Complex Networks
MATH 36201/M6201
Problem Sheet 9
Spring 2017
∗∗ Please hand in solutions to questions 1 and 2 on this sheet. ∗∗
1. Consider Kleinberg’s model for message routing on the cycle graph Cn , i.e., a ring on
n points. Each node is connected to its two nearest neighbours in the ring. In addition,
each node i generates a random shortcut, choosing node j as its long-range contact with
probability proportional to r(i, j)−s , where r(i, j) is the graph distance between nodes i
and j on the ring: in other words, r(i, j) = min{|i − j|, n − |i − j|}.
We consider the greedy algorithm for routing a message from a specified source node u to
a specified destination node v in the case when s = 1. If the message is at a node x, the
greedy algorithm routes to whichever of its neighbours, lattice or long-range, is closest to
the target v; if more than one is at the same distance, it chooses one of these at random.
(a) For any two nodes x and y, show that the probability that x chooses y as its long-range
contact is bounded below by cr(x, y)−1 / log n for some constant c > 0 that doesn’t
depend on n.
(b) Suppose that the message is currently at a node x, which is at distance D from the
target v. Define A = {y : r(y, v) ≤ D/2} to be the set of all nodes within distance
D/2 of the target. Show that the probability that x has a neighbour, either lattice or
long-range, within A is at least c/ log n for some constant c > 0.
Hint. Count the number of points within A, and use the answer to part (a).
(c) Show that the expected number of routing steps of the greedy algorithm before the
distance to the target halves is at most c log n, for some constant c that doesn’t depend
on n.
(d) Using the answer to part (c), argue that the expected number of steps that it takes the
greedy algorithm to deliver the message from u to v is bounded by c log2 n, for some
constant c.
∗2. Consider Kleinberg’s model again, but now with s = 4. In other words, we have a grid
of n2 points in 2 dimensions with each node connected to its four nearest neighbours. In
addition, each node i chooses another node j with probability proportional to r(i, j)−4 and
establishes an undirected long-range link to it. Here, r(i, j) is the lattice distance between
i and j.
We consider decentralised algorithms for routing a message from a specified source node
u to a specified destination node v. Suppose that r(u, v) ≥ n/2.
(a) Let x be an arbitrary node. Show that, for any node y, the probability that x chooses
y as its long-range contact is bounded above by r(x, y)−4 .
1
(b) We now compute the expected length of a shortcut. Let y be the long-range contact
chosen by x. Using the answer to the last part, show that there is a constant c that
does not depend on n such that E[r(x, y)] ≤ c. (The expectation is over the random
choice of y.)
(c) Each node has two
encountered
√
√ shortcuts on average. Hence, the number of shortcuts
within the first n steps of the routing algorithm is close to 2 n (maybe fewer if
some nodes are visited repeatedly, maybe slightly
√ more because of random fluctuations). Assuming that it is exactly equal to 2 n, compute an upper bound on the
expected distance traversed along short-cuts (even if every short-cut observed was
used by the algorithm).
(d) Use Markov’s inequality to show that the probability that the distance traversed along
short-cuts exceeds
√ n/4 is negligible. Hence argue that any decentralised algorithm
needs at least n steps for message delivery (if r(u, v) ≥ n/2), with probability
close to 1.
∗3. Consider Kleinberg’s model again, but now with s = 0. In other words, we have a grid
of n2 points in 2 dimensions with each node connected to its four nearest neighbours.
In addition, each node i chooses another node j uniformly at random and establishes an
undirected long-range link to it.
We consider decentralised algorithms for routing a message from a specified source node
u to a specified destination node v with r(u, v) ≥ n/2.
(a) For any two nodes x and y, show that the probability that x chooses y as its long-range
contact is bounded above by 1/(n2 − 1) ≈ 1/n2 .
√
(b) Let U denote the set of nodes within lattice distance n of the target node v. For any
node x, show that the probability that there is a shortcut between x and a node in U is
bounded by c/n, for some constant c that doesn’t depend on n.
(c) Using the answer to the last part, show that for any decentralised algorithm,
the prob√
ability that the algorithm sees a shortcut into U within the first 2 n steps is smaller
than a half. Hence conclude √
that the expected number of steps required by any decentralised algorithm is at least n.
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