Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.21474

Multiplicative Path Weights in Networks: From Katz Connectivity to Mean First-Passage Times

Robbert van der Burg·Bernd Heidergott·Thao Le·Alessandro Zocca
View original source ↗Read source document ↗Export metadata ↓

Abstract

Signal dispersion, stress propagation in interbank systems, and transmission probabilities in epidemic and reliability models are all captured by studying paths on directed graphs where the path weight is defined as the product of its edge weights. Connectivity in such networks is naturally quantified by aggregating these multiplicative path weights across nodes. This paper provides a comprehensive treatise on connectivity and ranking measures based on multiplicative path weights. We demonstrate how standard all-path connectivity, such as the standard Katz centrality, can be extended beyond its classical spectral threshold. In addition, we derive closed-form expressions for connectivity based on first-hit paths and introduce a new version of Katz centrality. Furthermore, for random edge weights, we derive resolvent-based expressions for the first and second moments, alongside variance bounds, under a path-indexed coupling framework. Finite irreducible Markov chains are a subclass of the models studied, and we show that the scaled matrix of first-hit path probabilities is the probability-generating function for first-passage and return times, and that differentiation yields the classical first-passage times. Numerical examples illustrate the application of our results in a variety of settings.

Explore related subjects

cs.SImath.PRphysics.soc-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Robbert van der Burg, Bernd Heidergott, Thao Le, Alessandro Zocca. 2026-10-15. Multiplicative Path Weights in Networks: From Katz Connectivity to Mean First-Passage Times. https://arxiv.org/abs/2610.21474

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ideological discrepancy between Facebook pages and shared news is associated with reaction homogeneity

Cross-cutting political content is often expected to mitigate polarization, yet encounters across ideological lines do not necessarily foster more constructive engagement. We analyzed 4,584 posts from 1,243 Facebook pages sharing 761 political news links in a dataset centered on Brazil's 2018 presidential election, and introduced Bias Discrepancy, which measures how far a shared link lies from the page's ideological sharing repertoire. We examined how Bias Discrepancy related to Content Reaction Positivity (how positively the same link was received in other posts), the Reaction Homogeneity Index (whether a post's own reactions were one-sided or mixed), and the toxicity of the page's accompanying text. Reaction homogeneity was lowest at both extremes of Bias Discrepancy, and posts furthest from the page's repertoire had the highest average toxicity. In a multivariable model, greater Bias Discrepancy was associated with lower reaction homogeneity, with a magnitude similar to the link's content bias and toxicity and smaller than Content Reaction Positivity. When a page shared news that departed from its repertoire, its accompanying text tended to be more toxic and its audience's reactions more mixed.

cs.SI↗
arXiv2026

Structural Properties of DomiRank Centrality: Symmetry Axioms, an Entropy Bound, and Spectral Interpolation

DomiRank is a node-importance algorithm for unweighted networks, defined by a dynamical model governed by a competition-strength parameter, a dominance threshold, and a decay rate. We establish three structural results. First, on symmetry. Automorphism-related vertices share the same DomiRank value, and a graph whose values are pairwise distinct must be asymmetric. The orbit count bounds the number of distinct values, and regular and vertex-transitive graphs admit closed forms. Second, on entropy. The maximum of the DomiRank entropy over connected graphs, log_2 N, is attained if and only if the graph is regular. The initial entropy derivative is the degree-log-degree edge covariance, making the decrease rigorous near vanishing competition for structurally disassortative graphs, and an assortative counterexample shows it does not extend to all graphs; a sufficient-condition theorem and a refined conjecture cover the rest. Third, on spectrum. The normalized distribution is sigma-invariant if and only if the degree vector is an adjacency-matrix eigenvector. The parameter sigma interpolates continuously between degree and smallest-eigenvector centrality within the resolvent (Bonacich-type) family, its sensitivity being a second-order suppression operator on the degree vector. Graph transformations are also characterized, including the m-product. Experiments on four real networks and synthetic families up to 10^4 vertices confirm the interpolation and exhibit monotone entropy decrease, continuous ranking interpolation, decaying positive fraction and ranking reversals; where the positive fraction has decayed far from one, the limit describes the solution vector rather than the induced ranking. DomiRank is thus a tunable complement to principal-eigenvector centrality. A companion study quantifies how much of its task-level edge over a degree baseline survives degree correction.

cs.SI↗
arXiv2026

Rewiring the Majority: Structural Median Opinion Control in the Friedkin-Johnsen Model

Opinion control under the Friedkin-Johnsen (FJ) model has traditionally focused on modifying individual users, for example by changing their opinions or susceptibility to social influence. We study a complementary problem: can collective outcomes be steered by modifying the interaction network instead? We formulate structural median control as an edge-addition problem aimed at moving the equilibrium median beyond a target threshold. We show that the problem is NP-hard even under highly restricted settings and propose both continuous optimization methods and computationally lighter opinion-aware heuristics for selecting edge additions. Experiments on synthetic and real-world networks reveal that intervention success depends jointly on network topology, the placement of favorable opinions, and user susceptibility. Structural interventions produce the largest shifts when opinion placement and network structure allow added edges to create useful influence pathways toward the target outcome. Finally, under a homogeneous reference setting, we compare structural and node-level interventions using normalized budgets and a degree-weighted cost model. We show that, although node interventions often appear stronger under normalized budgets, structural interventions become competitive and frequently more cost-effective once the higher cost of targeting high-reach users is taken into account.

cs.SI↗