4. Advantages of Cooperation 65
5.4. Analytical Approximations for the Uncorrelated Case
5.4.3. The Flow Distributions of the Links
0.0000 0.0010 0.0020
0100300500
generation
p(generation)
(a)κ= 1, generation
0.000 0.002 0.004
0100300500
generation
p(generation)
(b)κ= 2, generation
−0.00100 −0.00096 −0.00092
050000100000
load
p(load)
(c) κ= 1, consumption
Figure 5.10.:The data for the matched generation and load and the analytical approximations.
The black circles indicate the unmatched load and generation, the red circles the respective matched values. The green lines indicate the analytical derived distributions. The red vertical lines show the mean and the second raw moments for the data and the green the moments as derived analytical. For the consumption, the delta function is not shown and the unmatched load is indicated by one black circle. The figures are done forN = 1000,βg =βl= 0 and an exponential network with λ= 1.
5.4. Analytical Approximations for the Uncorrelated Case
data µ2′ analytical µ2′
5⋅10−5 2⋅10−4 5⋅10−5 2⋅10−4
(a)κ= 1
data µ2′ analytical µ2′
5⋅10−5 2⋅10−4 5⋅10−5 2⋅10−4 0.001
(b)κ= 2
0.96 0.98 1.00 1.02 1.04 1.06
05102030
analytical µ2′ data µ2′
probability
(c) κ= 1
1 2 3 4
05102030
analytical µ2′ data µ2′
probability
(d)κ= 2
Figure 5.11.:The second raw moment from the data (black) compared to the analytical derived results (red) shown in (a) and (b) for a network with exponential degree distribution with λ= 1andβL=βS= 1for104 fluctuation realizations. In (c) and (d) histograms of the analytical results relative to the second moments of the data are plotted. For details see text.
when neglecting the matching, see Equation (5.53). To incorporate the effects of the matching, the mean and variance of the matched in/out flow (see Equations (5.123) to (5.126)) are used for each vertex instead of the unmatched moments. This approach is a correction of the first two moments of the in/out flow, neglecting all higher order corrections.
In Figure 5.11, the second raw moments calculated analytically are compared to the second raw moments from the data. The black points represent the unmatched analytical calculated second moments against the second moments from the data, the red points are the analytical calculated second moments using the matching correction.
As can be seen in Figure 5.11a and Figure 5.11b, the red dots lie on the line x = y, indicating that the analytical second moments are close to being equal to the second moments from the data. In both cases, the matchinging correction improves the result, especially for κ= 2 in Figure 5.11b. The histograms of the analytical second moments normalized by the second moments calculated from the data are shown in Figure 5.11c and Figure 5.11d. Figure 5.11c shows the distributions of the relative deviations for κ= 1. The deviations are smaller than four percent when the matching corrections are used.
The first and third moments can only be calculated using approximations as derived in Section 5.4.1. Therefore, the deviation for the second moments is the best result that can be achieved with our approach, as no approximations were used. The case of κ= 2 in Figure 5.11d shows the large deviations for the unmatched flow and that the matching correction achieves the goal of correcting the moments. The plots of the moments are shown for N = 100 for clarity, the histograms are done with N = 1000, in both cases 10000 fluctuation realizations are used to obtain the simulation data.
0.96 0.98 1.00 1.02 1.04
0102030405060
analytical µ1′ data µ1′
probability
(a)first raw moment
0.95 1.00 1.05 1.10
05101520
analytical µ3′ data µ3′
probability
(b)third raw moment
Figure 5.12.:The relative deviations of the first and third raw moments to the moments of the data using the Gaussian approximation (red) and the mixed approximation (green) for a network with exponential degree distribution withN = 1000,λ= 1, and κ = 1 as well as βL = βS = 1. The dashed lines show the respective approximations without the matching correction, for comparison.
The approximations for the first and third raw moments, derived in Section 5.4.1.2 and 5.4.1.3, are tested against the moments of the data in Figure 5.12. Both approximations show only small deviations from the data, comparable to the deviations of the second moments in Figure 5.11c. However, the mixed approximation exhibits smaller deviations.
