# Reliable Design for a Network of Networks with Inspiration from Brain Functional Networks
## 2 Basic Principles of NoN Models
### B-NoN Model  
An image is recognized in the brain functional networks by the combination of the posterior occipital cortex, which is responsible for visual function, and the anterior cingulate cortex, which deals with recognition. These two regions compensate complementarily for the lack of information in each other; for instance, even if the posterior occipital cortex receives incomplete visual information from the eye, the anterior cingulate cortex can compensate for this lack and recognize the object. The B-NoN model reflects this as a _logical OR_-like dependence between the regions. A node is regarded as globally effective if _either_ its adjacent nodes of the same network _or_ those from other networks belong to the GC. For message passing in the B-NoN model, the variable \(\rho_{i\to j}\) is defined as information sent from node \(i\) to node \(j\) within the same network and the variable \(\varphi_{i\to j}\) is defined as information sent from node \(i\) to node \(j\) of a different network:  
\[\begin{split}\rho_{i\to j}&=\sigma_{i}\bigg{[}1- \prod_{k\in\mathcal{S}(i)\setminus j}(1-\rho_{k\to i})\prod_{l\in \mathcal{F}(i)}(1-\varphi_{l\to i})\bigg{]},\\ \varphi_{i\to j}&=\sigma_{i}\bigg{[}1-\prod_{k\in \mathcal{S}(i)}(1-\rho_{k\to i})\prod_{l\in\mathcal{F}(i)\setminus j}(1- \varphi_{l\to i})\bigg{]},\end{split} \tag{2}\]  
where \(\mathcal{S}(i)\) is the set of adjacent nodes of node \(i\) within the same network and \(\mathcal{F}(i)\) is the set of adjacent nodes of node \(i\) from other networks. By this formulation, \(\rho_{i\to j}\) and \(\varphi_{i\to j}\) become \(1\) if either \(\rho_{k\to i}\) or \(\varphi_{l\to i}\) is \(1\), reflecting the logical OR-like dependence. Starting with random configurations of\(\rho_{i\to j}\), \(\varphi_{i\to j}\in\{0,1\}\), the global effectiveness \(\rho_{i}\) as the converged probability of node \(i\) being connected to the GC through message passing is defined as  
\[\rho_{i}=\sigma_{i}\bigg{[}1-\prod_{k\in\mathcal{S}(i)}(1-\rho_{k\to i}) \prod_{l\in\mathcal{F}(i)}(1-\varphi_{l\to i})\bigg{]}. \tag{3}\]  
In contrast to the B-NoN model, a node in the C-NoN model is globally effective if _both_ its adjacent nodes of the same network _and_ those from other networks belong to the GC. This model reflects the _logical AND_-like dependence that the power network and the control network are not mutually replaceable functions, whereas the functions in the inter-areal brain networks are complementary. Based on this characteristic, the variables \(\rho_{i\to j}\) and \(\varphi_{i\to j}\) are defined as  
\[\begin{split}\rho_{i\to j}&=\sigma_{i}\bigg{[}1- \prod_{k\in\mathcal{S}(i)\setminus j}(1-\rho_{k\to i})\bigg{]}\bigg{[}1- \prod_{l\in\mathcal{F}(i)}(1-\varphi_{l\to i})\bigg{]},\\ \varphi_{i\to j}&=\sigma_{i}\bigg{[}1-\prod_{k\in \mathcal{S}(i)}(1-\rho_{k\to i})\bigg{]}\bigg{[}1-\prod_{l\in\mathcal{F}(i) \setminus j}(1-\varphi_{l\to i})\bigg{]}.\end{split} \tag{4}\]  
In contrast to the formulation of the B-NoN model, \(\rho_{i\to j}\) and \(\varphi_{i\to j}\) become 1 if both \(\rho_{k\to i}\) and \(\varphi_{l\to i}\) are 1, reflecting the logical AND-like dependence. As the converged probability of node \(i\) being connected to the GC, the global effectiveness \(\rho_{i}\) is then defined as  
\[\rho_{i}=\sigma_{i}\bigg{[}1-\prod_{k\in\mathcal{S}(i)}(1-\rho_{k\to i}) \bigg{]}\bigg{[}1-\prod_{l\in\mathcal{F}(i)}(1-\varphi_{l\to i}) \bigg{]}. \tag{5}\]