## 4 A Method for Configuring VWSNs to Use the Properties of Brain Networks
### VWSN Topology Construction Method  
As discussed above, a topology with the small-world properties will have high global communication efficiency, and a highly modular topology will have efficient segregated information processing and robust connectivity. We previously proposed a means of constructing a VWSN topology with small-world properties and high modularity [13]. Here, we integrate modules hierarchically to construct a VWSN topology. At the same time, we add long-distance links to a clustered topology to give the VWSN topology the small-world properties at each tier. Figure 2 shows an example of hierarchical VWSN construction. The first tier of the VWSN is a network in the minimum-unit module. The second-tier VWSN is constructed by integrating unit modules. The third-tier VWSN is constructed by integrating virtual sensor networks deployed for sub-applications. Figure 2: Example of a hierarchical VWSN topology. In this section, we briefly explain the method proposed in [13]. We assume that adding multi-hop wireless or wired links contributes to reachability between any two nodes. After this, a virtual link is mapped to the shortest physical path in an infrastructural topology. Note that we consider that the physical topology of WSN is the unit disk model and does not have small-world properties. We construct a virtual network having the small-world properties that contribute to a short average path length by embedding the small number of long-distance virtual links. First, a VWSN provider defines a set of nodes for a user application. The constructed VWSN must contain a set of source and destination nodes requested by the user, a set of relay nodes for communication between source and destination nodes and redundant nodes according to the user's demand for reliability. When these sets of nodes have been determined, the VWSN provider selects a set of modules to construct a VWSN and integrates them by embedding virtual links between them. Construction of an \(N\)-th-tier VWSN topology can be divided into two smaller subproblems. In the first problem, we regard each \((N-1)\)-th-tier VWSN as one module (\(M_{i}^{N-1}\)) in the \(N\)-th tier, where \(i\) indicates the specific module, and the choice to be made is how to connect pairs of modules. In the second problem, sensor nodes are to be mapped to the endpoints of \(N\)-th-tier virtual links. In this, \(M_{*}^{0}\) denotes a sensor node and \(M_{*}^{1}\) denotes a module clustered by applying the Newman algorithm [28]. For the first problem, we construct an \(N\)-th-tier virtual topology by adding virtual links between \((N-1)\)-th-tier modules. First, an initial virtual topology is constructed: when two nodes belonging to different \((N-1)\)-th-tier modules are connected by a physical link, an \((N-1)\)-th-tier virtual link is embedded between these \((N-1)\)-th-tier modules. Second, new \((N-1)\)-th-tier virtual links are embedded into the initial virtual topology according to a preferential attachment rule. The probability of adding an \((N-1)\)-th-tier virtual link between \(M_{i}^{N-1}\) and \(M_{j}^{N-1}\) is  
\[p_{\text{intra}}^{N}(M_{i}^{N-1},M_{j}^{N-1})=\frac{\frac{\frac{G^{intra}(k_{ M_{i}^{N-1}}k_{M_{j}^{N-1}})}{F(k_{M_{i}^{N-1}}M_{j}^{N-1})}}{\sum_{e_{M_{a}^{N-1}},M_{ b}^{N-1}\in E^{N}}\frac{G^{intra}(k_{M_{a}^{N-1}}k_{M_{b}^{N-1}}^{N-1})}{F(k_{M_{a}^{N- 1}},M_{b}^{N-1})}}. \tag{3}\]  
Here, \((N-1)\)-th-tier modules \(M_{i}^{N-1}\) and \(M_{j}^{N-1}\) belong to the same \(N\)-th-tier module, \(E^{N}\) is the set of virtual links in the graph complement of the \(N\)-th-tier initial virtual topology and \(F\) is a cost function \(F(d)=\text{e}^{d/d_{x}}\), where \(d_{x}\) is a constant parameter characterizing distance constraints. \(I_{M_{i}^{N-1},M_{j}^{N-1}}\) denotes the minimum hop count between \(M_{i}^{N-1}\) and \(M_{j}^{N-1}\) in the \(N\)-th-tier initial virtual topology, and \(k_{M_{i}^{N-1}}\) denotes the degree of \(M_{i}^{N-1}\).