
State space	binary

Connectivity

	A	B	C	D
A	·	·	·	·
B	1	·	·	·
C	1	·	·	·
D	1	·	·	·

fig3b

The substrate shown in Figure 3B of the 2014 IIT 3.0 paper.

Substrate4 units · A, B, C, D

Units	A, B, C, D
State space	binary

Connectivity

	A	B	C	D
A	·	·	·	·
B	1	·	·	·
C	1	·	·	·
D	1	·	·	·

fig4

The substrate shown in Figures 4, 6, 8, 9 and 10 of the 2014 IIT 3.0 paper.

Substrate3 units · A, B, C

Units	A, B, C
State space	binary

Connectivity

	A	B	C
A	·	1	1
B	1	·	1
C	1	1	·

fig5a

The substrate shown in Figure 5A of the 2014 IIT 3.0 paper.

Substrate3 units · A, B, C

Units	A, B, C
State space	binary

Connectivity

	A	B	C
A	·	·	·
B	1	·	1
C	1	1	·

fig5b

The substrate shown in Figure 5B of the 2014 IIT 3.0 paper.

Substrate3 units · A, B, C

Units	A, B, C
State space	binary

Connectivity

	A	B	C
A	·	1	1
B	·	·	1
C	·	1	·

Could not render: StateUnreachableForwardsError: The state (0, 0, 0) cannot be reached from any previous state (forward TPM check). IIT evaluates a system that arrived at its state through its own cause-effect power, so a state with no possible predecessor has no defined analysis.

fig16

The ABCDEFG portion of the substrate shown in Figure 16 of the 2014 IIT 3.0 paper.

Substrate7 units · A, B, C, D, E, F, G

Units	A, B, C, D, E, F, G
State space	binary

Connectivity

	A	B	C	D	E	F	G
A	·	1	1	·	1	·	·
B	1	·	1	·	·	·	·
C	1	1	·	·	·	·	·
D	·	1	·	1	1	·	·
E	·	·	·	1	1	·	·
F	·	·	·	·	·	1	1
G	·	·	·	·	·	1	1

actual_causation

The actual causation example substrate, consisting of an OR and AND gate with self-loops.

Substrate2 units · OR, AND

Units	OR, AND
State space	binary

Connectivity

	OR	AND
OR	1	1
AND	1	1

disjunction_conjunction

The disjunction-conjunction example from Actual Causation Figure 9.

Substrate4 units · A, B, C, D

Units	A, B, C, D
State space	binary

Connectivity

	A	B	C	D
A	·	·	·	1
B	·	·	·	1
C	·	·	·	1
D	·	·	·	·

gomez_p53_mdm2

The multi-valued p53-Mdm2 regulatory network from Gómez et al. (2020).

Substrate3 units · P, Mc, Mn

Units	P, Mc, Mn
State space	alphabets (3, 2, 2)

Connectivity

	P	Mc	Mn
P	1	1	1
Mc	1	1	1
Mn	1	1	1

frog

The substrate of a frog from Grasso et al. (2021).

Substrate8 units · SL, SC, SR, CL, CC, CR, ML, MR

Units	SL, SC, SR, CL, CC, CR, ML, MR
State space	binary

Connectivity

	SL	SC	SR	CL	CC	CR	ML	MR
SL	·	·	·	1	1	·	·	·
SC	·	·	·	1	1	1	·	·
SR	·	·	·	·	1	1	·	·
CL	·	·	·	·	·	·	1	·
CC	·	·	·	·	·	·	1	1
CR	·	·	·	·	·	·	·	1
ML	·	·	·	·	·	·	·	·
MR	·	·	·	·	·	·	·	·

iit4_2023_fig1a

The 3-unit logistic substrate of Fig 1A.

Substrate3 units · A, B, C

Units	A, B, C
State space	binary

Connectivity

	A	B	C
A	1	1	1
B	1	1	·
C	·	1	1

iit4_2023_fig6a

The 6-unit “bottleneck” architecture of Fig 6A.

