Kelly#
Antiquary, Reverence, Criticism#
Mozart/Crouche/Adulthood, Bach/Kelly/Maturity, Ludwig/Edmonds/Teenage
1. Phonetics
\
2. Temperament -> 4. Modes -> 5. NexToken -> 6. Emotion
/
3. Scales
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import networkx as nx
import matplotlib.pyplot as plt
import numpy as np
# Create a directed graph
G = nx.DiGraph()
# Add nodes representing different levels (subatomic, atomic, cosmic, financial, social)
levels = ['i: Audiencex', 'ii7♭5: Composer', 'V7: Performer']
# Add nodes to the graph
G.add_nodes_from(levels)
# Add edges to represent the flow of information (photons)
# Assuming the flow is directional from more fundamental levels to more complex ones
edges = [('ii7♭5: Composer', 'V7: Performer'),
('V7: Performer', 'i: Audiencex'),]
# Add edges to the graph
G.add_edges_from(edges)
# Define positions for the nodes in a circular layout
pos = nx.circular_layout(G)
# Set the figure size (width, height)
plt.figure(figsize=(10, 10)) # Adjust the size as needed
# Draw the main nodes
nx.draw_networkx_nodes(G, pos, node_color='lightblue', node_size=30000)
# Draw the edges with arrows and create space between the arrowhead and the node
nx.draw_networkx_edges(G, pos, arrowstyle='->', arrowsize=20, edge_color='grey',
connectionstyle='arc3,rad=0.2') # Adjust rad for more/less space
# Add smaller red nodes (photon nodes) exactly on the circular layout
for edge in edges:
# Calculate the vector between the two nodes
vector = pos[edge[1]] - pos[edge[0]]
# Calculate the midpoint
mid_point = pos[edge[0]] + 0.5 * vector
# Normalize to ensure it's on the circle
radius = np.linalg.norm(pos[edge[0]])
mid_point_on_circle = mid_point / np.linalg.norm(mid_point) * radius
# Draw the small red photon node at the midpoint on the circular layout
plt.scatter(mid_point_on_circle[0], mid_point_on_circle[1], c='lightpink', s=500, zorder=3)
# Draw a small lime green arrow inside the red node to indicate direction
arrow_vector = vector / np.linalg.norm(vector) * 0.1 # Scale down arrow size
plt.arrow(mid_point_on_circle[0] - 0.05 * arrow_vector[0],
mid_point_on_circle[1] - 0.05 * arrow_vector[1],
arrow_vector[0], arrow_vector[1],
head_width=0.03, head_length=0.05, fc='limegreen', ec='limegreen', zorder=4)
# Draw the labels for the main nodes
nx.draw_networkx_labels(G, pos, font_size=18, font_weight='normal')
# Add a legend for "Photon/Info"
plt.scatter([], [], c='lightpink', s=100, label='Chord Progression') # Empty scatter for the legend
plt.legend(scatterpoints=1, frameon=True, labelspacing=1, loc='upper right')
# Set the title and display the plot
plt.title('Emotional Arc', fontsize=15)
plt.axis('off')
plt.show()
These six topics cover all aspects of music. Challenge GPT-4o or any other ChatBot to find a issue that isn’t seamlessly subsumed by one of these headlines. Meanwhile, What could stop the Nvidia frenzy? Let’s test the resilience of this framework against an orthogonal topic! Lento (57 BPM) but “pocket” is Grave iambic meter (28.5 BPM) with a da-DUM “churchy” stamp, key scales are on B & E with G♯ minor/B Major modal interchange: VI-V7-i/v-I. One “must” see that these are transmutted ii-V7-i triadic progressions in two modes, with VI substitution in G♯ minor & ii deletion in E Major. Call-answer Gospel tokens are remeniscent of the cantor-audience in synagogue & these pockets are firmly rooted in the iambic meter. Given that the composer (R Kelly), performer (Marvin Sapp), and the audience (Black christians) are steeped in Gospel tradition, ministers of the word, and active recipients, this song is a tempest in a delicious teapot of soul & worship. Traditions are honored, craftsmanship is gifted, and Marvin Sapp is commended for reaching out to a “sinner”, through whom God has clearly worked a supreme miracle. (How did the framework perform?)#
