Emergence (coarse-graining and blackboxing)

Coarse-graining

We’ll use the macro module to explore alternate spatial scales of a network. The network under consideration is a 4-node non-deterministic network, available from the examples module.

>>> import pyphi
>>> network = pyphi.examples.macro_network()

The connectivity matrix is all-to-all:

>>> network.connectivity_matrix
array([[ 1.,  1.,  1.,  1.],
       [ 1.,  1.,  1.,  1.],
       [ 1.,  1.,  1.,  1.],
       [ 1.,  1.,  1.,  1.]])

We’ll set the state so that nodes are OFF.

>>> state = (0, 0, 0, 0)

At the “micro” spatial scale, we can compute the main complex, and determine the \(\Phi\) value:

>>> main_complex = pyphi.compute.main_complex(network, state)
>>> main_complex.phi
0.113889

The question is whether there are other spatial scales which have greater values of \(\Phi\). This is accomplished by considering all possible coarse-graining of micro-elements to form macro-elements. A coarse-graining of nodes is any partition of the elements of the micro system. First we’ll get a list of all possible coarse-grainings:

>>> grains = list(pyphi.macro.all_coarse_grains(network.node_indices))

We start by considering the first coarse grain:

>>> coarse_grain = grains[0]
>>> coarse_grain
CoarseGrain(partition=((0, 1, 2), (3,)), grouping=(((0, 1, 2), (3,)), ((0,), (1,))))

Each CoarseGrain specifies two fields: the partition of states into macro elements, and the grouping of micro-states into macro-states. Let’s first look at the partition:

>>> coarse_grain.partition
((0, 1, 2), (3,))

There are two macro-elements in this partiion: one consists of micro-elements (0, 1, 2) and the other is simply micro-element 3.

We must then determine the relationship between micro-elements and macro-elements. When coarse-graining the system we assume that the resulting macro-elements do not differentiate the different micro-elements. Thus any correspondence between states must be stated solely in terms of the number of micro-elements which are on, and not depend on which micro-elements are on.

For example, consider the macro-element (0, 1, 2). We may say that the macro-element is ON if at least one micro-element is on, or if all micro-elements are on; however, we may not say that the macro-element is ON if micro-element 1 is on, because this relationship involves identifying specific micro-elements.

The grouping attribute of the CoarseGrain describes how the state of micro-elements describes the state of macro-elements:

>>> grouping = coarse_grain.grouping
>>> grouping
(((0, 1, 2), (3,)), ((0,), (1,)))

The grouping consists of two lists, one for each macro-element:

>>> grouping[0]
((0, 1, 2), (3,))

For the first macro-element, this grouping means that the element will be OFF if zero, one or two of its micro-elements are ON, and will be ON if all three micro-elements are ON.

>>> grouping[1]
((0,), (1,))

For the second macro-element, the grouping means that the element will be OFF if its micro-element is OFF, and ON if its micro-element is ON.

One we have selected a partition and grouping for analysis, we can create a mapping between micro-states and macro-states:

>>> mapping = coarse_grain.make_mapping()
>>> mapping
array([0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 3])

The interpretation of the mapping uses the LOLI convention of indexing (see LOLI: Low-Order bits correspond to Low-Index nodes).

>>> mapping[7]
1

This says that micro-state 7 corresponds to macro-state 1:

>>> pyphi.convert.loli_index2state(7, 4)
(1, 1, 1, 0)
>>> pyphi.convert.loli_index2state(1, 2)
(1, 0)

In micro-state 7, all three elements corresponding to the first macro-element are ON, so that macro-element is ON. The micro-element corresponding to the second macro-element is OFF, so that macro-element is OFF.

The CoarseGrain object uses the mapping internally to create a state-by-state TPM for the macro-system corresponding to the selected partition and grouping

>>> coarse_grain.macro_tpm(network.tpm)
Traceback (most recent call last):
    ...
pyphi.exceptions.ConditionallyDependentError...

However, this macro-TPM does not satisfy the conditional independence assumption, so this particular partition and grouping combination is not a valid coarse-graining of the system. Constructing a MacroSubsystem with this coarse-graining will also raise ConditionallyDependentError.

Let’s consider a different coarse-graining instead.

