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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

Models for the processes by which ideas and influence propagate through a social network have been studied in a number of domains, including the diffusion of medical and technological innovations, the sudden and widespread adoption of various strategies in game-theoretic settings, and the effects of "word of mouth" in the promotion of new products. Recently, motivated by the design of viral marketing strategies, Domingos and Richardson posed a fundamental algorithmic problem for such social network processes: if we can try to convince a subset of individuals to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target?We consider this problem in several of the most widely studied models in social network analysis. The optimization problem of selecting the most influential nodes is NP-hard here, and we provide the first provable approximation guarantees for efficient algorithms. Using an analysis framework based on submodular functions, we show that a natural greedy strategy obtains a solution that is provably within 63% of optimal for several classes of models; our framework suggests a general approach for reasoning about the performance guarantees of algorithms for these types of influence problems in social networks.We also provide computational experiments on large collaboration networks, showing that in addition to their provable guarantees, our approximation algorithms significantly out-perform node-selection heuristics based on the well-studied notions of degree centrality and distance centrality from the field of social networks.

/pdf/maximizing-the-spread-of-influence-through-a-social-network-47y1ehuvwi.pdf

Maximizing the Spread of Influence through a Social Network

Tables of integrals

1 Introduction 2 Map 3 The Data Stream Phenomenon 4 Data Streaming: Formal Aspects 5 Foundations: Basic Mathematical Ideas 6 Foundations: Basic Algorithmic Techniques 7 Foundations: Summary 8 Streaming Systems 9 New Directions 10 Historic Notes 11 Concluding Remarks Acknowledgements References

/pdf/data-streams-algorithms-and-applications-1046wbaqok.pdf

Data Streams: Algorithms and Applications

We study the problem of testing the expansion of graphs with bounded degree d in sublinear time. A graph is said to be an @a-expander if every vertex set U@?V of size at most 12|V| has a neighborhood of size at least @a|U|. We show that the algorithm proposed by Goldreich and Ron [9] (ECCC-2000) for testing the expansion of a graph distinguishes with high probability between @a-expanders of degree bound d and graphs which are -far from having expansion at least @W(@a^2). This improves a recent result of Czumaj and Sohler [3] (FOCS-07) who showed that this algorithm can distinguish between @a-expanders of degree bound d and graphs which are -far from having expansion at least @W(@a^2/logn). It also improves a recent result of Kale and Seshadhri [12] (ECCC-2007) who showed that this algorithm can distinguish between @a-expanders and graphs which are -far from having expansion at least @W(@a^2) with twice the maximum degree. Our methods combine the techniques of [3], [9] and [12].

/pdf/testing-the-expansion-of-a-graph-oirn49nrsu.pdf

Testing the expansion of a graph

/pdf/exact-bounds-for-some-hypergraph-saturation-problems-1oq5smd1gv.pdf

Exact bounds for some hypergraph saturation problems

In the study of property testing, a particularly important role has been played by linear invariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed-Muller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linear-invariant properties, drawing on techniques from additive combinatorics and from graph theory.

A unified framework for testing linear-invariant properties

We show that if a graph G has the property that all subsets of vertices of size n/4 contain the “correct” number of triangles one would expect to find in a random graph G(n, 1/2), then G behaves like a random graph, that is, it is quasi-random in the sense of Chung, Graham, and Wilson [6]. This answers positively an open problem of Simonovits and Sos [10], who showed that in order to deduce that G is quasi-random one needs to assume that all sets of vertices have the correct number of triangles. A similar improvement of [10] is also obtained for any fixed graph other than the triangle, and for any edge density other than 1/2. The proof relies on a theorem of Gottlieb [7] in algebraic combinatorics, concerning the rank of set inclusion matrices.

/pdf/quasi-randomness-and-the-distribution-of-copies-of-a-fixed-3kvyuhym5z.pdf

Quasi-randomness and the distribution of copies of a fixed graph

Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a very small portion of the input. We discuss the types of answers that one can hope to achieve in this setting.

Asaf Shapira

Papers

Testing the expansion of a graph

Exact bounds for some hypergraph saturation problems

A unified framework for testing linear-invariant properties

Quasi-randomness and the distribution of copies of a fixed graph

Sublinear Time Algorithms.