Sharpness of the convergence rates

In this subsection, we examine how close our convergence rates are to the actual ones by simulating several quantities. For simplicity, we restrict ourselves to the pure birth case.

We setα=β=λ= 1 and begin with the case where the social indexS is constant and equal to 1.

In Figure 6.1, we consider the total variation distance between the degree distribution MixPo(Λ_T) and the asymptotic degree distribution MixPo(M). In order to estimate the total variation distance, we consider Corollary 3.1.6, which yields

d_{T V}(MixPo(Λ_T),MixPo(M)) = 1 2

∞

X

k=0

|f(k)−g(k)|,

wheref and g denote the probability mass functions of the MixPo(Λ_T) and the MixPo(M) distribu-tion, respectively. We computed the corresponding relative frequencies in order to approximate the probability mass functions. Note that the very natural estimator we used here is biased such that we need a huge number of iterations in order to obtain reliable results (see Figure 6.1). Note further

12345

T

−log(dtv(MixPo(ΛT),MixPo(M)))/(λT)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Figure 6.2: Estimated−log(d_{T V}(MixPo(Λ_T),MixPo(M)))/(λT) forS ∼Exp(1), α=β =λ= 1 and µ= 0 based on 250,000 simulated realizations of a linear birth and death process

that most of the differencesf(k)−g(k) are very small for largeT, which can easily lead to numerical problems.

Since the standard deviation σ_S is 0 in this case, Theorem 6.3.2 gives us the rate T e^−λT. In Figure 6.1, we consider the left-hand side in Theorem 6.3.2. More precisely, we consider the expression

−log(d_{T V}(MixPo(Λ_T),MixPo(M)))/T, which is of the order λ+O(log(T)/T) if our rate is sharp.

Indeed Figure 6.1 confirms that −log(dT V(MixPo(ΛT),MixPo(M)))/(λT) approaches a value close to 1, i.e. our rate is close to the actual one in this case. The gap between 1 and the value simulated (using 10⁶ iterations) is due to a small simulation error or terms of lower order.

In order to examine the case where σS > 0, we consider S ∼ Exp(1). Theorem 6.3.2 gives us the rate √

T e^−λ2T. In Figure 6.2, we consider the expression −log(d_{T V}(MixPo(Λ_T),MixPo(M)))/T again, which is of the orderλ/2 +O(log(T)/T) if our rate is sharp. However, Figure 6.1 suggests that

−log(d_{T V}(MixPo(Λ_T),MixPo(M)))/(λT) approaches a value close to 1 again, i.e. our rate does not seem to be sharp in this case.

6.3.22 Remark

Note that, due to the problems mentioned above, the simulations are very time-consuming and require

012345

T

−log(E(|ΛT−M|))/(λT)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0.5

Figure 6.3: Simulated−log(E(|Λ_T −M|))/(λT) for S ∼Exp(1), α =β =λ= 1 and µ= 0 based on 250,000 simulated realizations of a linear birth and death process

a high computing capacity, which makes our theoretical results even more valuable. Note further that the simulations for exponentially distributed social indices are computationally considerably more costly than in the case where the social indices are deterministic, in particular because the expression for ΛT given by (6.5) can be simplified in the latter case such that we do not need to simulate all ages (or birth times). As a consequence, we used a relatively small number of iterations for Figure 6.2, namely 250,000.

In the following, we examine which step of our derivation of our convergence rate leads to the different order. We bounded the total variation distance from above byE(|Λ_T−M|). In Figure 6.3, we consider the quantity−log(E(|Λ_T −M|))/T, which is approximately of the orderλif the rate that is given byE(|Λ_T−M|) is sharp. However, Figure 6.3 suggests that−log(E(|Λ_T−M|))/(λT) approaches a value close to 1/2, i.e. the corresponding rate is close to our rate but not to the actual one.

Theorem 3.4.1 implies that we also could have usedE(|√

Λ_T−√

M|) instead ofE(|Λ_T−M|) as upper bound for the total variation distancedT V(MixPo(ΛT),MixPo(M)). However, Figure 6.4 suggests that we would have obtained the same rate. Note that it is also plausible from a theoretical point of view that E(|√

Λ_T −√

M|) does not give us a better rate since we have that E(|Λ_T −M|) is equal to

012345

T

−log(E(|√ ΛT−√ M|))/(λT)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0.5

Figure 6.4: Estimated−log(E(|√

Λ_T −√

M|))/(λT) forS ∼Exp(1), α=β =λ= 1 andµ= 0 based on 250,000 simulated realizations of a linear birth and death process

E(|√

ΛT −√

M| ·(√

ΛT +√

M)). Note further that our proof would have become more difficult if we usedE(|√

Λ_T −√

M|) instead of E(|Λ_T −M|) as upper bound.

In order to examine further how our rate could be improved theoretically, we also consider the minimal coupling of ΛT and M, i.e. the expression

inf

Λ: ˜˜Λ=Λ^D M: ˜˜ M^D=M

E(|Λ˜_T −M|).˜ (6.79)

Obviously, it is smaller than or equal to E(|Λ_T −M|), and, by Remark 3.4.2, it is an upper bound ford_{T V}(MixPo(Λ_T),MixPo(M)). In order to be able to simulate the quantity (6.79) more effectively, we use the following theoretical background. By Example 3.1.3, (6.79) is the Wasserstein distance betweenL(Λ_T) and L(M). We also know from Example 3.1.3 that this Wasserstein distance can be expressed as

∞

Z

−∞

|P(ΛT ≤x)−P(M≤x)|dx.

0.20.40.60.8

T

−log(infE(|

˜ Λ^T

−

˜ M|))/(λT)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

Figure 6.5: Simulated−log(infE(|Λ˜_T −M|))/(λT˜ ) for S ∼Exp(1), α =β =λ= 1 and µ= 0 based on 500,000 simulated realizations of a linear birth and death process, where we take the infimum over all random variables ˜Λ and ˜M with ˜Λ= Λ and ˜^D M= M^D

In order to simulate this expression, we use the corresponding empirical distribution functions. Note that we need a large number of iterations in order to approximate the integral in a reasonable way.

Figure 6.5 suggests that the rate corresponding to (6.79) is smaller than one, i.e. that this expression decreases slower than the total variation distance. This indicates that the application of our mixed Poisson approximation result does not lead to a sharp upper bound here. However, since the class of mixed Poisson distributions is very large, it is difficult to find a better universal result, and even in our special case, we do not expect to find a more precise approximation since the mixing distribution of the MixPo(Λ_T) is very complex.

Note that Figure 6.5, which is based on 500,000 simulations of a linear birth and death process, is not very satisfying since it does not allow to draw a clear conclusion about the rate. Unfortunately, we cannot obtain a very convincing results here due to computational limitations.

We have reasons to doubt that our coupling is the optimal one since the situation in Figure 6.3 and Figure 6.5 looks different. However, our coupling is a very natural one, which allows us to derive an explicit upper bound.

6.3.23 Remark

In the pure birth case, we can simulate the linear birth and death process efficiently by using the distribution of the ages from Section 4.2. If nodes can die, we also need to consider the death times, which makes the simulations very time-consuming. Therefore, we do not examine this case numerically here. However, we note that a positive death rate µ leads to a higher variability in the degree distributions for finite T since in particular the death of a highly connected node (hub) can have a large impact. Thus we would not expect the same rate as in the pure birth case. However, the actual factor in the exponential rate may well be larger than the one stated in Theorem 6.3.4.

Im Dokument Convergence Rates in Dynamic Network Models (Seite 122-127)