1. Nodes
\
2. Edges -> 4. Nodes -> 5. Edges -> 6. Scale
/
3. Scale
ii \(\mu\) Single Note#
ii\(f(t)\) Phonetics: 42 43Fractals\(440Hz \times 2^{\frac{N}{12}}\), \(S_0(t) \times e^{logHR}\), \(\frac{S N(d_1)}{K N(d_2)} \times e^{rT}\)
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import numpy as np
import matplotlib.pyplot as plt
# Parameters
sample_rate = 44100 # Hz
duration = 20.0 # seconds
A4_freq = 440.0 # Hz
# Time array
t = np.linspace(0, duration, int(sample_rate * duration), endpoint=False)
# Fundamental frequency (A4)
signal = np.sin(2 * np.pi * A4_freq * t)
# Adding overtones (harmonics)
harmonics = [2, 3, 4, 5, 6, 7, 8, 9] # First few harmonics
amplitudes = [0.5, 0.25, 0.15, 0.1, 0.05, 0.03, 0.01, 0.005] # Amplitudes for each harmonic
for i, harmonic in enumerate(harmonics):
signal += amplitudes[i] * np.sin(2 * np.pi * A4_freq * harmonic * t)
# Perform FFT (Fast Fourier Transform)
N = len(signal)
yf = np.fft.fft(signal)
xf = np.fft.fftfreq(N, 1 / sample_rate)
# Modify the x-axis to represent a timeline from biblical times to today
timeline_labels = ['2000 BC', '1000 BC', 'Birth of Jesus', 'St. Paul', 'Middle Ages', 'Renaissance', 'Modern Era']
timeline_positions = np.linspace(0, 2024, len(timeline_labels)) # positions corresponding to labels
# Down-sample the y-axis data to match the length of timeline_positions
yf_sampled = 2.0 / N * np.abs(yf[:N // 2])
yf_downsampled = np.interp(timeline_positions, np.linspace(0, 2024, len(yf_sampled)), yf_sampled)
# Plot the frequency spectrum with modified x-axis
plt.figure(figsize=(12, 6))
plt.plot(timeline_positions, yf_downsampled, color='navy', lw=1.5)
# Aesthetics improvements
plt.title('Simulated Frequency Spectrum with Historical Timeline', fontsize=16, weight='bold')
plt.xlabel('Historical Timeline', fontsize=14)
plt.ylabel('Reverence', fontsize=14)
plt.xticks(timeline_positions, labels=timeline_labels, fontsize=12)
plt.ylim(0, None)
# Shading the period from Birth of Jesus to St. Paul
plt.axvspan(timeline_positions[2], timeline_positions[3], color='lightpink', alpha=0.5)
# Annotate the shaded region
plt.annotate('Birth of Jesus to St. Paul',
xy=(timeline_positions[2], 0.7), xycoords='data',
xytext=(timeline_positions[3] + 200, 0.5), textcoords='data',
arrowprops=dict(facecolor='black', arrowstyle="->"),
fontsize=12, color='black')
# Remove top and right spines
plt.gca().spines['top'].set_visible(False)
plt.gca().spines['right'].set_visible(False)
# Customize ticks
plt.xticks(timeline_positions, labels=timeline_labels, fontsize=12)
plt.yticks(fontsize=12)
# Light grid
plt.grid(color='grey', linestyle=':', linewidth=0.5)
# Show the plot
plt.tight_layout()
plt.show()
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V7\(S(t)\) Temperament: \(440Hz \times 2^{\frac{N}{12}}\)i\(h(t)\) Scales: 12 unique notes x 7 modes (Bach covers only x 2 modes in WTK)Soulja Boy has an incomplete Phrygian in PBS
Flamenco Phyrgian scale is equivalent to a Mixolydian
V9♭♯9♭13
V7 \(\sigma\) Chord Stacks#
\((X'X)^T \cdot X'Y\): Mode: \( \mathcal{F}(t) = \alpha \cdot \left( \prod_{i=1}^{n} \frac{\partial \psi_i(t)}{\partial t} \right) + \beta \cdot \int_{0}^{t} \left( \sum_{j=1}^{m} \frac{\partial \phi_j(\tau)}{\partial \tau} \right) d\tau\)
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import matplotlib.pyplot as plt
