4,797 research outputs found
Different approaches to community detection
A precise definition of what constitutes a community in networks has remained
elusive. Consequently, network scientists have compared community detection
algorithms on benchmark networks with a particular form of community structure
and classified them based on the mathematical techniques they employ. However,
this comparison can be misleading because apparent similarities in their
mathematical machinery can disguise different reasons for why we would want to
employ community detection in the first place. Here we provide a focused review
of these different motivations that underpin community detection. This
problem-driven classification is useful in applied network science, where it is
important to select an appropriate algorithm for the given purpose. Moreover,
highlighting the different approaches to community detection also delineates
the many lines of research and points out open directions and avenues for
future research.Comment: 14 pages, 2 figures. Written as a chapter for forthcoming Advances in
network clustering and blockmodeling, and based on an extended version of The
many facets of community detection in complex networks, Appl. Netw. Sci. 2: 4
(2017) by the same author
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
Maintaining Expander Decompositions via Sparse Cuts
In this article, we show that the algorithm of maintaining expander
decompositions in graphs undergoing edge deletions directly by removing sparse
cuts repeatedly can be made efficient. Formally, for an -edge undirected
graph , we say a cut is -sparse if . A
-expander decomposition of is a partition of into sets such that each cluster contains no -sparse cut
(meaning it is a -expander) with edges crossing
between clusters. A natural way to compute a -expander decomposition is
to decompose clusters by -sparse cuts until no such cut is contained in
any cluster. We show that even in graphs undergoing edge deletions, a slight
relaxation of this meta-algorithm can be implemented efficiently with amortized
update time . Our approach naturally extends to maintaining
directed -expander decompositions and -expander hierarchies and
thus gives a unifying framework while having simpler proofs than previous
state-of-the-art work. In all settings, our algorithm matches the run-times of
previous algorithms up to subpolynomial factors. Moreover, our algorithm
provides stronger guarantees for -expander decompositions. For example,
for graphs undergoing edge deletions, our approach is the first to maintain a
dynamic expander decomposition where each updated decomposition is a refinement
of the previous decomposition, and our approach is the first to guarantee a
sublinear bound on the total number of edges that cross
between clusters across the entire sequence of dynamic updates
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