# Application of Graph Theory for Blockchain Technologies
## 5 Graph Theory Application in Blockchain Systems
### Managing Scalability
#### 5.3.3. Shatding Blockchains  
Large networks can be partitioned into zones of dense intra-cluster and sparse inter-cluster connections via graph clustering methods. Transactions within shards process faster, enhancing capacity [83]. Figure 17 presents partitioning the network into shards like this, allowing transactions within each shard to be processed more quickly and efficiently, potentially enhancing the overall capacity and speed of the blockchain network. This graph-theoretical approach to sharding aims to improve blockchain scalability by optimizing the processing of transactions within densely connected communities. Nodes represent entities in the blockchain network. Edges represent transaction links between these entities. Different colors indicate different clusters or shards identified using graph clustering methods. Each color represents a different shard, nodes within the same shard (same color) are densely connected, indicating a high density of within-community transactions. Inter-cluster (inter-shard) connections are sparser, indicating less frequent transactions between different shards. **Assumption 16**. :
* \(G(V,E)\) _is a connected, undirected graph._
* \(C=\{C_{1},C_{2},\ldots,C_{k}\}\) _is the set of clusters identified within_ \(G\)_, where each_ \(C_{i}\) _is a subset of_ \(V\)_, representing a shard._  
**Application Statement 16**. :
* _Partitioning_ \(G\) _into_ \(k\) _shards such that transactions within each shard_ \(C_{i}\) _are processed more quickly and efficiently, thereby enhancing the overall capacity and speed of the blockchain network._  
**Proof. **  
Given a blockchain network modeled as a transaction graph \(G(V,E)\), where \(V\) represents the set of nodes (entities in the blockchain network) and \(E\) represents the set of edges (transaction links between entities), our objective is to partition \(G\) into subgraphs (\(G_{i}\)) that represent shards. This partitioning aims to maximize the density of intra-cluster transactions while minimizing the inter-cluster transactions, enhancing overall network throughput. 1. Cluster Identification: Apply a graph clustering algorithm, such as modularity optimization, to partition \(G\) into \(k\) clusters. The modularity \(Q\) of a partition is given by: \[Q\ =\ \frac{1}{2m}\sum_{ij}\bigg{[}A_{ij}-\frac{k_{i}k_{j}}{2m}\bigg{]} \delta(C_{i},C_{j})\] where \(m\) is the total number of edges, \(A_{ij}\) is the adjacency matrix of \(G\), \(k_{i}\) and \(k_{j}\) are the degrees of nodes \(i\) and \(j\), and \(\delta\) is the Kronecker delta function, indicating if nodes \(i\) and \(j\) are in the same cluster. The goal is to maximize \(Q\), which indicates a strong community structure with dense intra-cluster and sparse inter-cluster connections. * Shard Assignment: Each node \(v\in V\) is assigned to a shard based on its cluster membership determined in step 1. Formally, if \(v\in C_{i}\), then \(v\) belongs to shard \(i\). * Transaction Routing: Intra-shard transactions (those occurring between nodes within the same cluster) are routed internally within \(C_{i}\). This minimizes the path length and, by extension, the processing time for these transactions. Inter-shard transactions are minimized but when necessary, are processed through designated gateway nodes that facilitate communication between shards. * Throughput Enhancement: By confining the majority of transactions to densely connected clusters, each shard can independently process transactions in parallel, significantly increasing the network's throughput. The reduction in inter-shard transactions decreases the overall network load, further contributing to speed improvements. The application of graph clustering algorithms to partition a blockchain network into shards effectively segregates transactions into densely connected communities. This method enhances throughput by enabling parallel processing within shards and minimizing the load on the network, thereby increasing the capacity and speed of blockchain transactions.