Impact of constrained rewiring on network structure and node dynamics

In this paper, we study an adaptive spatial network. We consider a susceptible-infected-susceptible (SIS) epidemic on the network, with a link or contact rewiring process constrained by spatial proximity. In particular, we assume that susceptible nodes break links with infected nodes independently of distance and reconnect at random to susceptible nodes available within a given radius. By systematically manipulating this radius we investigate the impact of rewiring on the structure of the network and characteristics of the epidemic. We adopt a step-by-step approach whereby we first study the impact of rewiring on the network structure in the absence of an epidemic, then with nodes assigned a disease status but without disease dynamics, and finally running network and epidemic dynamics simultaneously. In the case of no labeling and no epidemic dynamics, we provide both analytic and semianalytic formulas for the value of clustering achieved in the network. Our results also show that the rewiring radius and the network's initial structure have a pronounced effect on the endemic equilibrium, with increasingly large rewiring radiuses yielding smaller disease prevalence.

Figure 1

The average degree distribution at the end of simulations starting from homogeneous (top) and heterogeneous (bottom) networks compared with the binomial distribution $X\sim B[N-1,\langle k\rangle /(N-1\left)\right]$ (black circles, corresponding to an Erdős-Rényi random network with $N$ nodes and connectivity $\langle k\rangle$). The left and right panels correspond to link- and node-based selection, respectively. The plots show the average of 100 simulations with $R=\sqrt{2N}/2$ (red solid line) and $R=\sqrt{6/\pi }$ (blue dashed line), with $N$ = 100 and $\langle k\rangle$ = 10.

Figure 2

Evolution of clustering during rewiring, starting from homogeneous (left) and heterogeneous (right) networks. The plots show the average of 100 simulations with $R=\sqrt{6/\pi }$, $\sqrt{10/\pi }$, $\sqrt{20/\pi }$, $\sqrt{30/\pi }$, and $R=\sqrt{2N}/2$ [green (a), blue (b), black (c), purple (d), and red (e) lines, respectively], where the solid and dotted ($☆$) lines correspond to link- and node-based selection, with $N$ = 100 and $\langle k\rangle$ = 10.

Figure 3

Evolution of the rewiring process, starting from homogeneous (left) and heterogeneous (right) networks with node-based selection. The plots show the average of 100 simulations with $R=\sqrt{6/\pi }$, $\sqrt{10/\pi }$, $\sqrt{20/\pi }$, $\sqrt{30/\pi }$, and $R=\sqrt{2N}/2$ [green (a), blue (b), black (c, o), purple (c, $\square$), and red (c, $☆$) lines, respectively], with $N$ = 100 and $\langle k\rangle$ = 10.

Figure 4

Clustering at the end of simulations starting from homogeneous networks with node-based selection. Simulation results (red $☆$) are compared with analytic formulas [Eq. (2) (black dotted) and Eq. (3) (blue dashed)], with $k=\langle k\rangle$. In the left panel we use the formulas with average $(B,k)$ values. In the right panel we use $(B,k)$'s joint distribution computed from simulations. The plots show the average of 100 simulations with $N$ = 100 and $\langle k\rangle$ = 10.

Figure 5

Distribution of distance between $i$ and $j$ if $g(i,j)=1$ and distribution of path length at the end of simulations starting from homogeneous networks with node-based selection. The plots show the average of 100 simulations for $n=7$ (top) and $n=18$ (bottom) with $N$ = 100 and $\langle k\rangle$ = 10.

Figure 6

The small-worldness index $(C/L)/(C_r/L_r)$ at the end of simulations starting from homogeneous networks with node-based selection. The plots show the average of 100 simulations with $N$ = 100 and $\langle k\rangle$ = 10.

Figure 7

The average degree distribution at the end of simulations starting from homogeneous (top, dashed line) and heterogeneous (bottom, solid line) networks with $N=100$, $\langle k\rangle =10$, and $R=\sqrt{10/\pi }$, compared with Eq. (4) $(☆)$ where $p=\pi R^2/N$. The left and right panels correspond to link- and node-based selection, respectively. The plots show the average of 100 simulations.

Figure 8

Average degree distribution of allw $\text{S}$ (blue solid line) and $\text{I}$ (red dashed line) nodes at the end of simulations, when starting from homogeneous (top) and heterogeneous (bottom) networks with node-based selection. The plots correspond to the average of 1000 simulations with $N$ = 100, $I_0$ = 20, $S_0=N-I_0$, and $\langle k\rangle$ = 10. In the left panel, $R=\sqrt{2N}/2$. In the right panel, $R=\sqrt{6/\pi }$. The blue and red ($☆$) markers correspond to Eqs. (9) and (10), respectively. The blue and red ($\circ$) markers correspond to Eqs. (7) and (8), respectively. We note that our analytic derivation needs the number of $\text{SI}$ links that have been cut by the end of the rewiring process. This is taken from the simulation.

Figure 9

Evolution of clustering during rewiring, starting from homogeneous (left) and heterogeneous (right) networks. The plots correspond to the average of 1000 simulations with $N$ = 100, $I_0$ = 20, $S_0=N-I_0$, and $\langle k\rangle$ = 10. Data for $R$ values of $\sqrt{6/\pi }$, $\sqrt{10/\pi }$, $\sqrt{20/\pi }$, $\sqrt{30/\pi }$, and $\sqrt{2N}/2$ are shown in green (a), blue (b), black (c), purple (d), and red (e), respectively.

Figure 10

Final values of clustering when starting from homogeneous (left) and heterogeneous (right) networks. The plots correspond to the average of 1000 simulations with $N$ = 100, and $\langle k\rangle$ = 10. Data are shown for $I_0=20$ (black dotted line), $I_0=50$ (blue dashed line), and $I_0=80$ (red solid line) with $S_0=N-I_0$.

Figure 11

Infection prevalence ($I/N$) starting from homogeneous (left) and heterogeneous (right) networks. The plots correspond the average of 200 simulations with $N$ = 100, $I_0=20$, $S_0=N-I_0$, $\langle k\rangle$ = 10, $\gamma =1$, $\tau =0.25$, and $w=0.2$. Data are shown for $n$ values of 5 (green, a), 10 (blue, b), 15 (black, c), 20 (purple, d), critical value $n^{*}$ = 26 for homogeneous network and $n$ = 27 (red, e, and pink, f, left panel), and critical value $n^{*}$ = 29 and $n$ = 30 (red, e, and pink, f, right panel).

Figure 12

Critical $n$ as a function of $\tau$ and $w$, starting from homogeneous (left) and heterogeneous (right) networks, and with $N$ = 100, $I_0=20$, $S_0=N-I_0$, $\langle k\rangle =10$, and $\gamma =1$.

Figure 13

Final value of clustering starting from homogeneous (left) and heterogeneous (right) networks with $N$ = 100 and $\langle k\rangle$ = 10. The dashed line shows the clustering in a network without any dynamics of the nodes and without node labeling. The dotted line denotes the clustering when the full model, coupled epidemic dynamics and rewiring, is considered, with $I_0=20$, $S_0=N-I_0$, $\gamma =1$, $\tau =0.25$, and $w=0.2$.