import numpy as np
# Clock settings; f(t) random disturbances making "paradise lost"
clock_face_radius = 1.0
number_of_ticks = 7
tick_labels = [
"Root-iADL (i)",
"Hunter-gather (ii7♭5)", "Peasant (III)", "Farmer (iv)", "Manufacturer (V7♭9♯9♭13)",
"Energy (VI)", "Transport (VII)"
]
# Calculate the angles for each tick (in radians)
angles = np.linspace(0, 2 * np.pi, number_of_ticks, endpoint=False)
# Inverting the order to make it counterclockwise
angles = angles[::-1]
# Create figure and axis
fig, ax = plt.subplots(figsize=(8, 8))
ax.set_xlim(-1.2, 1.2)
ax.set_ylim(-1.2, 1.2)
ax.set_aspect('equal')
# Draw the clock face
clock_face = plt.Circle((0, 0), clock_face_radius, color='lightgrey', fill=True)
ax.add_patch(clock_face)
# Draw the ticks and labels
for angle, label in zip(angles, tick_labels):
x = clock_face_radius * np.cos(angle)
y = clock_face_radius * np.sin(angle)
# Draw the tick
ax.plot([0, x], [0, y], color='black')
# Positioning the labels slightly outside the clock face
label_x = 1.1 * clock_face_radius * np.cos(angle)
label_y = 1.1 * clock_face_radius * np.sin(angle)
# Adjusting label alignment based on its position
ha = 'center'
va = 'center'
if np.cos(angle) > 0:
ha = 'left'
elif np.cos(angle) < 0:
ha = 'right'
if np.sin(angle) > 0:
va = 'bottom'
elif np.sin(angle) < 0:
va = 'top'
ax.text(label_x, label_y, label, horizontalalignment=ha, verticalalignment=va, fontsize=10)
# Remove axes
ax.axis('off')
# Show the plot
plt.show()
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i \(\%\) Predict NexToken#
\(\alpha, \beta, t\) NexToken: Attention, to the minor, major, dom7, and half-dim7 groupings, is all you need 44
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import matplotlib.pyplot as plt
import numpy as np
# Clock settings; f(t) random disturbances making "paradise lost"
clock_face_radius = 1.0
number_of_ticks = 9
tick_labels = [
"Sun-Genomics", "Chlorophyll-Transcriptomics", "Flora-Proteomics", "Animals-Metabolomics",
"Wood-Epigenomics", "Coal-Lipidomics", "Hydrocarbons-Glycomics", "Renewable-Metagenomics", "Nuclear-Phenomics"
]
# Calculate the angles for each tick (in radians)
angles = np.linspace(0, 2 * np.pi, number_of_ticks, endpoint=False)
# Inverting the order to make it counterclockwise
angles = angles[::-1]
# Create figure and axis
fig, ax = plt.subplots(figsize=(8, 8))
ax.set_xlim(-1.2, 1.2)
ax.set_ylim(-1.2, 1.2)
ax.set_aspect('equal')
# Draw the clock face
clock_face = plt.Circle((0, 0), clock_face_radius, color='lightgrey', fill=True)
ax.add_patch(clock_face)
# Draw the ticks and labels
for angle, label in zip(angles, tick_labels):
x = clock_face_radius * np.cos(angle)
y = clock_face_radius * np.sin(angle)
# Draw the tick
ax.plot([0, x], [0, y], color='black')
# Positioning the labels slightly outside the clock face
label_x = 1.1 * clock_face_radius * np.cos(angle)
label_y = 1.1 * clock_face_radius * np.sin(angle)
# Adjusting label alignment based on its position
ha = 'center'
va = 'center'
if np.cos(angle) > 0:
ha = 'left'
elif np.cos(angle) < 0:
ha = 'right'
if np.sin(angle) > 0:
va = 'bottom'
elif np.sin(angle) < 0:
va = 'top'
ax.text(label_x, label_y, label, horizontalalignment=ha, verticalalignment=va, fontsize=10)
# Remove axes
ax.axis('off')
# Show the plot
plt.show()
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\(SV_t'\) Emotion: How many degrees of freedom does a composer, performer, or audience member have within a genre? We’ve roped in the audience as a reminder that music has no passive participants
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import numpy as np
