DomiRank centrality
DomiRank centrality
is a measure designed to quantify the dominance of nodes within their local neighborhoods and to highlight structurally fragile regions whose integrity and functionality depend on these dominant nodes [2]. The evolution of node fitness is governed by two processes: (i)
natural relaxation
, where each node's fitness decays exponentially toward zero at rate \(β\); and (ii)
competition
, where nodes compete with neighbors for limited resources. A node's fitness increases when it is surrounded by neighbors whose fitness is below a domination threshold \(θ\), and decreases otherwise.
Formally, let \(\Gamma(t) \in \mathbb{R}^{N}\) denote the vector of evolving dominance scores. The dynamics follow:
\[
\frac{d\Gamma(t)}{dt} = α \, A \bigl(θ \mathbf{1} - \Gamma(t)\bigr) - β \, \Gamma(t),
\]
where \(A\) is the adjacency matrix of the network \(G\), \(α, β, θ\) are parameters controlling competition and relaxation dynamics, and \(\mathbf{1}\) is the all-ones vector. The domination threshold \(θ\) acts as a rescaling factor and is set to \(θ = 1\) without loss of generality. The parameter ratio \(σ = α / β\) determines the balance between local (nodal) and mesoscale (structural) information.
At steady state (\(\lim_{t \to \infty}\)), the dominance vector \(\Gamma\) satisfies
\[
\Gamma = θ \, σ \, (σ A + I)^{-1} A \, \mathbf{1},
\]
where \(I\) is the identity matrix. The convergence interval for \(σ\) is bounded as \(σ \in \left(0, -\frac{1}{λ_N}\right)\), where \(λ_N\) is the smallest (dominant negative) eigenvalue of \(A\).
Nodes with high DomiRank scores dominate many neighbors that themselves have low influence, thereby identifying fragile neighborhoods highly dependent on these nodes. Engsig
et al.
[2] demonstrate that DomiRank-based interventions can inflict more enduring network damage, impeding recovery and reducing overall system resilience.