Ranking Edges by their Impact on the Spectral Complexity of Information Diffusion over Networks

TL;DR

This work introduces a spectral-entropy–based edge centrality grounded in von Neumann entropy (VNE) to quantify how edge removals alter the complexity of information diffusion on networks. By combining diffusion-kernel VNE with first-order spectral perturbation theory, the authors derive a scalable approximate edge-ranking algorithm that reduces per-edge cost to $\mathcal{O}(N)$, enabling application to large networks. They validate the method on three real-world datasets (a polarized U.S. Senate voting network, a London multimodal transport network, and a multiplex brain network) and demonstrate pronounced multiscale changes in edge importance as the diffusion timescale $\beta$ varies, highlighting the influence of community structure, mode speeds, and interlayer coupling. The results offer a complementary, entropy-based perspective to centrality that captures the spectral complexity of diffusion and can inform targeted interventions, sparsification, or link-prediction tasks across social, physical, and biological systems.

Abstract

Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge's removal would change a system's von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks: e.g.\ the scaling is $\mathcal{O}(N^3)$ per edge for networks with $N$ nodes. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and fast to compute, scaling as $\mathcal{O}(N)$ per edge. Focusing on a form of VNE that is associated with a transport operator $e^{-β{ L}}$, where ${ L}$ is a graph Laplacian matrix and $β>0$ is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate and long timescales $β$ for diffusion.

Figures (9)

• Figure 1: Comparison of Algorithms \ref{['alg:actual']} and \ref{['alg:approximate']}.(A) We estimate the runtimes for $\beta = 1$ that are required to execute these algorithms for Erdős--Rényi $G_{NM}$ graphs with increasing size with $N$ nodes and $M=5N$ edges, so that mean degree remains fixed as $N$ increases. The inset is a logarithmic scale to reveal their runtimes' scaling behaviors (see text). Note that Algorithm \ref{['alg:actual']} would potentially take hundreds of hours for each $N$, so we extrapolated these times from that required to compute for a single edge, instead of running the algorithm to completion. In contrast, Algorithm \ref{['alg:approximate']} completed in under a minute for all values of $N$ plotted. (B) We compare the fractional overlap $|\mathcal{E}^{(\ell)} \cap \tilde{\mathcal{E}}^{(\ell)}| / \ell$ for the $\ell$ top-ranked edges for the two rankings $R_{pq}$ and $\tilde{R}_{pq}$ in a $G_{NM}$ graph with $N=100$ nodes and $M=1000$ edges at $\beta = 1$. The inset shows that they are strongly correlated: $R_{pq}\approx \tilde{R}_{pq}$ for all edges $(p,q)\in\mathcal{E}$. We computed a Pearson correlation coefficient of $0.996$.
• Figure 1: Rankings of inter- and intraparty edges for a polarized political network.(A) We visualize the rankings $\tilde{R}_{pq}$ of edges given by Algorithm \ref{['alg:approximate']} using grayscale for a graph that encodes voting similarity among the 117th U.S. Senate. The three columns depict three choices of timescale parameter $\beta$. Nodes represent Senators and the node colors red and blue indicate, respectively, the two major political parties: Republicans and Democrats (including independents). (B) For the same three values of $\beta$, we plot empirically measured distributions $p(H'(0))$ of the approximate change $H'(0)$ to diffusion-kernel VNE given by Eq. \ref{['eq:perturbation']}. In each panel, we show two distributions: one in which $p(H'(0))$ is measured across intraparty edges, and one in which $p(H'(0))$ is measured across interparty edges. (C) The solid black curve depicts the mean rank $\langle \tilde{R}_{pq}\rangle_{inter}$ across interlayer edges versus $\beta$. The colored curves indicate the values of $\langle \tilde{R}_{pq}\rangle_{inter}$ for a comparable random-graph model with two communities and different amounts of connectivity between communities (see text). Our main finding is that the interparty edges have the top rankings for larger $\beta$, the lowest rankings for intermediate $\beta$, and intermediate rankings for smaller $\beta$.
• Figure 1: Numerical validation of first-order perturbation theory for diffusion-kernel VNE.(A) Comparison of $h(L+ \Delta L)$ and our first-order prediction, $h(L) + H'(0)$, with $H'(0)$ given by Thm. \ref{['def:gen_FOP']}, for a perturbed graph. We consider an Erdős-Rényi $G_{NM}$ random graph with $N = 100$ nodes and $M = 1500$ unweighted edges, and each perturbation matrix $\Delta L$ encodes the removal of $k$ edges as described in taylor2016synchronization (which generalized Lemma \ref{['cor:spec_pert']} to edge sets). Observe that the first-order approximation becomes less accurate for larger perturbations (i.e., larger $k$). (B) Numerical validation that the approximation error $E(\epsilon) \equiv |h(L+\epsilon\Delta L) - [h(L) + \epsilon H'(0)]|$ vanishes in the limit of small $\epsilon$ as $\mathcal{O}(\epsilon^2)$, that is, $\log(E(\epsilon) ) \varpropto 2\log(\epsilon)$. We plot $E(\epsilon)$ for the same network as in panel (A) in a log-log scale and compute a least-squares linear fit to obtain empirically measured slopes $\{2.042, 2.022, 2.036\}$, respectively, for three choices of $k\in\{ 1, 10, 20\}$. These are all nearly equal to the predicted slope of 2, supporting our claim that Thm. \ref{['def:gen_FOP']} has second-error error.
• Figure 1: Comparison of Algorithms \ref{['alg:actual']} and \ref{['alg:approximate']} for U.S. Senate and multiplex brain networks. Extending the results in Fig. \ref{['fig:runtime']}(B), we now compare the two algorithms for two empirical networks: (A) a U.S. Senate voting network and (B) a multiplex brain network with interlayer coupling $\omega = 1$. See Secs. \ref{['sec:congress_network']} and \ref{['sec:brain_network']}, respectively, for their descriptions. In both panels, we plot the fractional overlap among the top-$\ell$ ranked edges for the two algorithms with $\beta=1$ and varying $\ell$. The insets show scatter plots that directly compare the rankings from the algorithms.
• Figure 1: Network encoding voting similarity among the 117th U.S. Senate.(A) Visualization of the adjacency matrix ${\bf A}$, where $A_{ij}$ encodes the fraction of bills in which Senators $i$ and $j$ vote identically (ignoring fractions less that $0.4$). By sorting node ids by the Senator's political affiliations so that nodes $\{1, \dots, 50\}$ are Republicans and nodes $\{51, \dots, 100\}$ are either Democrats or independents, the resulting 2-community structure manifests as two large 'blocks' in the matrix. (B) Histogram depicts the probability distributions for edge weights $\{A_{ij}\}$, which we measure separately across intraparty and interparty edges.

Theorems & Definitions (16)

• Definition 1: von Neumann Entropy for Graphs
• Remark 2.1
• Remark 2.2
• Theorem 2: Perturbation of Simple Eigenvalues atkinson2008introduction
• Proof 1
• Remark 2.3
• Proposition 3: Weighted Edge Perturbations for Laplacian matrices
• Lemma 4: First-order Spectral Impact of Edge Additions and Removals milanese2010approximating
• Definition 1: Edge Rankings by VNE Increases
• Remark 3.1