The basic concept of graph theory

Original address: http://blog.csdn.net/minenki/article/details/8606515

1 figure (graph), vertex (vertices), edge (edges)

A graph consists of vertices and edges, and is a means of representing the relationship between objects and objects (objects). In other terms, the graph is also called the Network,and the vertex is called the node (nodes), and the edge is called the link (links) .

The mathematical representation of the graph: where V is the vertex set:; E is the edge set:

2 vertices adjacent (adjacent)

two vertices are connected by an edge, saying that the two vertices are adjacent (in other words, if two vertices are adjacent, that is, there is an edge between the two vertices), the two vertices are neighbors vertex (neighbor vertices).

3 cluster coefficients (clustering coefficient)

Vertex of cluster factor : A vertex, which has a K neighbor vertex "neighbor vertex concept", see top "2 vertices adjacent ", the number of actual edges between k neighbors vertices is higher than The number of edges that the K-neighbor vertex may exist (that is , that is), the ratio of which is the cluster factor of the vertex.

2 (a) The blue vertex has 3 white neighbor vertices, the 3 neighbor vertices actually have edges 3 bars (shown in black with bold lines), and 3 neighbor vertices may have a maximum number of x (3-1)/2=3, so the final clustering coefficients cc=3/3=1

2 (b) The blue vertex has 3 white neighbor vertices, the 3 neighbor vertices actually have edges 0 bars (shown in black with bold lines), and 3 neighbor vertices may have a maximum number of x (3-1)/2=3, so the final clustering coefficients cc=0

The clustering coefficients of vertices are also called Local clustering coefficients (locallyclustering coefficient).

Figure 2: Clustering coefficients for vertices (a) cc=1 (b) cc=0

cluster coefficients of graphs: the average of the clustering coefficients of all the vertices in a graph is the cluster coefficients of the graph. Also known as the average clustering factor for networks (networkaverage clustering coefficient)

4 Vertex degree (degree), average (average degree), degree distribution (degree distribution)

degree of vertex: The degree of a vertex is the number of edges connected to the vertex. The degree of the vertex of a given graph can also be subdivided into degrees and degrees, which are the number of bars with the vertex as the head (head) , and the number of edges with that vertex as the tail (tail) .

the average degree of a graph (average degree) is:

where n is the number of vertices in the graph.

the degree distribution of a graph is the probability distribution of the degree of vertices in the entire graph.

For example, if a graph has N vertices, where the degree of Ni vertices is K, then the degree distribution of the graph is P (K) =ni/n.

The degree distribution of many social networks follows the Powerlaw, ie.

Power-law distributions are also known as scale-free distributions (distribution).

5 Full Graph (complete graph)

Each pair of vertices is connected to each other, that is, each pair of vertices is adjacent , then a graph is called a full graph.

A full graph of n endpoints in Kn , with n vertices and edges " any two vertices of the n vertex of the full graph just have an edge, and the number of edges is equal to a combinatorial problem ". Examples include:

Figure 3: Example of a complete picture

6 factions (clique)

A faction of a non-graph is defined as a set of sub-vertices in which the set of vertices of the non-direction graph has an edge connected to any of the two vertices in the set of sub-vertices (that is, any two vertices in the child vertex set are adjacent), then the graph of the child vertex set and its edges is a faction of this graph. If the faction has a vertex of K, it is called the Faction K- faction (k-clique).

In fact, a full picture of a non-graph is a faction of this graph .

The following figure shows a total of 23 1-faction (that is, the number of vertices of the graph), 42 2-factions (also all sides of the graph), 19 3-factions (triangles in all blue areas of the graph), 2 4-factions (quads in the dark blue region of the figure)

Figure 4: faction examples

7 path, simple path,shortest path (shortest path), Average Shortest path length (average shortest path lengths), Graph diameter (diameter), connected graph (connected graph), connected component (connected component)

Path: 1 (a), [a-b-e-c] is a path to vertex a to vertex C.

Simple path: A path in which there are no duplicate vertices is called a simple path, and [A-b-e-c] is a simple path.

Shortest path: 1 (a), the shortest path from vertex A to vertex C is [a-c]

Average Shortest path length: the average of the shortest path between any two points in a graph.

The diameter of the graph : In the shortest path between any two vertices in the graph, the longest shortest path is defined as the diameter of the graph .

Connected graphs: Any two vertices in the graph can be connected by a path, which is a connected graph.

Connected components: The largest connected sub-graph of a graph.

