1,898 research outputs found
A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem
Many graph mining applications rely on detecting subgraphs which are
near-cliques. There exists a dichotomy between the results in the existing work
related to this problem: on the one hand the densest subgraph problem (DSP)
which maximizes the average degree over all subgraphs is solvable in polynomial
time but for many networks fails to find subgraphs which are near-cliques. On
the other hand, formulations that are geared towards finding near-cliques are
NP-hard and frequently inapproximable due to connections with the Maximum
Clique problem.
In this work, we propose a formulation which combines the best of both
worlds: it is solvable in polynomial time and finds near-cliques when the DSP
fails. Surprisingly, our formulation is a simple variation of the DSP.
Specifically, we define the triangle densest subgraph problem (TDSP): given
, find a subset of vertices such that , where is the number of triangles induced
by the set . We provide various exact and approximation algorithms which the
solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to
the more general problem of maximizing the -clique average density. Finally,
we provide empirical evidence that the TDSP should be used whenever the output
of the DSP fails to output a near-clique.Comment: 42 page
Efficient External-Memory Algorithms for Graph Mining
The explosion of big data in areas like the web and social networks has posed big challenges to research activities, including data mining, information retrieval, security etc. This dissertation focuses on a particular area, graph mining, and specifically proposes several novel algorithms to solve the problems of triangle listing and computation of neighborhood function in large-scale graphs.
We first study the classic problem of triangle listing. We generalize the existing in-memory algorithms into a single framework of 18 triangle-search techniques. We then develop a novel external-memory approach, which we call Pruned Companion Files (PCF), that supports disk operation of all 18 algorithms. When compared to state-of-the-art available implementations MGT and PDTL, PCF runs 5-10 times faster and exhibits orders of magnitude less I/O.
We next focus on I/O complexity of triangle listing. Recent work by Pagh etc. provides an appealing theoretical I/O complexity for triangle listing via graph partitioning by random coloring of nodes. Since no implementation of Pagh is available and little is known about the comparison between Pagh and PCF, we carefully implement Pagh, undertake an investigation into the properties of these algorithms, model their I/O cost, understand their shortcomings, and shed light on the conditions under which each method defeats the other. This insight leads us to develop a novel framework we call Trigon that surpasses the I/O performance of both techniques in all graphs and under all RAM conditions.
We finally turn our attention to neighborhood function. Exact computation of neighborhood function is expensive in terms of CPU and I/O cost. Previous work mostly focuses on approximations. We show that our novel techniques developed for triangle listing can also be applied to this problem. We next study an application of neighborhood function to ranking of Internet hosts. Our method computes neighborhood functions for each host as an indication of its reputation. The evaluation shows that our method is robust to ranking manipulation and brings less spam to its top ranking list compared to PageRank and TrustRank
Dynamic Set Intersection
Consider the problem of maintaining a family of dynamic sets subject to
insertions, deletions, and set-intersection reporting queries: given , report every member of in any order. We show that in the word
RAM model, where is the word size, given a cap on the maximum size of
any set, we can support set intersection queries in
expected time, and updates in expected time. Using this algorithm
we can list all triangles of a graph in
expected time, where and
is the arboricity of . This improves a 30-year old triangle enumeration
algorithm of Chiba and Nishizeki running in time.
We provide an incremental data structure on that supports intersection
{\em witness} queries, where we only need to find {\em one} .
Both queries and insertions take O\paren{\sqrt \frac{N}{w/\log^2 w}} expected
time, where . Finally, we provide time/space tradeoffs for
the fully dynamic set intersection reporting problem. Using words of space,
each update costs expected time, each reporting query
costs expected time where
is the size of the output, and each witness query costs expected time.Comment: Accepted to WADS 201
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