Emergence (Macro/Micro)

For this example, we will use the pprint module to display lists in a way which makes them easer to read.

>>> from pprint import pprint

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 partitions:

>>> partitions = pyphi.macro.list_all_partitions(network)
>>> pprint(partitions)
[[[0, 1, 2], [3]],
 [[0, 1, 3], [2]],
 [[0, 1], [2, 3]],
 [[0, 1], [2], [3]],
 [[0, 2, 3], [1]],
 [[0, 2], [1, 3]],
 [[0, 2], [1], [3]],
 [[0, 3], [1, 2]],
 [[0], [1, 2, 3]],
 [[0], [1, 2], [3]],
 [[0, 3], [1], [2]],
 [[0], [1, 3], [2]],
 [[0], [1], [2, 3]],
 [[0, 1, 2, 3]]]

Lets start by considering the partition [[0, 1, 2], [3]]:

>>> partition = partitions[0]
>>> partition
[[0, 1, 2], [3]]

For this partition there are two macro-elements, one consisting 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. An assumption when coarse-graining the system, is that the resulting macro-elements do not differentiate the different micro-elements. Thus any correspondence between states must be stated soley in terms of the number of micro-elements which are on, and not depend on which micro-element 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.

To see a list of all possible groupings of micro-states into macro-states:

>>> groupings = pyphi.macro.list_all_groupings(partition)
>>> pprint(groupings)
[[[[0, 1, 2], [3]], [[0], [1]]],
 [[[0, 1, 3], [2]], [[0], [1]]],
 [[[0, 1], [2, 3]], [[0], [1]]],
 [[[0, 2, 3], [1]], [[0], [1]]],
 [[[0, 2], [1, 3]], [[0], [1]]],
 [[[0, 3], [1, 2]], [[0], [1]]],
 [[[0], [1, 2, 3]], [[0], [1]]]]

We will focus on the first grouping in the list.

>>> grouping = groupings[0]
>>> grouping
[[[0, 1, 2], [3]], [[0], [1]]]

The grouping contains 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 = pyphi.macro.make_mapping(partition, grouping)
>>> 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]
1.0

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.

Using the mapping, we can then create a state-by-state TPM for the macro-system corresponding to the selected partition and grouping:

>>> macro_tpm = pyphi.macro.make_macro_tpm(network.tpm, mapping)
>>> macro_tpm
array([[ 0.5838,  0.0162,  0.3802,  0.0198],
       [ 0.    ,  0.    ,  0.91  ,  0.09  ],
       [ 0.5019,  0.0981,  0.3451,  0.0549],
       [ 0.    ,  0.    ,  0.    ,  1.    ]])

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:

>>> pyphi.validate.conditionally_independent(macro_tpm)
False

In these cases, the object returned make_macro_network() function will have a boolean value of False:

>>> (macro_network, macro_state) = pyphi.macro.make_macro_network(network, state, mapping)
>>> bool(macro_network)
False

Lets consider a different partition instead.

>>> partition = partitions[2]
>>> partition
[[0, 1], [2, 3]]

>>> groupings = pyphi.macro.list_all_groupings(partition)
>>> grouping = groupings[0]
>>> grouping
[[[0, 1], [2]], [[0, 1], [2]]]

>>> mapping = pyphi.macro.make_mapping(partition, grouping)
>>> mapping
array([ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  1.,  2.,
        2.,  2.,  3.])

>>> (macro_network, macro_state) = pyphi.macro.make_macro_network(network, state, mapping)
>>> bool(macro_network)
True

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

>>> macro_main_complex = pyphi.compute.main_complex(macro_network, macro_state)
>>> macro_main_complex.phi
0.86905

The integrated information of the macro system (\(\Phi = 0.86905\)) 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.partition
[[0, 1], [2, 3]]
>>> M.grouping
[[[0, 1], [2]], [[0, 1], [2]]]
>>> M.emergence
0.755161

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.