Outward accessibility
The
outward accessibility
quantifies the ability of a node to reach other nodes in a network after a fixed number of steps along self-avoiding walks [2]. Let \( N \) be the total number of nodes in the network, and let \( P_h(i,j) \) denote the transition probability that an agent starting from node \( i \) reaches node \( j \) in exactly \( h \) steps along a self-avoiding walk (i.e., a simple path without revisiting nodes). The outward accessibility of node \( i \) after \( h \) steps is defined as
\begin{equation*}
c_{OA_h}(i)
= \frac{1}{N - 1}
e^{\left(
-\sum_{j:\, P_h(i,j) \neq 0}
P_h(i,j) \log P_h(i,j)
\right)}.
\end{equation*}
The expression inside the exponential represents the Shannon entropy of the transition probability distribution for node \( i \). A higher entropy indicates that node \( i \) can reach many other nodes through distinct paths with similar probabilities, reflecting greater accessibility. The normalization factor \( 1/(N - 1) \) ensures that the measure is comparable across networks of different sizes.