Elastic properties of small-world spring networks

We construct small-world spring networks based on a one-dimensional chain and study its static and quasistatic behavior with respect to external forces. Regular bonds and shortcuts are assigned linear springs of constant $k$ and $k^{\prime }$, respectively. In our models, shortcuts can only stand extensions less than ${\delta }_c$ beyond which they are removed from the network. First we consider the simple cases of a hierarchical small-world network and a complete network. In the main part of this paper we study random small-world networks (RSWN) in which each pair of nodes is connected by a shortcut with probability $p$. We obtain a scaling relation for the effective stiffness of RSWN when $k=k^{\prime }$. In this case the extension distribution of shortcuts is scale free with the exponent $-2$. There is a strong positive correlation between the extension of shortcuts and their betweenness. We find that the chemical end-to-end distance (CEED) could change either abruptly or continuously with respect to the external force. In the former case, the critical force is determined by the average number of shortcuts emanating from a node. In the latter case, the distribution of changes in CEED obeys power laws of the exponent $-\alpha$ with $\alpha ⩽\frac{3}{2}$.

Figure 1

Constructing a hierarchical small-world network.

Figure 2

Chemical end-to-end distance vs $a$ for a complete network of size $N=1000$ with $k=1$ and $k^{\prime }=0$.

Figure 3

$g\left(y\right)=2K∕\left(kpN\right)$ vs $y=p{N}^2$ for networks of size $N=100,500$ (averaged over 1000 realizations) and $N=1000$ (averaged over 300 realizations). The line represents the curve $1+2∕y$.

Figure 4

Extension distribution for RSWNs of size $N=1000$ and $k=k^{\prime }=1$ averaged over 5000 realizations. The line displays a power law of exponent $-2$.

Figure 5

$P\left(\Delta R\right)$ for single realizations with different $p$ when $k=k^{\prime }=1,N=100$. There are a few small changes in $R$ followed by a large change of order $N$ happened at critical force $F_c$.

Figure 6

$F_c$ vs $pN$ in RSWNs with $k=k^{\prime }=1$ for $N=50$ and $N=100$ averaged over 5000 and 1000 realizations, respectively. The solid line displays $F_c=kpN$. In each step of the quasistatic process we increase $F$ by 0.1. The inset shows more clearly the linear behavior of $F_c$ for large $pN$.

Figure 7

The CEED vs $F$ in RSWNs of size $N=1000,k^{\prime }=0$ and $k=1$ after 5000 (for $p=0.1$, 0.001) and 10000 (for $p=0.0001$) realizations.

Figure 8

$f\left(y\right)=R∕N$ vs $y=p{L}_F^2$ for RSWNs of different sizes (averaged over 10 000 realizations).

Figure 9

$P\left(\Delta R\right)$ for RSWNs of size $N=1000,k^{\prime }=0$ and $k=1$. The lines display $P\left(\Delta R\right)\propto {\left(\Delta R\right)}^{-\alpha }$. Number of realizations are the same as those of Fig. 6.

Figure 10

Extension distribution for RSWNs of size $N=1000,k=1$, and $k^{\prime }=0.1$ averaged over 5000 realizations.

Figure 11

Correlation coefficient of $\Delta x$ and $b$ for $N=100$ after averaging over 5000 realizations.

Figure 12

$P\left(\Delta R\right)$ for a network with $N=1000,p=0.001$, and $k=1$. The data are results of averaging over 10 000 $(k^{\prime }=0)$ and 5000 ($k^{\prime }=0.1$, 0.2) realizations. The lines represent power laws of exponent $-1$ and $-\frac{3}{2}$.

Figure 13

$F_{max}$ vs $k^{\prime }$ for $N=100$ and $k=1$ (averaged over 1000 realizations). The inset shows more clearly the linear behavior of data for large $k^{\prime }$. The lines display linear functions $F_{max}=a_0+a_1{k}^{\prime }$.

Figure 14

Schematic representation of phase diagram. Dashed region indicates where we have a continuous behavior of $R$ vs $F$. For $p<1∕N^2$ (black region) we cannot speak of continuous or discontinuous behavior of $R$. Note that as $N$ approaches to infinity the continuous region reduces to the line $k^{\prime }=0$. The inset shows the variation of $\alpha$ with $p$ when $N=1000,k=1$, and $k^{\prime }=0$. The errors in $\alpha$ are of order 0.01. The same behavior will be observed for $\alpha$ when we fix $p$ and increase $k^{\prime }$. The solid line in the inset indicates the expected value of $\alpha$ for large $p$, i.e., $\frac{3}{2}$.