An Efficient and Automatic Simplification Method for Arbitrary Complex Networks in Mine Ventilation

Abstract

The simplification of complex networks is a research field closely related to graph theory in discrete mathematics. The existing methods are typically limited to simplifying the series sub-networks, parallel sub-networks, diagonal sub-networks, and nested simple sub-networks. From the current perspective, there are no available methods that can handle complex sub-networks and nested complex sub-networks. In this paper, we innovatively propose an efficient and automatic equivalence simplification method for arbitrary complex ventilation networks. The method enables, for the first time, the maximum possible equivalence simplification of nested simple sub-networks and nested complex sub-networks. In order to avoid the NP-hard problem caused by the searching of simplifiable sub-networks, it is necessary to analyze the intrinsic topology relationship between simplifiable sub-networks and spanning sub-graphs to optimize the searching process. One of our main contributions is that we present an efficient searching method for arbitrarily nested reducible sub-networks based on the bidirectional traversal process of a directed tree. The method optimizes the searching process for simplifiable node pairs by combining the characteristics of a directed tree with the judgment rules of simplifiable sub-networks. Moreover, by deriving the formula of an equivalent air resistance calculation for complex sub-networks, another one of our main contributions is that we present an equivalent calculation and simplification method for arbitrarily complex sub-networks based on the principle of energy conservation. The basic idea of the method is to calculate the equivalent air resistance using the ventilation network resolution of the constructed virtual sub-networks. We realize the simplification method of arbitrarily complex mine ventilation networks, and we validate the reliability of the simplification method by comparing the air distribution results using the network solution method before and after simplification. It can be determined that, with appropriate modifications to meet specific requirements, the proposed method can also be applicable to equivalent simplification instances of other types of complex networks. Based on the results analysis of several real-world mine ventilation network examples, the effectiveness of the proposed method is further verified, which can satisfactorily meet the requirements for simplifying complex networks.

1. Introduction

Graph theory is a fundamental branch of discrete mathematics [1,2], while discrete mathematics is the theoretical foundation for computer science and network information science. Graph theory provides the theoretical foundation for studying complex networks, including fluid networks, social networks, biological networks, the Internet, and transportation networks [3,4,5,6,7,8]. A network can be represented as a graphical system consisting of nodes and edges that denote interactions between these nodes [9]. Modeling complex systems in the real world as a topology of points and edges makes complex systems easy to understand and manage [10]. Analyzing the topology of complex networks can reveal the global properties and behavioral patterns of these systems. However, large-scale networks contain an extremely high number of nodes and edges, making their topological network graphs difficult to understand, analyze, and visualize and requiring substantial computational resources to store and analyze their topological properties. Therefore, studying simplification methods for complex networks is of great significance.

Network simplification involves reducing the size of a network by decreasing the number of nodes and edges [11]. Numerous studies have focused on simplifying complex networks from various perspectives. To maintain maximum connectivity, Fang Zhou et al. proposed a generalized framework and method to simplify weighted graphs by pruning edges [12]. To address the task allocation problem, Bin Liao et al. proposed a novel network simplification algorithm called bid-based distributed broken-motifs, which simplifies the communication network by reducing the number of closed-loop triangles [13]. In summary, network simplification significantly reduces the complexity of networks while maintaining their basic features and functionality, providing strong support for research and applications in a variety of fields.

Research on mine ventilation typically abstracts the complex mine ventilation system into a ventilation network, and the analysis of this ventilation network can reveal various internal correlations, thus providing a theoretical basis for its design and network optimization. However, complex ventilation systems often contain thousands or even tens of thousands of branches, imposing high demands on both the operational response performance and real-time computational performance of these systems. To reduce the construction difficulty of an intelligent mine ventilation system [14,15], it is generally necessary to perform the equivalent simplification of the ventilation network based on the calculation of equivalent resistance.

For the simplification of ventilation networks, Fengliang Wu et al. proposed a method based on the isobaric point of diagonal branch simplification. However, this method is only effective for simple diagonal ventilation networks [16]. To achieve the automatic simplification of a ventilation network, Liangjian Wei proposed a “Path Tree Depth-first Growth” method, which searches all the paths of two arbitrary nodes [17]. To improve the accuracy and efficiency of simplification, Jinzhang Jia et al. proposed an optimization strategy for the set of node pairs requiring a depth search within the network, thereby reducing unnecessary depth searches for node pairs [18]. Although many scholars have conducted extensive research on the simplification of a mine ventilation network, further research is needed on the simplification principles of complex sub-networks within mine ventilation networks and rapid search methods for complex networks.

