# Kleinberg Navigation on Anisotropic Lattices

Authors: J. M. Campuzano; J. P. Bagrow; D. ben-Avraham
Journal: Research Letters in Physics
 (2008)

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## Abstract

We study the Kleinberg problem of navigation in small-world networks when the underlying
lattice is stretched along a preferred direction. Extensive simulations confirm that maximally efficient navigation is attained when the lengthr of long-range links is taken from the distribution P(r)∼r−α, when the exponent α is equal to 2, the dimension of the underlying lattice, regardless of the amount of anisotropy, but only in the limit of infinite lattice size, L→∞. For finite size
lattices we find an optimal α(L) that depends strongly on L. The convergence to α=2 as L→∞ shows interesting power-law dependence on the anisotropy strength.


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## Body

## 1. Introduction

The small-world phenomenon is one of the most
intriguing properties of human society. This describes the fact that unrelated
people in a society, who are a very large geographic distance apart from one
another, tend to be connected by surprisingly short chains of acquaintances.
This phenomenon was hypothesized in 1929 by Hungarian author  Karinthy
[1, 2] and was first observed
experimentally in the 1960s with sociologist Stanley Milgram's seminal
experiments [3],
wherein randomly chosen people were selected to mail a letter to an unknown
target person, but were only allowed to send the letter to a friend, who would pass
the letter along to another friend, and so forth, until the target was
reached. Successful transmissions took surprisingly few intermediate people,
lending credibility to the turn of phrase “six degrees of
separation,” popularized by Karinthy. Understanding this phenomenon is an
important sociological problem.To study the underlying mechanism that led to
Milgram's results, computer scientist Kleinberg modeled a society as
follows [4, 5]. Begin with a large, regular
square L×L lattice. Each node is connected to its nearest
lattice neighbors and to a single random node a large distance away. The
probability of nodes i and j being connected by such a long-range contact
is(1)Pij(α)=rij−α∑k≠irik−α,where rij is the Euclidian distance between the two
nodes and the sum runs over all nodes in  the network except i.
Physically, the local lattice connections represent associations with immediate
neighbors, fellow townspeople, and so forth, while long-range contacts might
model friends or relatives in another city or country.We seek to pass the message from a random starting
nodes to a random target t.
Of great importance is the fact that each node has no information beyond the
locations of its contacts and the target node t,
so the operational algorithm must be local in character. Kleinberg has proved
that no local algorithm can do better, functionally, than the greedy algorithm [4]: each message holder passes
the message along to whichever of its contacts is closest to t,
until the message reaches the target. Moreover, for α≠2, the delivery time T (number of intermediate steps) scales as a power of L,
while small-world behavior and the weakest dependence on lattice size emerges
for α=2,
where T∼ln2L.The distribution of nodes on a regular square lattice
is too rigid, failing to mimic important features of actual distributions of
populations (or computer routers, etc.). In an effort to account for these, we
have studied Kleinberg navigation in fractals [6], showing that the optimal
long-contact exponent is then α=df,
the fractal dimension of the lattice. In this letter, we study the effects of
anisotropy–-another commonly encountered distortion of the ideal Kleinberg
lattice. Our results indicate that in the limit of lattice size L→∞,
the optimal contact exponent for two-dimensional lattices is still α=2.
For finite L,
we find an optimal exponent α(L) quite different from the infinite limit. The
convergence to α=2 as L→∞ shows interesting power-law dependence on the
anisotropy strength.


## 2. Anisotropic Lattices

We wish to study the isolated effect of anisotropy on
Kleinberg navigation. To do this, we begin with a regular square lattice (d=2) and introduce one of two forms of
anisotropy.(i)Lattice anisotropy. The underlying lattice
is stretched horizontally, along the x -axis, by a factor b>0,
such that the area of each cell goes from 1×1 to b×1.(ii)Angular anisotropy. Long-range contacts
are chosen preferably along the vertical direction by a factor b>0.
More precisely, the random angle θ of each long-range contact vector (measured
counter-clockwise, from the x-axis) is modified to θ′:(2)θ′=arctan(btanθ).Both of these types of anisotropy tend to favor
connections in the verticaly-direction if b>1,
and along the x-direction if 0<b<1.


