Classification of Structural Complexity for Mine Ventilation Networks LIAN-JIANG WEI, 1 FU-BAO ZHOU, 1 JIAN-WEI CHENG, 1 XIN-RONG LUO, 1 AND XIAO-LIN LI 2 1
Faculty of Safety Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, People’s Republic of China; and 2School of Mines, China University of Mining and Technology, Xuzhou, Jiangsu 221116, People’s Republic of China
Received 15 February 2014; Revised 28 March 2014; accepted 8 April 2014
The mine ventilation system is most important and technical measure for ensuring safety production in mines. The structural complexity of a mine ventilation network can directly affect the safety and reliability of the underground mining system. Quantitatively justifying the degree of complexity can contribute to providing a deeper understanding of the essential characteristics of a network. However, so far, there is no such a model which is able to simply, practically, reasonably, and quantitatively determine or compare the structural complexity of different ventilation networks. In this article, by analyzing some typical parameters of a mine ventilation network, we conclude that there is a linear functional relationship among five key parameters (number of ventilation network branches, number of nodes, number of independent circuits, number of independent paths, and number of diagonal branches). Correlation analyses for the main parameters of ventilation networks are conducted based on SPSS. Based on these findings, a new evaluation model for the structural complexity of ventilation network (which is represented by C) has been proposed. By combining SPSS classification analyses results with the characteristics of mine ventilation networks, standards for the complexity classification of mine ventilation systems are put forward. Using the developed model, we carried out analyses and comparisons for the structural complexity of ventilation networks for C 2014 typical mines. Case demonstrations show that the classification results correspond to the actual situations. V Wiley Periodicals, Inc. Complexity 21: 21–34, 2015 Key Words: mine ventilation; network structure; complexity; quantitative; classification
1. INTRODUCTION
ventilation system for providing fresh air into mines,
A
reducing the concentration of explosive or toxic gas, and
s one of the main energy resources of society, coal plays an important role in the production and development of the global economy. When mining underground coal seams, it is a basic task to establish a mine
thus ensuring a safe underground working environment. Often referred to as the ‘‘heart’’ and ‘‘arteries’’ of mines, the mine ventilation system is the most important, fundamental, and economically technical measure to ensure
Correspondence to: F.-B. Zhou, E-mail:
[email protected]
Q2014 Wiley Periodicals, Inc., Vol. 21 No. 1 DOI 10.1002/cplx.21538 Published online 26 April 2014 in Wiley Online Library (wileyonlinelibrary.com)
safety production in mines [1].
C O M P L E X I T Y
21
The safety reliability of ventilation systems is directly influenced by the structural complexity of the ventilation network. Thus, the structure and complexity of complex systems or complex phenomena have become a hot research topic. The complexity of mine ventilation networks is not only affected by the parameter determination and flow rate regulation of the mine ventilation network, but is also closely related to economic investment, level and quantity of technical personnel, management and disaster prevention, and relief planning. Modern societies are characterized by a variety of complex networks, including networks of agents and computers [2]. Networks, commonly represented as graphs, are widely used in biology, communications, and other fields where the representation of complex relationships is central [3]. In recent years, many basic models and methods have been proposed [4, 5], such as the ‘‘small world’’ network model, scale-free network model, and network evolution theory, and researchers have explored a large number of complex natural and artificial networks, thus, the theoretical study on complex network has developed rapidly and become an effective tool for researching complex ventilation networks. At present, among the many indicators used to evaluate mine ventilation systems, only a small number of researches have evaluated the indicators of mine ventilation systems from the aspect of the topological structure of the ventilation network. The mine ventilation network is directed, and it is easier to control and manage simple ventilation network systems than complex ones. The reason for this is that the more complex the system structure is, the stronger the relation among system elements is, and thus the more difficult the analysis is [6]. Although the topological relation concept of the mine ventilation system has been proposed many years ago, research on the complexity of topological relation is still not sufficiently thorough [7, 8]. The structural complexity of the mine ventilation network is related to the number of roadways (i.e., number of branches), mine production allocation, the number and distribution of underground places where air is needed, as well as the complexity, the stability of underground air current, and flow rate regulation [9]. Therefore, it is significant for the safe production of mines to reasonably evaluate the complexity of mine ventilation networks, quantify simple, complex, and extremely complex mine ventilation systems, effectively guide the safe, economic, and reasonable investment in mines, allocate the appropriate level and quality of technical personnel, and determine a proper disaster rescue and relief plan. Structural complexity has been widely studied from many aspects, including supply chain network [10, 11], traffic network, global production network [12], brain structure [13], software structure [14, 15], mineral structure [16], ecosystem [17], and language structure [18]. A mine ventilation network is a complex network system,
