Local dimension (LD)
The
local dimension
(LD) quantifies the dimensionality of nodes in a network by examining how the volume of the neighborhood around each node scales with increasing topological distance [2]. In most spatially embedded real networks, which typically lack the small-world property, the distribution of link lengths follows a power law. Consequently, the number of nodes \(B_i(r)\) located within a topological distance \(r\) from a node \(i\) obeys the relationship
\[
B_i(r) \sim r^d,
\]
where the constant \(d\) characterizes the effective dimension of the network. Silva and Costa [2] refined this power-law relationship by allowing the dimensionality to vary locally, proposing that
\[
B_i(r) = α r^{D_i(r)},
\]
where \(D_i(r)\) represents the local dimension around node \(i\). The local dimension coefficient \(D_i(r)\) can be estimated from the slope of the \(B_i(r)\) curve on a double-logarithmic scale and is discretized as
\[
D_i(r) \simeq r \frac{n_i(r)}{B_i(r)},
\]
where \(n_i(r)\) denotes the number of nodes that are exactly at a topological distance \(r\) from the reference node \(i\). Pu et al. [3] further extended the local dimension measure by allowing the distance parameter \(r\) to vary across different nodes in the network.