A variety of ventilation network simplification methods have been proposed by related scholars, but the existing methods primarily rely on the simplification idea for a series sub-network, a parallel sub-network, and a diagonal sub-network for an equivalent air resistance calculation. In total, there are six kinds of simplifiable sub-networks: series sub-networks, parallel sub-networks, diagonal sub-networks, complex sub-networks, nested simple sub-networks, and nested complex sub-networks. Although it is possible to use nested sub-networks for the composite simplification of networks, the topological connectivity relations of actual complex ventilated networks often include more non-simplified sub-networks (series sub-networks, parallel sub-networks, and diagonal sub-networks), and the existing techniques are incapable of searching for and handling arbitrarily simplifiable sub-networks in complex networks.

In contrast, the proposed method is capable of handling all six types of sub-networks, making it a more versatile approach. The method no longer relies on the series, parallel, or diagonal relationships of sub-networks to calculate the equivalent air resistance. The method can be used to search for extremely simplifiable sub-networks in order to maximize the simplification of complex networks. Directly determining whether the sub-networks formed between individual pairs of nodes meet the criteria for simplifiable sub-networks is an NP-hard problem [19,20]. To enhance search efficiency, an important contribution of this paper is the proposal of a fast search method for any simplifiable nested sub-networks based on the bidirectional traversal of directed trees. The core idea of this method is to quickly search for simplifiable node pairs using the features of the side branches of simplifiable sub-networks (the branches that extend out from the interior of the sub-networks). Additionally, we propose, for the first time, an equivalent simplification method for arbitrarily complex passive sub-networks based on the principle of energy conservation. The core idea of this method is to calculate the equivalent air resistance by constructing a virtual sub-network for a ventilation network solution. The method no longer relies on the series, parallel, or diagonal relationships of sub-networks to calculate the equivalent air resistance.

The correctness of the proposed approach can be assessed by ensuring that the simplified network remains equivalent to the original network. To validate the accuracy of our method, we can analyze the airflow distribution results before and after the simplification using ventilation network solution methods. Unlike the traditional equivalent air resistance calculation methods based on series, parallel, and diagonal sub-networks, our proposed method applies to the simplification of arbitrarily complex passive sub-networks. This method can significantly reduce the branching ratio of the ventilation network and maximize its degree of simplicity. In addition, the simplified method serves as a critical tool for mine ventilation network analysis and calculation, which is conducive to separating the extremely simplifiable sub-network reflecting the most essential ventilation state of the ventilation system in the complex ventilation network, thereby assisting ventilation technicians in analyzing and managing the complex ventilation system. By establishing the mapping relationship between the simplified sub-network and the original network, the computational processes such as a pathway search [21], a network solution [22], air quantity regulation [23], and the reliability analysis of the ventilation network [24] can be optimized. This enhances the efficiency of managing the complex ventilation network and improves the operational performance of an intelligent mine ventilation system. In terms of efficiency, the differences between simplified and non-simplified ventilation systems can be evaluated by comparing the computational effort required to solve the networks and the ease of understanding the network structure. Our approach aims to maintain the essential characteristics of the network while reducing its complexity, thereby improving the efficiency of both the computational analysis and the practical management of the ventilation system.

The structure of this paper is organized as follows: Section 2 provides an overview of the network simplification method, introducing the basic steps and the algorithm framework. Section 3 delves into graph theory and some terms of network simplification, explaining the simplification rules used. Section 4 outlines the algorithm of network simplification, detailing the step-by-step process and key procedures. Section 5 presents the experiment and analysis, demonstrating the application of the algorithm and evaluating its reliability through real-world mines’ ventilation data. Finally, Section 6 presents the conclusion and discussion, summarizing the main findings and suggesting directions for future research.

2. Overview of the Method

We propose an equivalent simplification method for arbitrarily complex ventilation networks based on the ventilation network solution and the principle of energy conservation calculation. The ventilation network solution method is used to re-simulate the air distribution calculation of the simplified air network, which can ensure the same result as the air distribution calculation of the same roadway before and after simplification because the total air resistance remains consistent. The main algorithmic process of the method can be divided into the following three steps, as shown in Figure 1.

(1)

Searching for simplified sub-graphs.

The ventilation network graph is constructed, and simplifiable sub-networks that satisfy the first or second kinds of judgment rules for simplifiable sub-networks are identified using either the global simplification method or the local simplification method.

(2)

Calculation of the equivalent air resistance.

On the basis of determining the simplifiable sub-networks, the principle of energy conservation and the ventilation network solution are used to calculate the equivalent air resistance of each simplifiable sub-network.

