# Two-Level Sharing and Extraction of Sensing Information with Hybrid V2X Communications
## Appendix A Scalability Property  
We examined the general property of the scalability of the proposed scheme in extracting sensing information. Scalability in this paper refers to the system property whereby the number of global reports does not significantly increase as the number of vehicles increases in the network when there are many vehicles.  
To verify the general property of scalability of the proposed scheme, we examined the network by mathematical modeling and analysis. Let the service region of the edge server have a length of \(L_{R}\). We divide the region into multiple region blocks, each of which has the equal length of \(L_{r}\), as shown in Figure 11. Suppose that \(L_{r}\) is the length within which any two vehicles are in the local communication range of each other. That is, \(L_{r}\leq r\). If we ignore the width of lanes, the maximum length of \(L_{r}\) is \(r\). If we suppose that the width of lanes is \(w\), the maximum length of \(L_{r}\) is \(\sqrt{r^{2}-w^{2}}\).  
Suppose that the broadcast of a message in the exchange of sensing data is reliable without packet losses. We consider that there is one round of sensing-data exchange, since the proposed scheme employs at least one round of exchange. Suppose the sensing value at each vehicle is different from that of others. The MinFactor is the object information of extraction.  
**Proposition A1**.: _The total number of reports from the vehicles to the edge server is not larger than \(\dfrac{L_{R}}{L_{r}}+1\). Such a maximum number of reports is independent of the number of vehicles, indicating the property of being scalable with the number of vehicles._  
Proof.: Since any two vehicles in a region block with the length \(L_{r}\) are in the local communication range of each other for sensing-data exchange, there is no more than one report needed after a round of sensing-data exchange.  
Let each region block \(i\) have a set of vehicles Set[i] with a cardinality of \(|Set[i]|\). Given \(|Set[i]|>0\), let \(min[i]\) be the vehicle that has a minimum sensing value in region block \(i\). After the exchange broadcast from \(min[i]\), other vehicles, denoted by \(Set[i]\setminus min[i]\), in the \(set[i]\) are set to the state in which it is not necessary to report to the edge server. Hence, there is at most one report to the edge server in set[i]. If \(|Set[i]|=0\), there will be no report to the edge server at \(set[i]\). Therefore, after a round of local sharing of sensing data and local extraction of the MinFactor, no more than one vehicle in a region block needs to report to the edge server.  
Hence, the maximum global report number is not larger than \(\dfrac{L_{R}}{L_{r}}\) if \(\dfrac{L_{R}}{L_{r}}=k\), and k is an integer. If \(k<\dfrac{L_{R}}{L_{r}}<(k+1)\), and k is an integer, then the maximum global report number is not larger than \(\dfrac{L_{R}}{L_{r}}\).  
As a result, with the local sharing and extraction of sensing information, the maximum number of necessary reports to the edge server is only associated with the geographic region size and the local communication range. This leads to a scalability property whereby the maximum number of reports to the edge server is not affected by an increase in the number of vehicles when there are many vehicles.