# Agglomerative clustering with and without structure in Scikit-learn

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This example shows the effect of imposing a connectivity graph to capture local structure in the data. The graph is simply the graph of 20 nearest neighbors.

Two consequences of imposing a connectivity can be seen. First clustering with a connectivity matrix is much faster.

Second, when using a connectivity matrix, average and complete linkage are unstable and tend to create a few clusters that grow very quickly. Indeed, average and complete linkage fight this percolation behavior by considering all the distances between two clusters when merging them. The connectivity graph breaks this mechanism. This effect is more pronounced for very sparse graphs (try decreasing the number of neighbors in kneighbors_graph) and with complete linkage. In particular, having a very small number of neighbors in the graph, imposes a geometry that is close to that of single linkage, which is well known to have this percolation instability.

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### Version¶

```
import sklearn
sklearn.__version__
```

### Imports¶

This tutorial imports AgglomerativeClustering and kneighbors_graph.

```
import plotly.plotly as py
import plotly.graph_objs as go
from plotly import tools
import time
import matplotlib.pyplot as plt
import numpy as np
from sklearn.cluster import AgglomerativeClustering
from sklearn.neighbors import kneighbors_graph
```

### Calculations¶

Generate sample data

```
n_samples = 1500
np.random.seed(0)
t = 1.5 * np.pi * (1 + 3 * np.random.rand(1, n_samples))
x = t * np.cos(t)
y = t * np.sin(t)
X = np.concatenate((x, y))
X += .7 * np.random.randn(2, n_samples)
X = X.T
```

### Plot Results¶

Create a graph capturing local connectivity. Larger number of neighbors will give more homogeneous clusters to the cost of computation time. A very large number of neighbors gives more evenly distributed cluster sizes, but may not impose the local manifold structure of the data.

```
knn_graph = kneighbors_graph(X, 30, include_self=False)
```

```
fig = tools.make_subplots(rows=4, cols=3,
print_grid=False,
subplot_titles=('linkage=Average(time=0.22s)',
'n_cluster=30, connectivity=False<br>'+
'linkage=Complete(time=0.06s)',
'linkage=Ward(time=0.07s)',
'linkage=Average(time=0.22s)',
'n_cluster=3, connectivity=False<br>'+
'linkage=Complete(time=0.06s)',
'linkage=Ward(time=0.07s)',
'linkage=Average(time=0.22s)',
'n_cluster=30, connectivity=True<br>'+
'linkage=Complete(time=0.06s)',
'linkage=Ward(time=0.07s)',
'linkage=Average(time=0.22s)',
'n_cluster=3, connectivity=True<br>'+
'linkage=Complete(time=0.06s)',
'linkage=Ward(time=0.07s)'))
def matplotlib_to_plotly(cmap, pl_entries):
h = 1.0/(pl_entries-1)
pl_colorscale = []
for k in range(pl_entries):
C = map(np.uint8, np.array(cmap(k*h)[:3])*255)
pl_colorscale.append([k*h, 'rgb'+str((C[0], C[1], C[2]))])
return pl_colorscale
```

```
l = 0
row = 1
colors = matplotlib_to_plotly(plt.cm.spectral, 13)
for connectivity in (None, knn_graph):
for n_clusters in (30, 3):
for index, linkage in enumerate(('average', 'complete', 'ward')):
model = AgglomerativeClustering(linkage=linkage,
connectivity=connectivity,
n_clusters=n_clusters)
t0 = time.time()
model.fit(X)
elapsed_time = time.time() - t0
trace = go.Scattergl(x=X[:, 0], y=X[:, 1],
mode='markers',
showlegend=False,
marker=dict(color=X[:, 1],
colorscale=colors,
line=dict(color='black', width=1)
))
fig.append_trace(trace, row, index+1)
l=l+1
row = row+1
fig['layout'].update(height=1000,)
for i in map(str,range(1,13)):
y = 'yaxis'+i
x = 'xaxis'+i
fig['layout'][y].update(showticklabels=False, ticks='',
showgrid=False, zeroline=False)
fig['layout'][x].update(showticklabels=False, ticks='',
showgrid=False, zeroline=False)
py.iplot(fig)
```