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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

Let H be a k-uniform hypergraph whose vertices are the integers 1,...,N. We say that H contains a monotone path of length n if there are x_1 < x_2 < ... < x_{n+k-1} so that H contains all n edges of the form {x_i,x_{i+1},...,x_{i+k-1}}. Let N_k(q,n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of N_k(q,n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdos and Szekeres, the problem of bounding N_k(q,n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk. 
Our main contribution here is a novel approach for bounding the Ramsey-type numbers N_k(q,n), based on establishing a surprisingly tight connection between them and the enumerative problem of counting high-dimensional integer partitions. Some of the concrete results we obtain using this approach are the following: 
1. We show that for every fixed q we have N_3(q,n)=2^{\Theta(n^{q-1})}, thus resolving an open problem raised by Fox et al. 
2. We show that for every k >= 3, N_k(2,n)=2^{\cdot^{\cdot^{2^{(2-o(1))n}}}} where the height of the tower is k-2, thus resolving an open problem raised by Elias and Matousek. 
3. We give a new pigeonhole proof of the Erd\H{o}s-Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdos-Szekeres Lemma on increasing/decreasing subsequences.

Ramsey Theory, Integer Partitions and a New Proof of the Erdos-Szekeres Theorem

The arithmetic regularity lemma due to Green (GAFA 2005) is an analogue of the famous Szemeredi regularity lemma in graph theory. It shows that for any abelian group G and any bounded function f : G → (0,1), there exists a subgroup H ≤ G of bounded index such that, when restricted to most cosets of H, the function f is pseudorandom in the sense that all its nontrivial Fourier coefficients are s Quantitatively, if one wishes to obtain that for 1−ǫ fraction of the cosets, the nontrivial Fourier coefficients are bounded by ǫ, then Green shows that |G/H| is bounded by a tower of twos of height 1/ǫ 3 . He also gives an example showing that a tower of height (log1/ǫ) is necessary. Here, we give an improved example, showing that a tower of height (1/ǫ) is necessary.

An Improved Lower Bound for Arithmetic Regularity

We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well as graphs generated by various random processes (e.g. preferential attachment graphs). We say that $H$ is easy if there is a linear-time algorithm for counting the number of copies of $H$ in an input $G$ of bounded degeneracy. A seminal result of Chiba and Nishizeki from '85 states that every $H$ on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all $H$ on 5 vertices, and further proved that for every $k > 5$ there is a $k$-vertex $H$ which is not easy. They left open the natural problem of characterizing all easy graphs $H$. 
Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph $H$ to be easy. Here we show that this sufficient condition is also necessary, thus fully answering the Bera--Pashanasangi--Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms. Our proofs rely on several novel approaches for proving hardness results in the context of subgraph-counting.

Counting Subgraphs in Degenerate Graphs

Obtaining an efficient bound for the triangle removal lemma is one of the most outstanding open problems of extremal combinatorics. Perhaps the main bottleneck for achieving this goal is that triangle-free graphs can be highly unstructured. For example, triangle-free graphs might have only regular partitions (in the sense of Szemeredi) of tower-type size. And indeed, essentially all the graph properties $$\mathcal{P}$$
 for which removal lemmas with reasonable bounds were obtained, are such that every graph satisfying $$\mathcal{P}$$
 has a small regular partition. So in some sense, a barrier for obtaining an efficient removal lemma for property $$\mathcal{P}$$
 was not having an efficient regularity lemma for graphs satisfying $$\mathcal{P}$$
 . In this paper we consider the property of being induced C4-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for the induced C4 removal lemma. We thus obtain the first efficient removal lemma that does not rely on an efficient version of the regularity lemma. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox.

Efficient Removal Without Efficient Regularity

A graph property $\mathcal{P}$ is said to be testable if one can check whether a graph is close or far from satisfying $\mathcal{P}$ using few random local inspections. Property $\mathcal{P}$ is said to be non-deterministically testable if one can supply a “certificate” to the fact that a graph satisfies $\mathcal{P}$ so that once the certificate is given its correctness can be tested. The notion of non-deterministic testing of graph properties was recently introduced by Lovasz and Vesztergombi [9], who proved that (somewhat surprisingly) a graph property is testable if and only if it is non-deterministically testable. Their proof used graph limits, and so it did not supply any explicit bounds. They thus asked if one can obtain a proof of their result which will supply such bounds. We answer their question positively by proving their result using Szemeredi’s regularity lemma.

Asaf Shapira

Papers

Ramsey Theory, Integer Partitions and a New Proof of the Erdos-Szekeres Theorem

An Improved Lower Bound for Arithmetic Regularity

Counting Subgraphs in Degenerate Graphs

Efficient Removal Without Efficient Regularity

Deterministic vs Non-deterministic Graph Property Testing.