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research they have been professionally classified into 51

categories by a group of consultants. Therefore, we ob-

tain an undirected firm competition network that has

6597 edges. The network can be divided into 51 sub-

networks, in which all vertices connected to each other

obviously, according to the software product categories.

There are 479 firms among these 578 firms, which con-

strain to develop products in a single product category,

and thus the vertices representing them belong to one

single group only. On the other hand, other firms develop

products in several product categories, and thus the ver-

tices representing them belong to several groups simul-

taneously. The main network measurements often dis-

cussed in literature are presented briefly as follows.

1) Network density: The network density is defined as

the ratio between the number of actual links and the

maximum possible number of links. And the density is a

key network-level property that refers to the extent of

interconnection among the actors of the network. The

network density of this empirical network is 0.04, sparse

to some extent.

2) Vertex degree: The vertex degree is defined as the

number of edges connected to this vertex. And the aver-

age connectivity of vertices in the network is the mean

degree of all vertices. The average connectivity of verti-

ces in this network is 22.99 while the maximum degree is

144, meaning that an individual firm has about 30 com-

petitors in this industrial park on average and there exist

some firms which have 144 competitors at most.

3) Mean geodesic distance: According to [1], we use

the “harmonic mean” form to calculate this index. The

mean geodesic distance of this empirical network is 2.3,

which is indeed very small compared with the number of

vertices. It is certain that the so-called small world effect

exists in this empirical network.

4) Clustering coefficient: The clustering coefficient

measures the density of triangles in the network. In an-

other words, it measures the propensity for vertex pairs

Copyright © 2011 SciRes. JSSM

Characterizing and Modeling the Structure of Competition Networks

10

to be connected if they share a mutual neighbor. The

clustering coefficient for this network is 0.92. It has been

reported that for social networks, clustering appears to be

far greater than non-social networks [15], which can be

verified by our case very well.

2.2. Analysis of Four Structural Properties

1) Degree distribution: In current literature, there are two

forms of degree distribution, power-law form and expo-

nential form, commonly characterizing real networks

[1,2]. If the distribution would follow power-law form

then it would approximately fall on a straight line in a

log-log plot. Figure 1 shows the cumulative distributions

of degrees P(k) for this firm competition network in

log-log scale. One can see that there is a better fit to the

linear behavior in log-log scale and the solid line with

slope −2.15, which indicates that this network appears to

exhibit power-law degree distribution.

Reference [12] argued that power-law degree distribu-

tion is the consequence of two generic mechanisms.

Firstly, networks expand continuously by the addition of

new vertices. Secondly, new vertices attach preferentially

to sites that are already well-connected. In this firm

competition network, the new-added firm vertices will

construct competitive relationships with the existing

firms in limiting product categories. And obviously the

existing high-degree vertices have competitive relation-

ships with many firms in more product categories than

low-degree vertices. Therefore, it is reasonable to think

that the new-added firms will have higher probability to

construct competitive relationship with these high-degree

firms than with low-degree firms. It just follows prefer-

ential attachment mechanism, and thus power-law degree

distribution emerges as a result.

2) Degree correlation: It has been observed that the

degrees of adjacent vertices are positively correlated in

social networks and negatively correlated in most other

networks. Positive correlation is also called assortative

mixing that has been proposed to be distinctive feature of

social networks [1,15], which means a preference for

high-degree vertices to attach to other high-degree verti-

ces, and vice versa. For measuring degree correlation,

two quantifying ways are adopted usually in [1], includ-

ing plotting a one-parameter curve given by <knn> de-

pending on k, where <knn> is the average degree of

nearest neighbors of vertices with k links and calculating

the Pearson correlation coefficient r of the degrees at

either ends of an edge. If the fit to the curve follows

<knn> ~ k

where

> 0, or Pearson correlation coeffi-

cient r > 0 then the network is characterized by positive

degree correlation.

In this paper, we consider both of these measurements.

As shown in Figure 2, we could see that <knn> increases

Figure 1. Degree distribution of firm competition network.

Figure 2. Average degree of nearest neighbors.

with k first, which means that in the cases with small k,

the average degree of connected neighbors increases with

k. However, note that <knn> decreases with k in the tail

of the curve, which has also appeared in [14]. We think

that the appropriate explanation for this change is that the

number of high-degree vertices is very small compara-

tively. In addition, r is calculated to be 0.4976 (> 0).

Therefore, this empirical firm competition network

shows positive degree correlation between the degrees of

adjacent vertices, which is similar with other social net-

works.

3) Hierarchical modularity: Also of interest is the hi-

erarchical modularity, which has been found to be shared

by some real networks such as actor networks, language

networks, World Wide Web, and etc. Indeed, many net-

works are fundamentally modular [13], as one can easily

identify groups of vertices that are highly interconnected

Copyright © 2011 SciRes. JSSM

Characterizing and Modeling the Structure of Competition Networks11

with each other, but have only a few or no links to verti-

ces outside of the group to which they belong to.

This property is captured by the scaling law C(k) ~ k-

,

where C(k) is the average clustering coefficient of verti-

ces with k links. Figure 3 shows the C(k) curve for this

empirical network, while the value of C(k) is in the range

of [0.27, 1]. As the plot indicates, although the obtained

C(k) does not follow as closely the scaling law in almost

all the range of scale as observed in other networks (see

examples presented in [13]), it is clearly evident that

there is a linear fit to the real data in log-log scale for

values of k between 40 and 144. The slope of solid line in

the plot is −1.12 similar with −1 demonstrated to be

shown in other real networks. Therefore, it indicates that

in this network, many highly interconnected smaller ver-

tices coexist with a few larger vertices, which have lower

clustering coefficients, and thus the network exhibits the

hierarchical nesting topology. Note that, in cases with k

smaller than 40, C(k) seems independent with the in-

crease of k . We want to say that the need to satisfy the

scaling law in the whole plot is a little strict, and the

scale law existing in the tail of the plot is enough to in-

dicate hierarchical modularity [13], just as our case

shows.

4) Self-similarity: Recent research papers have pro-

posed that self-similarity is shared by a wide variety of

networks, from World Wide Web to cellular networks

[16]. This characteristic reflects a power-law relation

between the number of boxes needed to cover the net-

work NB and the size of the box lB, expressed as NB (lB) ~

lB −dB.

This paper adopts the covering algorithm based on the

breadth-first-search to investigate this property. Figure 4

shows the result. According to it, this empirical network

has the characteristic of self-similarity with dB = 1.

Roughly speaking, this network is tied together in the

same way across increasing levels in its hierarchical or-

ganizations, which means the links between clusters of

firms, and between clusters of clusters, and so on, obey

the same statistical trends as the links between individual

firms themselves. In short, the architecture of this firm

competition network is symmetrical.

2.3. Effect of Structural Properties

In the part, we give some discussions about the effect of

structural properties on dynamical processes on net-

worked systems. As mentioned above, the small world

effect has been found in this empirical network. It is well

known that the small world effect has obvious implica-

tions for the dynamics of processes taking place on net-

works. For example, if one considers the spread of in-

formation, or anything else across a network, the small

world effect implies that the spread will be fast. In this

Figure 3. Average clustering coefficient of vertices.

Figure 4. Demonstration of self-similarity.

firm competition network, that means any information

with respect to the competitors or any competition fluc-

tuations will spread very fast through the networks.

Moreover, high clustering and positive degree correla-

tion are highly distinctive statistical signatures common

to social networks, different from other types of networks.

Reference [15] conjectured that the fact that social net-

works are usually divided into groups or communities is

the appropriate explanation for both of these properties.

The characteristic of degree correlation has been sug-

gested to affect the resilience to damage of networks [17]

and the diffusion of innovations on networks [7]. Com-

pared with networks that are disassortative, it is not sur-

prising that the size of the giant components is smaller in

the assortative mixed networks since percolation will be

restricted to the sub-network. As for the competition

network, it is not clear yet how this structural property

Copyright © 2011 SciRes. JSSM

Characterizing and Modeling the Structure of Competition Networks

12

affects the competitive dynamics on it.

As for power-law degree distribution, the probability

of high degree vertices decays not so rapidly, compared

with the case with exponential degree distribution. It

means that there indeed exist a certain number of ex-

tremely high degree vertices, so-called hub vertices. Re-

lated to degree distribution is the property of resilience of

networks. In literature from different disciplines so far,

the importance of hub vertices has been mentioned fre-

quently. It has been observed that many networks with

power-law degree distribution are robust against random

vertex removal, but less robust to targeted removal of the

highest degree vertices. In this firm competition network,

the importance of high-degree firms is highlighted there-

fore, and the removal or any change of these types of

firms will have significant effects on the whole network.

There is little attention having been given to the effect

of hierarchical modularity and self-similarity on dy-

namical behaviors of networked systems yet. Reference

[15] have proposed that the presence of hierarchical

modularity reinterprets the role of the hubs in complex

networks, which are known to play a key role of increas-

ing robustness and spreading viruses for scale-free net-

works. In this firm competition network, the positions of

high-degree vertices, which full the structural holes

among communities, obviously have some important

implications for information spread. For self-similarity,

reference [18] has proposed that it can be used as a

benchmark for testing models of network structure,

which therefore is important to the research on dynamic

processes based on the network structural model.

3. Theoretical Model

3.1. Model of the Firm Competition Network

As mentioned above, the focus of network model research

is shifting away from the analysis of single structural

properties to consideration of multiple properties of net-

works. To the best of our knowledge, there is no existing

network model that can theoretically reproduce four char-

acteristics of the competitive network simultaneously al-

though they are proved to emerge at the same time.

Therefore there is a requirement to propose a theoretical

model that can reproduce them at the same time and pre-

dict the topology of firm competition network successfully,

which will full the gap of existing models to some extent.

As mentioned in the part of introduction, competition

behaviors are omnipresent behaviors, which could be

found everywhere. We found many valuable literatures

in the research field of competitive food web, studying

the structural properties of food webs and proposing sev-

eral theoretical network models [19,20]. Between the

competitions among species and the competitions among

firms, there indeed exists some essential similarity.

Drawing valuable ideas from this completely different

discipline, we form and describe our theoretical network

model as follows.

In fact, in firm competition networks, the features of

its developed products, such as the product function or

the customer target, can characterize each firm. And an

edge between two firms is constructed if their character-

istics are similar and thus two firms form competitive

relationship. It hints two possible factors affecting the

topology of the network. First is heterogeneity in vertices

characteristics. In fact, if all vertices characteristics are

completely homogeneous, that is, all firms in the real

system (such as in an industry or a park) produce the

similar products, then the corresponding network is fully

connected, which is not discussed in this paper. On the

other hand, in view of products features, there is a com-

peting range for each firm, for instance in reality the field

of e-commerce application platform or the field of enter-

prise management system. All firms in the system whose

developed products have the features falling in this range

are the competitors of this firm due to the similarity in

their products features. Other firms out of this range have

no direct competition with this firm. The ranges vary

depending on different products developed by firms,

which accordingly affect the connection of network.