(3)

Mapping the equivalent ventilation networks.

The shortest path method is used to determine the equivalent air resistance branch paths of the simplifiable sub-networks, generate the simplified network, and construct the mapping relationship between the simplified network and the original network.

3. Theory of Network Simplification

3.1. Basis of Graph Theory

The mine ventilation fluid network is the basis of mine ventilation system optimization theory; a complex ventilation system can be abstracted as a ventilation network graph $G(V,E)$ composed of nodes, branches, fans, and structures and other facilities [25]. If the direction of the airflow is considered the direction of branching, then the ventilation network graph is a connected graph, with the overall ventilation area belonging to a strongly connected graph (SCC) [26] and the local ventilation area belonging to a unidirectionally connected graph or a weakly connected graph.

A connected graph in the branch direction and flow must be consistent with the three basic laws of the fluid network (the air quantity balance law, the air pressure balance law, and the resistance law) expressed in the node branch constraints. Therefore, the ventilation network graph must be consistent with the inherent logic of the complex ventilation system.

The intrinsic logical relationship between nodes and branches in a fluid network can be adequately represented using various sub-graphs. Structural sub-graphs such as spanning trees, cotrees, loops, mesh holes, paths, cut-sets, and so on, are the basic concepts of the theory of ventilation networks, and they are of great significance for the optimal design, modification, and control of a ventilation network.

Let $G=(V,E)$ and $G^{\prime }=(V^{\prime },E^{\prime })$ be any two network graphs, and clarify the following concepts related to graph theory.

(1)

A graph, $G^{\prime }$, is said to be a subgraph of a graph, $G$, denoted as $G^{\prime }\subseteq G$, if $V^{\prime }\subseteq V$ and $E^{\prime }\subseteq E$, and $G$ is the parent graph (Supergraph) of $G^{\prime }$.

(2)

A graph, $G^{\prime }$, is said to be a spanning sub-graph (Spanning Subgraph) of a graph, $G$, if $G^{\prime }\subseteq G$ and $V^{\prime }\left(G^{\prime }\right)=V\left(G\right)$.

(3)

$G^{\prime }$ is said to be an Induced Subgraph derived from $V^{\prime }$ if $G^{\prime }\subseteq G$, and $V^{\prime }\left(G^{\prime }\right)\subseteq V\left(G\right)$, $V^{\prime }\left(G^{\prime }\right)\ne \mathrm{\varnothing }$, and $E^{\prime }\left(G^{\prime }\right)=\left\{e\right(v1,v2\left)\right|e\in E\left(G\right),v1\in V^{\prime }\left(G^{\prime }\right) \mathrm{and} v2\in V^{\prime }\left(G^{\prime }\right)\}$.

(4)

$G^{\prime }$ is said to be an Induced Subgraph derived from $E^{\prime }$ if $G^{\prime }\subseteq G$, and $E^{\prime }\left(G^{\prime }\right)\subseteq E\left(G\right)$, $E^{\prime }\left(G^{\prime }\right)=\left\{e\right(v1,v2\left)\right|e\in E^{\prime }\left(G^{\prime }\right),v1\in V^{\prime }\left(G^{\prime }\right) \mathrm{and} v2\in V^{\prime }\left(G^{\prime }\right)\}$.

Spanning trees are the most important spanning sub-graphs in graph theory, and they have many important applications in ventilation networks. Spanning trees are often used as the base sub-graphs for independent loops and independent cut-set circles, and the co-tree branches represent the independence features of the network. In network simplification applications, directed trees constructed based on a Breadth-First Search [27] are an important type of spanning sub-graphs for determining simplifiable conditions.

3.2. Term Definitions

The types of branches in a ventilation network can be classified as branches of the general-type airway, branches of the structure-type airway, branches of the fan-type airway, branches of the fixed air quantity-type airway, and branches of the fixed air pressure-type airway. In order to simplify the ventilation network, the following terms related to the simplification of the ventilation network are clarified.

Sub-networks: Sub-networks, in the simplification of ventilation networks, are sub-graphs in graph theory, but a simplifiable sub-network must be a directed sub-graph that is connected and contains only one source and one sink. In short, a simplifiable sub-network is a single source sink-directed, connected sub-graph in a ventilation network. A sub-network can be represented by a node pair consisting of an inlet node, $IN$, and an outlet node, $OUT$, with the node pair notation denoted as $(IN,OUT)$.

Air inlet node: the source point of the sub-network; the sub-network has one and only one air inlet node.

