# Econometric Information Recovery in Behavioral Networks
## 5 Network Information Recovery
### A Binary Network Example  
The network recovery Equations (5) and (6) in Section 5 are formalistic. The objective in this section is to clarify how entropy maximization is directly applicable to computing the behavioral network probabilities. To demonstrate the nature and applicability of information theoretic methods in network information recovery, we consider a simple binary directed network example. In this type of network, edges point in a particular direction between two vertices to represent, the flow of information, goods or traffic and the direction the edge is pointing may be represented by an asymmetric adjacency matrix with \(A_{ij}=1\). In an economic behavioral network, the efficiency of information flow is predicated on discovering or designing protocols that efficiently route information. In many ways, this is like a network where the emphasis is on design and efficiency in routing the flows (see for example, [21]). To carry the above behavioral routing information flow analogy a bit farther, consider the problem of determining least-time point-to-point flows between sub networks, when only _the average origin-destination rate flow volume is_ known. Given information about the network protocol in the form of a matrix composed of binary elements \(A_{ij}\) that encode the information about the possible connections, the problem is to recover the origin-destination routing flows from the noisy network data. The preferred probability distribution of dynamical pathways is that obtained by maximizing route-flow entropy over the possible ensemble of possible networks, subject to the known value of an observable average flow-flux rate. Since the unknown path probabilities are much larger than the single average flow data point, this results in an ill-posed linear inverse problem of the type first discussed in Section 3. To indicate the applicability and implications of the formulation and information recovery method, we follow [22] and use a transportation network problem with aggregate traffic volumes measured at \(T\) points in time, \(y_{it}\), where \(i\) denotes the sub network. The routing protocol matrix, \(\mathbf{A}\), encompasses the routing protocol, and \(\mathbf{A}_{ij}=1\), if the point-to-point traffic at sub network \(i\) is 1, and 0, otherwise. The network may at any one point in time, be expressed by the relationship:  
\[y_{i}=\sum\
olimits_{j=1}^{n}\mathbf{A}_{ij}x_{j}. \tag{7}\]  
If \(N=\sum\
olimits_{j=1}^{n}x_{j}\) is the total traffic at each sub network, then  
\[\frac{y_{i}}{N}=\frac{1}{N}\sum\
olimits_{j=1}^{n}\mathbf{A}_{ij}x_{j}=r_{i}= \sum\
olimits_{j=1}^{n}\mathbf{A}_{ij}p_{j}, \tag{8}\]  
In this problem, there are \(m+1\) constraints that involve an additivity constraint, and \(m\) other constraints,  
\[r_{i}=\sum\
olimits_{j=1}^{n}\mathbf{A}_{ij}p_{j}\quad i=1,\ldots,m. \tag{9}\]  
The problem is fundamentally underdetermined and indeterminate because there are more unknowns than data points on which to base a solution. If we let \(\gamma\to 0\), and \(q_{i}\), be a uniform distribution, the resulting Lagrangean function for the constrained maximization problem is  
\[\mathrm{I}\left(\mathbf{p},\mathbf{q},\ \gamma\to 0\right)=L=-\sum\limits_{j=1}^{n}p_{j} \mathrm{ln}p_{j}-\left(\lambda_{0}-1\right)\sum\limits_{j=1}^{n}\left(p_{j}-1 \right)-\sum\limits_{i=1}^{m}\lambda_{i}\sum\limits_{j=1}^{n}\left(A_{ij}p_{j} -r_{i}\right). \tag{10}\]  
The derivative of the Lagrangian function leads to,  
\[p_{j}=\exp\left(-\lambda_{0}-\sum\
olimits_{i=1}^{m}\lambda_{i}\mathbf{A}_{ij }\right)\quad j=1,2,\ldots,n, \tag{11}\]  
and yields a solution for the probabilities, \(p_{j}\), in terms of the Lagrange multipliers. In general, the solution for this fundamentally underdetermined problem does not have a closed-form expression, and the optimum values of the unknown parameters must be numerically determined (see [22]).