This finding is supported by Figure 5.13, where histograms of the absolute flows for selected links are shown. Similar to the work in Section 5.3, for each link the generalized gamma distribution is fitted to the moments (see Section A.3.2) using the moments
5.4. Analytical Approximations for the Uncorrelated Case
0.000 0.010 0.020 0.030
020406080120
f43a →80
probability
(a)Edge with small absolute excess kurto-sis (−0.06),κ= 1,βL=βS = 0
0.00 0.01 0.02 0.03 0.04
020406080
f37a →18
probability
(b) Edge with small absolute excess kurto-sis (−0.05), κ= 2,βL=βS = 1
0.000 0.005 0.010 0.015
020406080120
f10a →44
probability
(c) Edge with high absolute excess kurtosis (−1.2),κ= 1,βL=βS = 0
0.00 0.02 0.04 0.06
01020304050
f10a →44
probability
(d) Edge with high absolute excess kurtosis (−1.18), κ= 2,βL=βS= 1
Figure 5.13.:Typical distributions of the absolute flow and the analytical approximations.
The black dots indicate the histogram of the data, the black line the fit with the generalized gamma distribution based on the moments of the data, and the vertical dashed line the 99.9% quantile calculated directly from the data. The red lines indicate the generalized gamma distribution based on the Gaussian approximation and the99.9%quantile from this distribution, the green lines the corresponding using the mixed approximation. The dashed colored lines indicate the distributions based on the respective approximations without the matching correction. The plots are done for a network with exponential degree distribution with λ= 1 andN = 100with10000 fluctuation realizations.
approximated analytically. For some edges the fitted distribution describes the data very good and collapses to almost one line together with the generalized gamma distribution that is calculated from the moments of the data, see Figure 5.13a and Figure 5.13b. For
edges with a high excess kurtosis of the directed flow distribution, the approximation based on the Gaussian approximation shows larger deviations from the data. The mixed approximation is in good agreement with the data for the distribution and the quantile, as can be seen in Figure 5.13c and Figure 5.13d. The dashed lines indicate the respective approximations without the matching correction, showing the approach of correcting the mean and variance of the in/out flow of each vertex significantly improves the results.
−0.004 0.000 0.004
0200400600800
qapprox−qdata
probability
(a) Histogram of the deviations, κ = 1, βL =βS = 0 for a network with expo-nential degree distributions with λ= 1
−0.002 0.002 0.006
050010001500
qapprox−qdata
probability
(b)Histogram of the deviations, κ = 1, βL =βS = 0 for a network with scale degree distributions with α= 2.3
Figure 5.14.:Deviations of the99.9%quantile calculated from the data and from the generalized gamma distribution with parameters fitted to the moments obtained from the data (black), the Gaussian approximation (red) and the mixed approximation (green). The plots are done for a networks with N = 1000.
By inspecting all edges, we find that the generalized gamma distribution based on the moments of the Exact-Gaussian mix approximation is in very good agreement with the generalized gamma distribution based on the moments of the data. This is also supported by Figures 5.14a and 5.14b. The black curve illustrates the deviations between the simulation data quantiles and the quantiles calculated from the generalized gamma distribution directly fitted to the data. The deviations using the Exact-Gaussian mix approximation almost collapses with the deviations when the fitting is done directly to the similation results. To show that this does not only hold for networks with exponential degree distribution, results for a network with scalefree degree distribution are also shown in Figure 5.14.
5.4. Analytical Approximations for the Uncorrelated Case
0.65 0.75 0.85 0.95
0.750.850.95
q
non failure probability
Gaussian approximation Exact−Gauss mix
(a)Quantiles versus minimal non-failure probabilities,κ= 1,βL=βS = 0
1.0 1.5 2.0
0.650.750.850.95
investment
non failure probability (1+ α)mean
Gaussian approximation Exact−Gauss mix
(b) Investment versus minimal non-failure probabilities, κ= 1,βL=βS= 0
0.70 0.80 0.90 1.00
0.750.850.95
q
non failure probability
Gaussian approximation Exact−Gauss mix
(c) Quantiles versus minimal non-failure probabilities,κ= 2,βL=βS= 1
1.5 2.0 2.5 3.0 3.5
0.700.800.901.00
investment
non failure probability (1+ α)mean
Gaussian approximation Exact−Gauss mix
(d) Investment versus minimal non-failure probabilities, κ= 2,βL=βS= 1
Figure 5.15.:Non-failure probabilities for capacity layouts based on the Gaussian and the Exact-Gaussian mix approximation and the(1−α)·mean layout using the global factor matching policy on a network with exponential degree distributions with λ= 1.