Nonreciprocal random networks and their percolation properties

TL;DR

This work analyzes nonreciprocal directed random networks, where $P_{i\rightarrow j}\neq P_{j\rightarrow i}$, to understand how structure influences percolation. The authors develop a self-consistent integral-differential framework for reachability, enabling exact solutions for both unstructured (intransitive) and structured (transitive) networks and validating them with simulations. They derive degree-distribution properties, reveal anti-correlation between out- and in-degrees in the unstructured case, and show node-dependent Poisson degrees in the structured case, leading to nonfactorizable joint distributions. In percolation, intransitive networks recover the ordinary directed ER results with a threshold at $\lambda_p+\lambda_q=2$, while transitive networks exhibit a richer phase behavior with a nontrivial percolation line and $ ext{S}$ not generally equal to $\text{O}\text{I}$, highlighting how structure can suppress percolation. Overall, the paper provides a solvable framework linking nonreciprocity and structure to percolation phenomena, with potential generalisations to broader kernel/multi-type network models and practical implications for real-world directed systems.

Abstract

We study the effects of nonreciprocity and network structure on percolation. To this end, we investigate nonreciprocal random networks - directed networks for which the probability of a link occurring from node i to node j differs from the probability of the reverse link occurring from node j to node i. We analytically determine the degree and percolation properties of such networks with exactly two types of link probability, demonstrating that whether the networks are structured such that the nodes are not statistically indistinguishable has profound effects on these measures, both quantitively and in how such networks need to be approached. In particular, we develop a technique for solving the percolation problem which can be applied to both structured and unstructured networks. The method entails writing self-consistent integral and differential equations for the probability that each node will belong to the network's giant component. Exact solutions to these equations are obtained and simulations which confirm our analytic predictions are presented.

Figures (10)

• Figure 1: (a) A transitive structured nonreciprocal random network on 4 nodes. (b) An example of an intransitive unstructured nonreciprocal random network on 4 nodes. In each case, the blue links (long dashes) exist with probability $p$ while the red links (short dashes) exist with probability $q$.
• Figure 2: $(\lambda_p,\lambda_q)$ phase diagrams for (a) intransitive nonreciprocal random networks (b) transitive nonreciprocal random networks. The solid black lines in each plot correspond to the critical lines where the phase transitions occur and are given by Eqs. (\ref{['eq:percolation line intransitive']}) and (\ref{['eq:percolation line transitive']}).
• Figure 3: The mean reciprocity, $\left<\rho\right>$ as a function of $\lambda_q$ for $\lambda_p\in\{0,1/2,1,2,4\}$. The data points (symbols) are simulation data collected from $10^4$(a) intransitive networks and (b) transitive networks, each consisting of $N=10^2$ nodes. The solid lines are given by Eq. (\ref{['eq:mean reciprocity']}) and can be seen to predict the data superbly.
• Figure 4: The mean out-degree $\left<\overline{K_\mathrm{out}}\right>$ (which is identical to the mean in-degree $\left<\overline{K_\mathrm{in}}\right>$) as a function of $\lambda_q$ for $\lambda_p \in\{0,1/2,1,2,4\}$. The data points (symbols) are collected by averaging over $10^2$(a) intransitive networks and (b) transitive networks, each consisting of $N=10^4$ nodes. The solid lines are the theoretical values predicted by Eqs. (\ref{['eq:degree statistics intransitive']}) and (\ref{['eq:transitive mean degree']}), for plots (a) and (b) respectively, and can be seen to superbly match the data.
• Figure 5: The variance of the out-degree distribution, calculated in two ways, as a function of $\lambda_q$ for $\lambda_p\in\{0,1/2,1,2,4\}$. Plots (a) and (c) correspond to simulation data collected from $10^2$ intransitive networks with $N=10^4$ nodes each. Plots (b) and (d) are the same but for transitive networks. The solid lines in plots (a) and (c) are the predicted values given by Eq. (\ref{['eq:degree statistics intransitive']}). The solid lines in plot (b) are the values predicted by Eq. (\ref{['eq:transitive variance']}) while those in plot (d) are obtained by integrating Eq. (\ref{['eq:transitive mean and var vs alpha']}) over $\alpha$ from 0 to 1. In all cases, the theory perfectly matches the simulation results.