MINE: Quantifying Node and Edge Information

• MINE is a suite of entropy-based measures that assess how removing nodes or edges impacts the information dynamics of complex networks.
• It utilizes spectral methods, including von Neumann entropy and entanglement measures, to quantify structural complexity and diffusion processes.
• MINE employs both exact and approximate computational schemes, enabling scalable multiscale analysis in classical and quantum contexts.

The Measure of Information in Nodes and Edges (MINE) denotes a class of information-theoretic and spectral complexity measures that quantify, for each node or edge in a network, the extent to which its presence or removal influences the network's collective information dynamics or structure. MINE has appeared in both classical and quantum network contexts, including through von Neumann entropy-based edge and node centralities that characterize complexity of diffusion, and entanglement-entropy measures of quantum correlations. These approaches yield a matrix of node- and edge-level information values that generalize traditional degree- and path-based metrics, uncovering multi-scale dependencies and topological features that impact information propagation and storage.

1. Classical von Neumann-Entropy Framework for MINE

The von Neumann-entropy–based MINE approach defines, for an undirected, weighted, connected graph $G=(V,E)$ with Laplacian $L$, a diffusion density matrix

$\rho_{\beta}(L) = \exp(-\beta L)/Z_\beta$

with normalization $Z_\beta = \mathrm{Tr}[\exp(-\beta L)]$ and timescale parameter $\beta>0$, controlling the temporal resolution of diffusion. The spectral entropy at timescale $\beta$ is

$S_\beta(G) = -\sum_{k=1}^N s_k \log_2 s_k$

where $s_k = \exp(-\beta\lambda_k)/Z_\beta$ and $\lambda_k$ are the Laplacian eigenvalues. $S_\beta$ captures the spread of diffusion across eigenmodes; larger values reflect a more delocalized, complex dynamic (Kazimer et al., 2022).

MINE centrality is assigned by computing, for each edge $(p,q)$, the entropy shift

$$\Delta S_{pq} = S_\beta(L-\Delta L^{(pq)}) - S_\beta(L)$$

where $\Delta L^{(pq)}$ is the perturbation from removing edge $(p,q)$. Edges are ranked by $\Delta S_{pq}$; the highest-ranked edges most alter the spectral complexity and information diffusion structure.

A node-level contribution is similarly defined:

$$\Delta S_i = S_\beta(L-\Delta L_\text{node}(i)) - S_\beta(L)$$

with $\Delta L_\text{node}(i)$ the perturbation induced by removing all connections to node $i$. The “node entanglement” centrality (Kazimer et al., 2022) corresponds directly to this entropy shift.

First-order perturbation theory allows approximate computation of $\Delta S_{pq}$ and $\Delta S_{i}$, enabling efficient application to large networks by avoiding repeated full Laplacian diagonalization. The complexity per edge drops from $\mathcal{O}(N^3)$ to $\mathcal{O}(N)$ in the approximate (MINE-approx) scheme.

2. Quantum MINE: Entanglement Entropy and Mutual Information

In quantum oscillator networks, MINE is realized in terms of single-node von Neumann entropy and pairwise mutual information. For $N$ coupled harmonic oscillators (adjacency $A$, Laplacian $L$), the Hamiltonian is

$H = \frac{1}{2}[p^T p + x^T (I + 2cL)x]$

with coupling $c>0$. Diagonalization yields covariance matrices for the ground state, from which:

• The single-node (mode) entropy is $S_i = f(\nu_i)$, with $$\nu_i = \sqrt{\langle x_i^2\rangle\langle p_i^2\rangle}$$.
• The mutual information between nodes $i$ and $j$ is $I(i:j) = S_i + S_j - S_{ij}$, with $S_{ij}$ the joint entropy from the $4 \times 4$ covariance block (Cardillo et al., 2012).

These measures provide detailed information allocation profiles: $S_i$ quantifies the information node $i$ has about the rest of the system, while $I(i:j)$ forms the MINE matrix for pairwise quantum correlations.

Numerical and analytical studies reveal nontrivial dependencies: nodes with intermediate degree may maximize single-node entropy—a departure from monotonic degree-centrality paradigms. The full MINE matrix is sensitive to the spectrum and eigenstructure of $L$.