Air outlet node: the sink point of the sub-network; the sub-network has and only has one air outlet node.

Internal nodes: nodes in the sub-network other than the inlet node and the outlet node; there should be at least one internal node in the sub-network.

Air inlet branch: the branch adjacent to the air inlet node in the sub-network; there can be one or more air inlet branches in the sub-network.

Air outlet branch: the branch in the sub-network adjacent to the air outlet node; there may be one or more air outlet branches in the sub-network.

Active sub-network: a sub-network containing fan-type airway branches, fixed air quantity-type airway branches, or fixed air pressure-type airway branches.

Passive sub-network: a sub-network that does not contain fan-type airway branches, fixed air quantity-type airway branches, or fixed air pressure-type airway branches.

Simplifiable branches: General-type airway branches and structure-type airway branches that are allowed to be simplified. In order to avoid the simplification process affecting the ventilation system as much as possible, the ventilation technician can set any branch as a non-simplifiable branch as needed, and any sub-network containing any non-simplifiable branch is a non-simplifiable sub-network. For example, in practice, building-roadway branches are often set as non-simplifiable branches because of the frequent need to regulate structures.

Nested simplifiable sub-networks: sub-networks that contain other simplifiable sub-networks are called nested simplifiable sub-networks.

Directed Tree: A search tree graph obtained by traversing the network graph using a breadth-first-based directed graph. Figure 2 shows a schematic diagram of the directed tree search process.

Starting node: the root node of the directed tree.

Node depth: the number of layers of the node in the directed tree.

3.3. Simplification Rules

The common types of simplifiable sub-networks are classified into three basic types: series sub-network, parallel sub-network, and diagonal sub-network. These three basic types of simplifiable sub-network can be referred to as simple sub-networks. The more complex simplifiable sub-networks can be represented as nested sub-networks consisting of nested combinations of series, parallel, or diagonal connections with each other. The ventilation network-equivalent simplification method proposed in this paper, on the other hand, can handle arbitrarily complex simplifiable sub-networks. We classify the types of simplifiable sub-networks into six types: series sub-networks, parallel sub-networks, diagonal sub-networks, complex sub-networks, nested simple sub-networks, and nested complex sub-networks, as shown in Figure 3.

In the ventilation network, in order to be able to perform equivalent air resistance calculations on sub-graphs, the simplifiable sub-network should satisfy the following network characteristics [28].

(a) There is one and only one air inlet node;

(b) There is one and only one air outlet node;

(c) All branches in the sub-network are simplifiable branches;

(d) Branch independence, in which all branches associated with any node in the simplifiable sub-network other than the air inlet node and the air outlet node are within the sub-network;

(e) The number of in-degrees of the sub-network is equal to the number of out-degrees of the sub-network and equal to the number of branches in the sub-network;

(f) All branches in the sub-network are parent branches of the air outlet node, which can also be viewed as all branches in the sub-network are children of the air inlet node, i.e., there are no side branches in the sub-network. The network characteristics of the sub-network can be used as a sub-network determination rule for the simplified method of ventilation network.

Based on the above network characteristics of simplifiable sub-networks, the following judgment rule for a simplifiable sub-network and nested sub-network can be obtained.

(1) The first kind of judgment rule for simplifiable sub-networks.

The first kind of judgment rule for simplifiable sub-networks can be expressed as follows. There is one and only one inlet node and one outlet node in the sub-network, the number of inlets in the sub-network is equal to the number of outlets in the sub-network and equal to the number of branches in the sub-network, and all the branches in the sub-network are simplifiable branches. This rule can also be referred to as the in–out-degree calculation rule.

(2) The second kind of judgment rule for simplifiable sub-networks.

The second kind of judgment rule for simplifiable sub-networks can be expressed as follows. There is one and only one inlet node, and there is one outlet node, in the sub-network, all branches in the sub-network are parents of the outlet node, and all branches in the sub-network are simplifiable branches. This rule can also be referred to as the side-branch decision rule.

(3) Nested subnet determination rules.

The nested sub-network determination rule can be expressed as follows. When the node depth of the inlet node of sub-network $G_1$ is not greater than the node depth of the inlet node of sub-network $G_2$, and the node depth of the outlet node of sub-network $G_1$ is not less than the node depth of the inlet node of sub-network $G_2$, sub-network $G_2$ can be determined to be a sub-network of sub-network $G_1$, and sub-network $G_2$ can be referred to as a nested sub-network of sub-network $G_1$. In order to simplify the network as much as possible, the simplifiable sub-networks searched for in this paper are non-nested sub-networks. Therefore, the nesting relationship between simplifiable sub-networks can be judged in the directed tree based on the depth of the node pairs that represent the simplifiable sub-networks.