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and all its branches are correlated with each other; therefore, unexpected impact may be caused to the ventilation of other branches when changing the state of one branch [19, 20]. Preliminary study has been conducted for the structural complexity of ventilation networks before 2007 [8, 21]. Ushakov, a professor in the ventilation research laboratory of Moscow Mining Institute, when giving lectures in China in 1994 proposed the indicator used in the former Soviet Union to evaluate the reliability of mine ventilation systems by the structure method. The indicator is g [9], c 5 (n 2 m 1 1)/n (where n is the number of branches, m is the number of nodes, n 2 m 1 1 is the number of independent circuits, which conforms to the number of independent paths, i.e., n 2 m 1 2). Each circuit has at least one branch that the other circuits do not have, and this circuit is called the independent circuit. Each path also has at least one branch that the other paths do not have, and this path is called the independent path. The number of independent circuits and the number of independent paths are the basic parameters used to analyze the mine ventilation network. It is reasonable to select the number of independent circuits as the basic parameter to evaluate the complexity of the ventilation network in a sense, but a problem may occur when using indicator c: when the number of branches and number of nodes in different ventilation networks are the same as each other, the values of c are the same, but the complexity degrees of the two networks may be different. Some scholars have pron posed evaluating the complexity indicator W 5 nj using the ratio between the number of diagonal branches (nj) and the number of branches (n) in the network, and they considered that the larger the value of W was, the more complex the ventilation system was [9]. Yang Yunliang of Henan Polytechnic University proposed the indicator R5lg (where nj is the number of diagonal m3 n1nj branches) to reflect the structural complexity of mine ventilation networks [9]. This indicator gives consideration to diagonal branches and can reflect the complexity of the ventilation network, but further study is required to confirm whether or not m is related to (n 1 nj). Cheng et al. [21] analyzed and evaluated the complexity of mine ventilation networks based on the actual production and combining the four main influencing factors (namely number of nodes, number of branches, number of independent circuits, and number of diagonal branches), and classified the complexity of the mine ventilation network into four levels, namely simple, moderate, relatively complex, and complex. In some cases, the air distribution of a ventilation network must be adjusted manually, while programmers must debug the algorithm based on a program flow chart; therefore, the structural complexity of the ventilation network is similar to that of the software. The structural
Q 2014 Wiley Periodicals, Inc. DOI 10.1002/cplx
TABLE 1 Calculation and Comparison of Evaluation Models for the Complexities of the Four Ventilation Networks Name of Mine Simple diagonal connection Mine 1 Mine 2 Mine 3 Mine 4 Mine 5 Mine 6 Mine 7 Mine 8 Mine 9
n
m
n2m11
nj
n 1 nj
(n 1 nj)*m
c
W
R
V(G)
7 36 48 72 84 133 149 169 224 441
6 24 34 47 55 97 107 114 146 298
2 13 15 26 30 37 43 56 79 144
1 16 15 20 21 60 56 41 80 170
8 52 63 92 105 193 205 210 304 611
48 1248 2142 4324 5775 18721 21935 23940 44384 182078
0.29 0.36 0.31 0.36 0.36 0.28 0.29 0.33 0.35 0.33
0.14 0.44 0.31 0.28 0.25 0.45 0.38 0.24 0.36 0.39
1.68 3.10 3.33 3.64 3.76 4.27 4.34 4.38 4.65 5.26
3 14 16 27 31 38 44 57 80 145
complexity of software and task-structure complexity of a distributed system are represented by cyclomatic complexity_ENREF_22, which is the number of edges generated in the directed graph due to circulation in the program. The cyclomatic complexity is V(G), V(G) 5 E 2N 1 2, where E is the number of edges of in the program flow chart, and N is the number of nodes in the program flow chart. The meaning of cyclomatic complexity is the same as that of the number of independent paths (n 2 m 1 2), and the values are equal to each other. Further study must be conducted to confirm whether or not the structural complexity of the ventilation network can be calculated using the number of independent paths, as well as whether or not the structural complexity of the ventilation network is fully reflected. The existing research mainly explores the complexity of ventilation network from the perspective of network, but in practice the complexity of ventilation network is related to both the complexity of graph and the specific functional structure of the ventilation system [22], such as the ventilation disorder inside the network. To better quantitatively evaluate and compare the complexity of different ventilation networks, we must further analyze the related factors and evaluation model influencing the structural complexity of the ventilation network.
