## 3 System Model and Problem Formulation
### Blockchain Incentive Model  
Blockchain is a record list chained with consecutive timestamps. As training continues, the performance of the global model improves. Therefore, it is perfectly suitable to discriminate the value of global models according to timestamped records. Our proposed incentive model is illustrated in Figure 3.  
In this paper, we consider synchronized communication during DML, as illustrated in Figure 1. However, smartphones may only be allowed to continue synchronized DML until a specific round based on their data valuations. For any smartphone \(n\), it can download the up-to-date global model only when the phone is in the system. At the end of DML, smartphone \(n\) leaves the system with a global model reflecting its contribution to DML. All smartphones are required to contribute continually such that they can obtain the up-to-date global model as the model reward. As illustrated in Figure 3, any contributor, including free-riders, may access a set of global models. However, the length of the accessible model chain depends on the individual incremental contribution. Free-riders cannot access specific models that are beyond their access length. For any model \(\mathbf{w}\) recorded on blockchain, the length of the accessible model chain determines whether the access to \(\mathbf{w}\) is denied or permitted. In this context, weak contributors can still access a subset of on-chain models, while fake contributors may not have access to any model on the blockchain. As expected, weak contributors are not penalized at the same level as fake contributors.  
Let \(V\) and \(R_{n}\) be the valuation function in the system and the exit round of smartphone \(n\), respectively. Let \(\beta\) be a system parameter reflecting the valuation rate. For any smartphone \(n\), we formally introduce the model reward constraint as  
\[U_{n}(\mathbf{w}_{R_{n}})\leq\beta\sum_{r\in\mathcal{R}}V(a_{n,r}\mathbf{d}_{n,r}), \forall n\in\mathcal{N}. \tag{7}\]  
The above constraint ensures that any model reward will surely be based on data valuations contributed from any smartphone. Let \(V_{n,r}\in\mathcal{V}_{r}=\{V_{1,r},\,V_{2,r},\,\cdots,\,V_{N,r}\}\) denote the valuation of data contributed from smartphone \(n\) in round \(r\). Refer to Section 4 for the calculation of \(V_{n,r}\). To estimate actual data contributions \(\mathcal{C}_{r}\) in round \(r\), the data valuation of \(N\) distributed datasets are recorded on the blockchain.  
Our proposed blockchain structure is shown in Figure 4. Subscripts represent different rounds in DML. Let \(\mathbf{w}_{0}\) denote weights of the initial model for DML. _Public_ keys are exchanged among verified smartphones and servers. Note that blockchain-enabled DML for smart health needs to follow data regulations for data safety protection. Therefore, a DPoS [13] is used in our system to ensure the security and integrity of data in distributed ledgers. We assume stakeholders stake tokens for votes. Stakeholders (i.e., smartphones) vote for servers with their stakes. Let \(v_{m,r}\in\mathcal{V}_{r}=\{v_{1,r},\,v_{2,r},\,\cdots,\,v_{M,r}\}\) denote votes received by server \(m\) in round \(r\). Let \(M_{D}\) denote the number of delegated servers in DML. A detailed block generation workflow is described in Section 4. Note that the recorded data valuation can be verified by evaluating the prediction accuracy of local weights. Further discussions on blockchain-based data valuation are given in Section 4.  
Figure 3: Proposed time-dependent incentive mechanism atop blockchain: users gain access to varying sets of global models as rewards in each round, depending on their diverse contributions; free-riders have very limited or no access to global models recorded on the blockchain.