Fig. 3

The modification effect of cooperative coded forwarding on percolation graphs. Let \({G_j} = \left\langle {{V_j},{E_j}} \right\rangle\) denote the connected forwarding graph of the jth SCP \(y_j\), where \({V_j}\) is the set of nodes that has received \(y_j\), and \({E_j}\) contains the links that have successfully delivered \(y_j\). Consider two different forwarding graph \(G_a\) and \(G_b\), and a common node \(v \in V_a \cap V_b\). Denote the nodes covered by v in the two graphs as \({{\mathcal{N}}_a}\left( v \right) = {\mathcal{N}}\left( v \right) \cap {V_a}\) and \({{\mathcal{N}}_b}\left( v \right) = {\mathcal{N}}\left( v \right) \cap {V_b},\) respectively. Suppose v is going to encode \(y_a\) and \(y_b\) into a RCP \({z_{a,b}} = {y_a} \oplus {y_b}\) and forward it to its one-hop neighbors. A one-hop neighbor node \(u \in {\mathcal{N}}\left( v \right)\) of v can receive \({z_{a,b}}\) with probability \(1-\rho\) due to the packet erasure effect. If u has already received \(y_a\) and \(y_b\), i.e. \(u \in {\mathcal{N}}_a\left( v \right) \cap {\mathcal{N}}_b\left( v \right)\), the receiving of \({z_{a,b}}\) become redundant. If u has only received \(y_a\), the receiving of \({z_{a,b}}\) helps it to decode \(y_b\), and the node u is added to \({V_b}\). The link \(e\left( u,v\right)\) will also be involved in \({E_b}\). Similarly, the receiving of \(y_b\) and \({z_{a,b}}\) will add u into \({V_a}\). If u have received neither \(y_a\) nor \(y_b\), the receiving of \({z_{a,b}}\) will not change \(G_a\) or \(G_b\). The effect of this coded forwarding action on the broadcast performance can therefore be regarded as the modification of the two corresponding forwarding graph \(G_a\) and \(G_b\)