2. ANALYSIS OF THE PROPOSED EVALUATION MODEL Currently, four types of evaluation model can be used to evaluate the structural complexity of ventilation networks: c5ðn2m11Þ=n
(1)
W 5nj =n R5lg n1nj 3m
(2)
V ðGÞ5n2m12
(4)
(3)
To conduct rational analysis for the four evaluation models above, the corresponding complexities have been,
Q 2014 Wiley Periodicals, Inc. DOI 10.1002/cplx
respectively, calculated using the simple diagonal connection containing seven branches, along the actual data of the ventilation network in nine mines, as well as those based on the four evaluation models. The specific results are shown in Table 1. The simple diagonal connection and nine actual mines shown in Table 1 rank from simple to complex ventilation network structure. In terms of all aspects, mine 9 (with 441 branches and 144 independent circuits) is more complex than mine 1 (36 branches and 13 independent circuits); the complexities of mine 1 calculated by model g and model Ware 0.36 and 0.44, respectively, while the complexities of mine 9 are calculated by model g and model W are 0.33 and 0.39, respectively. It can be seen from the calculation results that mine 9 is easier than mine 1, which does not conform to the actual situation; therefore, the structural complexity of the ventilation network cannot be evaluated appropriately using model g and model W. The simple diagonal connection calculated by model R and model V(G), as well as the structural complexities of the ventilation networks in the nine actual mines are basically reasonable. Two basic parameters (n and (n 1 nj)) of the ventilation network are used in model R; we can analyze whether or not the two basic parameters are correlated with each other using the data in Table 1 and SPSS (A software named ‘‘Statistical Product and Service Solutions’’ for statistical analysis, data mining, predictive analysis and decision support), and the specific results are shown in Table 2. n is shown to be strongly correlated with m and (n 1 nj), and thus, it is not reasonable to calculate the structural complexity of the ventilation network using R5lg n1nj 3m . Based on the above analysis, it is shown that model g and model W cannot be used to effectively evaluate the structural complexity of the ventilation network. Model R and model V(G) can be used to evaluate the structural
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23
TABLE 2 Correlation Analysis of n and m(n 1 nj) Pearson
n m nj n 1 nj (n 1 nj)*m
n
m
nj
n 1 nj
(n 1 nj)*m
1.000 0.998 0.983 0.999 0.951
0.998 1.000 0.987 0.998 0.949
0.983 0.987 1.000 0.991 0.956
0.999 0.998 0.991 1.000 0.956
0.951 0.949 0.956 0.956 1.000
complexity of the ventilation network in a general manner, but the basic parameters of model V(G) are correlated, and the arrangement is not reasonable enough. The four evaluation models do not consider the influence of the number of intake shafts and the number of return shafts or ventilation disorder inside the network or other factors on the structural complexity of the ventilation network.
The number of all paths includes all possible paths from the start node to the end node of the ventilation network, and this parameter must be used in relief and escape routes and diagonal branch identification. The number of all paths is generally large; as shown in Table 6, mine 9 is a ventilation network with 441 branches, and the number of all paths is 20,185.
