In [1]:
from ei_net import * # import the .py file but you can find all the functions at the bottom of this notebook
from utilities import show_values
import matplotlib.pyplot as plt
%matplotlib inline
In [2]:
##########################################
############ PLOTTING SETUP ##############
EI_cmap = "Greys"
where_to_save_pngs = "../figs/pngs/"
where_to_save_pdfs = "../figs/pdfs/"
save = True
##########################################
##########################################
The emergence of informative higher scales in complex networks¶
Chapter 01: Effective Information in Networks¶
Networks and Causal Structure¶
Networks provide a powerful syntax for representing a wide range of systems, from the trivially simple to the highly complex. It is common to characterize networks based on structural properties like their degree distribution or whether they show community structure. While our understanding of these structural properties of networks has been crucial for the rapid rise of network science as a discipline, there is a distinct gap in our treatment of both dependencies between nodes and also higher scales in networks. This gap is especially pressing because networks often have an interpretation where links represent dependencies, such as contact networks in epidemiology, neuronal and functional networks in the brain, or interaction networks among cells, genes, or drugs, and these networks can often be analyzed at multiple different scales.
Previously, others have used directed acyclic graphs known as "causal diagrams" to represent causal relationships as dependencies in networks. But there has been little research on quantifying or broadly classifying such causation in networks, particularly those that have both weighted connections and feedback, which are hallmarks of complex systems across domains. Here we introduce information-theoretic measures designed to capture the information contained in the dependencies of networks and which can be used to identify when these networks possess informative higher scales.
Effective Information¶
Describing cause and effect implicitly invokes the idea of a network. For example, if a system in a particular state, A, always transitions to state B, the causal relationship between A and B can be represented by a node-link diagram wherein the two nodes---A and B---are connected by a directed arrow, indicating that B depends on A. In such a network, the out-weight vector, Wouti, of a node, vi, represents the possible transitions and their probabilities from that node. Specifically, Wouti consists of weights wij between node vi and its neighbors vj, where wij=0.0 if there is no edge from vi to vj. This means the edge weights wij can be interpreted as the probability pij that a random walker on vi will transition to vj in the next time step. We will refer to such a network as having a causal structure.
In the cases where links between nodes represent dependency in general, such as influence, strength, or potential causal interactions, but not explicitly transitions (or where details about transitions is lacking), for our analysis we create Wouti by normalizing each node's out-weight vector to sum to 1.0. This generalizes our results to multiple types of representations (although what sort of dependencies the links in the network represent should be kept in mind when interpreting the values of the measures we introduce below).
A network's causal structure can be characterized by the uncertainty in the relationships among the nodes' out-weights (possible effects) and in-weights (possible causes). The total information in the dependencies between nodes is a function of this uncertainty and can be derived from two fundamental properties. The first is the uncertainty of a node's effects, which can be quantified by the Shannon entropy of its out-weights, H(Wouti). The average of this entropy, ⟨H(Wouti)⟩, across all nodes is the amount of noise present in the network's causal structure. Only if ⟨H(Wouti)⟩ is zero is the network is deterministic.
The second fundamental causal property is how weight is distributed across the whole network, ⟨Wouti⟩. This vector ⟨Wouti⟩ consists of elements that are the sum of the in-weights wji to each node vi from each of its incoming neighbors, vj (then normalized by total weight of the network). Its entropy, H(⟨Wouti⟩), reflects how certainty is distributed across the network. If all nodes link only to the same node, then H(⟨Wouti⟩) is zero, and the network is totally degenerate since all causes lead to the same effect.
