Multi-local dimension (MLD) centrality
The
multi-local dimension (MLD) centrality
is a variant of the local dimension measure for identifying influential spreaders in complex networks [2]. MLD evaluates the structural information around a node by considering concentric boxes of increasing radius. For a given node \(i\), the box radius \(l\) ranges from 1 to the maximum shortest-path distance from \(i\).
The proportion of nodes within a box of radius \(l\) is
\begin{equation*}
μ_i(l) = \frac{N_i(l)}{N},
\end{equation*}
where \(N_i(l)\) is the number of nodes covered by the box, and \(N\) is the total number of nodes in the network. Based on \(μ_i(l)\), the generalized partition function \(Z_i(q,l)\) is defined as
\begin{equation*}
Z_i(q,l) =
\begin{cases}
μ_i(l)^q, & q \notin \{0,1\},\\[1mm]
1/μ_i(l), & q = 0,\\[1mm]
μ_i(l)\, \log_2 μ_i(l), & q = 1,
\end{cases}
\end{equation*}
where \(q \in \mathbb{R}\) is a tunable parameter controlling the emphasis on different structural scales.
The multi-local dimension \(c_{\textsc{MLD}}(i,q)\) of node \(i\) is then defined as
\begin{equation*}
c_{\textsc{MLD}}(i,q) =
\begin{cases}
\displaystyle \lim_{l \to 0} \frac{\log_2 Z_i(q,l)}{(q-1)\, \log_2 l}, & q \neq 1,\\[1mm]
\displaystyle \lim_{l \to 0} \frac{Z_i(q,l)}{\log_2 l}, & q = 1.
\end{cases}
\end{equation*}
In practice, \(c_{\textsc{MLD}}(i,q)\) is estimated numerically as the slope of a linear regression: if \(q \neq 1\), the regression is of \(\frac{\log_2 Z_i(q,l)}{q-1}\) versus \(\log_2 l\); if \(q = 1\), it is of \(Z_i(q,l)\) versus \(\log_2 l\). Wen et al. [2] show that \textsc{MLD} reduces to the local information dimensionality (LID) [3] when \(q=1\), and to the local dimension measure [4] when \(q=0\).