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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

A common thread in recent results concerning the testing of dense graphs is the use of Szemeredi's regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemeredi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemeredi-partitions. This means that in some sense, testing for Szemeredi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron [25] that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable.

/pdf/a-combinatorial-characterization-of-the-testable-graph-4ln9brali4.pdf

A combinatorial characterization of the testable graph properties: it's all about regularity

We consider the spread maximization problem that was defined by Domingos and Richardson [6,15] In this problem, we are given a social network represented as a graph and are required to find the set of the most "influential" individuals that by introducing them with a new technology, we maximize the expected number of individuals in the network, later in time, that adopt the new technology This problem has applications in viral marketing, where a company may wish to spread the rumor of a new product via the most influential individuals in popular social networks such as Myspace and Blogsphere

The spread maximization problem was recently studied in several models of social networks [10,11,13] In this short paper we study this problem in the context of the well studied probabilistic voter model We provide very simple and efficient algorithms for solving this problem An interesting special case of our result is that the most natural heuristic solution, which picks the nodes in the network with the highest degree, is indeed the optimal solution

/pdf/a-note-on-maximizing-the-spread-of-influence-in-social-km6yi0uc9e.pdf

A note on maximizing the spread of influence in social networks

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property testing. Our main result in this paper is a solution of an important special case of this general problem: Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property ${\cal P}$ has an oblivious one-sided error tester if and only if ${\cal P}$ is semihereditary. We stress that any “natural” property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the natural graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. More importantly, as a special case of our main result, we infer that some of the most well-studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well-known graph properties of being perfect, chordal, interval, comparability, permutation, and more. None of these properties was previously known to be testable.

/pdf/a-characterization-of-the-natural-graph-properties-testable-hezkck1adb.pdf

A Characterization of the (Natural) Graph Properties Testable with One-Sided Error

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property-testing. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property P has an oblivious one-sided error tester, if and only if P is (semi) hereditary. We stress that any "natural" property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the "natural" graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. These include the results of Goldreich et al., [1998] about testing k-colorability, the characterization of Goldreich and Trevisan [2001] of the graph-partition problems that are testable with 1-sided error, the induced vertex colorability properties of Alon et al., [2000], the induced edge colorability properties of Fischer [2001], a transformation from 2-sided to 1-sided error testing [Goldreich and Trevisan, 2001], as well as a recent result about testing monotone graph properties [Alon and Shapira, 2005]. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well known graph properties of being perfect, chordal, interval, comparability and more. None of these properties was previously known to be testable.

/pdf/a-characterization-of-the-natural-graph-properties-testable-3gr0vicij1.pdf

A characterization of the (natural) graph properties testable with one-sided error

Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least e n2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f(e,H) nh copies of H. This is proved by establishing a directed version of Szemeredi's regularity lemma, and implies that for every H there is a one-sided error property tester whose query complexity is bounded by a function of e only for testing the property PH of being H-free.

/pdf/testing-subgraphs-in-directed-graphs-1i7br83rpw.pdf

Asaf Shapira

Papers

A combinatorial characterization of the testable graph properties: it's all about regularity

A note on maximizing the spread of influence in social networks

A Characterization of the (Natural) Graph Properties Testable with One-Sided Error

A characterization of the (natural) graph properties testable with one-sided error

Testing subgraphs in directed graphs