

# Graph theory basics

Graphs are fundamental mathematical structures used to represent relationships between objects. In the context of community detection, graphs are used to model connections between individuals, groups, or any other entities. Let's start by defining some basic concepts in graph theory.

A graph $G$ consists of a set of vertices $V$ and a set of edges $E$. Vertices represent the entities, and edges represent the connections between them. The graph can be either directed (arrows indicate the direction of the connection) or undirected (no arrows, indicating bidirectional connections).

The degree of a vertex $v$ is the number of edges connected to it. The degree distribution of a graph is the frequency distribution of vertex degrees. It is an important characteristic of the graph and can provide insights into its structure and properties.

Consider a simple undirected graph with 5 vertices and 6 edges:

```
A -- B -- C
 \     /
  \   /
   \ /
    D
```

The degree distribution for this graph is:
- Vertex A: 2
- Vertex B: 3
- Vertex C: 2
- Vertex D: 3

## Exercise
Calculate the degree distribution for the following graph:

```
1 -- 2 -- 3 -- 4
 \     /     /
  \   /     /
   \ /     /
    5 -- 6
```

# Degree distribution and its importance

The degree distribution of a graph is a crucial characteristic that can provide insights into its structure and properties. It can help identify communities, hubs, and outliers within the graph.

The degree distribution can be characterized by its mean, median, and mode. The mean degree is the average degree of all vertices in the graph, the median degree is the degree that separates the graph into two equal parts, and the mode degree is the most frequent degree in the graph.

The distribution of degrees can be described using various statistical measures, such as the skewness and kurtosis. A graph with a high skewness and kurtosis has more extreme values, indicating a more uneven degree distribution.

## Exercise
Calculate the mean, median, mode, skewness, and kurtosis for the degree distribution of the graph from the previous exercise.

# Modularity and its properties

Modularity is a measure of the quality of community detection in a graph. It quantifies the density of connections within communities and the sparsity of connections between communities.

The modularity of a graph partition is defined as the sum of the products of the degrees of vertices within each community, minus the expected products if the vertices were randomly distributed across communities.

$$Q = \frac{1}{2m} \sum_{i \in C} \sum_{j \in C} (A_{ij} - \frac{k_i k_j}{2m})$$

where $C$ is a community, $A_{ij}$ is the number of edges between vertices $i$ and $j$, $k_i$ and $k_j$ are the degrees of vertices $i$ and $j$, and $m$ is the total number of edges in the graph.

Modularity has several properties:
- It is maximized when communities are well-defined and dense.
- It is minimized when communities are overlapping or when communities are too sparse.
- It is a global measure, meaning it takes into account the entire graph and not just individual communities.

## Exercise
Calculate the modularity of a graph partition with the following communities:

```
Community 1: A, B, C
Community 2: D, E, F
```

# Seed set expansion and its applications

Seed set expansion is an iterative algorithm that grows communities from a set of seed vertices. The algorithm starts with a set of seed vertices and expands the communities by adding vertices that are connected to the seed set. The expansion continues until a stopping criterion is met, such as a maximum number of iterations or a minimum modularity gain.

Seed set expansion can be applied to various types of graphs, including social networks, biological networks, and citation networks. It can help detect communities that are relevant to specific domains or topics.

## Exercise
Apply the seed set expansion algorithm to the graph from the previous exercise, starting with vertices A and B as seed vertices.

# Overlapping community detection

Overlapping community detection is a type of community detection that allows for partial membership of vertices in multiple communities. This can be useful in situations where entities can belong to multiple communities or when communities overlap.

One popular method for overlapping community detection is the Label Propagation Algorithm (LPA). In LPA, each vertex is assigned a label, and the algorithm iteratively propagates the labels to neighboring vertices. The algorithm stops when the labels do not change anymore.

## Exercise
Implement the Label Propagation Algorithm to detect communities in the graph from the previous exercise.

# Hierarchical clustering algorithms

Hierarchical clustering algorithms are a class of algorithms that build a hierarchy of clusters by successively merging or splitting clusters. This approach can be used to detect communities in a graph.

One popular hierarchical clustering algorithm is the Agglomerative Hierarchical Clustering (AHC). In AHC, clusters are formed by successively merging pairs of clusters with the smallest distance between them. The algorithm stops when the desired number of clusters is reached.

## Exercise
Implement the Agglomerative Hierarchical Clustering algorithm to detect communities in the graph from the previous exercise.

# Partitioning-based clustering algorithms

Partitioning-based clustering algorithms divide the graph into clusters by partitioning the vertices into disjoint sets. This approach can be used to detect communities in a graph.

One popular partitioning-based clustering algorithm is the K-means algorithm. In K-means, the algorithm iteratively assigns vertices to clusters based on their similarity and updates the cluster centers. The algorithm stops when the cluster assignments do not change anymore.

## Exercise
Implement the K-means algorithm to detect communities in the graph from the previous exercise.

# Spectral clustering and its properties

Spectral clustering is a graph-based clustering algorithm that leverages the spectral properties of the graph's adjacency matrix to detect communities. It is based on the idea that similar vertices have similar neighborhoods, and it can be used to detect non-convex clusters.

The spectral clustering algorithm can be divided into two main steps: (1) constructing the similarity matrix and (2) clustering the similarity matrix. The similarity matrix is typically constructed using the graph's Laplacian matrix or its eigenvectors.

## Exercise
Implement the spectral clustering algorithm to detect communities in the graph from the previous exercise.

# Seed set expansion-based clustering algorithms

Seed set expansion-based clustering algorithms are a class of algorithms that use seed set expansion to detect communities in a graph. These algorithms start with a set of seed vertices and iteratively expand the communities by adding vertices that are connected to the seed set.

One popular seed set expansion-based clustering algorithm is the Markov Cluster Algorithm (MCL). In MCL, the graph is first converted into a Markov chain, and the algorithm iteratively clusters the vertices based on the Markov chain. The algorithm stops when the clusters do not change anymore.

## Exercise
Implement the Markov Cluster Algorithm to detect communities in the graph from the previous exercise.

# Evaluating community detection algorithms

Evaluating the performance of community detection algorithms is crucial for selecting the most suitable algorithm for a given problem. Several metrics can be used to evaluate the performance of community detection algorithms, such as modularity, normalized cut, and conductance.

Modularity is a popular metric for evaluating community detection algorithms. It measures the density of connections within communities and the sparsity of connections between communities. Higher modularity values indicate better community detection.

Normalized cut and conductance are other metrics that can be used to evaluate community detection algorithms. Normalized cut measures the number of edges that are cut between communities, while conductance measures the number of edges that are cut divided by the number of edges within communities. Lower normalized cut and conductance values indicate better community detection.

## Exercise
Evaluate the performance of the community detection algorithms used in the previous exercises using modularity, normalized cut, and conductance.

# Applications of community detection in social networks

Community detection has numerous applications in social networks, including detecting communities of interest, identifying hubs and outliers, and studying the structure and dynamics of social networks.

In social networks, communities can represent groups of individuals with similar interests, affiliations, or behaviors. Detecting these communities can provide insights into the structure of the network and can be used for various purposes, such as recommendation systems, network analysis, and social network analysis.

## Exercise
Apply community detection algorithms to a social network dataset to detect communities of interest and identify hubs and outliers.