3. FACTORS INFLUENCING THE STRUCTURAL COMPLEXITY OF THE MINE VENTILATION NETWORK
3.2. Network Node Degree and Average Node Degree
The complexity of mine ventilation networks is not only correlated with the number of nodes, number of branches, number of independent circuits, number of all paths, number of diagonal branches, and clustering coefficient, but it is also closely related to the degree of disorder, number of intake shafts (sources), number of return shafts (sinks), and other factors. The analysis of the main factors influencing the complexity of the mine ventilation network is as follows.
3.1. Number of Branches, Nodes, Independent Circuits, Independent Paths, and All Paths Branches and nodes are fundamental components of the mine ventilation network, and the numbers of branches and nodes have a significant impact on the complexity of the ventilation network. In the mine ventilation network, the number of independent circuits is n 2 m 1 1. In general, the number of independent circuits is used as the basis for calculation when calculating the ventilation network. The greater the independent circuit is, the larger the calculation amount is. It can be seen from graph theory that the number of independent paths is n 2 m 1 2, and independent paths are generally taken as the basis when optimizing and adjusting the ventilation network. The larger the number of independent paths is, the more difficult the optimization and adjustment of ventilation system is. Therefore, the number of independent circuits (n 2 m 1 1) and number of independent paths (n 2 m 1 2) has a great significance in the quantitative evaluation of the structural complexity of the ventilation network.
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Node degree (ki) can be used to describe and measure the characteristics of a node, and indicates the number of edges connected to node i. In mine ventilation networks, node degree is the number of branches connected to this node. In general, the larger the node degree is, the more important the node is in the network, and the ki of node i is defined as the number of edges of this node connected to other nodes. The average degree of the network node is the ratio between the sum of all node degrees and the number of nodes. The average node degree can reflect the disorder degree inside the network. Logically, the larger the average degree value of network is, the closer the network connection is, and the stronger the ‘‘coupling’’ between the corresponding nodes is, thus reducing the traversal efficiency; in addition, the smaller the average degree value of the network is, the looser the network connection is, and the weaker the ‘‘coupling’’ between the corresponding nodes is, thus increasing the traversal efficiency. Therefore, the larger the average degree value of the network is, the more complex the network is, and the average node degree is positively correlated with the structural complexity of the ventilation network. The node degree distribution in mine ventilation networks is special in relation to other networks, because it has its own unique distribution law. The degree distribution of different types of nodes is shown in Table 3. If the number of branches (n) and number of nodes (m) are determined, then the average degree of this network node is a given value (D 5 2n/m). To produce better statistics for ventilation network degree distribution, the node degree distribution of mines 7–9 with a large
Q 2014 Wiley Periodicals, Inc. DOI 10.1002/cplx
TABLE 3 Degree Distribution of Different Types of Nodes in the Mine Ventilation Network Node Degree
Node Type in Mine
Description of Probability of Occurrence
1
Intake shafts and return shafts
2 3
Series Three-way intersection of parallel, wind joining and wind dividing Crossroad
4 5
The number of intake shafts and the number of return shafts is fixed, the number of intake shafts and the number of return shafts in the most common ventilation network is 1. Roadways in series (the number of such underground roadways is small). Such intersection is most common in the ventilation network. The crossroad is the point where two roadway planes under the mine intersect with each other (the number of such underground crossroad is small). The compound node means that one node is connected to more than five branches, which is very rare in ventilation networks and is generally caused by the simplification of the ventilation network.
