Scaling Expressions

To better understand the complexity in the scaling behavior of components with correlated and anti-correlated segments at different scales, we employ the superposition rule (see [61] and Appendix 7.1). For each component we have

$$F(n)/n=\sqrt{[F_{\rm corr}(n)/n]^2+[F_{\rm rand}(n)/n]^2},$$ (7)

where $F_{\rm corr}(n)/n$ accounts for the contribution of the correlated or anti-correlated non-zero segments, and $F_{\rm rand}(n)/n$ accounts for the randomness due to ``jumps'' at the borders between non-zero and zero segments in the component.

Components with correlated segments ($\alpha >0.5$)

At small scales $n<W$, our findings presented in Fig. 6(b) suggest that there is no substantial contribution from $F_{\rm rand}(n)/n$. Thus from Eq. (7),

$$F(n)/n\approx F_{\rm corr}(n)/n\sim b_0\sqrt{p}n^{\alpha},$$ (8)

where $b_0n^{\alpha}$ is the r.m.s. fluctuation function for stationary ($p=1$) correlated signals [Eq. (6) and [61]].
Similarly, at large scales $n\gg W$, we find that the contribution of $F_{\rm rand}(n)/n$ is negligible [see Fig. 7(a)], thus from Eq. (7) we have

$$F(n)/n\approx F_{\rm corr}(n)/n \sim b_0pn^{\alpha}.$$ (9)

However, in the intermediate scale regime, the contribution of $F_{\rm rand}(n)/n$ to $F(n)/n$ is substantial. To confirm this we use the superposition rule [Eq. (7)] and our estimates for $F_{\rm corr}(n)/n$ at small [Eq. (8)] and large [Eq. (9)] scales[65]. The result we obtain from

$$F_{\rm rand}(n)/n=\sqrt{[F(n)/n]^2-[b_0\sqrt{p}n^{\alpha}]^2-[b_0pn^{\alpha}]^2}$$ (10)

overlaps with $F(n)/n$ in the intermediate scale regime, exhibiting a slope of $\approx 0.5$: $F_{\rm rand}(n)/n\sim n^{0.5}$ [Fig. 9(a)]. Thus, $F_{\rm rand}(n)/n$ is indeed a contribution due to the random jumps between the non-zero correlated segments and the zero segments in the component [see Fig. 5(c)].

Figure 9: (a) Scaling behavior of components containing correlated segments ($\alpha >0.5$). $F(n)/n$ exhibits two crossovers and three scaling regimes at small, intermediate and large scales. From the superposition rule [Eq. (7)] we find that the small and large scale regimes are controlled by the correlations ($\alpha =0.9$) in the segments [ $F_{\rm corr}(n)/n$ from Eqs. (8) and (9)] while the intermediate regime [ $F_{\rm rand}(n)/n\sim n^{0.5}$ from Eq. (10)] is dominated by the random jumps at the borders between non-zero and zero segments. (b) The ratio $F_{\rm rand}(W_1=400)/F_{\rm rand}(W_2=20)$ in the intermediate scale regime for fixed $p$ and different values of $\alpha$, and the ratio $F_{\rm rand}(\alpha)/F_{\rm rand}(\alpha=0.5)$ for fixed $p$ and $W=W_1/W_2$. $F_{\rm rand}(n)/n$ is obtained from Eq. (10) and the ratios are estimated for all scales $n$ in the intermediate regime. The two curves overlap for a broad range of values for the exponent $\alpha$, suggesting that $F_{\rm rand}(n)/n$ does not depend on $h(\alpha )$ [see Eqs. (11) and (16)].

Further, our results in Fig. 8(b) suggest that in the intermediate scale regime $F(n)/n\sim W^{g_c(\alpha)}$ for fixed fraction $p$ [see Sec. 5.2.2], where the power-law exponent $g_c(\alpha)$ may be a function of the scaling exponent $\alpha$ characterizing the correlations in the non-zero segments. Since at intermediate scales $F_{\rm rand}(n)/n$ dominates the scaling [Eq. (10) and Fig. 9(a)], from Eq. (7) we find $F_{\rm rand}(n)/n \approx F(n)/n \sim W^{g_c(\alpha)}$. We also find that at intermediate scales, $F(n)/n\sim \sqrt{p(1-p)}$ for fixed segment size $W$ (see Appendix 7.2, Fig. 10). Thus from Eq. (7) we find $F_{\rm rand}(n)/n \approx F(n)/n \sim \sqrt{p(1-p)}$. Hence we obtain the following general expression

$$F_{\rm rand}(n)/n\sim h(\alpha)\sqrt{p(1-p)}W^{g_c(\alpha)}n^{0.5}.$$ (11)