3. Algorithmic Implementation and Computational Aspects

The exact MINE-centrality procedure (MINE-exact) involves, for each edge, Laplacian modification and re-diagonalization, yielding computational scaling of $\mathcal{O}(M N^3)$. The approximate scheme (MINE-approx) requires a single diagonalization of $L$ followed by $\mathcal{O}(N)$ operations per edge via first-order perturbation expressions:

$$\tilde{\Delta} S_{pq} = \sum_{i=1}^N \alpha_i (m - \lambda'_i)$$

where $\lambda'_i = -A_{pq}(u_p^{(i)} - u_q^{(i)})^2$ and $\alpha_i$ are spectral coefficients precalculated from $\{s_i\}$ (Kazimer et al., 2022). Node-level contributions are computed by analogous sums over all edges incident to node $i$.

In quantum networks, computation of $S_i$ and $I(i:j)$ requires spectral decomposition of the potential matrix $I + 2cL$ to obtain $\omega_k$, followed by construction of relevant covariance submatrices (Cardillo et al., 2012).

4. Empirical Characterization and Regime Transitions

MINE-based rankings exhibit pronounced dependence on the diffusion timescale parameter $\beta$ or, in quantum settings, coupling $c$:

• In social similarity networks (e.g., U.S. Senate roll-call), at low $\beta$, edges within dense communities are top-ranked; at high $\beta$, inter-community (bridging) edges dominate importance.
• In multimodal transit graphs, “shortcut” edges become critical at short times, with the role shifting toward alternative pathways at longer timescales.
• In multiplex brain networks, intralayer coherence edges are most informative at intermediate $\beta$, but interlayer edges dominate at high $\beta$ (Kazimer et al., 2022).

In quantum networks, star and scale-free topologies yield maximal single-node entropy at intermediate degree, revealing nuanced information allocation not captured by classical measures (Cardillo et al., 2012).

These transitions, controlled by β or c, uncover multiscale and functional regimes in information complexity and storage.

5. Theoretical Properties and Interpretation

The von Neumann entropy framework directly quantifies spectral complexity and multiscale mixing induced by diffusion processes. The entropy shift under edge or node removal captures the fragility or robustness of collective information flow to targeted perturbations—essential for network resilience and functional assessment (Kazimer et al., 2022).

In quantum networks, the entanglement entropy and mutual information MINE measures transcend classical centrality by defining an “information backbone”—the network substructure that maximizes nonlocal quantum correlations. Non-monotonicity with respect to degree reflects global spectral localization properties.

Both the classical and quantum MINE matrices rely fundamentally on the eigenvalues and eigenvectors of $L$, and thus encode not just local but also global and nonlocal structural information.

6. Connections and Distinctions with Related Measures

MINE is distinct from degree-, betweenness-, and even classical random-walk–based centralities by its foundation in information-theoretic quantities derived either from spectral (von Neumann or entanglement) entropy or from mutual information measures (Kazimer et al., 2022, Cardillo et al., 2012). Unlike effective information (EI), which is rooted in the mutual information between random-walk input and output distributions (Klein et al., 2019), MINE can resolve spectral complexity and quantum correlations at multiple scales and in both classical and quantum network settings.

A plausible implication is that MINE provides a rigorous and generalizable basis for identifying critical nodes and edges underpinning multiscale information diffusion, storage, and quantum coherence in complex networks.

7. Applications and Practical Relevance

MINE measures are applied in diverse domains, including:

• Structural resilience analysis (identifying critical nodes/edges for informational complexity in infrastructure, social, or biological networks)
• Neuroscience (characterizing functional communities and cross-frequency coupling in multiplex brain networks)
• Quantum network topology inference (using external probe entanglement entropy (Cardillo et al., 2012))
• Comparative analysis across network models and empirical systems to understand information allocation regimes and transitions

The adaptability of MINE to node or edge ranking, to both classical and quantum regimes, and its sensitivity to multiscale processes, establishes it as a central tool in spectral network analysis and quantum information science.