Substrate6 units · A, B, C, D, E, F

Units	A, B, C, D, E, F
State space	binary

Connectivity

	A	B	C	D	E	F
A	·	1	1	1	1	1
B	1	1	·	·	·	·
C	1	·	1	·	·	·
D	1	·	·	1	·	·
E	1	·	·	·	1	·
F	1	·	·	·	·	1

iit4_2023_fig6b

The 6-unit “modular” architecture of Fig 6B.

Substrate6 units · A, B, C, D, E, F

Units	A, B, C, D, E, F
State space	binary

Connectivity

	A	B	C	D	E	F
A	1	1	·	1	·	1
B	1	1	·	·	·	·
C	·	1	1	1	·	1
D	·	·	1	1	·	·
E	·	1	·	1	1	1
F	·	·	·	·	1	1

iit4_2023_fig6c

The 6-unit “copy” architecture of Fig 6C.

Substrate6 units · A, B, C, D, E, F

Units	A, B, C, D, E, F
State space	binary

Connectivity

	A	B	C	D	E	F
A	·	1	·	·	·	·
B	·	·	1	·	·	·
C	·	·	·	1	·	·
D	·	·	·	·	1	·
E	·	·	·	·	·	1
F	1	·	·	·	·	·

iit4_2023_fig6d

The 6-unit “specialized” architecture of Fig 6D.

Substrate6 units · A, B, C, D, E, F

Units	A, B, C, D, E, F
State space	binary

Connectivity

	A	B	C	D	E	F
A	1	1	1	1	1	1
B	1	1	1	1	1	1
C	1	1	1	1	1	1
D	1	1	1	1	1	1
E	1	1	1	1	1	1
F	1	1	1	1	1	1

iit4_2023_fig6e

The 6-unit “structured” architecture of Fig 6E.

Substrate6 units · A, B, C, D, E, F

Units	A, B, C, D, E, F
State space	binary

Connectivity

	A	B	C	D	E	F
A	1	1	1	1	1	1
B	1	1	1	1	1	1
C	1	1	1	1	1	1
D	1	1	1	1	1	1
E	1	1	1	1	1	1
F	1	1	1	1	1	1

iit4_2023_fig7

The 5-unit state-dependent network of Fig 7.

Substrate5 units · A, B, C, D, E

Units	A, B, C, D, E
State space	binary

Connectivity

	A	B	C	D	E
A	1	1	1	1	1
B	1	1	1	1	1
C	1	1	1	1	1
D	1	1	1	1	1
E	1	1	1	1	1

iit4_2023_fig7_inactivated

The Fig 7 network with unit E inactivated (Fig 7C).

Substrate5 units · A, B, C, D, E

Units	A, B, C, D, E
State space	binary

Connectivity

	A	B	C	D	E
A	1	1	1	1	1
B	1	1	1	1	1
C	1	1	1	1	1
D	1	1	1	1	1
E	·	·	·	·	·

mayner_2026_monad

A single binary unit that keeps its state with probability p (Mayner, Marshall & Tononi 2026, Fig 2A-B): an imperfect COPY for p > 0.5. Its φₛ is min{p·log₂(2p), −log₂ p} (2026, Eq. 27), maximal at p ≈ 0.744 where φₛ ≈ 0.427 (Fig 2C).

Substrate1 units · M

Units	M
State space	binary

Connectivity

	M
M	1

marshall_2023_fig1

The four-unit systems of Marshall et al. (2023), Fig 1.

Substrate4 units · A, B, C, D

Units	A, B, C, D
State space	binary

Connectivity

	A	B	C	D
A	1	1	1	1
B	1	1	1	1
C	1	1	1	1
D	1	1	1	1

marshall_2023_fig2

The four-unit sigmoid systems of Marshall et al. (2023), Fig 2.

Substrate4 units · A, B, C, D

Units	A, B, C, D
State space	binary

Connectivity

	A	B	C	D
A	1	1	1	1
B	1	1	1	1
C	1	1	1	1
D	1	1	1	1

marshall_2023_fig3

The eight-unit universe of Marshall et al. (2023), Fig 3.

