# Network Structure Model

## Cyberspace

The founder of network topology system research was recognized as Leonard Euler, a mathematician who solved a famous problem in 1736, called the Konigsberg Bridge Problem. Figure 3-16-a shows a general road map of the bridge. The problem is to find a circular road that passes through each of the bridges only once and finally returns to the starting point. Some experiments show that this task is impossible, but it is not so easy to go from thinking that there is no such route to explaining its steps.

Euler has succeeded in proving that this is an impossible task. In other words, there is no solution to this problem. To do this, he built a spatial model of the bridge, which abstracted the topological relationships between all the only bridges, as shown in Figure 3-16-b. Solid circles represent nodes or vertices. They are marked w, x, y, z and abstracted as land surface. Lines represent arcs or edges, which are abstracted as straight lines between land, and in each case a bridge is needed. A complete model is called a network or graph. Euler proved that it was impossible to start from a node, traverse each boundary only once along the graph boundary, and finally reach the first node. His argument is very simple, based on the odd/even number of edges passing through each node. We can see that besides the beginning and the end of the node, the path passing through one node must be along one boundary and out of the other. This shows that the two boundaries correspond to that node. Therefore, if the number of boundaries connected by each intermediate node must be even, of course, if the problem is solved. In Figure 3-15, the number of boundaries of no node is even. Therefore, the problem of graph theory is insoluble, and the initial problems related to Konigsberg bridge problem are also insoluble.

## network model

In the network model, objects are abstracted as chains, nodes and other objects, and the connection between them should be paid attention to. Network-based spatial models and factor-based models have some similarities in some respects, because they often deal with discrete objects, but the most basic feature is that they need the influence and interaction between multiple elements, usually along the channels connected with them. The exact shape of the related phenomena is not very important. It is important to measure the distance or resistance between specific phenomena. A typical example of a network model is the study of transportation, including land, sea and air routes, and the analysis of water, gasoline and electricity flows through pipelines and tunnels. For example, a power supply company may adopt both a factor-based perspective and a network-based perspective for their facility management, depending on whether they are concerned with replacing a particular pipeline, in which case a factor-based perspective may be appropriate; or, for example, they are concerned with the purpose of analyzing reconstructed lines. In this case, the network model will be appropriate.

The basic feature of the mesh model is that there is no clear subordination relationship between the node data, and one node can establish connections with other nodes. The mesh model organizes the data into a directed graph structure. Nodes in the structure represent data records, and links describe the relationship between data of different nodes. Digraph is formally defined as：

Digraph = Vertex, {Relation}

Vertex is a finite non-empty set of data elements (vertices) in a graph, and Relation is a set of relations between two vertices (vertexes).

Directed graph structure is more flexible and powerful than tree structure in data modeling. The network model can represent many-to-many relationships, and its data storage efficiency is higher than the hierarchical model. However, the complexity of its structure limits its application in spatial database.

The mesh model reflects the common many-to-many relationship in the real world, supports data reconstruction to a certain extent, has certain data independence and sharing characteristics, and has high operational efficiency. But it also has the following problems in its application：

The complexity of network structure increases the difficulty of user query and location. It requires users to be familiar with the logical structure of data and know where they are.

The network data operation command has a procedural nature.

It does not directly support the expression of hierarchy.

It has no deductive function.

There is no basic operation algebra.