Inward accessibility
The
inward accessibility
quantifies how likely a node is to be reached from 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 \( P_h(j,i) \) denote the transition probability that an agent starting from node \( j \) reaches node \( i \) in exactly \( h \) steps along a self-avoiding walk (i.e., a simple path without revisiting nodes). The inward accessibility of node \( i \) after \( h \) steps is defined as:
\begin{equation*}
c_{IA_h}(i)
= \frac{1}{N - 1}
e^{\left(
-\sum_{j:\, P_h(j,i) \neq 0}
\frac{P_h(j,i)}{N - 1}
\log\!\left(\frac{P_h(j,i)}{N - 1}\right)
\right)}.
\end{equation*}
The term inside the exponential represents the Shannon entropy of the distribution of probabilities that node \( i \) is reached from all other nodes after \( h \) steps. A higher entropy indicates that node \( i \) is reached with similar probabilities from many distinct sources, reflecting higher inward accessibility. The normalization factor \( 1/(N - 1) \) ensures comparability across networks of different sizes.