Compound nodes of close nodes after simplification
network scale has been selected, and the specific situation is shown in Table 4 and Figure 1. There is no node with a degree greater than 7, the number of nodes with a degree of 1, 2, 4, 5, or 6 is very small, and most of the node degrees are 3, which is basically consistent with the node degree distribution law shown in Table 3. The trends of each mine ventilation network node degree are generally similar to each other, and the average node degree of three mines is about 3, namely: D5
2n 3 m
(5)
In general, the actual mine average node degree (D) slightly fluctuates around 3, and thus, K is defined as the uneven coefficient of the average node degree: K5
D 2n 5 3 3m
(6)
K is proportional to D, and the value of K is about 1. The structural complexity of the ventilation network can be corrected using K value: K > 1 signifies that the complexity should be corrected toward complex whereas K < 1 signifies that the complexity should be corrected toward simple. The following equation can be achieved based on the previous equations:
2 m n 3
(7)
3.3. Number of Intake Shafts and Return Shafts The number and configuration of intake shafts and return shafts in the ventilation network of mines are different, thus, the flow rate distribution and stability of the ventilation network are also different. In general, the larger the number of intake shafts is, the greater the difference between the ventilation parameters of all intake shafts and return shafts is, and consequently, the more difficult the ventilation management and regulation is. In addition, the larger the number of return shafts in the ventilation network is, the closer it is to the partition ventilation, and the easier the ventilation management and regulation is. To facilitate the analysis of the ventilation network, the ventilation network of intake-return shafts can be changed into the type with one intake shaft and one return shaft, by adding a virtual branch and virtual node. The structural complexity of the ventilation network shows almost no change before and after its transformation. Although the number of return shaft has impacts on the complexity of ventilation network management, no effect is made on the structural complexity of the ventilation network.
TABLE 4 Node Degree Distribution of Mines 7-9
Mine 7 Mine 8 Mine 9
Q 2014 Wiley Periodicals, Inc. DOI 10.1002/cplx
1
2
3
4
5
6
Average Node Degree
6 4 11
16 17 30
87 104 227
12 15 26
1 4 4
0 2 0
3.02 3.16 3.04
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25
4. QUANTITATIVE EVALUATION MODEL BUILDING FOR STRUCTURAL COMPLEXITY OF VENTILATION NETWORK 4.1. Correlation Analysis of Influencing Factors
FIGURE 1
Node degree distribution of ventilation network in mines 7-9.
3.4. Number of Diagonal Branches Diagonal branch connects two parallel airflow paths [23] in the mine ventilation network (such as branch 5 in Figure 2). Breeze, no wind, and even airflow reversal may be caused by unstable airflow, thereby seriously affecting the stability of the ventilation system. Moreover, unstable airflow may also cause the gas in the roadways to gather, thus leading to gas and coal dust explosion during the mine production process, and even heavy casualties and property losses. A large number of diagonal branches make the connections between nodes or branches in the ventilation network more complex [24], thus the number of diagonal branches (nj) has a certain impact on the structural complexity of the ventilation network.
3.5. Network Structure Entropy Network structure entropy can reflect the disorder degree inside the network, as well as the impact of diagonal connection to a certain extent. When n 2 m 1 1 is constant and the connection methods between network nodes are different, the structural complexities of the mine ventilation network are different, which is caused by the internal network and can be represented by network structure entropy describing the disorder degree inside the network. The larger the network structure entropy is, the greater the disorder degrees inside and outside the network are, and the more complex the network is. Network structure entropy can be expressed as follows: E52
Xm
I i51 i
Ki I i 5 Pm
j51
ln Ii
Kj
C O M P L E X I T Y
FIGURE 2
(8)
(9)
where Ii is the importance of node i, m is the number of nodes in the network, and ki is the degree of the i node, thus it is clear that ki > 0.
26
It can be seen from the above analysis that the number of branches (n), number of nodes (m), number of independent circuits (n 2 m 1 1), number of independent paths (n 2 m 1 2), number of all paths (ap), number of diagonal branches (nj), sum (n 1 nj) of the number of branches (n), and number of diagonal branches (nj), as well as (n 1 nj)*m, may be correlated with the structural complexity of the ventilation network, and these factors can be taken as the basic parameters to evaluate the structural complexity of the ventilation network. The different values of the network structure entropy (e), clustering coefficient (cc), average degree of nodes, and average clustering coefficient (acc) have slight impacts on the structural complexity of the ventilation network, and all these parameters may be taken as the correction coefficient of the structural complexity of the ventilation network. To properly and reasonably select the basic parameters and correction parameters for the quantitative evaluation of the structural complexity of the ventilation network, correlation analysis must be conducted for the ventilation network parameters described above. The basic parameters for correlation analysis include one simple diagonal connection and the parameters related to the ventilation networks of the nine mines, as shown in Table 5. The correlation analysis has been conducted based on the related parameters in Table 5 and influencing factors of SPSS on the structural complexity of the ventilation network, and the analysis results are shown in Table 6. We conducted correlation analysis for the influencing Pearson factor of the structural complexity of the ventilation network, and the results show that the correlation among n, m, n 2 m 1 1, n 2 m 1 2, ap, e, nj, n 1 nj, and (n 1 nj)*m was strong. In addition, n, m, n – m 1 1, n 2 m 1 2, nj, and n 1 nj could be taken as the basic parameters to evaluate the structural complexity of the
Ventilation system network diagram of 12,403 working surface in a mine [23].