import matplotlib.pyplot as plt
# Define the total utility function U(Q)
def total_utility(Q):
return 100 * np.log(Q + 1) # Logarithmic utility function for illustration
# Define the marginal utility function MU(Q)
def marginal_utility(Q):
return 100 / (Q + 1) # Derivative of the total utility function
# Generate data
Q = np.linspace(1, 100, 500) # Quantity range from 1 to 100
U = total_utility(Q)
MU = marginal_utility(Q)
# Plotting
plt.figure(figsize=(14, 7))
# Plot Total Utility
plt.subplot(1, 2, 1)
plt.plot(Q, U, label=r'Total Utility $U(Q) = 100 \log(Q + 1)$', color='blue')
plt.title('Total Utility')
plt.xlabel('Quantity (Q)')
plt.ylabel('Total Utility (U)')
plt.legend()
plt.grid(True)
# Plot Marginal Utility
plt.subplot(1, 2, 2)
plt.plot(Q, MU, label=r'Marginal Utility $MU(Q) = \frac{dU(Q)}{dQ} = \frac{100}{Q + 1}$', color='red')
plt.title('Marginal Utility')
plt.xlabel('Quantity (Q)')
plt.ylabel('Marginal Utility (MU)')
plt.legend()
plt.grid(True)
# Adding some calculus notation and Greek symbols
plt.figtext(0.5, 0.02, r"$MU(Q) = \frac{dU(Q)}{dQ} = \lim_{\Delta Q \to 0} \frac{U(Q + \Delta Q) - U(Q)}{\Delta Q}$", ha="center", fontsize=12)
plt.tight_layout()
plt.show()
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import matplotlib.pyplot as plt
import numpy as np
from matplotlib.cm import ScalarMappable
from matplotlib.colors import LinearSegmentedColormap, PowerNorm
def gaussian(x, mean, std_dev, amplitude=1):
return amplitude * np.exp(-0.9 * ((x - mean) / std_dev) ** 2)
def overlay_gaussian_on_line(ax, start, end, std_dev):
x_line = np.linspace(start[0], end[0], 100)
y_line = np.linspace(start[1], end[1], 100)
mean = np.mean(x_line)
y = gaussian(x_line, mean, std_dev, amplitude=std_dev)
ax.plot(x_line + y / np.sqrt(2), y_line + y / np.sqrt(2), color='red', linewidth=2.5)
fig, ax = plt.subplots(figsize=(10, 10))
intervals = np.linspace(0, 100, 11)
custom_means = np.linspace(1, 23, 10)
custom_stds = np.linspace(.5, 10, 10)
# Change to 'viridis' colormap to get gradations like the older plot
cmap = plt.get_cmap('viridis')
norm = plt.Normalize(custom_stds.min(), custom_stds.max())
sm = ScalarMappable(cmap=cmap, norm=norm)
sm.set_array([])
median_points = []
for i in range(10):
xi, xf = intervals[i], intervals[i+1]
x_center, y_center = (xi + xf) / 2 - 20, 100 - (xi + xf) / 2 - 20
x_curve = np.linspace(custom_means[i] - 3 * custom_stds[i], custom_means[i] + 3 * custom_stds[i], 200)
y_curve = gaussian(x_curve, custom_means[i], custom_stds[i], amplitude=15)
x_gauss = x_center + x_curve / np.sqrt(2)
y_gauss = y_center + y_curve / np.sqrt(2) + x_curve / np.sqrt(2)
ax.plot(x_gauss, y_gauss, color=cmap(norm(custom_stds[i])), linewidth=2.5)
median_points.append((x_center + custom_means[i] / np.sqrt(2), y_center + custom_means[i] / np.sqrt(2)))
median_points = np.array(median_points)
ax.plot(median_points[:, 0], median_points[:, 1], '--', color='grey')
start_point = median_points[0, :]
end_point = median_points[-1, :]
overlay_gaussian_on_line(ax, start_point, end_point, 24)
ax.grid(True, linestyle='--', linewidth=0.5, color='grey')
ax.set_xlim(-30, 111)
ax.set_ylim(-20, 87)
# Create a new ScalarMappable with a reversed colormap just for the colorbar
cmap_reversed = plt.get_cmap('viridis').reversed()
sm_reversed = ScalarMappable(cmap=cmap_reversed, norm=norm)
sm_reversed.set_array([])
# Existing code for creating the colorbar
cbar = fig.colorbar(sm_reversed, ax=ax, shrink=1, aspect=90)
# Specify the tick positions you want to set
custom_tick_positions = [0.5, 5, 8, 10] # example positions, you can change these