4. Algorithm of Network Simplification

4.1. Two Kinds of Simplification Methods

The search methods for simplifiable sub-networks proposed in this paper are of two types: local ventilation network simplification and global ventilation network simplification.

(1) Local ventilation network simplification.

Local ventilation network simplification consists of the ventilation manager selecting local simplifiable areas in the ventilation network and then directly applying the first type of simplification judgment rule to determine whether the simplified areas satisfy the simplification conditions. In this simplification method, the user selects the simplifiable sub-network, and it has a low degree of automation. The first kind of judgment rule for simplifiable sub-networks analyzes whether the sub-graph constituted by pairs of nodes between any nodes satisfies the simplification conditions through the simplification judgment parameter. The simplification judgment parameter of the sub-graph constituted between node pairs is $\left(S_e,S_i,S_o\right)$, including the number of branches $S_e$, the number of in-degrees $S_i$ and the number of out-degrees $S_o$.

(2) Global ventilation network simplification.

Global ventilation network simplification is an automatic simplification method that can be divided into two methods. One method is applied by traversing the way of all node pairs in the ventilation network and then using the first type of simplification judgment rule to determine whether all node pairs satisfy the simplification conditions and ultimately identify all the simplifiable sub-networks after removing the nested simplifiable sub-networks. This traversal approach is less efficient in searching and cannot simplify the complex ventilation network. Another method is to quickly determine all simplifiable sub-networks according to the first type of simplification judgment rule, which can be called the bidirectional traversal method using a directed tree, and which can simplify the complex ventilation network. The following section focuses on the search process for the bidirectional traversal method using a directed tree.

4.2. Sub-Network Searching Method

In order to automatically search all simplifiable sub-networks of a ventilated network, we propose a bidirectional traversal method using a directed tree. The basic idea of the method is to gradually determine whether the node pairs consisting of the currently visited node and all its parents satisfy the second type of simplified decision rules during the backward traversal of the directed tree. Figure 4 shows the basic flow of the method. The specific execution process of the method is as follows.

Step 1: Initialization of a ventilation network. Construct the ventilation network topology relationship, establish node-branch adjacency, and create a ventilation network graph $G(Edge,Node)$, where $Edge$ is the set of roadway branches of the ventilation network, and $Node$ is the set of roadway nodes of the ventilation network. Set all inlet roadway branches and outlet roadway branches as non-simplifiable branches, and allow ventilation managers to set other non-simplifiable branches.

Step 2: Directed tree construction. Traverse the whole ventilation network in the downward direction using BFS method, and create a directed tree graph, $Tree$. In the process of directed tree traversal, according to the order of the directed tree breadth-first traversal of all the nodes of the ventilation network to sort, create a set of ordered nodes, ${Node}^{\prime }$; and record the node depth of all the nodes and the directly inherited parent branch (including the directed tree branch and the non-directed tree branch).

Step 3: Traversal of an ordered set of nodes. Starting from the last node, traverse the directed tree in the reverse direction to obtain the ordered node set ${Node}^{\prime }$. The node access state is divided into two types: an un-accessed state and an accessed state, and the access state of all nodes is initially marked as the un-accessed state. The branch access state is also divided into two types: an un-accessed state and an accessed state. Additionally, all branches are initially labeled as the un-accessed state.

Step 4: Node access state judgment. Traverse to the next node, ${Node}_i^{\prime }$, and judge the node access state until all nodes are traversed. If the node ${\mathrm{N}\mathrm{o}\mathrm{d}\mathrm{e}}_{\mathrm{i}}^{\prime }$ access state is accessed, then re-execute step 4; if the node ${Node}_i^{\prime }$ access state is not accessed, then set it to the accessed state, and execute step 5.

Step 5: Directed tree backtracking. Treat node ${Node}_i^{\prime }$ as an air outlet node, and use the BFS method to gradually backtrack in order to traverse all the parent branches of the directed tree graph Tree (including directed and non-directed tree branches). If a parent branch is a non-simplifiable branch or its branch access status is already accessed, then it will not continue to traverse its parent node.

Step 6: Root node counting. Record the number of root nodes of all parent branches during the backward and reverse traversals of the directed tree. As the number of root nodes is greater than 1, which must not satisfy the second type of simplified decision rule, the use of the number of root nodes to determine the conditions can be simplified and can greatly improve the efficiency of the search.