Substrate8 units · A, B, C, D, E, F, G, H

Units	A, B, C, D, E, F, G, H
State space	binary

Connectivity

	A	B	C	D	E	F	G	H
A	1	1	1	1	1	1	1	1
B	1	1	1	1	1	1	1	1
C	1	1	1	1	1	1	1	1
D	1	1	1	1	1	1	1	1
E	1	1	1	1	1	1	1	1
F	1	1	1	1	1	1	1	1
G	1	1	1	1	1	1	1	1
H	1	1	1	1	1	1	1	1

ac_2019_three_candidate_election

The three-candidate, seven-voter election of Albantakis et al. (2019), Fig 11: voters A-G each in state 0, 1, or 2 (candidates “1”, “2”, “3”), and W in state 1, 2, or 3 for the candidate with a strict majority of votes, or 0 for a tie. Voters repeat their state.

Substrate8 units · A, B, C, D, E, F, G, W

Units	A, B, C, D, E, F, G, W
State space	alphabets (3, 3, 3, 3, 3, 3, 3, 4)

Connectivity

	A	B	C	D	E	F	G	W
A	1	1	1	1	1	1	1	1
B	1	1	1	1	1	1	1	1
C	1	1	1	1	1	1	1	1
D	1	1	1	1	1	1	1	1
E	1	1	1	1	1	1	1	1
F	1	1	1	1	1	1	1	1
G	1	1	1	1	1	1	1	1
W	1	1	1	1	1	1	1	1

Could not render: StateUnreachableForwardsError: The state (0, 0, 0, 0, 0, 0, 0, 0) cannot be reached from any previous state (forward TPM check). IIT evaluates a system that arrived at its state through its own cause-effect power, so a state with no possible predecessor has no defined analysis.

Systems, TPMs and transitions

A system example is its substrate in the state used by the source it comes from. TPM examples return a bare transition probability matrix, and transition examples a Transition for actual causation.

Show code cell source

Hide code cell source

for category in ("system", "tpm", "transition"):
    items = "\n".join(
        f"- `{name}_{category}` — {summary(func)}"
        for name, func in EXAMPLES[category].items()
    )
    display(Markdown(f"### {category.capitalize()}s\n\n{items}"))

Systems

• grid3_system —

• basic_system — A system containing all the nodes of the :func:~pyphi.examples.basic_substrate.

• basic_noisy_selfloop_system — A system containing all the nodes of the :func:~pyphi.examples.basic_noisy_selfloop_substrate.

• residue_system — The system containing all the nodes of the :func:~pyphi.examples.residue_substrate.

• xor_system — The system containing all the nodes of the :func:~pyphi.examples.xor_substrate.

• macro_system — A system containing all the nodes of :func:~pyphi.examples.macro_substrate.

• rule110_system —

• rule154_system —

• fig4_system —

• fig5a_system —

• fig5b_system —

• differentiation_micro_1_system —

• iit4_2023_fig1a_system —

• iit4_2023_fig6a_system —

• iit4_2023_fig6b_system —

• iit4_2023_fig6c_system —

• iit4_2023_fig6d_system —

• iit4_2023_fig6e_system —

• iit4_2023_fig7_system —

• iit4_2023_fig7_inactivated_system —

Tpms

• cond_depend_tpm — A system of two general logic gates A and B such if they are in the same state they stay the same, but if they are in different states, they flip with probability 50%.

• cond_independ_tpm — A system of three general logic gates A, B and C such that: if A and B are in the same state then they stay the same; if they are in different states, they flip if C is ON and stay the same if C is OFF; and C is ON 50% of the time, independent of the previous state.

Transitions

• prevention_transition — The :class:~pyphi.actual.Transition for the prevention example from Actual Causation Figure 7D.

• frog_transition — The state transition analyzed for a frog from Grasso et al. (2021).