Growth dominates choice in network percolation

Vikram S. Vijayaraghavan, Pierre-André Noël, Alex Waagen, and Raissa M. D'Souza

Phys. Rev. E 88, 032141 – Published 30 September 2013

Abstract

The onset of large-scale connectivity in a network (i.e., percolation) often has a major impact on the function of the system. Traditionally, graph percolation is analyzed by adding edges to a fixed set of initially isolated nodes. Several years ago, it was shown that adding nodes as well as edges to the graph can yield an infinite order transition, which is much smoother than the traditional second-order transition. More recently, it was shown that adding edges via a competitive process to a fixed set of initially isolated nodes can lead to a delayed, extremely abrupt percolation transition with a significant jump in large but finite systems. Here we analyze a process that combines both node arrival and edge competition. If started from a small collection of seed nodes, we show that the impact of node arrival dominates: although we can significantly delay percolation, the transition is of infinite order. Thus, node arrival can mitigate the trade-off between delay and abruptness that is characteristic of explosive percolation transitions. This realization may inspire new design rules where network growth can temper the effects of delay, creating opportunities for network intervention and control.

Received 12 July 2013

DOI:https://doi.org/10.1103/PhysRevE.88.032141

Authors & Affiliations

Vikram S. Vijayaraghavan^1,2, Pierre-André Noël^2,3, Alex Waagen^2,4, and Raissa M. D'Souza^2,3,5,6

¹Department of Physics, University of California, Davis, California 95616, USA
²Complexity Sciences Center, University of California, Davis, California 95616, USA
³Department of Computer Science, University of California, Davis, California 95616, USA
⁴Department of Mathematics, University of California, Davis, California 95616, USA
⁵Department of Mechanical and Aerospace Engineering, University of California, Davis, California 95616, USA
⁶The Santa Fe Institute, Santa Fe, New Mexico 87501, USA

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Vol. 88, Iss. 3 — September 2013

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Figure 1
Example of the AE-CHKNS process. (a) At time $t = 11$ when a node (indicated by an arrow) is added to the graph but no edge is added. (b) The next time step, i.e., $t = 12$ , when both a node and an edge (indicated by arrows) are added. To add the edge, three nodes are chosen uniformly at random. The first node (indicated with the dashed circle) must form an edge with one of two other nodes (indicated by the dashed squares), always choosing to connect to the node in the smaller component. Thus, for this example, the edge indicated by the arrow is added.Reuse & Permissions
Figure 2
Fractional size of the largest component as a function of $δ$ for the CHKNS model, AE percolation, and AE-CHKNS. Solid line: Analytic behavior predicted for AE-CHKNS via Eq. (6) evaluated by numerically solving Eq. (5) up to $k_{m} = 2 \times 10^{4}$ . Circles are obtained from Monte Carlo simulations of the network with $N = 10^{5}$ nodes with each point averaged over 100 realizations. Error bars are smaller than the size of the marker. The results for AE percolation and CHKNS model are from Refs. [7] and [25], respectively.Reuse & Permissions
Figure 3
Plotted is $a_{k}$ versus $k$ , the fraction of nodes belonging to components of size $k$ , for three different values of $δ$ for the AE-CHKNS process. Solid lines: theoretical results obtained from solving Eqs. (4) and (5). Markers: simulation averaged over $10^{5}$ realizations. The error bars are smaller than the size of the marker.Reuse & Permissions
Figure 4
Fitting $a_{k} = A k^{- C} \exp (- B k)$ for different values of $δ$ and $k_{m}$ [the maximum value of $k$ solved for in Eq. (5)]. Plotted are $B$ values obtained from best fits and the associated error $E$ , with $k_{m}$ values indicated in parentheses. Both $B$ and $E$ show a sharp drop at $δ = 0.3223$ , indicating the distribution is an exact power-law in the vicinity of that point. While the point at which $B$ goes to zero with increasing $δ$ does not change with $k_{m}$ , the point at which $B$ again becomes nonzero shifts to the left as we increase the value of $k_{m}$ , indicating that in the truly infinite limit the exponential cutoff parameter, $B$ , will be zero only at the critical point.Reuse & Permissions
Figure 5
Susceptibility $χ$ (average size of a component to which a randomly chosen node belongs) as a function of edge density, $δ$ . Markers: numerical simulations averaged over 100 realizations for system sizes $N = 10^{4}$ (circles) and $N = 10^{5}$ (squares). Solid lines: analytical results obtained by truncating the sums of Eq. (7) at $k_{m} = 2 \times 10^{4}$ .Reuse & Permissions
Figure 6
The fractional size $S$ of the largest component close to the critical point. Eq. (6) was evaluated numerically to obtain $S$ for different values of $k_{m}$ in order to show the effect of truncation. (a) The absence of a linear segment in the plot of $\ln S$ vs. $\ln (δ - δ_{c})$ indicates that this phase transition is not second order. (b) The presence of a linear segment, best fit by the dashed line in the plot of $\ln (- \ln S)$ vs. $\ln (δ - δ_{c})$ is indicative of an infinite-order phase transition.Reuse & Permissions

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