Concluding Analysis
4.4 Reshuffling Models
A more theoretical direction to be pursued is exploring general mathematical properties of processes with reshuffling.
One possibility is to consider families of extensions
(X1) ,→ (X1, X2) ,→ (X1, X2, X3) ,→ · · · (4.4.1)
Intuitively, if we assume newly added parameters are never determined by the previous
ones then such a sequence increases the rigidity of the reshuffling step, becoming more and more similar to a fixed-graph process. However, in order to make sense of such a statement, a notion of closeness for different reshuffling models would be necessary. Instead of comparing to the fixed-graph scenario, we could then measure the changes caused by adding extra parameters to a model.
Question 4.4.1. Is there a natural notion of distance between a reshuffling model and its extension?
Another interesting theoretical direction to be explored is testing the framework on classes of parameters other than uniformly bounded and locally computable. Parameters that con-tain global information violate Proposition (1.3.1). In that case, if quadratic variations and covariations do not go to zero with n, the limit process, assuming it exists, might lead to more general stochastic differential equations.
Question 4.4.2. Are there choices of global graph parameters for which the n → ∞ limit of the corresponding reshuffling model is an analytically tractable SDE?
In summary, the reshuffling model framework is largely unexplored and, in our per-spective, it provides a flexible mathematical language to address conceptual questions in evolutionary dynamics.
Figure 4.4.1: Trajectories on ΩS,X for the fixed-graph process and corresponding coordinate S as a function of t. In all cases n = 12800 and trajectories start from lower region. Values of dd/re = 1, 2, 3.
Figure 4.4.2: Trajectories on ΩS,X for the fixed-graph process and corresponding coordinate S as a function of t. In all cases n = 12800 and trajectories start from lower region. Values of dd/re = 4, 5, 6.
Figure 4.4.3: Trajectories on ΩS,X for the fixed-graph process and corresponding coordinate S as a function of t. In all cases n = 12800 and trajectories start from upper region. Values of dd/re = 1, 2, 3.
Figure 4.4.4: Trajectories on ΩS,X for the fixed-graph process and corresponding coordinate S as a function of t. In all cases n = 12800 and trajectories start from upper region. Values of dd/re = 4, 5, 6.
Figure 4.4.5: Trajectories on ΩS,X for the fixed-graph process and corresponding coordinate S as a function of t. In all cases n = 12800 and trajectories start from x = υ(s). Values of dd/re = 1, 2, 3.
Figure 4.4.6: Trajectories on ΩS,X for the fixed-graph process and corresponding coordinate S as a function of t. In all cases n = 12800 and trajectories start from x = υ(s). Values of dd/re = 4, 5, 6.
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Appendices
Here we gather general results and bounds for expressions involving the binomial distribution as used in the theorems in the main text.
Lemma 4.4.1. Let p(k) = dkpk(1 − p)d−k. Let us assume that pd = Ω(d), (1 − p)d = Ω(d) and k = dp + a where a3/d2 = o(1). Then,
p(k) = 1
p2πp(1 − p)dexp
− a2 2p(1 − p)d
(1 + g(d)) (4.4.2)
where g(d) has the maximum asymptotic growth among 1/d, a/d and a3/d2. More explicitly, if 1 a d1/2, then
g(d) = − (1 − 2p)a
2p(1 − p)d+ O(a3/d2) (4.4.3) Otherwise, if d1/2 a d2/3, then
g(d) = (1 − 2p)a3
6p2(1 − p)2d2 + O(a/d) (4.4.4) If a = Θ(d1/2), then
g(d) = (1 − 2p)a 2p(1 − p)d
a2
3p(1 − p)d− 1
+ O(1/d) (4.4.5)
Proof. We start from Stirling’s approximation:
d! =√
2πd d e
d
(1 + O(1/d)) (4.4.6)
As k = dp(1 + o(1)) we know 1/k = O(1/d), so k! =√
2πk k e
k
(1 + O(1/d)) (4.4.7)
Similarly, d − k = d(1 − p)(1 + o(1)), so 1/(d − k) = O(1/d) and
Combining the expressions above we have p(k) =d The first term is seen to be
s d
Now we use the Taylor expansion of the logarithmic function, log(1+) = −2/2+O(3), to obtain: Plugging the two terms back together into (4.4.9) we obtain the result.
