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Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs called the rotor-router model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave in a remarkably similar way: Starting with the same initial configuration, the number of tokens in the rotor-router model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer [Combin. Probab. Comput., 15 (2006), pp. 815-822] showed that on Z(d), the single vertex discrepancy is only a constant c(d). Other authors also determined the precise value of c(d) for d = 1, 2. All of these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph Z(d). We show that this assumption is crucial by proving that, otherwise, the single vertex discrepancy can become arbitrarily large. For all dimensions d >= 1 and arbitrary discrepancies l >= 0, we construct configurations that reach a discrepancy of at least l.
Many complex systems that we encounter in the world can be formalized using networks. Consequently, they have been in the focus of computer science for decades, where algorithms are developed to understand and utilize these systems.
Surprisingly, our theoretical understanding of these algorithms and their behavior in practice often diverge significantly. In fact, they tend to perform much better on real-world networks than one would expect when considering the theoretical worst-case bounds. One way of capturing this discrepancy is the average-case analysis, where the idea is to acknowledge the differences between practical and worst-case instances by focusing on networks whose properties match those of real graphs. Recent observations indicate that good representations of real-world networks are obtained by assuming that a network has an underlying hyperbolic geometry.
In this thesis, we demonstrate that the connection between networks and hyperbolic space can be utilized as a powerful tool for average-case analysis. To this end, we first introduce strongly hyperbolic unit disk graphs and identify the famous hyperbolic random graph model as a special case of them. We then consider four problems where recent empirical results highlight a gap between theory and practice and use hyperbolic graph models to explain these phenomena theoretically. First, we develop a routing scheme, used to forward information in a network, and analyze its efficiency on strongly hyperbolic unit disk graphs. For the special case of hyperbolic random graphs, our algorithm beats existing performance lower bounds. Afterwards, we use the hyperbolic random graph model to theoretically explain empirical observations about the performance of the bidirectional breadth-first search. Finally, we develop algorithms for computing optimal and nearly optimal vertex covers (problems known to be NP-hard) and show that, on hyperbolic random graphs, they run in polynomial and quasi-linear time, respectively.
Our theoretical analyses reveal interesting properties of hyperbolic random graphs and our empirical studies present evidence that these properties, as well as our algorithmic improvements translate back into practice.
A standard approach to accelerating shortest path algorithms on networks is the bidirectional search, which explores the graph from the start and the destination, simultaneously. In practice this strategy performs particularly well on scale-free real-world networks. Such networks typically have a heterogeneous degree distribution (e.g., a power-law distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. <br /> To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is (O) over tilde (n(2-1/alpha) + n(1/(2 alpha)) + delta(max)) with high probability, where alpha is an element of (1/2, 1) controls the power-law exponent of the degree distribution, and dmax is the maximum degree. This bound is sublinear, improving the obvious worst-case linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.
Hyperbolic random graphs (HRGs) and geometric inhomogeneous random graphs (GIRGs) are two similar generative network models that were designed to resemble complex real-world networks.
In particular, they have a power-law degree distribution with controllable exponent beta and high clustering that can be controlled via the temperature T.
We present the first implementation of an efficient GIRG generator running in expected linear time.
Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs.
Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to T = 0 .
We also support parallelization, although this is not the focus of this paper.
Moreover, we note that our generators draw from the correct probability distribution, that is, they involve no approximation.
Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model.
This makes it possible to specify the desired expected average degree as input. Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straightforward inclusion does not hold in practice.
However, the difference is negligible for most use cases.