Abstract: We introduce a new model of
ad-hoc mobile networks, which we call hierarchical,
that are comprised of dense subnetworks of mobile
users (corresponding to highly populated
geographical areas, such as cities),
interconnected across access ports
by sparse but frequently used connections
(such as highways).
For such networks, we present
an efficient routing protocol which extends
the idea (introduced in WAE00) of exploiting the co-ordinated
motion of a small part of an ad-hoc mobile
network (the ``support'') to achieve
very fast communication between any two mobile users of the network.
The basic idea of the new protocol presented here is, instead
of using a unique (large) support for the whole network,
to employ a hierarchy of (small) supports (one for each city)
and also take advantage of the regular traffic
of mobile users across the interconnection highways to communicate
between cities.
We combine here theoretical analysis (average case estimations based on random walk properties) and experimental implementations (carried out using the LEDA platform) to claim and validate results showing that such a hierarchical routing approach is,
for this class of ad-hoc mobile networks, significantly more efficient than a simple extension of the
basic ``support'' idea presented in WAE00.

Abstract: We investigate the existence and efficient algorithmic
construction of close to optimal independent sets in random models
of intersection graphs. In particular, (a) we propose \emph{a new model} for random intersection graphs
($G_{n, m, \vec{p}}$) which includes the model of
\cite{RIG} (the ``uniform" random intersection graphs model) as an
important special case. We also define an interesting variation of
the model of random intersection graphs, similar in spirit to
randomregulargraphs. (b) For this model we derive \emph{exact formulae} for the mean
and variance of the number of independent sets of size $k$ (for
any $k$) in the graph. (c) We then propose and analyse \emph{three algorithms} for the
efficient construction of large independent sets in this model.
The first two are variations of the greedy technique while the
third is a totally new algorithm. Our algorithms are analysed for
the special case of uniform random intersection graphs.
Our analyses show that these algorithms succeed in finding
\emph{close to optimal} independent sets for an interesting range
of graph parameters.

Abstract: It was only recently shown by Shi and Wormald, using the differential equation method to analyze an appropriate algorithm, that a random 5-regulargraph asymptotically almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5-regulargraph is asymptotically almost surely equal to 3, provided a certain four-variable function has a unique maximum at a given point in
a bounded domain. We also describe extensive numerical evidence that strongly suggests that the latter condition holds. The proof applies the small subgraph conditioning method to the number of locally rainbow balanced 3-colorings, where a coloring is balanced if the number of vertices of each color is equal, and locally rainbow if every vertex is adjacent to at least one vertex of each of the other colors.

Abstract: In this work, we study protocols (i.e. distributed algorithms) so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol (i.e. the system is homogeneous). Moreover, we assume pairwise interactions between the processes that are scheduled by an adversary. The only constraint on the adversary scheduler is that it must be fair, intuitively meaning that it must assign to every reachable configuration of the system a non-zero probability to occur. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. In particular, in every interaction, the protocol may activate an inactive connection, deactivate an active one, or leave the state of a connection unchanged. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network (i.e. one that does not change any more). We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. We provide proofs of correctness for all of our protocols and analyze the expected time to convergence of most of them under a uniform random scheduler that selects the next pair of interacting processes uniformly at random from all such pairs. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. Our universality protocols use a subset of the population (waste) in order to distributedly construct there a TM able to decide a graph class in some given space. Then, the protocols repeatedly construct in the rest of the population (useful space) a graph equiprobably drawn from all possible graphs. The TM works on this and accepts if the presented graph is in the class. We additionally show how to partition the population into k supernodes, each being a line of log k nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions. Delicate composition and reinitialization issues have to be solved for these general constructions to work.

Abstract: This work extends what is known so far for a basic model of
evolutionary antagonis
m in undirected ne
tworks (graphs).
More specif-
ically, this work studies the generalized Moran process, as introduced
by Lieberman, Hauert, and Nowak [Nature, 433:312-316, 2005], where
the individuals of a population reside on the vertices of an undirected
connected graph. The initial population has a single
mutant
of a
fitness
value
r
(typically
r>
1), residing at some vertex
v
of the graph, while
every other vertex is initially occupied by an individual of fitness 1. At
every step of this process, an individual (i.e. vertex) is randomly chosen
for reproduction with probability proportional to its fitness, and then it
places a copy of itself on a random neighbor, thus replacing the individ-
ual that was residing there. The main quantity of interest is the
fixation
probability
, i.e. the probability that eventually the whole graph is occu-
pied by descendants of the mutant. In this work we concentrate on the
fixation probability when the mutant is initially on a specific vertex
v
,
thus refining the older notion of Lieberman et al. which studied the fix-
ation probability when the initial mutant is placed at a random vertex.
We then aim at finding graphs that have many “strong starts” (or many
“weak starts”) for the mutant. Thus we introduce a parameterized no-
tion of
selective amplifiers
(resp.
selective suppressors
)ofevolution.We
prove the existence of
strong
selective amplifiers (i.e. for
h
(
n
)=
Θ
(
n
)
vertices
v
the fixation probability of
v
is at least 1
−
c
(
r
)
n
for a func-
tion
c
(
r
) that depends only on
r
), and the existence of quite strong
selective suppressors. Regarding the traditional notion of fixation prob-
ability from a random start, we provi
de strong upper and lower bounds:
first we demonstrate the non-existence of “strong universal” amplifiers,
and second we prove the
Thermal Theorem
which states that for any
undirected graph, when the mutant starts at vertex
v
, the fixation prob-
ability at least (
r
−
1)
/
(
r
+
deg
v
deg
min
). This theorem (which extends the
“Isothermal Theorem” of Lieberman et al. for regulargraphs) implies
an almost tight lower bound for the usual notion of fixation probability.
Our proof techniques are original and are based on new domination ar-
guments which may be of general interest in Markov Processes that are
of the general birth-death type.

Abstract: We investigate the existence and efficient algorithmic construction
of close to optimal independent sets in random models of intersection
graphs. In particular, (a) we propose a new model for random
intersection graphs (Gn,m,p) which includes the model of [10] (the “uniform”
random intersection graphs model) as an important special case.
We also define an interesting variation of the model of random intersection
graphs, similar in spirit to randomregulargraphs. (b) For this
model we derive exact formulae for the mean and variance of the number
of independent sets of size k (for any k) in the graph. (c) We then propose
and analyse three algorithms for the efficient construction of large
independent sets in this model. The first two are variations of the greedy
technique while the third is a totally new algorithm. Our algorithms are
analysed for the special case of uniform random intersection graphs.
Our analyses show that these algorithms succeed in finding close to optimal
independent sets for an interesting range of graph parameters.

Abstract: We investigate the existence and efficient algorithmic construction of close to opti-
mal independent sets in random models of intersection graphs. In particular, (a) we
propose a new model for random intersection graphs (Gn,m,~p) which includes the
model of [10] (the “uniform” random intersection graphs model) as an important
special case. We also define an interesting variation of the model of random intersec-
tion graphs, similar in spirit to randomregulargraphs. (b) For this model we derive
exact formulae for the mean and variance of the number of independent sets of size
k (for any k) in the graph. (c) We then propose and analyse three algorithms for
the efficient construction of large independent sets in this model. The first two are
variations of the greedy technique while the third is a totally new algorithm. Our
algorithms are analysed for the special case of uniform random intersection graphs.
Our analyses show that these algorithms succeed in finding close to optimal in-
dependent sets for an interesting range of graph parameters.