293 research outputs found
Top-k overlapping densest subgraphs
International audienceFinding dense subgraphs is an important problem in graph mining and has many practical applications. At the same time, while large real-world networks are known to have many communities that are not well-separated, the majority of the existing work focuses on the problem of finding a single densest subgraph. Hence, it is natural to consider the question of finding the top-kdensest subgraphs. One major challenge in addressing this question is how to handle overlaps: eliminating overlaps completely is one option, but this may lead to extracting subgraphs not as dense as it would be possible by allowing a limited amount of overlap. Furthermore, overlaps are desirable as in most real-world graphs there are vertices that belong to more than one community, and thus, to more than one densest subgraph. In this paper we study the problem of finding top-koverlapping densest subgraphs, and we present a new approach that improves over the existing techniques, both in theory and practice. First, we reformulate the problem definition in a way that we are able to obtain an algorithm with constant-factor approximation guarantee. Our approach relies on using techniques for solving the max-sum diversification problem, which however, we need to extend in order to make them applicable to our setting. Second, we evaluate our algorithm on a collection of benchmark datasets and show that it convincingly outperforms the previous methods, both in terms of quality and efficiency
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
Dense packing on uniform lattices
We study the Hard Core Model on the graphs
obtained from Archimedean tilings i.e. configurations in with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.Comment: 18 page
Approximate Closest Community Search in Networks
Recently, there has been significant interest in the study of the community
search problem in social and information networks: given one or more query
nodes, find densely connected communities containing the query nodes. However,
most existing studies do not address the "free rider" issue, that is, nodes far
away from query nodes and irrelevant to them are included in the detected
community. Some state-of-the-art models have attempted to address this issue,
but not only are their formulated problems NP-hard, they do not admit any
approximations without restrictive assumptions, which may not always hold in
practice.
In this paper, given an undirected graph G and a set of query nodes Q, we
study community search using the k-truss based community model. We formulate
our problem of finding a closest truss community (CTC), as finding a connected
k-truss subgraph with the largest k that contains Q, and has the minimum
diameter among such subgraphs. We prove this problem is NP-hard. Furthermore,
it is NP-hard to approximate the problem within a factor , for
any . However, we develop a greedy algorithmic framework,
which first finds a CTC containing Q, and then iteratively removes the furthest
nodes from Q, from the graph. The method achieves 2-approximation to the
optimal solution. To further improve the efficiency, we make use of a compact
truss index and develop efficient algorithms for k-truss identification and
maintenance as nodes get eliminated. In addition, using bulk deletion
optimization and local exploration strategies, we propose two more efficient
algorithms. One of them trades some approximation quality for efficiency while
the other is a very efficient heuristic. Extensive experiments on 6 real-world
networks show the effectiveness and efficiency of our community model and
search algorithms
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