Q 2014 Wiley Periodicals, Inc. DOI 10.1002/cplx
TABLE 5 Simple Diagonal Connection and Basic Parameters of Ventilation Network in the Nine Actual Mines Name of Mine
n
Simple diagonal connection Mine 1 Mine 2 Mine 3 Mine 4 Mine 5 Mine 6 Mine 7 Mine 8 Mine 9
m
n2m11
n2m12
ap
e
7
6
2
3
3
0.99
36 48 72 84 133 149 169 224 441
24 34 47 55 97 107 114 146 298
13 15 26 30 37 43 56 79 144
14 16 27 31 38 44 57 80 145
31 32 262 480 206 166 2240 4774 20,185
2.72 2.76 3.03 3.73 3.73 3.77 4.51 4.72 5.45
cc
2n/m
2
2.33
1
8
48
0.29
3.00 2.82 3.06 3.05 2.74 2.79 2.96 3.07 2.96
16 15 20 21 60 56 41 80 170
52 63 92 105 193 205 210 304 611
1248 2142 4324 5775 18,721 21,935 23,940 44,384 182,078
0.10 0.02 0.05 0.05 0.01 0.02 0.06 0.05 0.02
3.7 0.93 3.4 4.2 1.73 2.56 9.93 11.2 9.37
nj
n 1 nj
(n 1 nj)*m
acc
the mine ventilation network, m5 23 n, thus producing the following equations:
ventilation network, because these factors were strongly correlated with each other. However, we must further explore which factors are reasonable, accurate, and intuitive; 2n/m and S are not correlated with any other factors and are positively correlated with the complexity of the ventilation network, and thus, these two factors can be taken as the correction coefficients for the structural complexity of the ventilation network.
n2m11 n2
2n n 11 3 3
(10)
n2m12 n2
2n n 12 3 3
(11)
The number of branches (n) and number of diagonal branches (nj) in the ventilation network of the nine actual mines are shown in Table 5. By conducting a regression analysis, it can be seen that the linear regression is the best mathematical expression to quantify the relationship between n and nj as following Eq. (12) and the results are shown in Figure 3:
4.2. Research Regarding the Essential Relationship Among Main Influencing Factors The six factors (n, m, n 2 m 1 1, n 2 m 1 2, nj, and n 1 nj) are strongly correlated with each other, and we must confirm the relationship among them. The specific analysis is as follows. It may be seen from Eq. (7) that in
TABLE 6 Correlation Analysis of Influencing Factor Pearson of Structural Complexity of Ventilation Network n
m
n 1 0.998 m 0.998 1 n2m11 0.992 0.983 n2m12 0.992 0.983 ap 0.925 0.916 e 0.860 0.854 cc 0.686 0.649 2n/m 20.350 20.314 0.983 0.987 nj 0.999 0.998 n 1 nj (n 1 nj)*m 0.951 0.949 acc 20.411 20.430 (n 2 m 1 1)/n 0.092 0.036
Q 2014 Wiley Periodicals, Inc. DOI 10.1002/cplx
n2m 1 1 n2m 1 2 0.992 0.983 1 1.000 0.932 0.863 0.752 20.418 0.964 0.988 0.946 20.367 0.205
0.992 0.983 1.000 1 0.932 0.863 0.752 20.418 0.964 0.988 0.946 20.367 0.205
ap
e
cc
2n/m
nj
n 1 nj (n 1 nj)*m
acc
0.925 0.860 0.686 20.350 0.983 0.999 0.951 20.411 0.916 0.854 0.649 20.314 0.987 0.998 0.949 20.430 0.932 0.863 0.752 20.418 0.964 0.988 0.946 20.367 0.932 0.863 0.752 20.418 0.964 0.988 0.946 20.367 1 0.658 0.607 20.256 0.920 0.927 0.991 20.182 0.658 1 0.726 20.656 0.804 0.848 0.692 20.685 0.607 0.726 1 20.571 0.578 0.658 0.572 20.069 20.256 20.656 20.571 1 20.273 20.330 20.233 0.527 0.920 0.804 0.578 20.273 1 0.991 0.956 20.414 0.927 0.848 0.658 20.330 0.991 1 0.956 20.413 0.991 0.692 0.572 20.233 0.956 0.956 1 20.245 20.182 20.685 20.069 0.527 20.414 20.413 20.245 1 0.165 0.245 0.555 20.800 0.006 0.068 0.090 0.074
(n 2 m 1 1)/n 0.092 0.036 0.205 0.205 0.165 0.245 0.555 20.800 0.006 0.068 0.090 0.074 1
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27
structural complexity of ventilation network, so as to build the evaluation model for the structural complexity of the ventilation network:
FIGURE 3
N 5nK
Linear fitting of number of branches and number of diagonal branches in the nine mines.
