# A Scheme to Design Community Detection Algorithms in Various Networks
## 3 Concrete Problem Reduction
### Transmission Network  
In this section, we discuss the relation derived from material transmission. The materials here can represent both material substance (like goods) or information (like data). Intuitively, the objects that can easily communicate with each other have strong relations. Then, these objects can somehow be considered as a community. We start our discussion by showing how a transmission network is constructed. Based on this, we derive the special relation strength measurement (SRSM) for measuring the relation of two nodes that have direct connections. Since SRSM cannot represent all the relations in a practical transmission network, we derived a generalized one called relation strength measurement for transmission network (RSMFTN). Finally, we prove that the shortest distance function is in fact an RSMFTN. Our discussion is based on the following definitions and assumptions. **Assumption 1**. : _For any material, there is a minimal transmission unit, which is named point._  
**Definition 1** (Node). : _The objects that receive and send point are nodes._  
**Definition 2** (Medium). : _The one through which the points are propagated is a medium._  
Nodes, media and points construct the transmission relations and determine their properties. We call a network based on the transmission relations _transmission network_. Two important characteristicsin this network type are _the number of points transferred_ and _the time consumed_ in a transmission process. Their ratio is termed as _speed_. **Definition 3** (Speed). : _Speed is the number of points transferred in a unit time interval._  
The transmission capability can be described by the speed function of time (denoted by \(f\)). We first consider the node pairs connected by media directly. The behaviour of \(f\) depends on the properties of the nodes, media, and points. Thus, its algebraic expression cannot be given unless all the properties are designated. However, some properties should be shared by all transmission networks. Figure 1 represents how the speed function \(f\) changes for a pair of nodes \((u,v)\) connected by media directly. In particular, the speed function \(f_{(u,v)}(t)\) has value zero when the transmission starts due to the latency of the point sending, propagation and reception. We call the moment that the function value becomes nonzero the _shortest transmission time (STT)_ (theoretically, \(STT=\text{sending time}+\text{transmission time}+\text{receiving time}\)). In addition, the speed function is upper bounded as the transmission speed cannot approach infinity in practice. We call the least upper bound _theoretically transmission speed (TTS)_. Remarkably, TTS may never be reached and is thus not very useful. To solve the problem, we can choose some proper threshold \(\delta\in\mathbb{R}^{*}\), the lowest acceptable speed. We call the time to reach the threshold the _critical moment (CM)_. **Remark 1**. : _Suppose two towns \(A\) and \(B\) are connected by a river. \(A\) is upstream of \(B\). For goods transported through the river, due to the water stream, the transformation speed from \(A\) to \(B\) is higher than the one in the opposite direction. Thus, \(f_{(A,B)}(t)\geq f_{(B,A)}(t)\). This example implies that, in general, \(f_{(u,v)}(t)\
eq f_{(v,u)}(t)\)._  
The speed function characterizes the relation properties in a transmission network. Based on the function and the objectives needed to be achieved in a concrete problem, we can devise a measurement of the relation strength for a pair of nodes connected by the medium directly. We call this measurement _special relation strength measurement_ (SRSM) which should have the following properties:  
1. The shorter STT is, the stronger the corresponding relation is. 2. The shorter CM is, the stronger the corresponding relation is. 3. The shorter the time to transfer a certain number of points is, the stronger the corresponding relation is. Although a proper SRSM cannot be constructed until a concrete problem is given, several key properties should be shared by all SRSMs.