From these two properties we can derive the amount of information in a network's causal structure, the effective information (EI), as:
EI=H(⟨Wouti⟩)−⟨H(Wouti)⟩Here, we use this measure to develop a general classification of networks. Networks with high EI contain more certainty in the relationships between nodes in the network (since the links represent greater dependencies), whereas networks with low EI contain less certainty.
In this work, we show how the connectivity and different growth rules of a network have a deep relationship to that network's EI. This also provides a principled means of quantifying the amount of information among the micro-, meso-, and macroscale dependencies in a network. We introduce a formalism for finding and assessing the most informative scale of a network: the scale that minimizes the uncertainty in the dependencies between nodes. For some networks, a macroscale description of the network can be more informative in this manner, demonstrating a phenomenon known as causal emergence.
1.0 Create a Few Example Transition-Probability Matrices¶
In [3]:
Copy_Copy = np.array([[1.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0],
[0.0, 1.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 1.0]])
And_And = np.array([[1.0, 0.0, 0.0, 0.0],
[1.0, 0.0, 0.0, 0.0],
[1.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 1.0]])
Or_Or = np.array([[1.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 0.0, 1.0],
[0.0, 0.0, 0.0, 1.0],
[0.0, 0.0, 0.0, 1.0]])
Or_Copy = np.array([[1.0, 0.0, 0.0, 0.0],
[0.0, 0.0, 1.0, 0.0],
[0.0, 0.0, 0.0, 1.0],
[0.0, 0.0, 0.0, 1.0]])
Star = np.array([[1.0, 0.0, 0.0, 0.0],
[1.0, 0.0, 0.0, 0.0],
[1.0, 0.0, 0.0, 0.0],
[1.0, 0.0, 0.0, 0.0]])
1.0.1 Plot these TPMs, showing their EI values¶
In [4]:
fig, (ax0, ax1, ax2, ax3, ax4) = plt.subplots(1, 5, figsize=(22,4))
c0 = ax0.pcolor(
np.arange(-0.5, Copy_Copy.shape[0], 1),
np.arange(-0.5, Copy_Copy.shape[0], 1),
Copy_Copy, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c1 = ax1.pcolor(
np.arange(-0.5, And_And.shape[0], 1),
np.arange(-0.5, And_And.shape[0], 1),
And_And, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c2 = ax2.pcolor(
np.arange(-0.5, Or_Or.shape[0], 1),
np.arange(-0.5, Or_Or.shape[0], 1),
Or_Or, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c3 = ax3.pcolor(
np.arange(-0.5, Or_Copy.shape[0], 1),
np.arange(-0.5, Or_Copy.shape[0], 1),
Or_Copy, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c4 = ax4.pcolor(
np.arange(-0.5, Star.shape[0], 1),
np.arange(-0.5, Star.shape[0], 1),
Star, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
show_values(c0, ax=ax0, fmt="%.1f", fontsize=16)
show_values(c1, ax=ax1, fmt="%.1f", fontsize=16)
show_values(c2, ax=ax2, fmt="%.1f", fontsize=16)
show_values(c3, ax=ax3, fmt="%.1f", fontsize=16)
show_values(c4, ax=ax4, fmt="%.1f", fontsize=16)
ax0.invert_yaxis()
ax1.invert_yaxis()
ax2.invert_yaxis()
ax3.invert_yaxis()
ax4.invert_yaxis()
xlabs = ylabs = ['0|0','0|1', '1|0', '1|1']
ax0.set_xticks(np.arange(0, Copy_Copy.shape[0], 1))
ax0.set_yticks(np.arange(0, Copy_Copy.shape[1], 1))
ax0.set_xticklabels(xlabs, fontsize=14)
ax0.set_yticklabels(ylabs, fontsize=14)
ax0.set_xticks(np.arange(-0.5, Copy_Copy.shape[0]-0.5, 1), minor=True)
ax0.set_yticks(np.arange(-0.5, Copy_Copy.shape[1]-0.5, 1), minor=True)
ax1.set_xticks(np.arange(0, And_And.shape[0], 1))
ax1.set_yticks(np.arange(0, And_And.shape[1], 1))
ax1.set_xticklabels(xlabs, fontsize=14)
ax1.set_yticklabels(ylabs, fontsize=14)
ax1.set_xticks(np.arange(-0.5, And_And.shape[0]-0.5, 1), minor=True)
ax1.set_yticks(np.arange(-0.5, And_And.shape[1]-0.5, 1), minor=True)
ax2.set_xticks(np.arange(0, Or_Or.shape[0], 1))
ax2.set_yticks(np.arange(0, Or_Or.shape[1], 1))
ax2.set_xticklabels(xlabs, fontsize=14)
ax2.set_yticklabels(ylabs, fontsize=14)
ax2.set_xticks(np.arange(-0.5, Or_Or.shape[0]-0.5, 1), minor=True)
ax2.set_yticks(np.arange(-0.5, Or_Or.shape[1]-0.5, 1), minor=True)
ax3.set_xticks(np.arange(0, Or_Copy.shape[0], 1))
ax3.set_yticks(np.arange(0, Or_Copy.shape[1], 1))
ax3.set_xticklabels(xlabs, fontsize=14)
ax3.set_yticklabels(ylabs, fontsize=14)
ax3.set_xticks(np.arange(-0.5, Or_Copy.shape[0]-0.5, 1), minor=True)
ax3.set_yticks(np.arange(-0.5, Or_Copy.shape[1]-0.5, 1), minor=True)
ax4.set_xticks(np.arange(0, Star.shape[0], 1))
ax4.set_yticks(np.arange(0, Star.shape[1], 1))
ax4.set_xticklabels(xlabs, fontsize=14)
ax4.set_yticklabels(ylabs, fontsize=14)
ax4.set_xticks(np.arange(-0.5, Star.shape[0]-0.5, 1), minor=True)
ax4.set_yticks(np.arange(-0.5, Star.shape[1]-0.5, 1), minor=True)
ax0.xaxis.tick_top()
ax1.xaxis.tick_top()
ax2.xaxis.tick_top()
