Research and Analysis of Structural Hole and Matching Coefficient 1081

Any node in the software network has direct contact

with other nodes. From the whole network view, it is

“no-hole” structure. This structure only exists in small-

scale software network and such groups are actually

closed, so the importance of each node in the networks is

basically equal. There are many nodes needed to be up-

dated. It is difficu lt to control them and update software.

In addition, the cost of maintaining this high redundancy

network is high.

Only the central node has direct link with every other

node in the network. The other nodes do not connect

with every node directly. From the whole view of the

network, the phenomenon of no direct contact or rela-

tionship breaking off is structural holes. There are no

direct connections among the rest nodes, which is

whole-hole structure.

2) The algorithm of structural holes

In the aspects of structural holes measurement, struc-

tural constraint algorithm and betweenness centrality

algorithm have been used. Structural constraint algorithm

uses closeness among nodes as measure targets, depend-

ence among nodes as the evaluation criteria. It can de-

termine the degree of software network structural holes.

At the same time if nodes across more structural holes,

they have less redundant connections, can access more

non-redundant in formation and are used more frequently.

Betweenness centrality algorithm largely determines the

centering level of the nodes. Therefore, the paper uses

structural constraint algorithm to compute structural

holes.

Definition 2.1 Network Constraint index: This index

describes direct or indirect closeness between a node and

other nodes. If the network Constraint index is higher,

the network is closer and the structural holes are fewer.

The concrete calculating steps is as follows [4]:

()

ij ji

ij ik ki

k

dd

pdd

(1)

ij

is the ratio of the shortest path length between

node i and node j to the sum of the shortest path length

about all the neighboring nodes of node i. is the

shortest pat h length bet ween node i and node j. ij

d

2

,,

(

ijijik kj

kk ikj

cp pp

)

ij

c

(2)

ij is the binding level between node i and node j.

When node j is the only adjacent node of node i, ij

gets maximal value 1.When node j is indirectly con-

nected with node i through other nodes, ij gets mini-

mum value.Node k is the adjacent node of node i.

cc

c

2

ij

p

By formula (2) and formula (3) we can calculate net-

work constraint index of node i.

i

j

C

(3)

Structural holes are used to describe a node in de-

pendence on other nodes. Few structural holes show

strong dependence on other nodes. Network constraint

index is the quantization of structural ho les. By calculat-

ing the network constraint index of structural holes, we

can understand the degree of structural holes in the soft-

ware network.

3. Matching Coefficient

In 2002, Newman put forward another important statis-

tical parameter used to mark the network, which is as-

sortativity. Assortativity is represented by r. It is chang-

ing between –1 and 1 that means nodes are prior to es-

tablish side connection with similar nodes in the network

[5,6]. When r is greater than zero, nodes are prior to con-

nect with similar nodes. Such network is called assorta-

tive mixing. When r is less than zero, nodes are prior to

connect with dissimilar nodes. Such network is called

disassortative m i xi ng.

Definition 3.1 assortative coefficient: Incidence rela-

tion between nodes in the network can be described by

assortative coefficient [7,8]:

11

1221

1

[()]

2

11

()[ ()

22

iii i

ii

ii ii

ii

EjkE jk

rEjkEjk

2

2

]

(4)

i and i are the degree of the i side’s two vertices.

E is the number of sides in the network.

j k

If assortative coefficient is greater than 0, the network

is assortative mixing; if assortative coefficient is less

than 0, the network is disassortative mixing; if assorta-

tive coefficient is equal to 0, the network is randomized.

Assortative coefficient reflects the connectivity of net-

work nodes. In the assortative mixing network, nodes of

a high degree tend to connect with nodes of a high de-

gree. In the disassortative mixing network, nodes of a

high degree tend to connect with nodes of a low degree.

In Figure 2, it is a network composed by 10 nodes. In

Figure 2(a), 0.372881r

1r

. Node 1’s degree is 5,

which is a high degree node and connect with nodes (de-

gree is 2 or 1). Such network is disassortative mixing. In

Figure 2(b),

. Degrees of all nodes are similar, that

is assortative mixing.

4. The Law and Analysis of Metrics in the

Network Software

4.1. Correlation Analysis of Degree and

Structural Holes

Degree is used to describe the connected complexity of a

C

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