Chapter 3 - Basic topological properties of the network (complex network study notes)

Degree nodes and the average degree of

Of: \ (node degree k i refers to the number of edges directly connected to node i \)
The degree: $ I $ node pointing to other nodes in the number of edges
Penetration: other node points to the node number of sides $ I $
Average degree: The average of all the network nodes
\ (K_i \) : the degree of node i
\ (<K> \) : the average of the network

If the network is weighted G, then the degree of the node can be defined as the weighted 出强度and入强度

Sparse and dense network of

Density of the network: a containing density network $ N $ nodes $ \ Rho $ defined network actually exists in the number of edges $ M $ maximum possible ratio of the number of edges, i.e., \ (\ rho = \ frac { M } {\ frac {1} { 2} N (N-1)} \) for the Internet, the above formula can be removed 1/2.

If and when N tends to infinity and is a constant density of the network, the network indicates the actual number of sides and $ N ^ 2 $ is the same order, it is said that the network is dense.

Average degree: \ (<K> = \ 2M FRAC {{N}} \)
Density: \ (\ Rho = \ FRAC {M} {\ FRAC. 1} {2} {N (. 1-N)} \)
And the relationship between the average density of: \ (<K> = (. 1-N) \ Rho \ approx N \ Rho \)

Average path length and diameter

Average path length

The shortest path: between two nodes in the network 边数最少path is called shortest path
From $ d_ $: is defined as the node i, j is the number of edges in the shortest path.
Average path length $ L $: defined as the average distance of any two nodes in the network \ (L = \ frac {1 } {\ frac {1} {2} N (N-1)} \ sum_ {i> = j} d_ {ij} \)

Network diameter

Network diameter D: the network is defined as the maximum distance of any two nodes \ (D = max (d_ { ij}) \)

In fact, we may be more concerned about is the distance between the vast majority of network users, so the first gives the following definition:

\ (f (d) \) : network statistics from 等于$ d $ of 连通的节点对the total network in 连通的节点对proportion
\ (G (d \) ): network statistics from 不超过$ d $ of 连通的节点对the total network in 连通的节点对proportion

Generally, if the diameter $ D $ satisfies \ (g (D-1) <0.9, g (D) \ ge0.9 \) then said effective diameter D for the network.

Shortest path algorithm

Dijkstra's algorithm: A weighted directed generally used network (nonnegative weight) of the shortest path between the nodes
bellman-ford algorithm: for the presence of a negative weight value

Clustering coefficient (clustering coefficient)

A node 聚类系数depicts the node 邻居节点in 任意一对节点with even edges 概率. \ (C_i = clustering coefficient of a point = \ {the number of edges between neighboring nodes actually present point} frac {these neighbor nodes may exist the maximum number of edges} \) \ (C_i = \ {FRAC E_i} {K_i (-K_i. 1) / 2} = \ FRAC {} {2E_i K_i (-K_i. 1)} \)

among them

\ (E_i \) : the number of edges between neighboring nodes actually present point
$ K_i (k_i-1) / 2 $: the maximum number of edges which may exist neighbors

Distribution (degree distribution)

There will have a network connection, we are naturally concerned about the distribution of nodes in the network degrees.

Gaussian distribution (distribution is too / bell-shaped distribution)

Distribution is too positive for a continuous random variable, its corresponding discrete random variable, the most common is the Poisson distribution (Poisson Distribution) \ (P (K) = \ {FRAC \ ^ KE the lambda ^ {- \} the lambda } {k!} \)