ax3.xaxis.tick_top()
ax4.xaxis.tick_top()
ax0.set_title('Copy-Copy logic gate\n $EI = %.3f$ \n'%
effective_information(Copy_Copy), fontsize=20, pad=10)
ax1.set_title('And-And logic gate\n $EI = %.3f$ \n'%
effective_information(And_And), fontsize=20, pad=10)
ax2.set_title('Or-Or logic gate\n $EI = %.3f$ \n'%
effective_information(Or_Or), fontsize=20, pad=10)
ax3.set_title('Or-Copy logic gate\n $EI = %.3f$ \n'%
effective_information(Or_Copy), fontsize=20, pad=10)
ax4.set_title('Star-like logic gate\n $EI = %.3f$ \n'%
effective_information(Star), fontsize=20, pad=10)
if save:
plt.savefig(where_to_save_pngs+"Example1_LogicGates.png", bbox_inches='tight', dpi=425)
plt.savefig(where_to_save_pdfs+"Example1_LogicGates.pdf", bbox_inches='tight')
plt.show()
1.1 Add noise to the transition probability matrices¶
In [5]:
noise = np.random.uniform(0.0,0.1,size=Copy_Copy.shape)
Copy_Copy_noise = Copy_Copy + noise
Copy_Copy_noise = Copy_Copy_noise / Copy_Copy_noise.sum(axis=1)
noise = np.random.uniform(0.0,0.1,size=And_And.shape)
And_And_noise = And_And + noise
And_And_noise = And_And_noise / And_And_noise.sum(axis=1)
noise = np.random.uniform(0.0,0.1,size=Or_Or.shape)
Or_Or_noise = Or_Or + noise
Or_Or_noise = Or_Or_noise / Or_Or_noise.sum(axis=1)
noise = np.random.uniform(0.0,0.1,size=Or_Copy.shape)
Or_Copy_noise = Or_Copy + noise
Or_Copy_noise = Or_Copy_noise / Or_Copy_noise.sum(axis=1)
noise = np.random.uniform(0.0,0.1,size=Star.shape)
Star_noise = Star + noise
Star_noise = Star_noise / Star_noise.sum(axis=1)
1.1.1 Plot these TPMs, showing their EI values¶
In [6]:
fig, (ax0, ax1, ax2, ax3, ax4) = plt.subplots(1, 5, figsize=(22,4))
c0 = ax0.pcolor(
np.arange(-0.5, Copy_Copy_noise.shape[0], 1),
np.arange(-0.5, Copy_Copy_noise.shape[0], 1),
Copy_Copy_noise, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c1 = ax1.pcolor(
np.arange(-0.5, And_And_noise.shape[0], 1),
np.arange(-0.5, And_And_noise.shape[0], 1),
And_And_noise, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c2 = ax2.pcolor(
np.arange(-0.5, Or_Or_noise.shape[0], 1),
np.arange(-0.5, Or_Or_noise.shape[0], 1),
Or_Or_noise, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c3 = ax3.pcolor(
np.arange(-0.5, Or_Copy_noise.shape[0], 1),
np.arange(-0.5, Or_Copy_noise.shape[0], 1),
Or_Copy_noise, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c4 = ax4.pcolor(
np.arange(-0.5, Star_noise.shape[0], 1),
np.arange(-0.5, Star_noise.shape[0], 1),
Star_noise, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
show_values(c0, ax=ax0, fmt="%.2f", fontsize=16)
show_values(c1, ax=ax1, fmt="%.2f", fontsize=16)
show_values(c2, ax=ax2, fmt="%.2f", fontsize=16)
show_values(c3, ax=ax3, fmt="%.2f", fontsize=16)
show_values(c4, ax=ax4, fmt="%.2f", fontsize=16)
ax0.invert_yaxis()
ax1.invert_yaxis()
ax2.invert_yaxis()
ax3.invert_yaxis()
ax4.invert_yaxis()
xlabs = ylabs = ['0|0','0|1', '1|0', '1|1']
ax0.set_xticks(np.arange(0, Copy_Copy_noise.shape[0], 1))
ax0.set_yticks(np.arange(0, Copy_Copy_noise.shape[1], 1))
ax0.set_xticklabels(xlabs, fontsize=14)
ax0.set_yticklabels(ylabs, fontsize=14)
ax0.set_xticks(np.arange(-0.5, Copy_Copy_noise.shape[0]-0.5, 1), minor=True)
ax0.set_yticks(np.arange(-0.5, Copy_Copy_noise.shape[1]-0.5, 1), minor=True)
ax1.set_xticks(np.arange(0, And_And_noise.shape[0], 1))
ax1.set_yticks(np.arange(0, And_And_noise.shape[1], 1))
ax1.set_xticklabels(xlabs, fontsize=14)
ax1.set_yticklabels(ylabs, fontsize=14)
ax1.set_xticks(np.arange(-0.5, And_And_noise.shape[0]-0.5, 1), minor=True)
ax1.set_yticks(np.arange(-0.5, And_And_noise.shape[1]-0.5, 1), minor=True)
ax2.set_xticks(np.arange(0, Or_Or_noise.shape[0], 1))
ax2.set_yticks(np.arange(0, Or_Or_noise.shape[1], 1))
ax2.set_xticklabels(xlabs, fontsize=14)
ax2.set_yticklabels(ylabs, fontsize=14)
ax2.set_xticks(np.arange(-0.5, Or_Or_noise.shape[0]-0.5, 1), minor=True)
ax2.set_yticks(np.arange(-0.5, Or_Or_noise.shape[1]-0.5, 1), minor=True)
ax3.set_xticks(np.arange(0, Or_Copy_noise.shape[0], 1))
ax3.set_yticks(np.arange(0, Or_Copy_noise.shape[1], 1))
ax3.set_xticklabels(xlabs, fontsize=14)
ax3.set_yticklabels(ylabs, fontsize=14)
ax3.set_xticks(np.arange(-0.5, Or_Copy_noise.shape[0]-0.5, 1), minor=True)
ax3.set_yticks(np.arange(-0.5, Or_Copy_noise.shape[1]-0.5, 1), minor=True)
ax4.set_xticks(np.arange(0, Star_noise.shape[0], 1))
ax4.set_yticks(np.arange(0, Star_noise.shape[1], 1))
ax4.set_xticklabels(xlabs, fontsize=14)
ax4.set_yticklabels(ylabs, fontsize=14)
ax4.set_xticks(np.arange(-0.5, Star_noise.shape[0]-0.5, 1), minor=True)
ax4.set_yticks(np.arange(-0.5, Star_noise.shape[1]-0.5, 1), minor=True)
ax0.xaxis.tick_top()
