Indexing Distances in Large Graphs and Applications in Search Tasks

This thesis elaborates on the problem of preprocessing a large graph so that single-pair shortest-path queries can be answered quickly at runtime. Computing shortest paths is a well studied problem, but exact algorithms do not scale well to real-world huge graphs in applications that require very short response time. The focus is on approximate methods for distance estimation, in particular in landmarks-based distance indexing. This approach involves choosing some nodes as landmarks and computing (offline), for each node in the graph its embedding, i.e., the vector of its distances from all the landmarks. At runtime, when the distance between a pair of nodes is queried, it can be quickly estimated by combining the embeddings of the two nodes. Choosing optimal landmarks is shown to be hard and thus heuristic solutions are employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the techniques presented in this thesis is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach which considers selecting landmarks at random. Finally, they are applied in two important problems arising naturally in large-scale graphs, namely social search and community detection.

On the Performance of Polynomial-time CLIQUE Approximation Algorithms on Very Large Graphs

The performance of a randomized version of the subgraph-exclusion algorithm (called Ramsey) for CLIQUE by Boppana and Halldorsson is studied on very large graphs. We compare the performance of this algorithm with the performance of two common heuristic algorithms, the greedy heuristic and a version of simulated annealing. These algorithms are tested on graphs with up to 10,000 vertices on a workstation and graphs as large as 70,000 vertices on a Connection Machine. Our implementations establish the ability to run clique approximation algorithms on very large graphs. We test our implementations on a variety of different graphs. Our conclusions indicate that on randomly generated graphs minor changes to the distribution can cause dramatic changes in the performance of the heuristic algorithms. The Ramsey algorithm, while not as good as the others for the most common distributions, seems more robust and provides a more even overall performance. In general, and especially on deterministically generated graphs, a combination of simulated annealing with either the Ramsey algorithm or the greedy heuristic seems to perform best. This combined algorithm works particularly well on large Keller and Hamming graphs and has a competitive overall performance on the DIMACS benchmark graphs.

Small-World Internet Topologies: Possible Causes and Implications on Scalability of End-System Multicast

Recent work has shown the prevalence of small-world phenomena [28] in many networks. Small-world graphs exhibit a high degree of clustering, yet have typically short path lengths between arbitrary vertices. Internet AS-level graphs have been shown to exhibit small-world behaviors [9]. In this paper, we show that both Internet AS-level and router-level graphs exhibit small-world behavior. We attribute such behavior to two possible causes–namely the high variability of vertex degree distributions (which were found to follow approximately a power law [15]) and the preference of vertices to have local connections. We show that both factors contribute with different relative degrees to the small-world behavior of AS-level and router-level topologies. Our findings underscore the inefficacy of the Barabasi-Albert model [6] in explaining the growth process of the Internet, and provide a basis for more promising approaches to the development of Internet topology generators. We present such a generator and show the resemblance of the synthetic graphs it generates to real Internet AS-level and router-level graphs. Using these graphs, we have examined how small-world behaviors affect the scalability of end-system multicast. Our findings indicate that lower variability of vertex degree and stronger preference for local connectivity in small-world graphs results in slower network neighborhood expansion, and in longer average path length between two arbitrary vertices, which in turn results in better scaling of end system multicast.

Swarming on optimized graphs for n-way broadcast

In an n-way broadcast application each one of n overlay nodes wants to push its own distinct large data file to all other n-1 destinations as well as download their respective data files. BitTorrent-like swarming protocols are ideal choices for handling such massive data volume transfers. The original BitTorrent targets one-to-many broadcasts of a single file to a very large number of receivers and thus, by necessity, employs an almost random overlay topology. n-way broadcast applications on the other hand, owing to their inherent n-squared nature, are realizable only in small to medium scale networks. In this paper, we show that we can leverage this scale constraint to construct optimized overlay topologies that take into consideration the end-to-end characteristics of the network and as a consequence deliver far superior performance compared to random and myopic (local) approaches. We present the Max-Min and MaxSum peer-selection policies used by individual nodes to select their neighbors. The first one strives to maximize the available bandwidth to the slowest destination, while the second maximizes the aggregate output rate. We design a swarming protocol suitable for n-way broadcast and operate it on top of overlay graphs formed by nodes that employ Max-Min or Max-Sum policies. Using trace-driven simulation and measurements from a PlanetLab prototype implementation, we demonstrate that the performance of swarming on top of our constructed topologies is far superior to the performance of random and myopic overlays. Moreover, we show how to modify our swarming protocol to allow it to accommodate selfish nodes.

Nearest-neighbor Queries in Probabilistic Graphs

Large probabilistic graphs arise in various domains spanning from social networks to biological and communication networks. An important query in these graphs is the k nearest-neighbor query, which involves finding and reporting the k closest nodes to a specific node. This query assumes the existence of a measure of the "proximity" or the "distance" between any two nodes in the graph. To that end, we propose various novel distance functions that extend well known notions of classical graph theory, such as shortest paths and random walks. We argue that many meaningful distance functions are computationally intractable to compute exactly. Thus, in order to process nearest-neighbor queries, we resort to Monte Carlo sampling and exploit novel graph-transformation ideas and pruning opportunities. In our extensive experimental analysis, we explore the trade-offs of our approximation algorithms and demonstrate that they scale well on real-world probabilistic graphs with tens of millions of edges.

Comparison of K-ary N-cube and de Bruijn Overlays in QoS-Constrained Multicast Applications

Research on the construction of logical overlay networks has gained significance in recent times. This is partly due to work on peer-to-peer (P2P) systems for locating and retrieving distributed data objects, and also scalable content distribution using end-system multicast techniques. However, there are emerging applications that require the real-time transport of data from various sources to potentially many thousands of subscribers, each having their own quality-of-service (QoS) constraints. This paper primarily focuses on the properties of two popular topologies found in interconnection networks, namely k-ary n-cubes and de Bruijn graphs. The regular structure of these graph topologies makes them easier to analyze and determine possible routes for real-time data than complete or irregular graphs. We show how these overlay topologies compare in their ability to deliver data according to the QoS constraints of many subscribers, each receiving data from specific publishing hosts. Comparisons are drawn on the ability of each topology to route data in the presence of dynamic system effects, due to end-hosts joining and departing the system. Finally, experimental results show the service guarantees and physical link stress resulting from efficient multicast trees constructed over both kinds of overlay networks.

Anisotropic Interpolation on Graphs: The Combinatorial Dirichlet Problem

The combinatorial Dirichlet problem is formulated, and an algorithm for solving it is presented. This provides an effective method for interpolating missing data on weighted graphs of arbitrary connectivity. Image processing examples are shown, and the relation to anistropic diffusion is discussed.

The Graph Analysis Toolbox: Image Processing on Arbitrary Graphs

Office of Naval Research (N00014-01-1-0624)