Local information dimensionality (LID)
The
local information dimensionality
(LID) is an entropy-based, semi-local centrality measure derived from the concept of local dimension [2]. It characterizes the structural complexity of the neighborhood around each node by quantifying how local information changes with scale.
For a given node \(i\), the local information dimensionality \(D_{I_i}\) is defined as
\[
D_{I_i} = - \frac{dI_i(r)}{d \ln r},
\]
where the derivative is taken with respect to the logarithm of the topological distance \(r\), which represents the scale of locality around node \(i\). The information content \(I_i(r)\) is computed based on the number of nodes \(B_i(r)\) within a topological distance \(r\) from node \(i\) as
\[
I_i(r) = - \frac{B_i(r)}{N}\ln \frac{B_i(r)}{N},
\]
where \(N\) is the total number of nodes in the network.
The local information dimensionality \(D_{I_i}\) can be estimated from the slope of the relationship between \(I_i(r)\) and \(\ln r\) on a logarithmic scale, and discretized as
\[
D_{I_i} \simeq \frac{n_i(r)\left[1 + \ln \frac{B_i(r)}{N}\right]r}{N},
\]
where \(n_i(r)\) denotes the number of nodes located exactly at distance \(r\) from the reference node \(i\).
The scale of locality \(r\) is typically chosen as half of the maximum shortest-path distance from node \(i\), defined as
\[
r = \lceil d_{\max}(i)/2 \rceil,
\]
where \(d_{\max}(i)\) is the maximum geodesic distance from node \(i\), and \(\lceil \cdot \rceil\) denotes the ceiling function.