Basic concepts and representation methods of 002 diagram

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1. Composition of the graph

• Graph (graph, G) consists of nodes (nodes, N) and connections (edges, E).

2. Ontology diagram

2.1 What is ontology diagram

• Designing the ontology diagram is the first step in designing the diagram.
• That is, before designing the diagram, it is necessary to clarify the possible node types and connection types.
• The picture below shows the ontology diagram of a certain medical knowledge graph.
  
• After designing the ontology diagram, import the data to generate the diagram.
• The picture below is the medical knowledge graph (part) generated based on the picture above.
  

2.2 How to design ontology diagram

• First, the principle is that it depends on what problem we want to solve;
• Secondly, the general ontology graph is unique and unambiguous. For example, in a network of interpersonal relationships, the nodes are the characters, and the connections are whether there are relationships;
• Again, like the previous example of the medical knowledge graph, there are many types of nodes and many types of relationships;
• In short, the ontology diagram is flexibly designed according to the target tasks.

3. Types of pictures

3.1 According to whether the connection is directed

• Directed graph: such as: subway line map.
• Undirected graph: such as: Weibo follow graph.

3.2 According to ontology diagram

• Ordinary graph: There is only one type of nodes and connections;
• Heterogeneous graph: There is more than one type of nodes and connections;
• Bipartite graph: a special heterogeneous graph with two node types.

Note: The bipartite graph can be expanded into two graphs for separate processing.

3.3 Divide according to whether the connection is weighted

• Connect weighted graphs: Literally connect with weights.
• If there are multiple paths between two nodes: the weight is the sum of the weights of each path.

4. Number of node connections (degree of node)

• The degree of a node can be used as an indicator of the importance of the node.

4.1 Degree of nodes in undirected graph

• The number of connections that exist for a node is the degree of the node.
• The average degree of an undirected graph is $\bar{K}=\frac{2}{N}$. Among them, E is the total number of connections, and N is the number of summary points.

4.2 Degree of directed graph node

• The degree of a directed graph node is divided into in-degree and out-degree.

• In-degree: is the number of connections pointing to the node.
• Out-degree: is the number of connections issued by the node.

• The degree of the entire node is the sum of in-degree and out-degree.
• A node with an in-degree of 0 is called a source node, and a node with an out-degree of 0 is called a sink node.
• The average degree of a directed graph is $\bar{K}=\frac{E}{N}$, the average out-degree and the average in-degree are the same.

5. Representation method of graph

5.1 Adjacency matrix

• Where there is a connection it is 1, where there is no connection it is 0.
• Features: The adjacency matrix of an undirected graph is a symmetric matrix, and the adjacency matrix of a directed graph is an asymmetric matrix.
• For undirected graphs, the total number of connections is half the element-wise sum of the adjacency matrix; the degrees of nodes can be summed along rows or columns.

Note: If there are self-connections, the total number of self-connections does not need to be divided by two when calculating the total number of connections.

• For directed graphs, the total number of connections is the element-wise sum of the adjacency matrices; the in-degree of a node is the column-wise sum, and the out-degree is the row sum.
  

5.2 Connection list, adjacency list

邻接矩阵多为稀疏矩阵，这造成了存储空间的浪费。

• Connection list: Only record node pairs with connections.
• Adjacency list: only records connections issued by each node.
• Adjacency lists further compress storage requirements based on connection lists.
  

6. Graph connectivity

• For an undirected graph, if any two nodes can reach each other, it is called a connected graph; otherwise, it is called a non-connected graph, and the maximum connected subgraph of a non-connected graph is called a connected domain.
• For a directed graph, if any two nodes can reach each other, it is called a strongly connected graph; if a non-strongly connected graph is a connected graph after removing the direction, it is called a weakly connected graph.
• Strongly Connected Domain (SCC): It is a maximal subgraph that meets the definition of a strongly connected graph. It is very necessary to do SCC decomposition for a picture.