8 diagram type: Binary graph (bipartite graphs), random (randomlygraphs), regular (Regular graphs), no scale graph (scale-free graphs), Small World map (small-world graphs)

Dichotomy: A set of vertices of a non-graph can be divided into two disjoint subsets, and the two vertices associated with each edge of the graph belong to the two different subset of vertices (i.e., the vertices in each vertex subset are not adjacent ), this graph can be called a binary graph.

is a binary graph, because the set of vertices of this graph can be divided into this way: {4,5,6,7} and {three}, vertex subset {x-i} is not adjacent, four vertices in the subset of vertices {4,5,6,7} are not adjacent, conforming to the dichotomy condition.

Figure 5: two-part diagram

Random graphs: The edges in the graph are randomly generated, that is, the probability of having an edge between any two vertices is p.

Regular graphs: If all vertices in the graph have equal degrees, this graph is called a regular graph.

Scale-free graph: The degree distribution of graphs obeys the graph of power law distribution without scale.

Small World: A graph with a very small path between any two vertices in the graph (for example, the average distance between any two people on the Earth is 6, and the interpersonal graph is a Small world map).

92 Plot: Cartesian plot ( Cartesian Product graph ), Inline plot ( Direct product Graph ), Span style= "font-family: ' Microsoft Yahei '; font-size:14px; " > The composition of the graph ( lexicographical Product Graph )

The product of two graphs is the operation of generating a new figure H by two figures G1 and G2: The set of vertices that require the new figure H (vertex set) is the Cartesian product of the set of G1 and G2 vertices, that is, if and only if the u1,u2,v1,v2 satisfies a particular condition, the two vertices in the new figure H (U1, U2) and (V1,v2) are connected to each other .

Figure 5: G1 and G2

5 shown in Fig. Two, G1 and G2 are defined as follows:

Cartesian plot: For any two vertices in the vertex set v=  u= (U1,U2) and  v= (V1,v2), when u1=v1 and in G2 U2 adj v2   or  u2=v2  and

(note:adj  is adjacent adjacent abbreviation, U1 adj v1u1  and v1 are adjacent in Figure G1, that is, U1 and V1 have edges connected in the diagram G1. )

Example: By definition, the vertex set of the Cartesian plot of G1 and G2 is ={(1,3), (1,4), (1,5), (2,3), (2,4), (2,5) }, according to conditions such as vertex (1,3) and (1,4), satisfies 1=1, and 3 and 4 The conditions connected in figure G2, so that there is an edge between the vertex (1,3) and (1,4), and so on, the Cartesian plot of G1 and G2 is shown below.

Figure 6: Cartesian plot of G1 and G2

straight plot diagram : for any of the two vertices u= (U1,U2) and v= (V1,V2) in the vertex set v=, connect the vertices u and v when G1 U1 adj in v1 and G2 U2 adj in v2 , the new figure so obtained is called G1 and G2 's straight plot.

Example: By definition, the vertex set of G1 and G2 's histogram is ={(1,3), (1,4), (1,5), (2,3), (2,4), (2,5)}, depending on the conditions such as vertex (1,3) and (2,4), satisfies 1 and 2 in Figure G1 and 3 and 4 The conditions connected in figure G2, so there are edges between vertices (1,3) and (2,4), and so on, and so on, G1 and G2 are shown below.

Figure 7: G1 and G2 of the vertical plot

< Span style= "font-family: ' Microsoft Yahei '; font-size:14px; " > compositing : for vertex sets V=    U= (U1,U2) and  v= (V1,V2), When g1   U1 adj v1 or u1= v1, when the G2 U2 adj v2 , connect the vertices u and V, so that the new figure is called G1 and G2 of the composite diagram G1[g2] .

Example: By definition, the vertex set of G1 and G2 is ={(1,3), (1,4), (1,5), (2,3), (2,4), (2,5) }, depending on the condition, such as vertex (1,3) and (2,4), meet 1 and 2 Connected in Figure G1 , and (1,3) and (1,4) satisfy 1=1,3 and 4 are connected in diagram G2, so there are edges between vertices (1,3) and (1,4), and so on, G1 and G2 of synthesis diagram G1[g2] as shown below.

Figure 8: Synthetic drawings of G1 and G2 G1[g2]

Reference

"Complex network theory and its application"

"Graph-based NATURAL LANGUAGE Processing and information retrieval"

Wikipedia:clustering coefficient,Path(Social network), completegraph,clique (graph theory),degree distribution, graph product

The basic concept of graph theory