Step 7: Simplified conditional judgment. When the number of root nodes is 1, the root node Root is treated as an inlet node, and the node pair formed by it and the outlet node ${Node}_i^{\prime }$ is judged as to whether it satisfies the second type of simplification decision rule. If satisfied, the node pair $(Root,{Node}_i^{\prime })$ is taken as an alternative simplifiable sub-network.

Step 8: Nested sub-network removal. At the end of the backward traversal of the directed tree, for all the alternative simplifiable sub-networks, remove all the nested sub-networks by using the nested sub-network decision rule, and determine all the simplifiable sub-networks with node ${\mathrm{N}\mathrm{o}\mathrm{d}\mathrm{e}}_{\mathrm{i}}^{\prime }$ as the outgoing node. Set the branch access state of all branches in the above simplifiable sub-network set to the accessed state.

Step 9: Simplifiable sub-network set output. After identifying all the simplifiable sub-network sets with node ${Node}_i^{\prime }$ as the outgoing node, re-execute the fourth step to judge all the simplifiable sub-network sets with the following node as the outgoing node until all the nodes are traversed, and finally, identify all the simplifiable sub-network sets of the whole ventilation network $\left\{G_i\right\}$.

4.3. Equivalent Simplification Theory

In order to obtain the same ventilation network regulation, the simplified equivalent branch air resistance should be made equal to the total air resistance of the simplifiable sub-network. For the set ${\{G}_i\}$ of simplifiable sub-networks that satisfy the simplifiable condition, the principle of energy conservation is used to calculate the equivalent air resistance, $R_{simplfy}$, of the sub-networks between node pairs $({IN}_i,{OUT}_i)$, where ${IN}_i$ denotes the inlet node of the $ith$ simplifiable sub-network $G_i$, and $OU{T}_i$ denotes the outlet node of the ith simplifiable sub-network $G_i$.

As shown in Figure 5, the ventilation network-equivalent simplified calculation process is as follows.

• Corresponding to the $ith$ simplifiable sub-network $G_i$, construct a virtual branch from the sub-network outlet node ${OUT}_i$ to the sub-network inlet node ${IN}_i$, and set the value of the air resistance of this branch to zero value.
• Set the virtual branch as a branch of the fixed air quantity-type airway; the fixed air quantity value is $100 {\mathrm{m}}^3/\mathrm{s}$ (can also be set to other non-negative constants).
• The connected network consisting of virtual branches and simplifiable sub-networks is regarded as a new ventilation network, ${\tilde{G}}_i$, and an airflow iteration error is specified (in order to obtain a high accuracy of equivalent air resistance, it is recommended to set a high iteration accuracy, e.g., $0.0001 {\mathrm{m}}^3/\mathrm{s}$), and the ventilation network solutions, such as the loop airflow method [29], are used to calculate the airflow allocation to it. In order to improve the speed of the complex ventilation network solution, we use the minimum independent closed loop-based circuit airflow method [30] to perform the ventilation network solution process.
• Calculate the equivalent air resistance according to the relationship of energy conservation before and after the simplification of the ventilation network, i.e., the energy-gain power of the fixed air quantity branch should be equal to the loss power of the equivalent air resistance branch.

The incremental power $P_{power}$ for a fixed airflow branch is

$$P_{power}=h_{total}{Q}_{total}/1000$$

(1)

where $h_{total}$ is the value of unbalanced air pressure in the virtual branch (fixed airflow branch), and $Q_{total}$ is the value of airflow in the virtual branch (fixed airflow branch).

The lost power, $P_{loss}$, in the equivalent air resistance branch is

$$P_{loss}=R_{simplify}{Q}_{simplify}^3/1000$$

(2)

where $R_{simplify}$ is the equivalent air resistance value of the simplified branch, and $Q_{simplify}$ is the air quantity value of the simplified branch, which satisfies $Q_{simplify}=Q_{total}$.

Therefore, the equivalent air resistance, $R_{simplify}$, for the simplified branch is calculated as

$$R_{total}={\displaystyle \frac{h_{total}{Q}_{total}}{Q_{simplify}^3}}={\displaystyle \frac{h_{total}}{Q_{total}^2}}$$

(3)

4.4. Equivalent Mapping Method

For the set of simplifiable sub-networks $\left\{G_i\right\}$, after calculating the equivalent air resistance, each sub-network is replaced with the corresponding branch of equivalent air resistance, respectively, and the equivalent mapping relationship of the corresponding branch before and after the simplification of the ventilation network is established.

As shown in Figure 6, the ventilation network-equivalent mapping is executed as follows.

