iv class="t m0 x3 ha y5e ff3 fs0 fc0 sc0 ls0 ws1">the involved products is not stated here. Notice that in the
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
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
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
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
and
, we can find out the
optimal network that predicts those empirical character-
istics better than all other model-generated networks.
Characterizing and Modeling the Structure of Competition Networks
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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Characterizing and Modeling the Structure of Competition Networks