add: from directed to undirected modularity

Author: BeaMarton13

doc/source/community_detection_guide/notebooks/modularity.ipynb

@@ -199,7 +199,7 @@
     "\n",
     "The directed modularity formula is:\n",
     "\n",
-    "$$Q_{dir} = \\frac{1}{m} \\sum_{i,j} \\left[ A_{ij} -  \\gamma \\frac{k_{i}^{out} k_{j}^{in}}{m} \\right] \\delta(c_i, c_j)$$\n",
+    "$$Q = \\frac{1}{m} \\sum_{i,j} \\left[ A_{ij} -  \\gamma \\frac{k_i^\\text{out} k_j^\\text{in}}{m} \\right] \\delta(c_i, c_j)$$\n",
     "\n",
     "Where:\n",
     "* $m$: Total number of directed edges in the graph.\n",
@@ -213,6 +213,23 @@
     "* **Flipping a single edge:** Reversing a single edge (e.g., from A → B to B → A) will change the modularity score. This is because the out-degree of A and the in-degree of B would change, altering the null model's calculation and, consequently, the overall score.\n",
     "* **Flipping all edges:** If you reverse the direction of **every single edge** in the network, the modularity score will **remain the same**. This is due to a symmetry property of the formula. The set of in-degrees becomes the new set of out-degrees, and vice versa. When the formula is applied to this completely reversed network, the total modularity score is unchanged. This is a fascinating property of directed modularity.\n",
     "\n",
+    "# From directed to undirected formula\n",
+    "Start with the directed formula:\n",
+    "\n",
+    "$$Q = \\frac{1}{m} \\sum_{i,j} \\left[ A_{ij} - \\gamma \\frac{k_i^\\text{out} k_j^\\text{in}}{m} \\right] \\delta(c_i, c_j)$$\n",
+    "\n",
+    "Substitute $(k_i^\\text{out} = k_i)$ and $(k_j^\\text{in} = k_j)$:\n",
+    "\n",
+    "$$Q = \\frac{1}{m} \\sum_{i,j} \\left[ A_{ij} - \\gamma \\frac{k_i k_j}{m} \\right] \\delta(c_i, c_j)$$\n",
+    "\n",
+    "However, this sum is **over ordered pairs** \\((i,j)\\), so each undirected edge is **counted twice**.  \n",
+    "To avoid double counting, we divide by $2$:\n",
+    "\n",
+    "\n",
+    "$$Q = \\frac{1}{2m} \\sum_{i,j} \\left[ A_{ij} - \\gamma \\frac{k_i k_j}{2m} \\right] \\delta(c_i, c_j)$$\n",
+    "\n",
+    "This matches the undirected formula exactly.\n",
+    "\n",
     "\n",
     "\n",
     "# Why the resolution parameter is important\n",
@@ -253,6 +270,14 @@
     "\n",
     "In this formulation, the null model assumes **uniform edge probability**, so communities are favored if their **internal density** is higher than the global density.\n"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1b9c3fa0-05b7-461f-acf8-8348f344a2d9",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {