# Adjacency-Information-Entropy-Based Cooperative Name Resolution Approach in ICN
## 3 Model and Theory Fundamental
### Local Name Mapping and Resolution System  
We use an undirected graph, \(G(V,E,\,W)\), to model the structure of LNMRS. \(V=\{v_{1},v_{2}\ldots,v_{N}\}\) is the set of nodes in the physical network, representing the network entities with the potential to place name resolution servers. \(N=|V|\) is the total number of network entities. \(E\) is the set of edges, and \(e_{ij}\in E\) means that \(v_{i}\) and \(v_{j}\) can communicate with each other without going through any other nodes in \(V\). \(W\) is the weight set of \(E\), and \(w_{ij}\in W\) is the transmission latency between \(v_{i}\) and \(v_{j}\). We used a matrix, \(D\), to denote the distances for all pairs of nodes in \(V\), where \(d\big{(}v_{i},v_{j}\big{)}\in D\) represents the shortest path latency between \(v_{i}\) and \(v_{j}\). The set of users in the network is denoted as \(U=\{u_{1},u_{2},\ldots\}\). In our model, users are each connected to the network through one of the nodes in \(G\), which can be a router, a base station, etc.; the access node of user \(u\) is denoted as \(AN(u)\), and the transmission latency between them is denoted as \(AL(u)\). \(T=\{t_{1},t_{2},\ldots,t_{L}\}\) is a set that consists of the upper bounds of each layer's constraint latencies, and \(L=|T|\) is the number of scenarios with different latency requirements. For all \(0\leq l1<l2\leq L\), \(t_{l1}<t_{l2}\) holds, which means that the name resolution service provided by a higher-level MN has a higher latency upper bound. \(T\) is usually determined by application scenarios in the network.  
Based on this model, we can accurately describe the structure using exact formulas. At a layer with latency upper bound of \(t_{l}\), we assume that \(G\) is divided into several name resolution regions. The nodes and the edges in each region can form a subgraph of \(G\), the set of these subgraphs is donated as \(Gs=\left\{G_{i}^{l}\Big{(}V_{i}^{l},E_{i}^{l},W_{i}^{l}\Big{)}\ \middle|\ i=1,2,\ldots,M^{l}\right\}\) and \(MN_{i}^{l}\) represents the MN corresponding to \(G_{i}^{l}\). The user set of \(MN_{i}^{l}\) is donated as \(U_{i}^{l}\). In summary, the structural characteristics of LNMRS can be formulated as follows:  
\[\mathop{\cup}\limits_{i=1}^{M^{l}}V_{i}^{l}=V,\ \forall l\in\{1,2,\ldots,L\} \tag{1}\]  
\[V_{i}^{l}\cap V_{j}^{l}=\varnothing,\ \forall\ i,j\in\left\{1,2,\ldots,M^{l} \right\},\ i\
eq j,\ l\in\{1,2,\ldots,L\} \tag{2}\]  
\[V_{i}^{l1}\subseteq V_{j}^{l2},\ if\ \exists v\in V_{i}^{l1}\ and\ v\in V_{j}^{l2},\]  
\[i\in\left\{1,2,\ldots,M^{l1}\right\},j\in\left\{1,2,\ldots,M^{2}\right\},\ 0\leq l1<l2\leq L \tag{3}\]  
\[d\Big{(}v,MN_{i}^{l}\Big{)}<t_{l},\ \forall v\in V_{i}^{l},\ i\in\left\{1,2, \ldots,M^{l}\right\},\ \forall\ t_{l}\in T \tag{4}\]  
\[d\Big{(}AN(u),MN_{i}^{l}\Big{)}+AL(u)<t_{l},\ \forall u\in U_{i}^{l},\ i\in \left\{1,2,\ldots,M^{l}\right\},\ \forall\ t_{l}\in T \tag{5}\]  
Equation (1) means that all the nodes are divided into a subgraph at each level. The constraint of Equation (2) ensures that there is no overlap between name resolution regions in the same level, so the tree structure can be formed. Nested relationships between name resolution regions from different levels are indicated in Expression (3). In Expression (4), the upper bounds of name resolution response latencies at each level are guaranteed. Expression (5) ensures that users can obtain deterministic name resolution latency from its MN.