>>> coarse_grain = grains[14]
>>> coarse_grain.partition
((0, 1), (2, 3))
>>> coarse_grain.grouping
(((0, 1), (2,)), ((0, 1), (2,)))
>>> mapping = coarse_grain.make_mapping()
>>> mapping
array([0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 2, 3])
>>> coarse_grain.macro_tpm(network.tpm)
array([[[ 0.09,  0.09],
        [ 1.  ,  0.09]],

       [[ 0.09,  1.  ],
        [ 1.  ,  1.  ]]])

We can now construct a MacroSubsystem using this coarse-graining:

>>> macro_subsystem = pyphi.macro.MacroSubsystem(network, state, network.node_indices, coarse_grain=coarse_grain)
>>> macro_subsystem
MacroSubsystem((n0, n1))

We can then consider the integrated information of this macro-network and compare it to the micro-network.

>>> macro_mip = pyphi.compute.big_mip(macro_subsystem)
>>> macro_mip.phi
0.597212

The integrated information of the macro subsystem (\(\Phi = 0.597212\)) is greater than the integrated information of the micro system (\(\Phi = 0.113889\)). We can conclude that a macro-scale is appropriate for this system, but to determine which one, we must check all possible partitions and all possible groupings to find the maximum of integrated information across all scales.

>>> M = pyphi.macro.emergence(network, state)
>>> M.emergence
0.483323
>>> M.system
(0, 1, 2, 3)
>>> M.coarse_grain.partition
((0, 1), (2, 3))
>>> M.coarse_grain.grouping
(((0, 1), (2,)), ((0, 1), (2,)))

The analysis determines the partition and grouping which results in the maximum value of integrated information, as well as the emergence (increase in \(\Phi\)) from the micro-scale to the macro-scale.

Blackboxing

The macro module also provides tools for studying the emergence of systems using blackboxing.

>>> import pyphi
>>> network = pyphi.examples.blackbox_network()

We consider the state where all nodes are off:

>>> state = (0, 0, 0, 0, 0, 0)
>>> all_nodes = (0, 1, 2, 3, 4, 5)

The system has minimal \(\Phi\) without blackboxing:

>>> subsys = pyphi.Subsystem(network, state, all_nodes)
>>> pyphi.compute.big_phi(subsys)
0.215278

We will consider the blackbox system consisting of two blackbox elements, \(ABC\) and \(DEF\), where \(C\) and \(F\) are output elements and \(AB\) and \(DE\) are hidden within their respective blackboxes.

Blackboxing is done with a Blackbox object. As with CoarseGrain, we pass it a partition of micro-elements:

>>> partition = ((0, 1, 2), (3, 4, 5))
>>> output_indices = (2, 5)
>>> blackbox = pyphi.macro.Blackbox(partition, output_indices)

Blackboxes have a few convenience methods. The hidden_indices property returns the elements which are hidden within blackboxes:

>>> blackbox.hidden_indices
(0, 1, 3, 4)

The micro_indices property lists all the micro-elements in the box:

>>> blackbox.micro_indices
(0, 1, 2, 3, 4, 5)

The macro_indices property generates a set of indices which index the blackbox macro-elements. Since there are two blackboxes in our example, and each has one output element, there are two macro-indices:

>>> blackbox.macro_indices
(0, 1)

The macro_state method converts a state of the micro elements to the state of the macro-elements. The macro-state of a blackbox system is simply the state of the system’s output elements:

>>> micro_state = (0, 0, 0, 0, 0, 1)
>>> blackbox.macro_state(micro_state)
(0, 1)

Let us also define a time scale over which to perform our analysis:

>>> time_scale = 2

As in the coarse-graining example, the blackbox and time scale are passed to MacroSubsystem:

>>> macro_subsystem = pyphi.macro.MacroSubsystem(network, state, all_nodes, blackbox=blackbox, time_scale=time_scale)

We can now compute \(\Phi\) for this macro system:

>>> pyphi.compute.big_phi(macro_subsystem)
0.638888

We find that the macro subsystem has greater integrated information (\(\Phi = 0.638888\)) than the micro system (\(\Phi = 0.215278\))—the system demonstrates emergence.