ax1.xaxis.tick_top()
ax2.xaxis.tick_top()
ax3.xaxis.tick_top()
ax4.xaxis.tick_top()
ax0.set_title('Copy-Copy logic gate\n $EI = %.3f$ \n'%
effective_information(Copy_Copy_noise), fontsize=20, pad=10)
ax1.set_title('And-And logic gate\n $EI = %.3f$ \n'%
effective_information(And_And_noise), fontsize=20, pad=10)
ax2.set_title('Or-Or logic gate\n $EI = %.3f$ \n'%
effective_information(Or_Or_noise), fontsize=20, pad=10)
ax3.set_title('Or-Copy logic gate\n $EI = %.3f$ \n'%
effective_information(Or_Copy_noise), fontsize=20, pad=10)
ax4.set_title('Star-like logic gate\n $EI = %.3f$ \n'%
effective_information(Star_noise), fontsize=20, pad=10)
if save:
plt.savefig(where_to_save_pngs+"Example2_LogicGates.png", bbox_inches='tight', dpi=425)
plt.savefig(where_to_save_pdfs+"Example2_LogicGates.pdf", bbox_inches='tight')
plt.show()
1.2 Random Matrices¶
In [7]:
rand0 = np.random.rand(4,4)
rand0 = np.array([rand0[i]/sum(rand0[i])
for i in range(rand0.shape[0])])
rand1 = np.random.rand(4,4)
rand1 = np.array([rand1[i]/sum(rand1[i])
for i in range(rand1.shape[0])])
rand2 = np.random.rand(4,4)
rand2 = np.array([rand2[i]/sum(rand2[i])
for i in range(rand2.shape[0])])
rand3 = np.random.rand(4,4)
rand3 = np.array([rand3[i]/sum(rand3[i])
for i in range(rand3.shape[0])])
rand4 = np.random.rand(4,4)
rand4 = np.array([rand4[i]/sum(rand4[i])
for i in range(rand4.shape[0])])
1.2.1 Plot these TPMs, showing their EI values¶
In [8]:
fig, (ax0, ax1, ax2, ax3, ax4) = plt.subplots(1, 5, figsize=(22,4))
c0 = ax0.pcolor(
np.arange(-0.5, rand0.shape[0], 1),
np.arange(-0.5, rand0.shape[0], 1),
rand0, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c1 = ax1.pcolor(
np.arange(-0.5, rand1.shape[0], 1),
np.arange(-0.5, rand1.shape[0], 1),
rand1, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c2 = ax2.pcolor(
np.arange(-0.5, rand2.shape[0], 1),
np.arange(-0.5, rand2.shape[0], 1),
rand2, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c3 = ax3.pcolor(
np.arange(-0.5, rand3.shape[0], 1),
np.arange(-0.5, rand3.shape[0], 1),
rand3, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
c4 = ax4.pcolor(
np.arange(-0.5, rand4.shape[0], 1),
np.arange(-0.5, rand4.shape[0], 1),
rand4, edgecolors='#999999', linewidths=3.0, cmap=EI_cmap)
show_values(c0, ax=ax0, fmt="%.2f", fontsize=16)
show_values(c1, ax=ax1, fmt="%.2f", fontsize=16)
show_values(c2, ax=ax2, fmt="%.2f", fontsize=16)
show_values(c3, ax=ax3, fmt="%.2f", fontsize=16)
show_values(c4, ax=ax4, fmt="%.2f", fontsize=16)
ax0.invert_yaxis()
ax1.invert_yaxis()
ax2.invert_yaxis()
ax3.invert_yaxis()
ax4.invert_yaxis()
xlabs = ylabs = ['0|0','0|1', '1|0', '1|1']
ax0.set_xticks(np.arange(0, rand0.shape[0], 1))
ax0.set_yticks(np.arange(0, rand0.shape[1], 1))
ax0.set_xticklabels(xlabs, fontsize=14)
ax0.set_yticklabels(ylabs, fontsize=14)
ax0.set_xticks(np.arange(-0.5, rand0.shape[0]-0.5, 1), minor=True)
ax0.set_yticks(np.arange(-0.5, rand0.shape[1]-0.5, 1), minor=True)
ax1.set_xticks(np.arange(0, rand1.shape[0], 1))
ax1.set_yticks(np.arange(0, rand1.shape[1], 1))
ax1.set_xticklabels(xlabs, fontsize=14)
ax1.set_yticklabels(ylabs, fontsize=14)
ax1.set_xticks(np.arange(-0.5, rand1.shape[0]-0.5, 1), minor=True)
ax1.set_yticks(np.arange(-0.5, rand1.shape[1]-0.5, 1), minor=True)
ax2.set_xticks(np.arange(0, rand2.shape[0], 1))
ax2.set_yticks(np.arange(0, rand2.shape[1], 1))
ax2.set_xticklabels(xlabs, fontsize=14)
ax2.set_yticklabels(ylabs, fontsize=14)
ax2.set_xticks(np.arange(-0.5, rand2.shape[0]-0.5, 1), minor=True)
ax2.set_yticks(np.arange(-0.5, rand2.shape[1]-0.5, 1), minor=True)
ax3.set_xticks(np.arange(0, rand3.shape[0], 1))
ax3.set_yticks(np.arange(0, rand3.shape[1], 1))
ax3.set_xticklabels(xlabs, fontsize=14)
ax3.set_yticklabels(ylabs, fontsize=14)
ax3.set_xticks(np.arange(-0.5, rand3.shape[0]-0.5, 1), minor=True)
ax3.set_yticks(np.arange(-0.5, rand3.shape[1]-0.5, 1), minor=True)
ax4.set_xticks(np.arange(0, rand4.shape[0], 1))
ax4.set_yticks(np.arange(0, rand4.shape[1], 1))
ax4.set_xticklabels(xlabs, fontsize=14)
ax4.set_yticklabels(ylabs, fontsize=14)
ax4.set_xticks(np.arange(-0.5, rand4.shape[0]-0.5, 1), minor=True)
ax4.set_yticks(np.arange(-0.5, rand4.shape[1]-0.5, 1), minor=True)
ax0.xaxis.tick_top()
ax1.xaxis.tick_top()
ax2.xaxis.tick_top()
ax3.xaxis.tick_top()
ax4.xaxis.tick_top()
ax0.set_title('Random TPM 0\n $EI = %.3f$ \n'%
effective_information(rand0), fontsize=20, pad=10)
ax1.set_title('Random TPM 1\n $EI = %.3f$ \n'%
effective_information(rand1), fontsize=20, pad=10)
ax2.set_title('Random TPM 2\n $EI = %.3f$ \n'%
effective_information(rand2), fontsize=20, pad=10)
