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when they choose distinct actions. Problems with
similar characteristics have been studied in many
research areas. Here we list some of them:
1. Load balancing: a set of tasks is to be
assigned to a set of machines, and to minimise
the overall processing time, the load need to be
distributed as evenly as possible (see, e.g.,
Azar et al., 1999).
2. Niche selection: this problem has been studied
in economics as well as in ecology. In
economics, it models a situation in which
producers wish to occupy one of a number of
different market niches, and they prefer to
occupy niches with fewer competitions. A few
examples are the Santa Fe problem (Arthur,
1994) and the class of minority games (Challet
and Zhang, 1997). In ecology, this problem is
also called habitat selection, and an example of
this is how animals choose feeding patches
with low population density with respect to
food supply (see, e.g., Houston and
McNamara, 1997). In particular, Fretwell and
Lucas (1969) introduced the equilibrium notion
of the ideal free distribution for habitat
distribution in birds. That is a particular
analogue of our notion of dispersion, where
some nodes have greater capacity (habitat is
more supportive of foragers).
3. Congestion games: proposed initially by
Rosenthal (1973), in this class of games
individuals seek facilities or locations of low
population density, due to the monotonically
increasing cost of using the facilities. One of
the subclasses of congestion games that has
received significant attention is routing games,
in which a set of players in a network choose
the route with low congestion level (see, e.g.,
Qiu et al. 2003).
4. Multi-agent area coverage: a team of agents
seek to cover an entire area (which may or may
not be known a priori), and therefore it is
preferred that each agent covers different part
of the area. This problem has been studied in
several research communities, including
robotics/agents, sensor networks, and
computational geometry. For a recent survey in
the fields of robotics, see Galceran and
Carreras (2013).
3 OBJECTIVES
Given a graph
G
with k nodes, we want to see
which (Markovian) stunted random walk strategies,
if simultaneously adopted by all agents, lead to the
least expected time
T
to dispersion? In all the
problems considered, the same optimization
criterion is used. We consider both (i) a random
initial placement of the n agents onto the
k nodes
of
,G and (ii) an initial placement of all the agents
onto a single node
of
.G
In the latter case, it is of
interest to see where the best place to introduce the
agents is.
We also will consider two types of stunted
random walk strategies, and compare them. First, a
simple type, where there is a common parameter
representing the probability of staying still for the
next period. Second, a probability vector
p
in which
i
p (the i-th entry in the vector) gives the probability
of staying still (rather than randomly moving) that
depends on the number
i
of agents currently at your
node (called the agent population of the node).
A conjecture of Alpern and Reyniers (2002) in a
related context says that if you are alone at your
node then you should not move and no one should
move to your node. While this might work in the
context of a complete graph, it clearly fails on the
line graph
.
n
L In the case when there are many
agents at node 1 and one agent at node 2, the ones at
node 1 could never disperse. However the weaker
property of setting
1
1p
seems to hold for some
graphs, which does not have the added assumption
that no one goes to a node with current population of
1.
Our objective is to determine the minimum
dispersal time and optimal levels of
or of
,p
for
several classes of networks, including line networks
,
n
L cycle networks ,
n
C and complete graphs .
n
We also wish to compare on
n
L the efficiency of an
initial placement of all agents at a node
,
as
varies. It would seem likely that it is better to place
them as close as possible to the centre.
From now on we assume that the graphs we
consider have n nodes and that there are n agents.
4 METHODOLOGY
To evaluate the expected dispersal time T on a
graph
G
as a function of the stationary probability
(or of the vector
1
( ,..., )
n
pp
p ) we use two
distinct methodologies. For small graphs (small n ),
we use the theory of absorbing time for Markov
chains. The states of the chain are the population
ANetworkDispersionProblemforNon-communicatingAgents
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