Shortest-path percolation on scale-free networks

TL;DR

The paper investigates how resource depletion through shortest paths drives percolation on scale-free networks. Using extensive Monte Carlo simulations and event-based finite-size scaling, it shows that finite budgets (C>1) yield mean-field ERN-like critical exponents independent of the degree exponent λ, while unbounded budgets (C≈N) produce ERN-like exponents with different values, implying a homogenization of network structure before the transition. The results identify a mechanism—drastic homogenization via path-based edge deletions—that explains the universality, and they quantify this via betweenness-centrality and maximum-degree scaling. These findings advance our understanding of robustness and failure in heterogeneous networks and have implications for transportation, communication, and supply-chain systems where resources are consumed along shortest paths.

Abstract

The shortest-path percolation (SPP) model aims at describing the consumption and eventual exhaustion of a network's resources. Starting from a network containing a macroscopic connected component, random pairs of nodes are sequentially selected, and if the length of the shortest path connecting the node pairs is smaller than a tunable budget parameter, then all edges along such a path are removed from the network. As edges are progressively removed, the network eventually breaks into multiple microscopic components, undergoing a percolation-like transition. It is known that SPP transition on Erdős-Rényi networks (ERNs) belongs to same universality class as of the ordinary bond percolation if the budget parameter is finite; for unbounded budget, instead, the SPP transition becomes more abrupt than the ordinary percolation transition. By means of large-scale numerical simulations and finite-size scaling analysis, here we study the SPP transition on random scale-free networks (SFNs) characterized by power-law degree distributions. We find, in contrast with ordinary percolation, that the transition is identical to the one observed on ERNs, denoting independence from the degree exponent. Still, we distinguish finite- and infinite-budget SPP universality classes. Our findings follow from the fact that the SPP process drastically homogenizes the heterogeneous structure of SFNs before the SPP transition takes place.

Figures (38)

• Figure 1: Estimation of $\beta/\bar{\nu}$ and $\gamma/\bar{\nu}$. We plot $P_c(N)$ as a function of $N$ for SFNs with (a) $\lambda=2.1$, (b) $\lambda=2.7$, (c) $\lambda=3.5$, and (d) $\lambda=4.5$. The slope of the straight lines on the log-log plot corresponds to the best estimate of the ratio $-\beta/\bar{\nu}$. Similarly, we plot $S_c(N)$ as a function of $N$ for SFNs with (e) $\lambda=2.1$, (f) $\lambda=2.7$, (g) $\lambda=3.5$, and (h) $\lambda=4.5$. The slope of the straight lines on the log-log plot corresponds to the best estimate of the ratio $\gamma/\bar{\nu}$. Different symbols display results of the SPP model with $C=1$ (square), $C=2$ (circle), $C=3$ (triangle), and $C=N$ (diamond). Different line styles are used to display the best power-law fitting of the SPP model with $C=1$ (solid), $C=2$ (dashed), $C=3$ (dotted), and $C=N$ (dashed-dotted). Symbols are the average values and error bars are the standard deviations. We report the number of realizations in Table \ref{['tab:simulation']} of Ref. supplementary.
• Figure 2: Data collapse of probability density functions. We plot the probability density function of the rescaled critical observables (a) $P_c(N)N^{\beta/\bar{\nu}}$ and (b) $S_c(N)N^{-\gamma/\bar{\nu}}$ for the SPP model with $C=2$ on SFNs with $\lambda=2.7$. We also plot similar results for (c-d) $C=3$, and (e-f) $C=N$. We use the best estimates of the critical exponent ratios $\beta/\bar{\nu}$ and $\gamma/\bar{\nu}$ reported in Table \ref{['tab:summary']}. Note that we use 30 bins to plot the data.
• Figure 3: Scaling of the $k_{\text{max}}$ ratios. Plot of $\phi_{1}(N)$ as a function of $N$ for different values of $C$. Results are valid for SFNs with $\lambda=2.1$ using (a) $q=0.7$ (c) $q=0.8$, and (e) $q=0.9$. We also plot $\phi_{2}(N)$ as a function of $N$ for different values of $C$ using (b) $q=0.7$ (d) $q=0.8$, and (f) $q=0.9$. Plateau values and exponents are reported in Table \ref{['tab:k_max_slope']}. Symbols are the average values and error bars are the standard deviations. We report the number of realizations used for the scaling analysis in Table \ref{['tab:k_ratio_realization']} of Ref. supplementary.
• Figure 4: Estimation of $1/\bar{\nu}$. We plot $|r_c - r_c(N)|$ (square symbol) and $\sigma_{r^*_c}$ (circle symbol) as functions of $N$. The slope in the double-log scale corresponds to $1/\bar{\nu}_{1}$ (solid line) and $1/\bar{\nu}_{2}$ (dashed line), respectively. Results are valid for SPP with $C=2$ on SFNs with (a) $\lambda=2.1$, (b) $\lambda=2.7$, (c) $\lambda=3.5$, and (d) $\lambda=4.5$. Symbols are the average values and error bars are the standard deviations. Note that error bars for $\sigma_{r^*_c}$ do not exist so they are omitted in the figure. Estimated critical exponents are reported in Table \ref{['tab:summary']}.
• Figure 5: Data collapse of $P$ and $S$. We plot the rescaled observable $PN^{\beta/\bar{\nu}}$ as a function of $[r-r_c(N)]N^{1/\bar{\nu}}$ for different values of network size $N$. Results are valid for SFNs with $\lambda=2.7$ and $k_{\min}=4$ and SPP model with (a) $C=2$, (c) $C=3$, and (e) $C=N$. Similarly, we plot the rescaled observable $SN^{-\gamma/\bar{\nu}}$ as a function of $[r-r_c(N)]N^{1/\bar{\nu}}$ for (b) $C=2$, (d) $C=3$, and (f) $C=N$. Note that for SFNs with $\lambda=2.7$, we omit the subscript in $\bar{\nu}_{2}$ and used $1/\bar{\nu}$ since $1/\bar{\nu}_{1}\approx 1/\bar{\nu}_{2}$. The critical exponent ratios used in the figure are reported in Table \ref{['tab:summary']}. The range of abscissa is adjusted to display 200 bins of the curve for $N=2^{20}$. We report the number of bins used for each network size in Table \ref{['tab:binning']} of Ref. supplementary for clarity.