• Corresponding to the $ith$ simplifiable sub-network $G_i$, taking the inlet node ${IN}_i$ as the starting node and the outlet node ${OUT}_i$ as the end point, Dijkstra’s shortest path algorithm is used to determine a directed shortest path, and the ventilation management personnel are allowed to adjust the path according to the needs of ventilation management.
• Construct an equivalent air resistance branch pointing from the sub-network inlet node to the sub-network outlet node, such that the branch passes through the above path, and set the branch air resistance value to the equivalent air resistance value.
• Equivalent air resistance branches are used to replace the sub-networks, and an equivalent mapping relationship between the equivalent air resistance branches and their corresponding sub-networks is established.
• The network solutions for the ventilation networks before and after simplification are carried out separately, and the results of the airflow distribution obtained from the same branch calculations are equal within a certain error range, which indicates that the automatic equivalent simplification method of the ventilation network proposed in this paper is reliable.

5. Experiment and Analysis

To test the reliability of the complex network simplification method, we realized the proposed simplification algorithm suitable for an arbitrary complex ventilation network based on a 3D mine ventilation software platform (iVent 2023_V1.0.14) using the C++ programming language. By conducting an experimental analysis using several network examples of real-world mines, the effectiveness of the simplification method was verified. The method can satisfactorily meet the requirements for simplifying complex networks.

5.1. Computation Analysis

Figure 7 shows examples of the global and local simplification of the #1 ventilation network in the mine. Figure 7(1) shows the diagram of the ventilation network before simplification, with 272 branches. Figure 7(2) shows the diagram of the ventilation network after simplification using the method proposed in this paper, with 169 branches in the simplified ventilation network. The black branch in Figure 7(2) represents the simplified branch, while the red branch represents the non-simplifiable branch. This example demonstrates that the proposed method can simplify complex sub-networks.

The rectangular box in Figure 7(1) shows a simplifiable, complex sub-network with network structures that is not diagonal, a series, or parallel. Due to that special network structure, traditional simplifying methods based on series and parallel ventilation networks cannot achieve the best simplification of the ventilation network. The proposed method can simplify a complex sub-network consisting of 22 branches into one equivalent sub-network. The equivalent simplified calculation process of the sub-network is as follows. Construct a virtual branch (black branch in Figure 7(3)) from the air outlet node to the air inlet node of the sub-network, and set it as a fixed-flow branch with a fixed air quantity of 100 ${\mathrm{m}}^3/\mathrm{s}$ to obtain a complex sub-network that can be used for ventilation network calculation. Set the iteration accuracy of the air quantity to 0.0001 ${\mathrm{m}}^3/\mathrm{s}$, and use the circuit airflow method based on the minimum independent closed loops (MICL) to calculate the air quantity distribution results of this complex sub-network. The numbering of each airway in the simplified sub-network is shown in

Table 1. Calculation results of a complex sub-network using the network solution method.

Airway No	Air Resistance (N·s2/m8)	Air Quantity (m3/s)	Air Velocity (m/s)	Air Pressure (Pa)
1	0.00945	100.000	16.949	−267.735
2	0.00153	77.788	13.412	9.247
3	0.02395	22.212	3.830	11.816
4	0.02349	10.459	1.803	2.569
5	0.00383	67.329	11.609	17.342
6	0.00463	57.034	9.833	15.063
7	0.03490	10.295	1.775	3.699
8	0.01731	32.671	5.633	18.472
9	0.00544	42.966	7.408	10.041
10	0.03006	6.633	1.144	1.323
11	0.00923	48.650	8.388	21.845
12	0.03142	15.017	2.589	7.087
13	0.00637	36.333	6.264	8.410
14	0.00561	51.350	8.853	14.783
15	0.02774	0.951	0.164	0.025
16	0.00519	47.699	8.224	11.802
17	0.02714	14.822	2.556	5.963
18	0.00649	52.301	9.017	17.740
19	0.00555	32.877	5.668	5.999
20	0.02597	32.877	5.668	28.074
21	0.00624	67.123	11.573	28.111
22	0.02184	100.000	17.241	218.364

. The unbalanced wind pressure value of the fixed flow branch is 362.245 Pa. The final equivalent wind resistance corresponding to the simplified branch of the complex sub-network can be determined to be 0.03622 N·s²/m⁸.

5.2. Mine Examples

Figure 8 shows two simplified examples of mine ventilation networks. The red lines in the figure represent the branches of the ventilation network, and the green lines represent the nodes of the ventilation network. Figure 8(1,2) demonstrate the networks of Mine #2 before simplification and after simplification, respectively. Figure 8(3,4) demonstrate the networks of Mine #3 before simplification and after simplification, respectively.

