Results 1  10
of
129
Graph mining: laws, generators, and algorithms
 ACM COMPUT SURV (CSUR
, 2006
"... How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M: N relation in ..."
Abstract

Cited by 131 (7 self)
 Add to MetaCart
How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M: N relation in database terminology can be represented as a graph. A lot of these questions boil down to the following: “How can we generate synthetic but realistic graphs? ” To answer this, we must first understand what patterns are common in realworld graphs and can thus be considered a mark of normality/realism. This survey give an overview of the incredible variety of work that has been done on these problems. One of our main contributions is the integration of points of view from physics, mathematics, sociology, and computer science. Further, we briefly describe recent advances on some related and interesting graph problems.
All Pairs Almost Shortest Paths
 SIAM Journal on Computing
, 1996
"... Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive onesided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time ..."
Abstract

Cited by 95 (7 self)
 Add to MetaCart
(Show Context)
Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive onesided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time algorithm APASP 2 for computing all distances in G with an additive onesided error of at most 2. The algorithm APASP 2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k ? 2, we describe an g) time algorithm APASP k for computing all distances in G with an additive onesided error of at most k.
Quantum algorithms for the triangle problem
 PROCEEDINGS OF SODA’05
, 2005
"... We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is b ..."
Abstract

Cited by 93 (10 self)
 Add to MetaCart
(Show Context)
We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.
Efficient semistreaming algorithms for local triangle counting in massive graphs
 in KDD’08, 2008
"... In this paper we study the problem of local triangle counting in large graphs. Namely, given a large graph G = (V, E) we want to estimate as accurately as possible the number of triangles incident to every node v ∈ V in the graph. The problem of computing the global number of triangles in a graph ha ..."
Abstract

Cited by 70 (4 self)
 Add to MetaCart
(Show Context)
In this paper we study the problem of local triangle counting in large graphs. Namely, given a large graph G = (V, E) we want to estimate as accurately as possible the number of triangles incident to every node v ∈ V in the graph. The problem of computing the global number of triangles in a graph has been considered before, but to our knowledge this is the first paper that addresses the problem of local triangle counting with a focus on the efficiency issues arising in massive graphs. The distribution of the local number of triangles and the related local clustering coefficient can be used in many interesting applications. For example, we show that the measures we compute can help to detect the presence of spamming activity in largescale Web graphs, as well as to provide useful features to assess content quality in social networks. For computing the local number of triangles we propose two approximation algorithms, which are based on the idea of minwise independent permutations (Broder et al. 1998). Our algorithms operate in a semistreaming fashion, using O(V ) space in main memory and performing O(log V ) sequential scans over the edges of the graph. The first algorithm we describe in this paper also uses O(E) space in external memory during computation, while the second algorithm uses only main memory. We present the theoretical analysis as well as experimental results in massive graphs demonstrating the practical efficiency of our approach. Luca Becchetti was partially supported by EU Integrated
Parameterized complexity and approximation algorithms
 Comput. J
, 2006
"... Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We ..."
Abstract

Cited by 60 (2 self)
 Add to MetaCart
(Show Context)
Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research. 1.
Parameterized graph separation problems
 In Proc. 1st IWPEC, volume 3162 of LNCS
, 2004
"... We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminal ..."
Abstract

Cited by 56 (3 self)
 Add to MetaCart
(Show Context)
We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminals is separated, (c) exactly ℓ vertices are cut away from the graph, (d) exactly ℓ connected vertices are cut away from the graph, (e) the graph is separated into at least ℓ components. We show that if both k and ℓ are
Fast Sparse Matrix Multiplication
, 2004
"... Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multi ..."
Abstract

Cited by 56 (3 self)
 Add to MetaCart
Let A and B two n n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m ) algebraic operations (i.e., multiplications, additions and subtractions) over R. The naive matrix multiplication algorithm, on the other hand, may need to perform #(mn) operations to accomplish the same task. For , the new algorithm performs an almost optimal number of only n operations. For m the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses O(n ) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast rectangular matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices.
Doulion: Counting Triangles in Massive Graphs with a Coin
 PROCEEDINGS OF ACM KDD,
, 2009
"... Counting the number of triangles in a graph is a beautiful algorithmic problem which has gained importance over the last years due to its significant role in complex network analysis. Metrics frequently computed such as the clustering coefficient and the transitivity ratio involve the execution of a ..."
Abstract

Cited by 53 (16 self)
 Add to MetaCart
Counting the number of triangles in a graph is a beautiful algorithmic problem which has gained importance over the last years due to its significant role in complex network analysis. Metrics frequently computed such as the clustering coefficient and the transitivity ratio involve the execution of a triangle counting algorithm. Furthermore, several interesting graph mining applications rely on computing the number of triangles in the graph of interest. In this paper, we focus on the problem of counting triangles in a graph. We propose a practical method, out of which all triangle counting algorithms can potentially benefit. Using a straightforward triangle counting algorithm as a black box, we performed 166 experiments on realworld networks and on synthetic datasets as well, where we show that our method works with high accuracy, typically more than 99 % and gives significant speedups, resulting in even ≈ 130 times faster performance.
Fast Counting of Triangles in Large Real Networks: Algorithms and Laws
"... How can we quickly find the number of triangles in a large graph, without actually counting them? Triangles are important for real world social networks, lying at the heart of the clustering coefficient and of the transitivity ratio. However, straightforward and even approximate counting algorithms ..."
Abstract

Cited by 52 (10 self)
 Add to MetaCart
(Show Context)
How can we quickly find the number of triangles in a large graph, without actually counting them? Triangles are important for real world social networks, lying at the heart of the clustering coefficient and of the transitivity ratio. However, straightforward and even approximate counting algorithms can be slow, trying to execute or approximate the equivalent of a 3way database join. In this paper, we provide two algorithms, the EigenTriangle for counting the total number of triangles in a graph, and the EigenTriangleLocal algorithm that gives the count of triangles that contain a desired node. Additional contributions include the following: (a) We show that both algorithms achieve excellent accuracy, with up to ≈ 1000x faster execution time, on several, real graphs and (b) we discover two new power laws ( DegreeTriangle and TriangleParticipation laws) with surprising properties. Figure 1. Speedup ratio versus accuracy for the Wikipedia web graph ( ≈ 3, 1M nodes, ≈ 37M edges). Proposed method achieves 1021x faster time, for 97.4 % accuracy, compared to a typical competitor, the Node Iterator method. 1
The parameterized complexity of counting problems
 SIAM Journal on Computing
, 2002
"... We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W[t], for t ≥ 1, that corresponds to Downey and Fellows’s Whierarchy [13] and show that a few central Wcompleteness results for de ..."
Abstract

Cited by 49 (0 self)
 Add to MetaCart
We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W[t], for t ≥ 1, that corresponds to Downey and Fellows’s Whierarchy [13] and show that a few central Wcompleteness results for decision problems translate to #Wcompleteness results for the corresponding counting problems. Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, is #W[1]complete. This makes it highly unlikely that any of these problems is fixedparameter tractable, even though their decision versions are fixedparameter tractable. More explicitly, our result shows that most likely there is no f(k) · n calgorithm for counting cycles or paths of length k in a graph of size n for any computable function f: N → N and constant c, even though there is a 2 O(k) · n 2.376 algorithm for finding a cycle or path of length k [2]. 1