Emmanuelle Lebhar

Unit disk graph and physical interference model: Putting pieces together

Modeling communications in wireless networks is a challenging task, since it requires a simple mathematical object on which efficient algorithms can be designed but which must also reflect the complex physical constraints inherent in wireless networks, such as interferences, the lack of global knowledge, and purely local computations. As a tractable mathematical object, the unit disk graph (UDG) is a popular model that has enabled the development of efficient algorithms for crucial networking problems. In a ρ-UDG, two nodes are connected if and only if their distance is at most ρ, for some ρ ≪ 0. However, such a connectivity requirement is basically not compatible with the reality of wireless networks due to the environment of the nodes as well as the constraints of radio transmission. For this purpose, the signal interference plus noise ratio model (SINR) is the more commonly used model. The SINR model focuses on radio interferences created over the network depending on the distance to transmitters. Nevertheless, due to its complexity, this latter model has been the subject of very few theoretical investigations and lacks of good algorithmic features. In this paper, we demonstrate how careful scheduling of the nodes enables the two models to be combined to give the benefits of both the algorithmic features of the UDG and the physical validity of the SINR. Precisely, we show that it is possible to emulate a 1/√n ln n-UDG that satisfies the constraints of the SINR over any set of n wireless nodes distributed uniformly in a unit square, with only a O(ln3 n) time and power stretch factor. The main strength of our contribution lies in the fact that the scheduling is set in a fully distributed way and considers non-uniform power ranges, and it can therefore fit the sensor network setting. Moreover, our scheduling is optimal up to a polylogarithmic factor in terms of throughput capacity according to the lower bound of Gupta and Kumar.

Opportunistic spatial gossip over mobile social networks

This paper investigates how the principles underlying online social network services could be used to take advantage of node mobility in an opportunistic manner. As an example, we show how to take advantage of opportunistic contacts between mobile phones that run an online social network service. Our model includes static nodes, and mobile nodes which follow random walks. As in an online network service, we assume that each node can only communicate with a small subset of others nodes (called its mates) in addition to its geographical neighbors. Here we prove that, in such context, a simple connection scheme enables to execute sophisticated tasks (e.g., routing) and mechanisms (e.g., spatial gossip), while using only opportunistic communication and communication between mates. In other words, our results show that future online social networks can exploit mobility as long as they forget connections appropriately.

Networks Become Navigable as Nodes Move and Forget

We propose a dynamic process for network evolution, aiming at explaining the emergence of the small world phenomenon, i.e., the statistical observation that any pair of individuals are linked by a short chain of acquaintances computable by a simple decentralized routing algorithm, known as greedy routing. Our model is based on the combination of two dynamics: a random walk (spatial) process, and an harmonic forgetting (temporal) process. Both processes reflect natural behaviors of the individuals, viewed as nodes in the network of inter-individual acquaintances. We prove that, in k-dimensional lattices, the combination of these two processes generates long-range links mutually independently distributed as a k-harmonic distribution. We analyze the performances of greedy routing at the stationary regime of our process, and prove that the expected number of steps for routing from any source to any target in any multidimensional lattice is a polylogarithmic function of the distance between the two nodes in the lattice. Up to our knowledge, these results are the first formal proof that navigability in small worlds can emerge from a dynamic process for network evolution. Our dynamica process can find practical applications to the design of spatial gossip and resource location protocols.

Local MST Computation with Short Advice

We use the recently introduced advising scheme framework for measuring the difficulty of locally distributively computing a Minimum Spanning Tree (MST). An (m,t)-advising scheme for a distributed problem ℘ is a way, for every possible input I of ℘, to provide an “advice” (i.e., a bit string) about I to each node so that: (1) the maximum size of the advices is at most m bits, and (2) the problem ℘ can be solved distributively in at most t rounds using the advices as inputs. In case of MST, the output returned by each node of a weighted graph G is the edge leading to its parent in some rooted MST T of G. Clearly, there is a trivial (⌈log n⌉,0)-advising scheme for MST (each node is given the local port number of the edge leading to the root of some MST T), and it is known that any (0,t)-advising scheme satisfies

A Lower Bound for Network Navigability

t\geq\tilde{\Omega}(\sqrt{n})

Graph Augmentation via Metric Embedding

. Our main result is the construction of an (O(1),O(log n))-advising scheme for MST. That is, by only giving a constant number of bits of advice to each node, one can decrease exponentially the distributed computation time of MST in arbitrary graph, compared to algorithms dealing with the problem in absence of any a priori information. We also consider the average size of the advices. On the one hand, we show that any (m,0)-advising scheme for MST gives advices of average size Ω(log n). On the other hand we design an (m,1)-advising scheme for MST with advices of constant average size, that is one round is enough to decrease the average size of the advices from log n to constant.

Forget him and keep on moving

In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining whether such an augmentation is possible for all graphs. In this paper, we answer this question negatively by exhibiting an infinite family of graphs that cannot be augmented to become navigable whatever the distribution of random edges is. Precisely, it was known that graphs of doubling dimension at most

Local MST computation with short advice

O(\log\log n)

Universal augmentation schemes for network navigability: overcoming the √ n -barrier

are navigable. We show that for doubling dimension

Recovering the long-range links in augmented graphs

\gg\log\log n