ax3.set_title('Random TPM 3\n $EI = %.3f$ \n'%
effective_information(rand3), fontsize=20, pad=10)
ax4.set_title('Random TPM 4\n $EI = %.3f$ \n'%
effective_information(rand4), fontsize=20, pad=10)
if save:
plt.savefig(where_to_save_pngs+"Example3_RandomTPMs.png", bbox_inches='tight', dpi=425)
plt.savefig(where_to_save_pdfs+"Example3_RandomTPMs.pdf", bbox_inches='tight')
plt.show()
1.3 Example calculation figure (Supplemental Information, Figure 1)¶
In [9]:
############ PLOTTING SETUP ##############
from matplotlib import gridspec
plt.rc('axes', linewidth=3)
font = {'family': 'serif',
'color': 'k',
'weight': 'normal',
'size': 28}
plt.rc('text', usetex=True)
plt.rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
##########################################
In [10]:
TPM = np.array([[0.0, 0.0, 0.0, 0.5, 0.5],
[1/3, 0.0, 1/3, 1/3, 0.0],
[0.0, 0.5, 0.0, 0.5, 0.0],
[0.0, 0.0, 0.0, 0.0, 1.0],
[0.5, 0.0, 0.0, 0.5, 0.0]])
In [11]:
fig = plt.figure(figsize=(7.3, 16))
gs = gridspec.GridSpec(4, 1, height_ratios=[7, 10, 1.2, 1.2])
xlabs = ylabs = ['$A$', '$B$', '$C$', '$D$', '$E$']
ax0 = plt.subplot(gs[1])
ax0.set_xticks(np.arange(0, TPM.shape[0], 1))
ax0.set_yticks(np.arange(0, TPM.shape[0], 1))
ax0.set_xticklabels(xlabs, fontsize=32)
ax0.set_yticklabels(ylabs, fontsize=32)
ax0.set_xticks(np.arange(-0.5, TPM.shape[0]-0.5, 1), minor=True)
ax0.set_yticks(np.arange(-0.5, TPM.shape[0]-0.5, 1), minor=True)
ax0.tick_params(axis='y', which='major', pad=7)
c0 = plt.pcolor(
np.arange(-.5, TPM.shape[0], 1),
np.arange(-.5, TPM.shape[1], 1),
TPM, edgecolors='k', linewidths=3.0, cmap='Blues', vmin=-.05, vmax=1.2)
show_values(c0, ax=ax0, fmt="%.2f", fontsize=26)
ax0.invert_yaxis()
ax0.xaxis.set_label_position("top")
ax0.xaxis.tick_top()
ax0.set_xlabel(r'$t + 1$', size=28, labelpad=8.0)
ax0.set_ylabel(r'$t$', size=28, rotation=0, labelpad=27.0)
ax0.xaxis.label.set_position((0.5,5.0))
ax0.text(4.65, 0.20, '$=W_{A}^{out}$', ha='left',
rotation=0, wrap=True, size=32)
ax0.text(4.65, 1.20, '$=W_{B}^{out}$', ha='left',
rotation=0, wrap=True, size=32)
ax0.text(4.65, 2.20, '$=W_{C}^{out}$', ha='left',
rotation=0, wrap=True, size=32)
ax0.text(4.65, 3.20, '$=W_{D}^{out}$', ha='left',
rotation=0, wrap=True, size=32)
ax0.text(4.65, 4.20, '$=W_{E}^{out}$', ha='left',
rotation=0, wrap=True, size=32)
ms = 78
ax1 = plt.subplot(gs[2])
Win_j = TPM.sum(axis=0).reshape(1,TPM.shape[0])
Win = W_in(TPM).reshape(1,TPM.shape[0])
c1 = plt.pcolor(
np.arange(-.5, Win_j.shape[1], 1),
np.arange(0.0, 1.5, 1),
Win_j, edgecolors='k', linewidths=3.0, cmap='Oranges', vmin=0, vmax=3.0)
show_values(c1, ax=ax1, fmt="%.2f", fontsize=26)
ax1.set_xlabel("")
ax1.set_ylabel("")
ax1.set_xticks([])
ax1.set_yticks([])
ax1.set_xticklabels([''])
ax1.set_yticklabels([''])
ax2 = plt.subplot(gs[3])
c2 = plt.pcolor(
np.arange(-.5, Win.shape[1], 1),
np.arange(0.0, 1.5, 1),
Win, edgecolors='k', linewidths=3.0, cmap='Oranges', vmin=0, vmax=0.75)
show_values(c2, ax=ax2, fmt="%.2f", fontsize=26)
ax2.set_xlabel("")
ax2.set_ylabel("")
ax2.set_xticks([])
ax2.set_yticks([])
ax2.set_xticklabels([''])
ax2.set_yticklabels([''])
string10 = r'$= \displaystyle\sum_{i=1}^N w_{ij}$'
string20 = r'$= \langle W_{i}^{out} \rangle$'
ax1.text(4.65, -.15, string10, ha='left', rotation=0, wrap=True, size=28)
ax2.text(4.65, 0.15, string20, ha='left', rotation=0, wrap=True, size=32)
plt.subplots_adjust(wspace=0, hspace=0.05)
if save:
plt.savefig(where_to_save_pngs+"Example4_ExampleTPM.png", bbox_inches='tight', dpi=425)
plt.savefig(where_to_save_pdfs+"Example4_ExampleTPM.pdf", bbox_inches='tight')
plt.show()
/usr/local/lib/python3.7/site-packages/matplotlib/font_manager.py:1241: UserWarning: findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans. (prop.get_family(), self.defaultFamily[fontext])) /usr/local/lib/python3.7/site-packages/matplotlib/font_manager.py:1241: UserWarning: findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans. (prop.get_family(), self.defaultFamily[fontext])) /usr/local/lib/python3.7/site-packages/matplotlib/font_manager.py:1241: UserWarning: findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans. (prop.get_family(), self.defaultFamily[fontext]))
The adjacency matrix of a network with 1.158 bits of effective information. The rows correspond to Wouti, a vector of probabilities that a random walker on node vi at time t transitions to vj in the following time step, t+1. ⟨Wouti⟩ represents the (normalized) input weight distribution of the network, that is, the probabilities that a random walker will arrive at a given node vj at t+1, after a uniform introduction of random walkers into the network at t.