cbar.set_ticks(custom_tick_positions)
# Specify custom labels for those tick positions
custom_tick_labels = ['5', '3', '1', '0'] # example labels, you can change these
cbar.set_ticklabels(custom_tick_labels)
# Label for the colorbar
cbar.set_label(r'♭', rotation=0, labelpad=15, fontstyle='italic', fontsize=24)
# Label for the colorbar
cbar.set_label(r'♭', rotation=0, labelpad=15, fontstyle='italic', fontsize=24)
cbar.set_label(r'♭', rotation=0, labelpad=15, fontstyle='italic', fontsize=24)
# Add X and Y axis labels with custom font styles
ax.set_xlabel(r'Principal Component', fontstyle='italic')
ax.set_ylabel(r'Emotional State', rotation=0, fontstyle='italic', labelpad=15)
# Add musical modes as X-axis tick labels
# musical_modes = ["Ionian", "Dorian", "Phrygian", "Lydian", "Mixolydian", "Aeolian", "Locrian"]
greek_letters = ['α', 'β','γ', 'δ', 'ε', 'ζ', 'η'] # 'θ' , 'ι', 'κ'
mode_positions = np.linspace(ax.get_xlim()[0], ax.get_xlim()[1], len(greek_letters))
ax.set_xticks(mode_positions)
ax.set_xticklabels(greek_letters, rotation=0)
# Add moods as Y-axis tick labels
moods = ["flow", "control", "relaxed", "bored", "apathy","worry", "anxiety", "arousal"]
mood_positions = np.linspace(ax.get_ylim()[0], ax.get_ylim()[1], len(moods))
ax.set_yticks(mood_positions)
ax.set_yticklabels(moods)
# ... (rest of the code unchanged)
plt.tight_layout()
plt.show()
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Emotion & Affect as Outcomes & Freewill. And the predictors \(\beta\) are MQ-TEA: Modes (ionian, dorian, phrygian, lydian, mixolydian, locrian), Qualities (major, minor, dominant, suspended, diminished, half-dimished, augmented), Tensions (7th), Extensions (9th, 11th, 13th), and Alterations (♯, ♭) 45#
1. f(t)
\
2. S(t) -> 4. Nxb:t(X'X).X'Y -> 5. b -> 6. df
/
3. h(t)
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import pandas as pd
# Data for the table
data = {
"Thinker": [
"Heraclitus", "Plato", "Aristotle", "Augustine", "Thomas Aquinas",
"Machiavelli", "Descartes", "Spinoza", "Leibniz", "Hume",
"Kant", "Hegel", "Nietzsche", "Marx", "Freud",
"Jung", "Schumpeter", "Foucault", "Derrida", "Deleuze"
],
"Epoch": [
"Ancient", "Ancient", "Ancient", "Late Antiquity", "Medieval",
"Renaissance", "Early Modern", "Early Modern", "Early Modern", "Enlightenment",
"Enlightenment", "19th Century", "19th Century", "19th Century", "Late 19th Century",
"Early 20th Century", "Early 20th Century", "Late 20th Century", "Late 20th Century", "Late 20th Century"
],
"Lineage": [
"Implicit", "Socratic lineage", "Builds on Plato",
"Christian synthesis", "Christianizes Aristotle",
"Acknowledges predecessors", "Breaks tradition", "Synthesis of traditions", "Cartesian", "Empiricist roots",
"Hume influence", "Dialectic evolution", "Heraclitus influence",
"Hegelian critique", "Original psychoanalysis",
"Freudian divergence", "Marxist roots", "Nietzsche, Marx",
"Deconstruction", "Nietzsche, Spinoza"
]
}
# Create DataFrame
df = pd.DataFrame(data)
# Display DataFrame
print(df)
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Thinker Epoch Lineage
0 Heraclitus Ancient Implicit
1 Plato Ancient Socratic lineage
2 Aristotle Ancient Builds on Plato
3 Augustine Late Antiquity Christian synthesis
4 Thomas Aquinas Medieval Christianizes Aristotle
5 Machiavelli Renaissance Acknowledges predecessors
6 Descartes Early Modern Breaks tradition
7 Spinoza Early Modern Synthesis of traditions
8 Leibniz Early Modern Cartesian
9 Hume Enlightenment Empiricist roots
10 Kant Enlightenment Hume influence
11 Hegel 19th Century Dialectic evolution
12 Nietzsche 19th Century Heraclitus influence
