Indirect Influence on Network Diffusion

TL;DR

This study quantifies indirect influence and reveals a structural phase transition associated with altered diffusion dynamics, providing new insights into network behavior and applications in information spreading, disease propagation, or transport dynamics across physical and biological systems.

Abstract

Models of network diffusion typically rely on the Laplacian matrix, capturing interactions via direct connections. Beyond direct interactions, information in many systems can also flow via indirect pathways, where influence typically diminishes over distance. In this work, we analyze diffusion dynamics incorporating such indirect connections using the $d$-path Laplacian framework. We introduce a parameter, the indirect influence, based on the change in the second smallest eigenvalue of the generalized path Laplacian, to quantify the impact of these pathways on diffusion timescales relative to direct-only models. Using perturbation theory and mean-field approximations, we derive analytical expressions for the indirect influence in terms of structural properties of random networks. Theoretical predictions align well with numerical simulations, providing a phase diagram for when indirect influence becomes significant. We also identify a structural phase transition governed by the emergence of $d$-paths and derive the critical connection probability above which they dramatically alter diffusion. This study provides a quantitative understanding of how indirect pathways shape network dynamics and reveals their collective structural onset.

Figures (6)

• Figure 1: Diagrams of $\bm{d}$-path networks and the path network. The figure shows the $d$-path networks derived from a base network and the corresponding path network that combines all $d$-paths.
• Figure 2: Expected connection probabilities of $\bm{d}$-path networks relative to the 1-path network. Expected connection probabilities for different $d$-path networks as a function of the expected 1-path connection probability, with ${N = 1000}$. a Erdős-Rényi (ER) networks and b Random Regular (RR) networks. Each point corresponds to the average over 50 different networks with error bars of one standard deviation, and the solid lines correspond to their respective analytical expression. The gray shaded range indicates the values of ${p < \log N / N}$, corresponding to the values where the network is almost surely disconnected erdHos1961strength, and therefore are not considered.
• Figure 3: Average indirect influence across parameter space. Average indirect influence computed over all possible $d = 1, \dots, D$ paths for networks with $N = 1000$. a Erdős-Rényi (ER) networks. b Random Regular (RR) networks. Solid white lines correspond to the contour levels of the color map that corresponds to the simulations, and the dashed white lines correspond to the theoretical contour levels. We compute 100 different networks for each combination of $\alpha$ ranging from 0 to 4 in steps of 0.1 and $p$ ranging from $0$ to $1$ in steps of 0.001.
• Figure 4: Degree distribution of the path network. Degree distribution of the path network, $k = k^{(1)} + k^{(2)}$, which includes 1-path and 2-path links, averaged over 50 realizations of a ER networks and b RR networks. In both cases, $N = 1000$ and $\langle k^{(1)} \rangle = 30$. Solid lines represent the theoretical binomial distribution that an ER network should have, with $N$ nodes and a connection probability of $p + p^{(2)}$.
• Figure 5: Average indirect influence of the uniform weight limit. Average indirect influence for a Erdős-Rényi networks and b Random Regular networks as a function of the expected connection probability for the path Laplacian truncated at $d_{\max} = 2$ and $d_{\max} = D$. Each point corresponds to the average over 50 different networks with error bars of one standard deviation, and the solid lines correspond to the theoretical curves for the respective truncation. The gray shaded range indicates the values of $p < \log N / N$, corresponding to the values where the network is almost surely disconnected, and therefore are not considered.