The K is approximately 1, and the number of branches (n) is corrected, so as to allow the structural complexity of the ventilation network to reflect the actual situation effectively. N may fluctuate up and down slightly, and N may be referred to as the number of equivalent branches. The following equation can be established after substituting Eq. (6) into Eq. (15): N5
nj 50:3889n25:3716 n1nj 5n10:3889n25:371651:3389n25:3716
(12)
(13)
The following equation can be established based on Eqs. (7), (10), and (13): 3m 3ðn2m11Þ 3ðn2m12Þ 2:57nj 113:81 2 0:72 n1nj 13:87
n
(14)
It can be seen from Eqs. (10–14) that essentially there is a linear function relationship among the six factors (namely n, m, n 2 m 1 1, n 2 m 1 2, nj, and n 1 nj), thus explaining why these six factors are strongly correlated with each other. The analysis results show that any one of these six factors can be selected as the basic parameter to evaluate the structural complexity of the ventilation network, and the effect is equivalent, thus further explaining why c 5 lg[(n 1 nj) 3 m] and V(G) 5 n 2 m 1 2 can be used to evaluate the structural complexity of the ventilation network. However, further analysis is required to determine which factor is more intuitive and acceptable. Among the six factors (n, m, n 2 m 1 1, n 2 m 1 2, nj, and n 1 nj), n is the number of branches of the ventilation network (i.e., the number of roadways in the mine ventilation system), and it is the simplest and most intuitive parameter. In addition, it can be seen from the correlation analysis of Pearson in Table 6 that the correlation between n and the other five factors is the strongest. Taking all factors into account, it is reasonable to take n as the basic parameter to select the structural complexity of the ventilation network.
4.3. Quantitative Evaluation Model Building for Structural Complexity of Ventilation Networks Based on the above analysis, n is selected as the basic parameter to evaluate the structural complexity of the ventilation network, while the uneven coefficient of the average node degree (K) is selected as the correction efficient for the influence of the average node degree on the
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(15)
2n 2 3m
(16)
We calculated the number equivalent branches of simple diagonal connections and the nine mines, and the results are shown in Table 7. The number of equivalent branches may more accurately reflect the structural complexity of the ventilation network. With strong operability, this factor (N) can intuitively and quantitatively determine the structural complexity of the ventilation network. It can be seen from Eq. (16) that the number of equivalent branches (N) is only correlated with the number of branches (n) and number of nodes (m). The parameters are easy to obtain and convenient for calculation.
