Section: New Results

Models and Algorithms for Networks

Analysis of Multiple Random Walks on Paths and Grids

In [22], we derive several new results on multiple random walks on “low-dimensional” graphs. First, inspired by an example of a weighted random walk on a path of three vertices given by Efremenko and Reingold, we prove the following dichotomy: as the path length n tends to infinity, we have a super-linear speed-up w.r.t. the cover time if and only if the number of walks $k$ is equal to 2. An important ingredient of our proofs is the use of a continuous-time analogue of multiple random walks, which might be of independent interest. Finally, we also present the first tight bounds on the speed-up of the cover time for any $d$-dimensional grid with $d\ge 2$ being an arbitrary constant, and reveal a sharp transition between linear and logarithmic speed-up.

Decomposing a Graph into Shortest Paths with Bounded Eccentricity

In [11], we introduce the problem of hub-laminar decomposition which generalizes that of computing a shortest path with minimum eccentricity (MESP). Intuitively, it consists in decomposing a graph into several paths that collectively have small eccentricity and meet only near their extremities. The problem is related to computing an isometric cycle with minimum eccentricity (MEIC). It is also linked to DNA reconstitution in the context of metagenomics in biology. We show that a graph having such a decomposition with long enough paths can be decomposed in polynomial time with approximated guaranties on the parameters of the decomposition. Moreover, such a decomposition with few paths allows to compute a compact representation of distances with additive distortion. We also show that having an isometric cycle with small eccentricity is related to the possibility of embedding the graph in a cycle with low distortion.

Individual versus collective cognition in social insects

The concerted responses of eusocial insects to environmental stimuli are often referred to as collective cognition at the level of the colony. To achieve collective cognition, a group can draw on two different sources: individual cognition and the connectivity between individuals. Computation in neural networks, for example, is attributed more to sophisticated communication schemes than to the complexity of individual neurons. The case of social insects, however, can be expected to differ. This is because individual insects are cognitively capable units that are often able to process information that is directly relevant at the level of the colony. Furthermore, involved communication patterns seem difficult to implement in a group of insects as they lack a clear network structure. In [5] we discusses links between the cognition of an individual insect and that of the colony. We provide examples for collective cognition whose sources span the full spectrum between amplification of individual insect cognition and emergent group-level processes.