Here we assume that $F_{\rm rand}(n)/n$ also depends directly on the type of correlations in the segments through some function $h(\alpha )$.
To determine the form of $g_c(\alpha)$ in Eq. (11), we perform the following steps:
(a) We fix the values of $p$ and $\alpha$, and from Eq. (10) we calculate the value of $F_{\rm rand}(n)/n$ for two different values of the segment size $W$, e.g., we choose $W_1=400$ and $W_2=20$.
(b) From the expression in Eq. (11), at the same scale $n$ in the intermediate scale regime we determine the ratio:

$$F_{\rm rand}(W_1)/F_{\rm rand}(W_2)=(W_1/W_2)^{g_c(\alpha)}.$$ (12)

(c) We plot $F_{\rm rand}(W_1)/F_{\rm rand}(W_2)$ vs. $\alpha$ on a linear-log scale [Fig. 9(b)]. From the graph and Eq. (12) we obtain the dependence

$$g_c(\alpha)=\frac{\log[F_{\rm rand}(W_1)/F_{\rm rand}(W_2)]}... ... \leq 1$}\\ 0.50, \mbox{ for $\alpha>1$}, \end{array} \right.$$ (13)

where $C=0.87\pm0.06$. Note that $g_c(0.5)=0$.
To determine if $F_{\rm rand}(n)/n$ depends on $h(\alpha )$ in Eq. (11), we perform the following steps:
(a) We fix the values of $p$ and $W$ and calculate the value of $F_{\rm rand}(n)/n$ for two different values of the scaling exponent $\alpha$, e.g., $0.5$ and any other value of $\alpha$ from Eq. (10).
(b) From the expression in Eq. (11), at the same scale $n$ in the intermediate scale regime we determine the ratio:

$$\frac{F_{\rm rand}(\alpha)}{F_{\rm rand}(0.5)}=\frac{h(\alph... ...c(\alpha)-g_c(0.5)}= \frac{h(\alpha)}{h(0.5)} W^{g_c(\alpha)},$$ (14)

since $g_c(0.5)=0$ from Eq. (13).
(c) We plot $F_{\rm rand}(\alpha)/F_{\rm rand}(0.5)$ vs. $\alpha$ on a linear-log scale [Fig. 9(b)] and find that when $W\equiv W_1/W_2$ [in Eqs. (12) and (14)] this curve overlaps with $F_{\rm rand}(W_1)/F_{\rm rand}(W_2)$ vs. $\alpha$ [Fig. 9(b)] for all values of the scaling exponent $0.5\leq \alpha\leq 1.5$. From this overlap and from Eqs. (12) and (14), we obtain

$$W^{g_c(\alpha)}=\frac{h(\alpha)}{h(0.5)}W^{g_c(\alpha)}$$ (15)

for every value of $\alpha$, suggesting that $h(\alpha)=const$ and thus $F_{\rm rand}(n)/n$ can finally be expressed as:

$$F_{\rm rand}(n)/n\sim \sqrt{p(1-p)}W^{g_c(\alpha)}n^{0.5}.$$ (16)

Components with anti-correlated segments ($\alpha <0.5$)

Our results in Fig. 6(a) suggest that at small scales $n<W$ there is no substantial contribution of $F_{\rm rand}(n)/n$ and that:

$$F(n)/n\approx F_{\rm corr}(n)/n\sim b_0\sqrt{p}n^{\alpha},$$ (17)

a behavior similar to the one we find for components with correlated segments [Eq. (8)].
In the intermediate and large scale regimes ($n\geq W$), from the plots in Fig. 7(b) and Fig. 8(a) we find the scaling behavior of $F(n)/n$ is controlled by $F_{\rm rand}(n)/n$ and thus

$$F(n)/n\approx F_{\rm rand}(n)/n\sim \sqrt{p(1-p)}W^{g_a(\alpha)}n^{0.5},$$ (18)

where $g_a(\alpha)=C\alpha-C/2$ for $0<\alpha<0.5$ [see Fig. 9(b)] and the relation for $F_{\rm rand}(n)/n$ is obtained using the same procedure we followed for Eq. (16).