General expression for the component size distribution in infinite configuration networks

In the infinite configuration network the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. This work presents a simple equation that gives for an arbitrary degree distribution the corresponding size distribution of connected components. This equation is suitable for fast and stable numerical computations up to the machine precision. The analytical analysis reveals that the asymptote of the component size distribution is completely defined by only a few parameters of the degree distribution: the first three moments, scale, and exponent (if applicable). When the degree distribution features a heavy tail, multiple asymptotic modes are observed in the component size distribution that, in turn, may or may not feature a heavy tail.

Figure 1

Examples of component size distributions (solid lines) that feature fast (1) and slow (2) convergence to their asymptotes (dashed lines). Both asymptotes are covered by Case A, Table 2. (1) $u\left(k\right)=C{e}^{-1.05k},$ all three: the analytical expression (see Table 1), numerical values [according to Eq. (8)] and the asymptote practically coincide. (2) $u\left(k\right)$ is nonzero in three points $u\left(1\right)=0.97,u\left(2\right)=0.015,u\left(10\right)=0.015;$ the component size distribution features oscillations before it converges to the asymptote.

Figure 2

Component-size distributions (solid lines) and their asymptotes (dashed lines) as obtained for degree distributions with exponent $\beta =3.5\left(\alpha =1.5\right)$ and various values of scale parameter. Three distinct asymptotic modes are illustrated: (1) $s=0.066:\theta <0,\eta =\frac{5}{2}$; (2) $s=0.644:\theta \approx 0$, $\eta =\frac{5}{3}$; (3) $s=0.8:\theta >0$.

Figure 3

Component-size distributions (solid line) corresponding to degree distributions with exponent $\beta =2.6\left(\alpha =0.6\right)$. In this case, the component size distributions cannot feature a heavy tail; however, depending upon the scale parameter $s$ a transient asymptote with exponent $-1.6$ (dashed line) emerges: (1) $s=8.3\times {10}^{-2},$ fast convergence to the exponential asymptote. (2) $s=8.3\times {10}^{-5},$ the distribution transiently follows what seems to be a heavy tail for $n<{10}^6,$ whereas for larger $n$ the theory predicts no heavy tail.

Figure 4

Examples of component size distributions (solid lines) that are associated with heavy-tailed degree distributions with $\beta =6\left(\alpha =4\right)$. The dashed lines represent the asymptotes in accordance with Case B in Table 2. Depending on the sign of $\theta ,$ three asymptotic modes are distinguished: (1) $s=1.93:\theta =-0.8$, (1a) $s=9.42:\theta =-0.027,$ (2) $s=9.69:\theta =1.6\times {10}^{-7}$, (3) $s=10.05:\theta =0.038.$ When $\theta$ is a small negative number (curve 1a), $w\left(n\right)$ first decays as $n^{-3/2}$ but eventually switches to asymptote $n^{-\alpha -1}=n^{-5}.$

Figure 5

The effect of a cutoff imposed on a heavy-tailed degree distribution with $\beta =3.3\left(\alpha =1.3\right)$ and $s=7.73.$ The solid curves correspond to component size distributions with: (1) no cutoff, $\theta >0$; (2) cutoff at $k=1000,\theta =0$; (3) cutoff at $k=100,\theta <0.$ The asymptote for 1 is covered by Case D, Table 2; while due to the cutoffs, the asymptotes for 2 and 3 are covered by Case A.

Figure 6

The effect of a cutoff imposed on a heavy-tailed degree distribution with exponent $\beta =1\left(\alpha =-1\right)$ and $s=-2\times {10}^{-4}$. The following values of the cutoff are considered: (1) $k_{\text{cut}}=80$, corresponds to $\theta <0$; (2) $k_{\text{cut}}=100,$ corresponds to $\theta \approx 0;\left(3\right)k_{\text{cut}}=150$ corresponds to $\theta >0,$ (4) $k_{\text{cut}}={10}^3;$ (5) $k_{\text{cut}}={10}^5.$

Figure 7

The correspondence between the exponent $\beta$ in a heavy-tailed degree distribution, and the exponent $\eta$ in the associated heavy-tailed component size distribution. The critical brunch corresponds to $\theta =0,$ subcritical branch to $\theta <0.$ Positive $\theta$ is not associated with heavy-tailed component size distributions.