Geometry of wireless networks: A bit of topological algebra

Once again, there are several notable introductions to topological algebra and especially to simplicial complexes , so I just here underline the basic ideas, at least, as far as I understood them.

For instance, consider again the following situation

In the proximity graph, we would put an edge between two sensors each time they are sufficiently close to each other: each time, their coverage balls do intersect. This graph is connected if and only if each sensor can send a message to any other sensor, so that a one of the question is answered. Unfortunately, we can easily imagine where the proximity graph does not give a satisfactory answer to the coverage problem. Take an equilateral triangle of length [math] a[/math]. The circumscribed radius is [math ] asqrt{3}/3sim 0.58 a[/math], hence if the radius of the coverage balls is between [math] 0.5 a[/math] and [math] 0.58 a[/math], the three edges of the triangle are connected because their balls intersect two by two but at least, the center of the circumscribed cercle is not covered. This means that the proximity graph is too crude to reveal coverage properties. The rest is algebraic topology, see these slides.