# Structural and Algorithmic Properties of Static and Mobile Random Geometric Graphs

### Alexandre Stauffer

### EECS Department, University of California, Berkeley

### Technical Report No. UCB/EECS-2011-39

### May 5, 2011

### http://www2.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-39.pdf

We study fundamental problems for static and mobile networks. First, we consider the random geometric graph model, which is a well-known model for static wireless networks. In this model, n nodes are distributed independently and uniformly at random in the d-dimensional torus of volume n and edges are added between pairs of nodes whose Euclidean distance is at most some parameter r. We consider the case where r is a sufficiently large constant so that a so-called giant component (a connected component with \Theta(n) nodes) exists with high probability. In this setting, we show that the graph distance between every pair of nodes whose Euclidean distance is sufficiently large is only a constant factor larger than their Euclidean distance. This result gives, as a corollary, that the diameter of the giant component is \Theta(n^{1/d}/r). Then, we apply this result to analyze the performance of a broadcast algorithm known as the push algorithm. In this algorithm, at each discrete time step, each informed node chooses a neighbor independently and uniformly at random and informs it. We show that the push algorithm informs all nodes of the giant component of a random geometric graph within a number of steps that is only a constant factor larger then the diameter of the giant component.

In the second part of the thesis, we consider a model of mobile graphs that we call mobile geometric graphs, and which is an extension of the random geometric graph model to the setting where nodes are not static but are moving in space in continuous time. In this model, we start with a random geometric graph and let the nodes move as independent Brownian motions. Then, at any given time, there exists an edge between every pair of nodes whose Euclidean distance at that time is at most r. This model has been recently used as a model for mobile wireless networks. We study four fundamental problems in this model: detection (the time until a target point---fixed or moving---is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); percolation (the time until a given node belongs to the giant component of the graph) and broadcast (the time until all nodes of the graph receive a piece of information that was initially known by a single node). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.

Finally, in the last part of the thesis, we revisit the push algorithm described above and study its performance in general regular graphs. Our goal is to understand the relation between the performance of the push algorithm and the vertex expansion of the graph. We prove an upper bound for the runtime of this algorithm that depends on the vertex expansion of the graph and is tight up to polylogarithmic factors. Then, we show that there exists a substantial difference between the impact of vertex expansion and edge expansion on the performance of the push algorithm. In particular, we prove the existence of regular graphs (which are also vertex transitive) that have constant vertex expansion and for which the runtime of the push algorithm is a factor of \Omega(\log n) slower than on any regular graph with constant edge expansion.

**Advisors:** Alistair Sinclair

BibTeX citation:

@phdthesis{Stauffer:EECS-2011-39, Author= {Stauffer, Alexandre}, Title= {Structural and Algorithmic Properties of Static and Mobile Random Geometric Graphs}, School= {EECS Department, University of California, Berkeley}, Year= {2011}, Month= {May}, Url= {http://www2.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-39.html}, Number= {UCB/EECS-2011-39}, Abstract= {We study fundamental problems for static and mobile networks. First, we consider the random geometric graph model, which is a well-known model for static wireless networks. In this model, n nodes are distributed independently and uniformly at random in the d-dimensional torus of volume n and edges are added between pairs of nodes whose Euclidean distance is at most some parameter r. We consider the case where r is a sufficiently large constant so that a so-called giant component (a connected component with \Theta(n) nodes) exists with high probability. In this setting, we show that the graph distance between every pair of nodes whose Euclidean distance is sufficiently large is only a constant factor larger than their Euclidean distance. This result gives, as a corollary, that the diameter of the giant component is \Theta(n^{1/d}/r). Then, we apply this result to analyze the performance of a broadcast algorithm known as the push algorithm. In this algorithm, at each discrete time step, each informed node chooses a neighbor independently and uniformly at random and informs it. We show that the push algorithm informs all nodes of the giant component of a random geometric graph within a number of steps that is only a constant factor larger then the diameter of the giant component. In the second part of the thesis, we consider a model of mobile graphs that we call mobile geometric graphs, and which is an extension of the random geometric graph model to the setting where nodes are not static but are moving in space in continuous time. In this model, we start with a random geometric graph and let the nodes move as independent Brownian motions. Then, at any given time, there exists an edge between every pair of nodes whose Euclidean distance at that time is at most r. This model has been recently used as a model for mobile wireless networks. We study four fundamental problems in this model: detection (the time until a target point---fixed or moving---is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); percolation (the time until a given node belongs to the giant component of the graph) and broadcast (the time until all nodes of the graph receive a piece of information that was initially known by a single node). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis. Finally, in the last part of the thesis, we revisit the push algorithm described above and study its performance in general regular graphs. Our goal is to understand the relation between the performance of the push algorithm and the vertex expansion of the graph. We prove an upper bound for the runtime of this algorithm that depends on the vertex expansion of the graph and is tight up to polylogarithmic factors. Then, we show that there exists a substantial difference between the impact of vertex expansion and edge expansion on the performance of the push algorithm. In particular, we prove the existence of regular graphs (which are also vertex transitive) that have constant vertex expansion and for which the runtime of the push algorithm is a factor of \Omega(\log n) slower than on any regular graph with constant edge expansion.}, }

EndNote citation:

%0 Thesis %A Stauffer, Alexandre %T Structural and Algorithmic Properties of Static and Mobile Random Geometric Graphs %I EECS Department, University of California, Berkeley %D 2011 %8 May 5 %@ UCB/EECS-2011-39 %U http://www2.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-39.html %F Stauffer:EECS-2011-39