In [12]:
Win = W_in(TPM)
vals = Win
Win_cols = plt.cm.Oranges(Win+0.15)
WoutA_cols = plt.cm.Blues(max(TPM[0]))
WoutB_cols = plt.cm.Blues(max(TPM[1]))
WoutC_cols = plt.cm.Blues(max(TPM[2]))
WoutD_cols = plt.cm.Blues(max(TPM[3])*0.66)
WoutE_cols = plt.cm.Blues(max(TPM[4]))
In [13]:
tpm0 = TPM
Gtpm0 = check_network(tpm0)
fig, ((ax00, ax01), (ax02, ax03), (ax04, ax05)) = plt.subplots(
3, 2, figsize=(16*1.3,9*1.3))
plt.subplots_adjust(left=None, bottom=0.1, right=None,
top=None, wspace=0.17, hspace=0.3)
ax00.bar(
np.linspace(0.0, 5.5, 5),
TPM[0]+0.01, color=WoutA_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$W_{A}^{out}$')
x_wij = np.linspace(0.0, 5.5, 5)
y_wij = tpm0[0]
for i in range(len(x_wij)):
ax00.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax00.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='royalblue', alpha=0.6)
ax00.bar(
np.linspace(-.6, 4.9, 5),
vals, color=Win_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$\langle W_{i}^{out} \rangle$')
x_wij = np.linspace(-.6, 4.9, 5)
y_wij = vals
for i in range(len(x_wij)):
ax00.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax00.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#e56c13', alpha=0.9)
ax00.set_ylim(0.0,1.19)
ax00.set_xlim(-0.95,5.85)
ax00.set_xticks(np.linspace(-.3, 5.2, 5))
ax00.set_yticks(np.linspace(0, 1, 6))
ax00.set_yticklabels(np.round(np.linspace(0, 1, 6), 2), size=16)
ax00.set_xticklabels(xlabs, size=22)
ax00.set_ylabel(r'$w_{ij}$', fontsize=30, rotation='horizontal', labelpad=25)
ax00.set_xlabel('out-neighbor', fontsize=20, labelpad=0)
ax00.set_axisbelow(True)
ax00.grid(which='major', linestyle='-',
color='#999999', linewidth=2.5, alpha=0.3)
ax00.legend(loc=2, fontsize=22, framealpha=0.7)
strax1 = r'$D_{KL}[W^{out}_{A}||\langle W_{i}^{out} \rangle] = %.3f $'%\
effect_information_i(Gtpm0, node_i=0)
ax00.text(3.12, 0.802, strax1, ha='center', fontsize=28, color='k')
ax02.bar(
np.linspace(0.0, 5.5, 5),
tpm0[1]+0.01, color=WoutB_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$W_{B}^{out}$')
x_wij = np.linspace(0.0, 5.5, 5)
y_wij = tpm0[1]
for i in range(len(x_wij)):
ax02.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax02.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='royalblue', alpha=0.6)
ax02.set_xticklabels(["A", "B", "C", "D", "E"])
ax02.bar(
np.linspace(-.6, 4.9, 5),
vals, color=Win_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$\langle W_{i}^{out} \rangle$')
x_wij = np.linspace(-.6, 4.9, 5)
y_wij = vals
for i in range(len(x_wij)):
ax02.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax02.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#e56c13', alpha=0.9)
ax02.set_ylim(0.0,1.19)
ax02.set_xlim(-0.95,5.85)
ax02.set_xticks(np.linspace(-.3, 5.2, 5))
ax02.set_yticks(np.linspace(0, 1, 6))
ax02.set_yticklabels(np.round(np.linspace(0, 1, 6), 2), size=16)
ax02.set_xticklabels(xlabs, size=22)
ax02.set_ylabel(r'$w_{ij}$', fontsize=30, rotation='horizontal', labelpad=25)
ax02.set_xlabel('out-neighbor', fontsize=20, labelpad=0)
ax02.set_axisbelow(True)
ax02.grid(which='major', linestyle='-', color='#999999', linewidth=2.5, alpha=0.3)
ax02.legend(loc=2, fontsize=22, framealpha=0.7)
strax1 = r'$D_{KL}[W^{out}_{B}||\langle W_{i}^{out} \rangle] = %.3f $'%\
effect_information_i(Gtpm0, node_i=1)
ax02.text(3.12, 0.802, strax1, ha='center', fontsize=28, color='k')
ax04.bar(
np.linspace(0.0, 5.5, 5),
tpm0[2]+0.01, color=WoutC_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$W_{C}^{out}$')
x_wij = np.linspace(0.0, 5.5, 5)
y_wij = tpm0[2]
for i in range(len(x_wij)):
ax04.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax04.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='royalblue', alpha=0.6)
ax04.set_xticklabels(["A", "B", "C", "D", "E"])
ax04.bar(
np.linspace(-.6, 4.9, 5),
vals, color=Win_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$\langle W_{i}^{out} \rangle$')
x_wij = np.linspace(-.6, 4.9, 5)
y_wij = vals
for i in range(len(x_wij)):
ax04.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax04.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#e56c13', alpha=0.9)
ax04.set_ylim(0.0,1.19)
ax04.set_xlim(-0.95,5.85)
ax04.set_xticks(np.linspace(-.3, 5.2, 5))
ax04.set_yticks(np.linspace(0, 1, 6))
ax04.set_yticklabels(np.round(np.linspace(0, 1, 6), 2), size=16)
ax04.set_xticklabels(xlabs, size=22)
ax04.set_ylabel(r'$w_{ij}$', fontsize=30, rotation='horizontal', labelpad=25)
ax04.set_xlabel('out-neighbor', fontsize=20, labelpad=0)
ax04.set_axisbelow(True)
ax04.grid(which='major', linestyle='-', color='#999999', linewidth=2.5, alpha=0.3)
ax04.legend(loc=2, fontsize=22, framealpha=0.7)
strax1 = r'$D_{KL}[W^{out}_{C}||\langle W_{i}^{out} \rangle] = %.3f $'%\
effect_information_i(Gtpm0, node_i=2)
ax04.text(3.12, 0.802, strax1, ha='center', fontsize=28, color='k')
ax01.bar(
np.linspace(0.0, 5.5, 5),
tpm0[3]+0.01, color=WoutD_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$W_{D}^{out}$')
x_wij = np.linspace(0.0, 5.5, 5)
y_wij = tpm0[3]
for i in range(len(x_wij)):
ax01.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax01.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='royalblue', alpha=0.6)
ax01.set_xticklabels(["A", "B", "C", "D", "E"])
ax01.bar(
np.linspace(-.6, 4.9, 5),
vals, color=Win_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$\langle W_{i}^{out} \rangle$')
x_wij = np.linspace(-.6, 4.9, 5)
y_wij = vals
for i in range(len(x_wij)):
ax01.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax01.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#e56c13', alpha=0.9)
ax01.set_ylim(0.0,1.19)
ax01.set_xlim(-0.95,5.85)
ax01.set_xticks(np.linspace(-.3, 5.2, 5))
ax01.set_yticks(np.linspace(0, 1, 6))
ax01.set_yticklabels(np.round(np.linspace(0, 1, 6), 2), size=16)
ax01.set_xticklabels(xlabs, size=22)
ax01.set_ylabel(r'$w_{ij}$', fontsize=30, rotation='horizontal', labelpad=25)
ax01.set_xlabel('out-neighbor', fontsize=20, labelpad=0)
ax01.set_axisbelow(True)
ax01.grid(which='major', linestyle='-', color='#999999', linewidth=2.5, alpha=0.3)
ax01.legend(loc=2, fontsize=22, framealpha=0.7)
strax1 = r'$D_{KL}[W^{out}_{D}||\langle W_{i}^{out} \rangle] = %.3f $'%\
effect_information_i(Gtpm0, node_i=3)
ax01.text(3.12, 0.802, strax1, ha='center', fontsize=28, color='k')
ax03.bar(
np.linspace(0.0, 5.5, 5),
tpm0[4]+0.01, color=WoutE_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$W_{E}^{out}$')
x_wij = np.linspace(0.0, 5.5, 5)
y_wij = tpm0[4]
for i in range(len(x_wij)):
ax03.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax03.text(x_wij[i], y_wij[i]+0.040, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='royalblue', alpha=0.6)
ax03.set_xticklabels(["A", "B", "C", "D", "E"])
ax03.bar(
np.linspace(-.6, 4.9, 5),
vals, color=Win_cols, linewidth=2.0, edgecolor='k',
width=0.6, label=r'$\langle W_{i}^{out} \rangle$')
x_wij = np.linspace(-.6, 4.9, 5)
y_wij = vals
for i in range(len(x_wij)):
ax03.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#262626')
ax03.text(x_wij[i], y_wij[i]+0.035, "%.2f"%y_wij[i], ha='center',
fontsize=21, color='#e56c13', alpha=0.9)
ax03.set_ylim(0.0,1.19)
ax03.set_xlim(-0.95,5.85)
ax03.set_xticks(np.linspace(-.3, 5.2, 5))
ax03.set_yticks(np.linspace(0, 1, 6))
ax03.set_yticklabels(np.round(np.linspace(0, 1, 6), 2), size=16)
ax03.set_xticklabels(xlabs, size=22)
ax03.set_ylabel(r'$w_{ij}$', fontsize=30, rotation='horizontal', labelpad=25)
ax03.set_xlabel('out-neighbor', fontsize=20, labelpad=0)
ax03.set_axisbelow(True)
ax03.grid(which='major', linestyle='-', color='#999999', linewidth=2.5, alpha=0.3)
ax03.legend(loc=2,fontsize=22, framealpha=0.7)
strax1 = r'$D_{KL}[W^{out}_{E}||\langle W_{i}^{out} \rangle] = %.3f $'%\
effect_information_i(Gtpm0, node_i=4)
ax03.text(3.12, 0.802, strax1, ha='center', fontsize=28, color='k')
string1 = r'$EI = \displaystyle\frac{1}{N}$'+\
r'$\displaystyle\sum_{i=1}^N D_{KL}[W_{i}^{out}||$'+\
r'$\langle W_{i}^{out} \rangle]$'
string2 = r'$EI = \displaystyle\frac{1}{5}$'+\
r'$\hspace{0.5cm} [0.592 + 1.061 + 1.385 + 1.737 + 1.016]$'
string3 = r'$EI = 1.158 \hspace{0.5cm}$' r'$\rm bits$'
ax05.text(-.02, 0.590, string1, ha='left', rotation=0, wrap=False, size=28)
ax05.text(-.02, 0.250, string2, ha='left', rotation=0, wrap=False, size=28)
ax05.text(-.02, -0.01, string3, ha='left', rotation=0, wrap=False, size=28)
ax05.axis('off')
if save:
plt.savefig(where_to_save_pngs+\
"Example4_ExampleCalculation.png", bbox_inches='tight', dpi=425)
plt.savefig(where_to_save_pdfs+\
"Example4_ExampleCalculation.pdf", bbox_inches='tight')
plt.show()
/usr/local/lib/python3.7/site-packages/matplotlib/font_manager.py:1241: UserWarning: findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans. (prop.get_family(), self.defaultFamily[fontext])) /usr/local/lib/python3.7/site-packages/matplotlib/font_manager.py:1241: UserWarning: findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans. (prop.get_family(), self.defaultFamily[fontext]))
Each node's contribution to the EI (EIi) is the KL divergence of its Wouti vector from the network's ⟨Wouti⟩, known as the effect information.
EI=1NN∑i=1DKL[Wouti||⟨Wouti⟩]where EI is the average of the effect information, EIi, of each node. This is equivalent to our derivation of EI from first principles above since
EI=1NN∑i=1DKL[Wouti||⟨Wouti⟩]=1NN∑i=1N∑j=1wijlog2(wijWj)=1NN∑i=1(N∑j=1wijlog2(wij)−N∑j=1wijlog2(Wj))=1NN∑i=1N∑j=1wijlog2(wij)−1NN∑i=1N∑j=1wijlog2(Wj)- Note that for a given node, vi, the term in the first summation above, ∑Nj=1wijlog2(wij), is equivalent to the negative entropy of the out-weights from vi, −H(Wouti). Also note that Wj, the jth element in the ⟨Wouti⟩ vector, is the normalized sum of the incoming weights to vj from its neighbors, vi, such that Wj=1N∑Ni=1wij. We substitute these two terms into the equation above such that:
This is equivalent to the formulation of EI above, since H(⟨Wouti⟩)=−∑Nj=1Wjlog2(Wj):
EI=H(⟨Wouti⟩)−⟨H(Wouti)⟩In this figure, we adopt the relative entropy formulation of EI for ease of derivation.
1.4 Network motifs and effective information (N = 3)¶
In [14]:
G01 = nx.DiGraph()
G02 = nx.DiGraph()
G03 = nx.DiGraph()
G04 = nx.DiGraph()
G05 = nx.DiGraph()
G06 = nx.DiGraph()
G07 = nx.DiGraph()
G08 = nx.DiGraph()
G09 = nx.DiGraph()
G10 = nx.DiGraph()
G11 = nx.DiGraph()
G12 = nx.DiGraph()
G13 = nx.DiGraph()
G01.add_nodes_from([0,1,2])
G02.add_nodes_from([0,1,2])
G03.add_nodes_from([0,1,2])
G04.add_nodes_from([0,1,2])
G05.add_nodes_from([0,1,2])
G06.add_nodes_from([0,1,2])
G07.add_nodes_from([0,1,2])
G08.add_nodes_from([0,1,2])
G09.add_nodes_from([0,1,2])
G10.add_nodes_from([0,1,2])
G11.add_nodes_from([0,1,2])
G12.add_nodes_from([0,1,2])
G13.add_nodes_from([0,1,2])
G01.add_edges_from([(0,1),(0,2)])
G02.add_edges_from([(0,1),(2,0)])
G03.add_edges_from([(0,1),(0,2),(2,0)])
G04.add_edges_from([(1,0),(2,0)])
G05.add_edges_from([(1,0),(1,2),(0,2)]) # e. coli
G06.add_edges_from([(1,0),(1,2),(0,2),(0,1)])
G07.add_edges_from([(1,0),(0,2),(2,0)])
G08.add_edges_from([(1,0),(0,1),(0,2),(2,0)])
G09.add_edges_from([(1,0),(0,2),(2,1)])
G10.add_edges_from([(1,0),(2,0),(1,2),(0,1)])
G11.add_edges_from([(1,0),(2,0),(2,1),(0,1)])
G12.add_edges_from([(0,1),(1,0),(1,2),(2,1),(0,2)])
G13.add_edges_from([(0,1),(1,0),(1,2),(2,1),(0,2),(2,0)])
motif_dict = {"Motif 01": {"G":G01, "edges":str(list(G01.edges())), "EI":effective_information(G01)},
"Motif 02": {"G":G02, "edges":str(list(G02.edges())), "EI":effective_information(G02)},
"Motif 03": {"G":G03, "edges":str(list(G03.edges())), "EI":effective_information(G03)},
"Motif 04": {"G":G04, "edges":str(list(G04.edges())), "EI":effective_information(G04)},
"Motif 05": {"G":G05, "edges":str(list(G05.edges())), "EI":effective_information(G05)},
"Motif 06": {"G":G06, "edges":str(list(G06.edges())), "EI":effective_information(G06)},
"Motif 07": {"G":G07, "edges":str(list(G07.edges())), "EI":effective_information(G07)},
"Motif 08": {"G":G08, "edges":str(list(G08.edges())), "EI":effective_information(G08)},
"Motif 09": {"G":G09, "edges":str(list(G09.edges())), "EI":effective_information(G09)},
"Motif 10": {"G":G10, "edges":str(list(G10.edges())), "EI":effective_information(G10)},
"Motif 11": {"G":G11, "edges":str(list(G11.edges())), "EI":effective_information(G11)},
"Motif 12": {"G":G12, "edges":str(list(G12.edges())), "EI":effective_information(G12)},
"Motif 13": {"G":G13, "edges":str(list(G13.edges())), "EI":effective_information(G13)}}
ei_heights = np.array([list(motif_dict.values())[i]['EI']
for i in range(len(list(motif_dict.values())))]) + 0.005
ei_bars = np.array(range(len(list(motif_dict.values()))))
colors = ["#486164","#9094c9","#ab4e53","#fa8d11","#74d76c",
"#bc7dc6","#db453b","#cad24b","#8f52d2","#00aaff",
"#c2843a","#4f5435","#d05185"]
bar_labels = list(motif_dict.keys())
In [15]:
import numpy as np
import networkx as nx
from matplotlib import gridspec
colors = ["#45af9c","#5b91cb","#9f8448","#bf6d8c","#9876c0","#bb6eac","#5ea05c",
"#cf5c57","#c24864","#c69932","#d3468f","#ce5c2f","#6b6cd9","#78b43d","#ba58c2"]
i = 10
ns = 250
ew = 3.5
nc = 'w'
ec = '#333333'
oc = '#e4c600'
nc_o = '#333333'
fig, ax = plt.subplots(1,1,figsize=(17,8))
plt.subplots_adjust(wspace=0.10, hspace=0.1)
plt.rc('axes', axisbelow=True)
plt.rc('axes', linewidth=1.5)
gs = gridspec.GridSpec(2, len(ei_bars), height_ratios=[7,1.7])
ax0 = plt.subplot(gs[0, :])
cols_i = 'grey'
ax0.bar(ei_bars, ei_heights, color=colors, width=0.75,
edgecolor='#333333', linewidth=3, alpha=1)
ax0.set_xlim(min(ei_bars)-0.5,max(ei_bars)+0.5)
ax0.set_xticks(ei_bars)
ax0.set_xticklabels([""]*13)
ax0.set_yticklabels(np.round(np.linspace(0,1.6,num=9),2), size=18)
ax0.set_ylim(0, max(ei_heights)+0.04)
ax0.set_xlim(-0.5,12.5)
ax0.set_ylabel("$EI$", size=28)
ax0.grid(linestyle='-', color='#999999', linewidth=2.5, alpha=0.35)
for q, Q in enumerate(list(motif_dict.values())):
g = Q['G']
ax0 = plt.subplot(gs[-1, q])
pos = nx.circular_layout(g)
nx.draw_networkx_nodes(g, pos, node_size=ns, node_color=colors[q],
linewidths=3, edgecolors="#333333", ax=ax0)
nx.draw_networkx_edges(g, pos, width=ew*0.9, edge_color="#3F3F3F",#colors[q],
arrowsize=19, alpha=1, ax=ax0)
ax0.set_axis_off()
posy = np.array(list(zip(*list(pos.values())))[1])
posx = np.array(list(zip(*list(pos.values())))[0])
ax0.set_ylim(min(posy)*1.35, max(posy)*1.5)
ax0.set_xlim(min(posx)*1.69, max(posx)*1.6)
title = list(motif_dict.keys())[q]
ax0.set_title(title, fontsize=16, pad=-0.35)
if save:
plt.savefig("../figs/pngs/EffectiveInformation_NetworkMotifs.png", bbox_inches='tight', dpi=425)
plt.savefig("../figs/pdfs/EffectiveInformation_NetworkMotifs.pdf", bbox_inches='tight', dpi=425)
plt.show()
/usr/local/lib/python3.7/site-packages/matplotlib/font_manager.py:1241: UserWarning: findfont: Font family ['sans-serif'] not found. Falling back to DejaVu Sans. (prop.get_family(), self.defaultFamily[fontext]))
End of Chapter 01. In Chapter 02, we will look at the EI of common networks¶
References:¶
- Hoel, E. P. (2017). When the Map Is Better Than the Territory. Entropy, 19(5), 188. doi: 10.3390/e19050188
- Hoel, E. P., Albantakis, L., & Tononi, G. (2013). Quantifying causal emergence shows that macro can beat micro. Proceedings of the National Academy of Sciences, 110(49), 19790–5. doi: 10.1073/pnas.1314922110
- Tononi, G. (2001). Information measures for conscious experience. Archives Italiennes de Biologie,139(4), 367–371. doi: 10.4449/aib.v139i4.51