Figure 9 shows two simplified examples of complex mine ventilation networks. The red lines in the figure represent the branches of the ventilation network, and the green lines represent the nodes of the ventilation network. Figure 9(1,2) demonstrate the networks of Mine #4 before simplification and after simplification, respectively. Figure 9(3,4) demonstrate the networks of Mine #5 before simplification and after simplification, respectively.

To test the effectiveness of ventilation network simplification,

Table 2. Comparison results of simplification examples.

Real-World Mines	Before Simplification		After Simplification
	Number of Edges	Number of Nodes	Number of Edges	Number of Nodes
Mine #1	272	215	169	121
Mine #2	594	453	475	338
Mine #3	660	481	499	326
Mine #4	1058	552	736	267
Mine #5	8650	4241	6777	2566

demonstrates the comparison results of several real-world network examples. Figure 10 demonstrates the comparison results of several simplified branch numbers of real-world mine ventilation networks. Figure 11 demonstrates the comparison results of the number of simplified node numbers of several real-world mine ventilation networks. The above comparison results indicate that the proposed method can extremely simplify mine ventilation networks.

6. Conclusions and Discussion

Complex network simplification is a very meaningful graph theory research direction in discrete mathematics. In this paper, we have innovatively proposed, for the first time, a method that can efficiently and automatically simplify any complex ventilation network. We have revealed the most essential features of simplifiable sub-networks based on directed tree-spanning sub-graphs, and we have proposed simplified sub-network decision rules and nested sub-network judgment rules. Among them, the first kind of judgment rule for simplifiable sub-networks is applicable to local ventilation network assisted simplification, and the second kind of judgment rule for simplifiable sub-networks is applicable to global ventilation network automatic simplification. The nested sub-network determination rules can be used to search for extremely simplifiable sub-networks in order to maximize the simplification of complex networks. Through the analysis of several real mine ventilation network simplification examples, it has been shown that the proposed method can minimize the branching ratio of the ventilation network and improve the degree of ventilation network simplification. By analyzing the characteristics of the existing techniques, it was found that this is the best practice among all the current ventilation network simplification methods.

This complex network simplification method has the following advantages. Firstly, this method can automatically perform the equivalent simplification of any complex ventilation network without manually setting simplification parameters, improving the efficiency and accuracy of simplification. Secondly, this method is suitable for simplifying any complex passive sub-network, and it can handle non-simple sub-networks (series sub-networks, parallel sub-networks, and diagonal sub-networks). It can greatly reduce the branching ratio of the ventilation network, greatly improve the simplicity of the ventilation network, and have higher applicability and more flexibility than traditional methods. Finally, by establishing a mapping relationship between the simplified sub-network and the original network, this method can optimize the calculation processes of complex ventilation networks, such as searches, solutions, and regulation. This can improve the operational performance of intelligent ventilation systems in mines, which is crucial for the safe production and economic operation of mines. Overall, this method can effectively simplify complex ventilation networks and optimize the calculation process, which is beneficial for improving the operational performance of intelligent ventilation systems in mines. It has important theoretical significance and practical application value.

However, there are still some limitations to our work that need to be improved. One of the main limitations is that the simplification method proposed in this paper applies to directed sub-graphs with only one source point and one sink point. In the field of mine ventilation networks, it is hard to compute ebullient air resistance for sub-networks with multiple source points or sink points. For other complex network domains, to adapt to simplifiable sub-networks with multiple source points and sink points, it is necessary to adjust or optimize the search process based on the corresponding sub-network features. Another limitation of this approach is that the search process can be further improved to enhance the robustness and adaptability of the algorithm.

The simplification of complex networks, as an important branch of graph theory research, has been widely studied and applied in interdisciplinary fields because of its theoretical and practical value. Due to the intricate internal topological relationships of complex networks, it is difficult to directly calculate and analyze them. The equivalent sub-graph reconstructing by simplifying a complex network can effectively reduce computational complexity or extract multi-level network structural features, which have important application value. The fast simplification method for complex networks proposed in this paper can be applied not only to the field of ventilation networks but also to other types of complex network fields, such as water networks and social networks. It can be determined that, with appropriate modifications to meet specific requirements, the proposed method can also be applicable to equivalent simplification instances for other types of complex networks, such as water networks. However, the equivalent condition requires specialized knowledge in a specific field background. In practical application expansion, simplified sub-networks that meet specific conditions can be obtained by setting simplified branches. And the equivalent branches, after simplification, should be calculated according to reliable equivalent formulas.