Add gallery example: Clique Percolation Method for overlapping communities

Adds a new example to the sphinx-gallery demonstrating the Clique
Percolation Method (CPM) for overlapping community detection.
The example shows both k=3 and k=4, visualized with VertexCover.

Closes #819 (partially — clique percolation item)

Author: Jacopo888

doc/examples_sphinx-gallery/clique_percolation.py

@@ -0,0 +1,126 @@
+"""
+.. _tutorials-clique-percolation:
+
+==========================
+Clique Percolation Method
+==========================
+
+This example shows how to detect **overlapping communities** using the
+Clique Percolation Method (CPM). Unlike partitioning algorithms, CPM allows
+a node to belong to more than one community simultaneously.
+"""
+
+import itertools
+from collections import Counter
+
+import igraph as ig
+import matplotlib.pyplot as plt
+
+# %%
+# We construct a graph with three dense cliques that share individual nodes,
+# creating natural *overlapping* community boundaries:
+g = ig.Graph(9)
+g.add_edges(list(itertools.combinations([0, 1, 2, 3], 2)))  # clique A
+g.add_edges(list(itertools.combinations([3, 4, 5, 6], 2)))  # clique B, shares node 3 with A
+g.add_edges(list(itertools.combinations([6, 7, 8], 2)))     # clique C, shares node 6 with B
+
+# %%
+# The CPM algorithm works in three steps:
+#
+# 1. Find all k-cliques (complete subgraphs of exactly k nodes).
+# 2. Build a clique graph : two k-cliques are adjacent when they share k-1 nodes.
+# 3. Each connected component of the clique graph is a community — the union
+#    of all its k-cliques. A node shared between cliques in *different*
+#    components belongs to multiple communities simultaneously.
+k = 3
+cliques = [set(c) for c in g.cliques(k, k)]
+
+clique_graph = ig.Graph(len(cliques))
+clique_graph.add_edges([
+    (i, j)
+    for i, j in itertools.combinations(range(len(cliques)), 2)
+    if len(cliques[i] & cliques[j]) >= k - 1
+])
+
+communities = []
+for component in clique_graph.connected_components():
+    members = set()
+    for idx in component:
+        members |= cliques[idx]
+    communities.append(sorted(members))
+
+# %%
+# Nodes that appear in more than one community are *overlapping nodes*:
+overlap = [
+    v for v, count in Counter(v for comm in communities for v in comm).items()
+    if count > 1
+]
+print(f"Communities (k={k}): {communities}")
+print(f"Overlapping nodes:   {overlap}")
+
+# %%
+# We visualize the result using :class:`igraph.VertexCover`, which draws a
+# shaded hull around each community and handles overlapping nodes naturally:
+cover = ig.VertexCover(g, communities)
+palette = ig.RainbowPalette(n=len(communities))
+
+fig, ax = plt.subplots(figsize=(6, 5))
+ig.plot(
+    cover,
+    mark_groups=True,
+    palette=palette,
+    vertex_size=25,
+    vertex_label=list(range(g.vcount())),
+    vertex_label_size=10,
+    edge_width=1.5,
+    target=ax,
+)
+ax.set_title(
+    f"Clique Percolation Method (k={k})\n"
+    f"{len(communities)} communities  —  overlapping nodes: {overlap}"
+)
+plt.show()
+
+# %%
+# Advanced: effect of k
+# ----------------------
+# Raising k to 4 requires larger cliques. The 3-clique {6, 7, 8} no longer
+# qualifies, so community C disappears and node 6 is no longer in any community:
+k = 4
+cliques_4 = [set(c) for c in g.cliques(k, k)]
+
+clique_graph_4 = ig.Graph(len(cliques_4))
+clique_graph_4.add_edges([
+    (i, j)
+    for i, j in itertools.combinations(range(len(cliques_4)), 2)
+    if len(cliques_4[i] & cliques_4[j]) >= k - 1
+])
+
+communities_4 = []
+for component in clique_graph_4.connected_components():
+    members = set()
+    for idx in component:
+        members |= cliques_4[idx]
+    communities_4.append(sorted(members))
+
+print(f"Communities (k=4): {communities_4}")
+
+cover_4 = ig.VertexCover(g, communities_4)
+palette_4 = ig.RainbowPalette(n=max(len(communities_4), 1))
+
+fig, ax = plt.subplots(figsize=(6, 5))
+ig.plot(
+    cover_4,
+    mark_groups=True,
+    palette=palette_4,
+    vertex_size=25,
+    vertex_label=list(range(g.vcount())),
+    vertex_label_size=10,
+    edge_width=1.5,
+    target=ax,
+)
+ax.set_title(
+    f"Clique Percolation Method (k=4)\n"
+    f"{len(communities_4)} communities  —  nodes 6, 7, 8 no longer qualify"
+)
+plt.show()