## 3. Simulations

To simulate Kleinberg navigation efficiently, we use
several tricks and approximations. First, rather than testing a finite-size
squareL×L lattice, we consider an infinite lattice and
place the source and target at distance L from one another. Since the message always
progresses toward the target, by the greedy algorithm, the message holder
remains within a disc of radius L centered on the target node, so in practice
only a finite number of sites would be explored
anyway.
Second, the computation of the normalizing sum in the denominator of (1) is
dependent (in finite lattices) on the location of the node i,
and can be very time-consuming. The infinite lattice circumvents this problem,
as the normalizing constant is the same for all nodes. Note, however, that ∑krik−α does not converge for α<d.
In that case, we imagine a lattice larger than the L-disc of activity, with periodic boundary
conditions, such that the normalizing factor is still the same for all sites.Because of the monotonic progression toward the target
no site is ever revisited in the process. Moreover, as observed by Kleinberg
[5], one can think of
the long-contact link-out of node i as being created at the very instant that the
message arrives at i.
Thus, the full lattice is unnecessary, and we need keep in memory only the
current location of the message holder (and the location of the target). When
the message arrives at i,
we create a random long contact, compare the distances of all five neighbors of i (the four lattice neighbors and the long
contact) to the target, and move the message to the site closest to the target.The long contact is created by choosing a random
distance and angle,(r,θ).
In order to reproduce the correct P(rij)∼rij−α, the distance r is taken from the distribution P(r)∼r−α−1,
to account for the linear growth of the area of the ring where the contact
might fall. The angle θ is distributed uniformly between 0 and 2π.
In the case of angular anisotropy, θ is replaced by θ′,
according to (2). Finally, a vector (r,θ) is drawn from site i,
and the contact is placed on the site j closest to the vector tip.Because of the anisotropy, the angular displacement
from the source to the target makes a difference. It is sufficient to test only
the two extremes ofθ=0 and θ=π/2,
where the target is either parallel or perpendicular to the anisotropy
direction. We note, however, that anisotropy strength b and a target at θ=0 is equivalent to anisotropy 1/b and target at θ=π/2.
For this reason, we simply set the source and target at (0,0) and (L,0),
respectively, throughout, and let b vary both below and above the isotropic divide
of b=1.Simulations were performed for various values ofb over a large range of α and L,
each averaged 1000 times. For each b and L,
the minimum α was computed by first fitting a fifth-order
polynomial to the averaged data, then using Newton's Method on the polynomial's
derivative. (A parabola could be fitted to the data closest to the minimum, but
we must first know what is “closest.” A higher-order polynomial overcomes this
difficulty, similar to including higher order terms in a series expansion near
the minmum of a function.)
Finally, αmin was plotted as a function of 1/ln2L for each chosen value of b.
These are shown in Figure 1 and indicate that αmin→2 as L→∞,
regardless of b.
In Figure 2, we show detailed results of the extrapolation to L→∞.Simulations for lattice and
angle anisotropies. A horizontal scale of1/ln2L is used throughout. All curves approach α(∞),
regardless of b.
For the lattice case, there is a crossover effect where curves for b>1 dip below the b=1 curve. For the angular case, curves for b>1 approach the infinite limit at differing
rates, while curves for b<1 eventually collapse onto the b=1 curve. These phenomena are further explored in
Figure 3. See Figure 2 for the extrapolated α(∞).
(a)
Lattice(b)
AngleExtrapolation results for (a) lattice and (b) angular anisotropies.
Extrapolating to1/In2L→0 with a linear least-squares fit to the curves
in Figure 1 shows excellent convergence of α(∞) to the expected value of d=2.
Good values should occur when the curves are flattest, which happens roughly
around 0.25. A more robust fitting procedure could be used, but the accuracy of
these results implies that it is unnecessary. The
horizontal lines at α=2 provide a guide for the eye.
(a)
Lattice(b)
AngleTo further clarify the behavior shown in Figure1, the
following procedure was performed. First, fit a cubic polynomial pb,
using least squares, to each b's curve. Then, subtract that polynomial from
the isotropic case, pb−p1.
This maps b=1 to the horizontal axis and gives the behavior
of the b≠1 curves ”relative” to the isotropic curve.
These are shown in Figure 3. The different behavior for each type of anisotropy
is clear: for both types of anisotropy, the results for b>1 show dramatic differences from the isotropic case
of b=1 (the differences for b<1 and large L are negligible). For lattice anisotropy, the b>1 curves start above the b=1 curve and cross below until they eventually
converge at a similar rate, as L→∞.
For the angular anisotropy, the b>1 curves approach α(∞) at a different rate than the b=1 curve, resulting in distinctly different
slopes in the plots of Figure 1(b).Curves relative to the
isotropic divideb=1,
for (a) lattice and (b) angular cases. To provide a measure of smoothing,
cubic polynomials pb were fitted to the curves in Figure 1. To
clarify the impact of anisotropy, we show the behavior relative to the
isotropic case by subtracting p1   from each pb.
This maps the isotropic curve to a horizontal line and introduces only minor
distortion. The crossover behavior for b>1 is clearly displayed.
(a)
Lattice(b)
AngleThe observed “crossover” behavior present in the
lattice anisotropy is somewhat unexpected. The crossover point,Lcrossover(b),
is explored by finding the zero of each pb−p1.
These are plotted in Figure 4(a), and seem to indicate a power-law relationship, Lcrossover(b)∼b2.
Likewise, the different slopes for angular anisotropy, plotted in Figure  4(b),
show power-law behavior and seem to increase roughly as b1/4.
What is responsible for these phenomena remains an open question.Dependence of
results on anisotropy strengthb for (a) lattice  and (b) angular  anisotropies.
The straight lines are of slopes 2 and 1/4,
respectively.
(a)
Lattice(b)
Angle


## 4. Conclusions

We have shown, by extensive numerical simulations,
that Kleinberg navigation in two-dimensional lattices with two types of
anisotropy displays the same gross characteristics as navigation in isotropic
lattices. In particular, the optimal long-contact exponent in the limit of
infinitely distant source and target remainsα=2,
even in the presence of anisotropy.It is worthwhile to note that the actual values for
the optimal exponentα(L) for finite L can differ considerably from the limit α=2,
even for reasonably large lattices. Thus, for practical applications, the
optimal exponent ought to be evaluated on a case-by-case basis.The modes of convergence to the limitL→∞ show intriguing power-law dependence upon the
strength of the anisotropy. A theoretical explanation for this behavior remains
the subject of future work.



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*Source: 346543-2008-11-16.xml*