5. CLASSIFICATION AND REASONABILITY TEST OF STRUCTURAL COMPLEXITY OF VENTILATION NETWORK 5.1. Classification of Structural Complexity of Ventilation Network To better describe the complexity of the ventilation network and compare the structural complexities of the different mine ventilation networks, the structural
TABLE 7 Simple Diagonal Connections and Number of Equivalent Branches in the Nine Mines Name of Mine
n
m
N
Simple diagonal connection Mine 1 Mine 2 Mine 3 Mine 4 Mine 5 Mine 6 Mine 7 Mine 8 Mine 9
7
6
5
36 48 72 84 133 149 169 224 441
24 34 47 55 97 107 114 146 298
36 45 74 86 122 138 167 229 435
Q 2014 Wiley Periodicals, Inc. DOI 10.1002/cplx
TABLE 8 Final Cluster Center for the Number of Equivalent Branches from 1 to 1000 Level N Cluster center
1
2
3
4
5
6
20.50
75.00
171.50
331.00
561.00
847.00
3 < C 4; the ventilation network is defined as complex when 400 < N 800, and the complexity is defined as level V, 4 < C 5; finally, the ventilation network is defined as extremely complex when N > 800, and the complexity is defined as level VI, C > 5. Based on the classification standards, we can compare the complexity of the ventilation networks in different mines, as well as the complexity of the same ventilation network in different periods, or before or after the transformation.
complexity of the ventilation network must be classified. There are dozens of branches in simple mine ventilation networks, 400–500 branches in complex ventilation networks, and about 1000 in some very complex ventilation networks. To reasonably and accurately evaluate the structural complexity of the ventilation network, we classified it into six levels, namely simple, relatively simple, moderate, relatively complex, complex, and extremely complex. Cluster analysis has been conducted for the number of equivalent branches (N) from 1 to 1000, based on SPSS, and the analysis results of the cluster center with six clusters are shown in Table 8. We slightly adjusted the cluster center in Table 8 combining the actual situation and usage of the ventilation network in Table 7, and classified the structural complexity of the ventilation network into six levels according to the cluster center after adjustment. The cluster center after adjustment and corresponding number range of equivalent branches are shown in Table 9. We further proposed the structural complexity of the ventilation network (C) according to the classification in Table 10, and the meaning of complexity C is the same as that using the number of equivalent branches to evaluate the structural complexity of the ventilation network. Through evaluation, it is shown that the values correspond to each other. Constant interval division can be conducted for the structural complexity of the ventilation network using value C, as shown in Table 10. The ventilation network is defined as simple when N 50, and the complexity is defined as level I, C 1; the ventilation network is defined as relatively simple when50 < N 100, and the complexity is defined as level II, 1 < C 2; the ventilation network is defined as moderate when 100 < N 200, and the complexity is defined as level III, 2 < C 3; the ventilation network is defined as relatively complex when 200 < N 400, and the complexity is defined as level IV,
5.2. Calculation Model for Structural Complexity of Ventilation Network (C) We can summarize the corresponding relationship between the number of equivalent branches (N) and structural complexity of ventilation network (C) based on Table 10, and the results are shown in Table 11. To determine the corresponding relationship between the number of equivalent branches (N) and structural complexity of ventilation network (C), we conducted four types of fitting experiments (namely linear fitting, quadratic multinomial fitting, power fitting, and logarithmic fitting) for the two factors, using Excel, as shown in Figure 4. The analysis results of the four types of fitting show that the effect of logarithmic fitting is the best, thus the functional relationship between N and C is as follows: C51:4427 ln ðN Þ24:6439
(17)
By substituting Eq. (16) into Eq. (17), the complete equation of C is as follows: C51:4427 ln
2n2 24:6439 3m
(18)
As can be seen from Eq. (18), the structural complexity of mine ventilation network (C) is only correlated with the
TABLE 9 Cluster Center and Range After Adjustment of the Number of Equivalent Branches from 1 to 1000 Level N Cluster center Range
Q 2014 Wiley Periodicals, Inc. DOI 10.1002/cplx
1
2
3
4
5
6
25.00 N 50
75.00 50 < N 100
150.00 100 < N 200
300.00 200 < N 400
600.00 400 < N 800
850 N > 800
C O M P L E X I T Y
29
TABLE 10 Quantitative Classification Standards for Structural Complexity of the Ventilation Network Number of Equivalent Branches N N 50 50 < N 100 100 < N 200 200 < N 400 400 < N 800 N > 800
Complexity C C1 1 < C 2 2
