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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

Given a fixed k-uniform hypergraph F , the F -removal lemma states that every hypergraph with few copies of F can be made F -free by the removal of few edges. Unfortunately, for general F , the constants involved are given by incredibly fast growing Ackermann-type functions. It is thus natural to ask for which F can one prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when F is k-partite. Alon proved that when k = 2 (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and Rödl conjectured in 2002 that Alon’s result can be extended to all k > 2, namely, that the only k-graphs F for which the hypergraph removal lemma has polynomial bounds are the trivial cases when F is k-partite. In this paper we prove this conjecture.

Hypergraph removal with polynomial bounds

We consider the following Tur\'an-type problem: given a fixed tournament $H$, what is the least integer $t=t(n,H)$ so that adding $t$ edges to any $n$-vertex tournament, results in a digraph containing a copy of $H$. Similarly, what is the least integer $t=t(T_n,H)$ so that adding $t$ edges to the $n$-vertex transitive tournament, results in a digraph containing a copy of $H$. Besides proving several results on these problems, our main contributions are the following: 
(1) Pach and Tardos conjectured that if $M$ is an acyclic $0/1$ matrix, then any $n \times n$ matrix with $n(\log n)^{O(1)}$ entries equal to $1$ contains the pattern $M$. We show that this conjecture is equivalent to the assertion that $t(T_n,H)=n(\log n)^{O(1)}$ if and only if $H$ belongs to a certain (natural) family of tournaments. 
(2) We propose an approach for determining if $t(n,H)=n(\log n)^{O(1)}$. This approach combines expansion in sparse graphs, together with certain structural characterizations of $H$-free tournaments. 
Our result opens the door for using structural graph theoretic tools in order to settle the Pach-Tardos conjecture.

A tournament approach to pattern avoiding matrices

Let Q ( n ) denote the number of integers 1 ≤ q ≤ n whose prime factorization q = satisﬁes a 1 ≥ a 2 ≥ . . . ≥ a t . Hardy and Ramanujan proved that Before proving the above precise asymptotic formula, they studied in great detail what can be obtained concerning Q ( n ) using purely elementary methods, and were only able to obtain much cruder lower and upper bounds using such methods. In this paper we show that it is in fact possible to obtain a purely elementary (and much shorter) proof of the Hardy–Ramanujan Theorem. Towards this goal, we ﬁrst give a simple combinatorial argument, showing that Q ( n ) satisﬁes a (pseudo) recurrence relation. This enables us to replace almost all the hard analytic part of the original proof with a short inductive argument.

An Elementary Proof of a Theorem of Hardy and Ramanujan

For fixed $m$ and $R\subseteq \{0,1,\ldots,m-1\}$, take $A$ to be the set of positive integers congruent modulo $m$ to one of the elements of $R$, and let $p_A(n)$ be the number of ways to write $n$ as a sum of elements of $A$. Nathanson proved that $\log p_A(n) \leq (1+o(1)) \pi \sqrt{2n|R|/3m}$ using a variant of a remarkably simple method devised by Erdős in order to bound the partition function. In this short note we describe a simpler and shorter proof of Nathanson's bound.

On Erd\H{o}s's Method for Bounding the Partition Function

For fixed m and R⊆{0,1,…,m−1} , take A to be the set of positive integers congruent modulo m to one of the elements of R, and let pA(n) be the number of ways to write n as a sum of elements of A. N...

Asaf Shapira

Papers

Hypergraph removal with polynomial bounds

A tournament approach to pattern avoiding matrices

An Elementary Proof of a Theorem of Hardy and Ramanujan

On Erd\H{o}s's Method for Bounding the Partition Function

On Erdős’s Method for Bounding the Partition Function