We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Social Network Analysis

In this overview paper we motivate the need for and research issues arising from a new model of data processing. In this model, data does not take the form of persistent relations, but rather arrives in multiple, continuous, rapid, time-varying data streams. In addition to reviewing past work relevant to data stream systems and current projects in the area, the paper explores topics in stream query languages, new requirements and challenges in query processing, and algorithmic issues.

/pdf/models-and-issues-in-data-stream-systems-3aa6f5lsfv.pdf

Models and issues in data stream systems

https://www.cse.unsw.edu.au/~cs9314/07s1/lectures/Lin_CS9314_References/cm-latin.pdf

An improved data stream summary: the count-min sketch and its applications

Most statistical and machine-learning algorithms assume that the data is a random sample drawn from a stationary distribution. Unfortunately, most of the large databases available for mining today violate this assumption. They were gathered over months or years, and the underlying processes generating them changed during this time, sometimes radically. Although a number of algorithms have been proposed for learning time-changing concepts, they generally do not scale well to very large databases. In this paper we propose an efficient algorithm for mining decision trees from continuously-changing data streams, based on the ultra-fast VFDT decision tree learner. This algorithm, called CVFDT, stays current while making the most of old data by growing an alternative subtree whenever an old one becomes questionable, and replacing the old with the new when the new becomes more accurate. CVFDT learns a model which is similar in accuracy to the one that would be learned by reapplying VFDT to a moving window of examples every time a new example arrives, but with O(1) complexity per example, as opposed to O(w), where w is the size of the window. Experiments on a set of large time-changing data streams demonstrate the utility of this approach.

/pdf/mining-time-changing-data-streams-a9kghvc13p.pdf

Mining time-changing data streams

Massive data sets often arise as physically distributed, parallel data streams. We present algorithms for estimating simple functions on the union of such data streams, while using only logarithmic space per stream. Each processor observes only its own stream, and communicates with the other processors only after observing its entire stream. This models the set-up in current network monitoring products. Our algorithms employ a novel coordinated sampling technique to extract a sample of the union; this sample can be used to estimate aggregate functions on the union. The technique can also be used to estimate aggregate functions over the distinct “labels” in one or more data streams, e.g., to determine the zeroth frequency moment (i.e., the number of distinct labels) in one or more data streams. Our space and time bounds are the best known for these problems, and our logarithmic space bounds for coordinated sampling contrast with polynomial lower bounds for independent sampling. We relate our distributed streams model to previously studied non-distributed (i.e., merged) streams models, presenting tight bounds on the gap between the distributed and merged models for deterministic algorithms.

/pdf/estimating-simple-functions-on-the-union-of-data-streams-nb4ttsgalz.pdf

Estimating simple functions on the union of data streams

This paper presents a new space-efficient algorithm for counting and sampling triangles--and more generally, constant-sized cliques--in a massive graph whose edges arrive as a stream. Compared to prior work, our algorithm yields significant improvements in the space and time complexity for these fundamental problems. Our algorithm is simple to implement and has very good practical performance on large graphs.

/pdf/counting-and-sampling-triangles-from-a-graph-stream-3py2bgfmfc.pdf

Counting and sampling triangles from a graph stream

This paper presents algorithms for estimating aggregate functions over a "sliding window" of the N most recent data items in one or more streams. Our results includeFor a single stream, we present the first e-approximation scheme for the number of 1's in a sliding window that is optimal in both worst case time and space. We also present the first e for the sum of integers in [0..R] in a sliding window that is optimal in both worst case time and space (assuming R is at most polynomial in N). Both algorithms are deterministic and use only logarithmic memory words.In contrast, we show that an deterministic algorithm that estimates, to within a small constant relative error, the number of 1's (or the sum of integers) in a sliding window over the union of distributed streams requires Ω(N) space. We present the first randomized (e,δ)-approximation scheme for the number of 1's in a sliding window over the union of distributed streams that uses only logarithmic memory words. We also present the first (e,δ)-approximation scheme for the number of distinct values in a sliding window over distributed streams that uses only logarithmic memory words. Our results are obtained using a novel family of synopsis data structures.

/pdf/distributed-streams-algorithms-for-sliding-windows-287nurxg6j.pdf

Distributed streams algorithms for sliding windows

A b s t r a c t We discuss a graph-algorithmic approach to comparing shapes. We focus in this paper on comparing simple closed curves in the plane. Our approach is to (1) represent such a shape by its skeleton, which is a tree embedded in the plane, and (2) compare two shapes by comparing their skeletons via tree edit-distance. In this paper, we define our version of tree edit-distance (it differs from that previously described in the literature), and give a polynomial-time algorithm to compute the distance between two trees.

/pdf/a-tree-edit-distance-algorithm-for-comparing-simple-closed-4wlenttayf.pdf

A tree-edit-distance algorithm for comparing simple, closed shapes

http://home.engineering.iastate.edu/~sumitc/documents/orgelectron2012.pdf

Srikanta Tirthapura

Papers

Estimating simple functions on the union of data streams

Counting and sampling triangles from a graph stream

Distributed streams algorithms for sliding windows

A tree-edit-distance algorithm for comparing simple, closed shapes

A graph-based formulation for computational characterization of bulk heterojunction morphology