## 5 Hierarchical Blockchain Model
### Consensus Mechanism  
Let \(\texttt{BC}_{l}^{h}\) be a blockchain maintained by \(l\)-th peer-to-peer network \(\texttt{PP}_{l}^{h}\) at hierarchy level \(h\), where \(l=1,\ldots,n_{l}\). At lower level \(h=1\), the set of networks is \(\texttt{PP}^{1}=\{\texttt{PP}_{1}^{1},\texttt{PP}_{2}^{1},\ldots,\texttt{PP }_{n_{1}}^{1}\}\). Given \(k\)-th subset \(S_{k}\subseteq\texttt{PP}^{1}\), one representative member (who acts as leader) from each network in \(S_{k}\) together forms a network \(\texttt{PP}_{k}^{2}\) at the next level of hierarchy \(h=2\), where \(k=1,\ldots,n_{2}\). Therefore, at level \(h=2\), the set of networks is \(\texttt{PP}^{2}=\{\texttt{PP}_{1}^{2},\texttt{PP}_{2}^{2},\ldots,\texttt{PP }_{n_{2}}^{2}\}\). Following similar approach, networks at subsequent levels are formed. Each network follows traditional consensus algorithm in building and maintaining local blockchain.  
Role of Leader Participating in Multilevel Blockchain  
Multilevel blockchains are useful in dealing with large volume of data generated across the globe. Since leaders from a subset of networks at level \(h\) is also participating into another network at level \(h+1\), such nodes are fully responsible to maintain data-consistency among blockchains across the levels. To preserve data semantics, a leader should listen a number of transactions \(\{\tau_{i}\}\) (which are mined in the blockchain at level \(h\)) for wait-time \(\delta\) and then broadcast a transaction \(\overline{\tau}\) (which is a sound approximation of \(\{\tau_{i}\}\)) in the (\(h+1\))-level network. However, \(\delta\) can be varied to obtain different precision depending on the applications or the network latency and throughput.  
The most important feature in such multilevel hierarchy is the ability to ensure soundness of the information as we move towards the higher level of the hierarchy compromising the precision of information. Consider \(\texttt{BC}_{l}^{h_{i}}\) and \(\texttt{BC}_{k}^{h_{j}}\) such that \(h_{i}\leqslant h_{j}\) and there exists a common node (leader) participating in both the networks at levels \(h_{i}\) and \(h_{j}\). The soundness is guaranteed iff:  
\[\forall x\in\texttt{BC}_{l}^{h_{i}},\exists y\in\texttt{BC}_{k}^{h_{j}}:x \subseteq\gamma(y) \tag{1}\]There are many consensus mechanisms which are currently in use by blockchains, including PoW (Proof of Work), PoS (Proof of Stake), DPoS (Delegated Proof of Stake), PBFT (Practical Byzantine Fault Tolerance), PoB (Proof of Bandwidth), PoET (Proof of Elapsed Time), PoA(Proof of Authority) and many more [44; 45; 46] to validate blocks. Two most popular blockchain systems--Bitcoin and Ethereum--use the PoW mechanism. Due to huge energy consumption in PoW, Ethereum is now trying to adopt the alternative PoS mining approach. Moreover, Ethereum also incorporates PoA mechanism (i.e., Kovan public test chain). The existing consensus algorithms are powerful enough to deal with the situation in presence of nodes-/links-failure or even in presence of Byzantine faults. In case of our proposed multilevel blockchains systems, existing consensus algorithms can easily be adopted. In fact, the algorithm will prefer to adopt a mechanism to change leaders participating across the levels in the hierarchy, in case of suspicious behaviours of the current leaders.