add: resolution limit and modularity formula update

Author: BeaMarton13

doc/source/community_detection_guide/notebooks/modularity.ipynb

@@ -9,13 +9,14 @@
     "\n",
     "Modularity is a quantitative metric used to evaluate the strength of a network's division into modules (or communities). It measures how well the network is partitioned by comparing the density of edges within communities to the expected density of such edges in a randomized network that preserves the original degree distribution. The formula for modularity is given below.\n",
     "\n",
-    "$$Q = \\frac{1}{2m} \\sum_{i,j} \\left[ A_{ij} - \\frac{k_i k_j}{2m} \\right] \\delta(c_i, c_j)$$\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",
     "Where:\n",
     "* $m$: Total number of edges in the graph (sum of all edge weights for a weighted graph).\n",
     "* $A_{ij}$: Element of the adjacency matrix, representing the weight of the edge between node $i$ and node $j$ (0 if no edge).\n",
     "* $k_i$: Degree of node $i$ (sum of weights of edges connected to node $i$).\n",
     "* $k_j$: Degree of node $j$.\n",
+    "* $\\gamma$ is the resolution parameter. The default value is 1.0.\n",
     "* $\\delta(c_i, c_j)$: Kronecker delta function, which is 1 if node $i$ and node $j$ belong to the same community ($c_i = c_j$), and 0 otherwise.\n",
     "\n",
     "### Example graph\n",
@@ -190,7 +191,18 @@
    "id": "f3981bc5-f6bf-4b41-a168-cae9c17ec764",
    "metadata": {},
    "source": [
-    "*Note:* Based on our previous analysis, the \"good\" partitioning yields a significantly higher modularity score. It is important to note, however, that a high modularity score is not always a definitive indicator of a better community partitioning, as was previously demonstrated with the Grid Graph [here](test_significance_of_community.ipynb)."
+    "*Note:* Based on our previous analysis, the \"good\" partitioning yields a significantly higher modularity score. It is important to note, however, that a high modularity score is not always a definitive indicator of a better community partitioning, as was previously demonstrated with the Grid Graph [here](test_significance_of_community.ipynb).\n",
+    "\n",
+    "### Why the Resolution Parameter is Important\n",
+    "\n",
+    "The resolution parameter addresses a fundamental limitation of the original modularity measure, known as the **\"resolution limit\"**. This is the tendency of the original formula (where $\\gamma=1$) to fail at detecting small communities, especially in large graphs. It often merges smaller, distinct communities into a single larger one to maximize the modularity score.\n",
+    "\n",
+    "By adjusting the value of $\\gamma$, you can control the granularity of the community detection:\n",
+    "\n",
+    "* **$\\gamma > 1$**: Increasing the resolution parameter penalizes connections more heavily. This encourages the algorithm to find **fewer and larger communities**, as it becomes harder for smaller, highly connected groups to form their own separate communities.\n",
+    "* **$\\gamma < 1$**: Decreasing the resolution parameter reduces the penalty. This allows the algorithm to find **more and smaller communities**, as it becomes easier for closely-knit groups to be identified as their own communities.\n",
+    "\n",
+    "In essence, the resolution parameter provides a flexible way to explore the community structure of a network at different scales, moving beyond the limitations of a single, fixed-scale partition."
    ]
   }
  ],