2,234 research outputs found
Core Decomposition in Multilayer Networks: Theory, Algorithms, and Applications
Multilayer networks are a powerful paradigm to model complex systems, where
multiple relations occur between the same entities. Despite the keen interest
in a variety of tasks, algorithms, and analyses in this type of network, the
problem of extracting dense subgraphs has remained largely unexplored so far.
In this work we study the problem of core decomposition of a multilayer
network. The multilayer context is much challenging as no total order exists
among multilayer cores; rather, they form a lattice whose size is exponential
in the number of layers. In this setting we devise three algorithms which
differ in the way they visit the core lattice and in their pruning techniques.
We then move a step forward and study the problem of extracting the
inner-most (also known as maximal) cores, i.e., the cores that are not
dominated by any other core in terms of their core index in all the layers.
Inner-most cores are typically orders of magnitude less than all the cores.
Motivated by this, we devise an algorithm that effectively exploits the
maximality property and extracts inner-most cores directly, without first
computing a complete decomposition.
Finally, we showcase the multilayer core-decomposition tool in a variety of
scenarios and problems. We start by considering the problem of densest-subgraph
extraction in multilayer networks. We introduce a definition of multilayer
densest subgraph that trades-off between high density and number of layers in
which the high density holds, and exploit multilayer core decomposition to
approximate this problem with quality guarantees. As further applications, we
show how to utilize multilayer core decomposition to speed-up the extraction of
frequent cross-graph quasi-cliques and to generalize the community-search
problem to the multilayer setting
Fully Dynamic Algorithm for Top- Densest Subgraphs
Given a large graph, the densest-subgraph problem asks to find a subgraph
with maximum average degree. When considering the top- version of this
problem, a na\"ive solution is to iteratively find the densest subgraph and
remove it in each iteration. However, such a solution is impractical due to
high processing cost. The problem is further complicated when dealing with
dynamic graphs, since adding or removing an edge requires re-running the
algorithm. In this paper, we study the top- densest-subgraph problem in the
sliding-window model and propose an efficient fully-dynamic algorithm. The
input of our algorithm consists of an edge stream, and the goal is to find the
node-disjoint subgraphs that maximize the sum of their densities. In contrast
to existing state-of-the-art solutions that require iterating over the entire
graph upon any update, our algorithm profits from the observation that updates
only affect a limited region of the graph. Therefore, the top- densest
subgraphs are maintained by only applying local updates. We provide a
theoretical analysis of the proposed algorithm and show empirically that the
algorithm often generates denser subgraphs than state-of-the-art competitors.
Experiments show an improvement in efficiency of up to five orders of magnitude
compared to state-of-the-art solutions.Comment: 10 pages, 8 figures, accepted at CIKM 201
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
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