Impact of boundaries on fully connected random geometric networks

Many complex networks exhibit a percolation transition involving a macroscopic connected component, with universal features largely independent of the microscopic model and the macroscopic domain geometry. In contrast, we show that the transition to full connectivity is strongly influenced by details of the boundary, but observe an alternative form of universality. Our approach correctly distinguishes connectivity properties of networks in domains with equal bulk contributions. It also facilitates system design to promote or avoid full connectivity for diverse geometries in arbitrary dimension.

Figure 1

Isolated nodes shown as black balls concentrate at the boundaries of the domain and particularly near corners at higher densities. Nodes are placed randomly in a cube, with lighter colors indicating a higher probability of being in the largest connected component. We use $\eta =2$, while the side length of the cube is $L=10r_0$. There are 500 nodes in (a) and 700 nodes in (b).

Figure 2

(a) Comparison of the full analytic prediction of Eq. (7) (solid curve) with direct numerical simulation of the random network in a cube of side length $7r_0$ (jagged curve). The dashed line corresponds to the bulk contribution (previous theory). (b) Contributions from the bulk (dotted blue line, left), faces (red line), edges (yellow line), and corners (green line, right), together with the total (solid blue line) and numerical simulation (black jagged curve), showing the dominance of the corners at the highest densities and good agreement between the theory and simulation at moderate to high densities. Here it is convenient to plot the outage probability $P_{\text{out}}=1-P_{fc}$.

Figure 3

Corner contributions in triangles with equal area and perimeter: Comparison of theory with direct simulation, as in Fig. 2. The red triangle has side lengths of 26.88, 15.44, and 15.44 in units of the connectivity length scale $r_0$, while the blue triangle has side lengths of 8.40, 24.68, and 24.68. The black dashed lines correspond to the equal bulk (left curve) and bulk $+$ edge (right curve) contributions while neglecting corner contributions. The colored curves give the total (including crucial corner) contributions for each triangle. Both theory and simulation are plotted, showing excellent agreement with the numerical simulations (jagged curves) which cover them completely for $\rho >4$.