Therefore, the theoretical model can be constructed as

follows. Given N vertices, each vertex i is assigned a

characteristic value xi that is drawn from a beta distribu-

tion

1,Beta

where

1,

quantifies the het-

erogeneity level in nodes’ characteristic values. Con-

cretely, the heterogeneity level is increasing as

de-

creases towards 1. Especially, when

= 1 all nodes’

characteristic values are drawn from the uniform distri-

bution. As mentioned above, in view of some practical

limits, the vertices would be constrained to compete with

vertices within a certain range. Expressed by formula, if

|(xi − xj)|≤ (ri + rj)/2 then vertex i and j are connected, or

else two vertices are not connected. Here, ri is assumed to

be a random variable from a beta distribution

Be 1,ta

where

represents the variability of ri. It is known that

11Ex

for

~1,xBeta

and

11Er

for

~1,rBeta

. In order to assure

the model-generated network’s average connectivity close

to the empirical network’s, we choose

and

in si-

mulation to satisfy

11 1

4C where the left

item is just

rEx E and C equals

21LN N

that is just the network density being the ratio between

the number of actual links and the possible number

L

12NN in the empirical undirected network. By

adjusting the parameter

and

, we can find out the

optimal network that predicts those empirical character-

istics better than all other model-generated networks.

Copyright © 2011 SciRes. JSSM

Characterizing and Modeling the Structure of Competition Networks

Copyright © 2011 SciRes. JSSM

13

3.2. Analysis of Simulation Results theoretical insights into the topological complexity of

firm competition networks. In looking forward to future

directions in this area, it is clear that there is much to be

done.

In our simulation, according to the empirical firm compe-

tition network, the parameters and in the model

are respectively 574 and 0.04. In order to obtain the opti-

mal predicted network,

N C

is chosen to be 1.5 and then

can be calculated by solving

1 2.5 10.04

.

Firstly, the structure analysis of firm competition net-

works is only the first step. In a sense, our ultimate goal

is to understand the behaviors and functions of this spe-

cial competition networks. Therefore, the question to be

explored next is how these observed structural properties

affect the competitive dynamics on earth. So far, research-

ers from different disciplines have developed a variety of

techniques and models, helping us understand or predict

the behaviors of networked system, however, studies of

the effect of structure on system behavior are still in their

infancy. The next thing we need to do is to study the effect

of structural properties mentioned in this paper on the

competitive dynamics of firm competition networks.

Figure 5 presents the comparison of structural proper-

ties between the empirical network and the model-gen-

erated network. The value in the bracket is the slope of

corresponding solid line. As it indicates, our proposed

model generates the network displaying multiple proper-

ties simultaneously, including power-law degree distri-

bution, hierarchical modularity, positive degree correla-

tion and self-similarity, as similar as the empirical network.

Meanwhile, all four properties are predicted successfully.

Especially, as shown in the plot (a), the full predicted data

of degree distribution displays good fit with the empirical

data. Therefore, this simple network model has the power

to provide mechanistic explanations for the structural

complexity of firm competition networks.

In addition, although the theoretical model proposed in

this paper could reproduce the topology of the firm

competition network to some extents, there are many

levels of sophistication one can add to this model to

make it more appropriate for real competitive networks.

For example, the model should not be static, but may

4. Concluding Remarks

In summary, this paper has thrown some empirical and

Figure 5. Comparison of the empirical network (red circle) and the model-generated network (blue diamond).

Characterizing and Modeling the Structure of Competition Networks

14

evolve over time with vertices or edges appearing or

disappearing, or values defined on them changing.

Moreover, in the future research, we can use this theo-

retical model as a research platform to explore a vast

variety of complex and poorly understood competitive

phenomena in the field of industry organization. For

example, how do the competition networks evolve? How

does the different position in the networks influence the

individual firm’s control ability of competition? How do

the fluctuations spread on the competition networks?

It is worthy to note that although the research in this

paper is constructed on the basis of firm competition

networks, it may be extended to the analysis of the gen-

eral competition systems. As mentioned above, competi-

tive phenomena are omnipresent in real socio-economi-

cal systems. We hope that our preliminary investigations

in the firm competition will stimulate other researchers to

pursue more extensions.

5. Acknowledgements

The authors would like to thank CHEN Xiao-rong for her

fruitful discussions during the development of this paper.

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