Download 4.2 Path Lengths

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4.2
Path Lengths
In this section we are concerned with estimating the average path length between two randomly chosen sites `(n, p) as a function of the number of nodes n, the fraction of shortcuts
p, and the range of interaction k. For this problem and the others we will consider below, we
will consider Newman and Watts (1999) version of the model in which no edges are removed
but each vertex has a shortcut to a randomly chosen neighbor with probability p.
To quote Albert and Barbási (2002), “it is now widely accepted that the characteristic
path length obeys the general scaling form
`(n, p) ∼
n
f (pkn)
k
where f (u) is a universal scaling function that obeys
(
1/4
if u << 1
f (u) =
ln(u)/u if u >> 1”
Newman, Moore, and Watts (2000) have taken a “mean-field approach” to computing
this quantity. They write differential equations for the number of sites within distance r of
a fixed point and the number of clusters of occcupied sites, by assuming for example that
gaps have the sizes of a randomly broken stick. They conclude that
1
u
−1
√
f (u) = √
tanh
(4.2.1)
2 u2 + 2u
u2 + 4u
Simulations show that this formula agrees with simulations for small u or large u, but “as
expected, there is some disagreement when u ≈. See Figure 3 in Newman (2000). Using the
identity
1+y
1
−1
tanh y = log
2
1−y
we have
tanh−1
u
√
u2 + 4u
1
= log
2
!
√
1 + u/ u2 + 2u
√
1 − u/ u2 + 2u
Inside the logarithm the numerator → 2 as u → ∞. The denominaotr
1
1
=1− p
≈ 1/u
≈1−
1 + 1/u
1 + 2/u
combining our calculations
log(2u)
4u
which matches (21) in Newman, Moore, and Watts (2000).
f (u) ∼
4.2. PATH LENGTHS
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Barbour and Reinert (2001) have done a rigorous analysis of the average distance between
points in a continuum model in which there is a circle of circumfrence L and a Poisson mean
Lρ/2 of random chords. The chords are the short cuts and have length 0. The first step
in their analysis is to consider a model that ignores intersections of growing arcs and that
overlapping arcs see independent Poisson processes of shortcut endpoints. Let L(t) be the
Lebesgue measure of the set of points within distance t of a chosen point and let M(t) be
the number of intervals. Under our assumptions
L0 (t) = 2M(t)
while M(t) is a branching process in which there are no deaths and births occur at rate 2ρ.
Branching processes tells us that EM(t) = e2ρt and e−2ρt M(t) → W almost surely where
W has an exponential distribution with mean 1. Integrating gives
EL(t) =
Z
t
0
1
2e2ρs ds = (e2ρt − 1)
ρ
At time t = (2ρ)−1 (1/2) log(Lρ), L(t) = (Lρ)1/2 − 1. Ignoring the −1 if we run the process
starting from a second randomly chosen point and call the result L̄(t) then L(t)L̄(t) =
L/ρ and by analogy with the birthday problem we can expect a positive probability of an
intersection.
The strategy is similar to the one we adopted to compute the average distance in the
Erdos-Renyi model. There we grew the two clusters until they had size n1/2+ . The details
are more complex in Barbour and Reinert’s computation since the intervals have different
sizes. In Section 3 of their paper they use what they call “the local version of the Stein-Chen
method” to estimate intersection probabilities. The result is
Theorem 4.2.1. Suppose Lρ → ∞. Let O be a fixed point of the circle, choose P at random,
and let D be the distance from O to P . Then
Z ∞
e−y
1 1
log(Lρ) + x
→
dy
P D>
ρ 2
1 + 2e2x
0
After the fact the form of the answer is reasonable. The spacing between endpoints is 1/ρ
and one will need of order log(Lρ) shortcuts to be able to navigate between two chosen
endpoints.
According to Barbour and Reinert (2001), the Newman, Moore, and Watts heuristic
leads to
1
1 1
log(Lρ) + x
→
P D>
ρ 2
1 + e2x
“the logistic distribution with mean 0, whereas the true asymptotic distribution given in 4.2.1
has mean ≈ −.0058 and has a much wider spread.” See Figure 1 in Barbour and Reinert
(2001) for the picture.
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Albert, R., and Barbási, A.L. (2002) Statistical mechanics of complex networks. Reviews of
Modern Physics. 74, 47–97
Barbour, A.D., and Reinert, G. (2001) Small worlds. Random. Struct. Alg. 19, 54–74
Newman, M.E.J. (2000) Models of the small world. J. Stat. Phys. 101, 819–841
Newman, M.E.J., Moore, C., and Watts, D.J. (2000) Mean-field solution of the small-world
network model. Phys. Rev. Letters. 84, 3201–3204
Newman, M.E.J., and Watts, D.J. (1999) Renormalization group analysis of the small-world
nnetwork model. —it Physics Letters A. 263, 341–346