13 Marx 19th Century Hegelian critique
14 Freud Late 19th Century Original psychoanalysis
15 Jung Early 20th Century Freudian divergence
16 Schumpeter Early 20th Century Marxist roots
17 Foucault Late 20th Century Nietzsche, Marx
18 Derrida Late 20th Century Deconstruction
19 Deleuze Late 20th Century Nietzsche, Spinoza
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# Edited by X.AI
import networkx as nx
import matplotlib.pyplot as plt
def add_family_edges(G, parent, depth, names, scale=1, weight=1):
if depth == 0 or not names:
return parent
children = names.pop(0)
for child in children:
# Assign weight based on significance or relationship strength
edge_weight = weight if child not in ["Others"] else 0.5 # Example: 'Others' has less weight
G.add_edge(parent, child, weight=edge_weight)
if child not in ["GPT", "AGI", "Transformer", "Google Brain"]:
add_family_edges(G, child, depth - 1, names, scale * 0.9, weight)
def create_extended_fractal_tree():
G = nx.Graph()
root = "God"
G.add_node(root)
adam = "Adam"
G.add_edge(root, adam, weight=1) # Default weight
descendants = [
["Seth", "Cain"],
["Enos", "Noam"],
["Abraham", "Others"],
["Isaac", "Ishmael"],
["Jacob", "Esau"],
["Judah", "Levi"],
["Ilya Sutskever", "Sergey Brin"],
["OpenAI", "AlexNet"],
["GPT", "AGI"],
["Elon Musk/Cyborg"],
["Tesla", "SpaceX", "Boring Company", "Neuralink", "X", "xAI"]
]
add_family_edges(G, adam, len(descendants), descendants)
# Manually add edges for "Transformer" and "Google Brain" as children of Sergey Brin
G.add_edge("Sergey Brin", "Transformer", weight=1)
G.add_edge("Sergey Brin", "Google Brain", weight=1)
# Manually add dashed edges to indicate "missing links"
missing_link_edges = [
("Enos", "Abraham"),
("Judah", "Ilya Sutskever"),
("Judah", "Sergey Brin"),
("AlexNet", "Elon Musk/Cyborg"),
("Google Brain", "Elon Musk/Cyborg")
]
# Add missing link edges with a lower weight
for edge in missing_link_edges:
G.add_edge(edge[0], edge[1], weight=0.3, style="dashed")
return G, missing_link_edges
def visualize_tree(G, missing_link_edges, seed=42):
plt.figure(figsize=(12, 10))
pos = nx.spring_layout(G, seed=seed)
# Define color maps for nodes
color_map = []
size_map = []
for node in G.nodes():
if node == "God":
color_map.append("lightblue")
size_map.append(2000)
elif node in ["OpenAI", "AlexNet", "GPT", "AGI", "Google Brain", "Transformer"]:
color_map.append("lightgreen")
size_map.append(1500)
elif node == "Elon Musk/Cyborg" or node in ["Tesla", "SpaceX", "Boring Company", "Neuralink", "X", "xAI"]:
color_map.append("yellow")
size_map.append(1200)
else:
color_map.append("lightpink")
size_map.append(1000)
# Draw all solid edges with varying thickness based on weight
edge_widths = [G[u][v]['weight'] * 3 for (u, v) in G.edges() if (u, v) not in missing_link_edges]
nx.draw(G, pos, edgelist=[(u, v) for (u, v) in G.edges() if (u, v) not in missing_link_edges],
with_labels=True, node_size=size_map, node_color=color_map,
font_size=10, font_weight="bold", edge_color="grey", width=edge_widths)
# Draw the missing link edges as dashed lines with lower weight
nx.draw_networkx_edges(
G,
pos,
edgelist=missing_link_edges,
style="dashed",
edge_color="lightgray",
width=[G[u][v]['weight'] * 3 for (u, v) in missing_link_edges]
)
plt.axis('off')
# Save the plot as a PNG file in the specified directory
plt.savefig("../figures/ultimate.png", format="png", dpi=300)
plt.show()
# Generate and visualize the tree
G, missing_edges = create_extended_fractal_tree()
visualize_tree(G, missing_edges, seed=2)
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