We will proceed to find bounds for p(l) = dlpl(1 − p)d−l and F (l) = Pl
j=0p(j) in the range of indices of the form
l = dp + ηp
p(1 − p)d log d + O(1) (4.4.13)
particularly using
Particularly, in the range given by (4.4.14) we have p(l) =
so that Now we look at the upper tail
1 − F (l) = p(l + 1) + p(l + 2) + · · · + p(d) = p(l) (R(l) + R(l)R(l + 1) + · · · + R(l) . . . R(d − 1))
is monotonically decreasing in l, so that in particular
1 − F (l) ≤ p(l) R(l) + R(l)2+ · · · = p(l) R(l)
To obtain a lower bound we can use a cutoff l0 = dp + η0pp(1 − p)d log d + O(1) where Using l0 we get the lower bound
1 − F (l) ≥ p(l)
R(l0) + R(l0)2+ · · · R(l0)l0−l
= p(l)R(l0)(1 − R(l0)l0−l)
1 − R(l0) (4.4.32) The same results obtained for R(l) in (4.4.25) work for R(l0) so that
R(l0) = 1 − η0
on the other hand for c2 > 2:
R(l0)l0−l= 1
√c1(log d)c2/2 1 + O((log log d)2/ log d) = O(1/(log d)c2/2) (4.4.36)
Correspondingly, we have for c2 ≤ 2:
F (l) ≤ 1 − p(l) Comparing to (4.4.28) we see that in either case
F (l) = 1 − p(l)
Two particularly useful choices of cutoff indices are
l0 = dp + s
2 −log 9π(log d)3 log d
pp(1 − p)d log d + O(1) (4.4.42)
corresponding to c1 = 9π and c2 = 3 and
l1 = dp +p
2p(1 − p)d log d + O(1) (4.4.43)
corresponding to c1 = 1 and c2 = 0. In that case, we can conclude from the lemma above
p(l0) = O((log d/d)3/2), F (l0) = 1 + O(log d/d)
Ocasionally, we might need to use a cutoff given by
l2 = dp + 2p
p(1 − p)d log d + O(1) (4.4.46)
that is, when η = 2, in which case the lemma gives us
p(l2) = O(1/d5/2) (4.4.47)
As an application, we have the following useful result:
Lemma 4.4.3. Let p(l) and F (l) be the PMF and CDF for the binomial distribution as above.
Then,
are, respectively, the lower and upper Riemann sum for the integral Z 1
0
xd−1dx = 1
d (4.4.51)
with respect to P . So, in particular,
d−1
Thus, to obtain the result we only need to bound the difference
d
Thus the difference between upper and lower sum is O((log d/d)3/2), concluding the result.
We can eliminate the indices below l0 since
l0
For the next few lemmas we will compare distributions with two distinct parameters pC and pD related by
pC− pD = ζ
d (4.4.58)
where we will assume ζ3 d2. For generality, let us fix a parameter s between pC and pD,
We will denote the PMF and CDF of the binomial distribution for l wins out of d trials and probability parameters pC and pD by pC(l), FC(l), pD(l) and FD(l), respectively.
Under the extra assumption that |ζ| √
d |a| and |aζ| d we have:
Proof. We start from
logpD(l)
We then apply the expansion log(1 + ) = − 2/2 + O(3) to get
If we assume |a| > |(1 − 2t)ζ|/2 the expression above has the same sign as −ζa, from which the result follows.
Under the assumptions |ζ| √
d |a| and |aζ| d we have aζ/d a2ζ2/d2 ζ2/d aζ3/d2 aζ2/d. Thus, we can use the expansion of the exponential function to obtain:
pC(l) − pD(l) = pC(l)
Now we combine some of the previous results to obtain bounds on the difference between the CDFs.
Proof. Using the cutoff
l2 =j
dpC + 2p
pC(1 − pC)d log dm
(4.4.68)
clearly we have l ≤ l2 so we can write:
clearly we have l ≤ l2 so we can write: