7,105 research outputs found
Regression and Singular Value Decomposition in Dynamic Graphs
Most of real-world graphs are {\em dynamic}, i.e., they change over time.
However, while problems such as regression and Singular Value Decomposition
(SVD) have been studied for {\em static} graphs, they have not been
investigated for {\em dynamic} graphs, yet. In this paper, we introduce,
motivate and study regression and SVD over dynamic graphs. First, we present
the notion of {\em update-efficient matrix embedding} that defines the
conditions sufficient for a matrix embedding to be used for the dynamic graph
regression problem (under norm). We prove that given an
update-efficient matrix embedding (e.g., adjacency matrix), after an update
operation in the graph, the optimal solution of the graph regression problem
for the revised graph can be computed in time. We also study dynamic
graph regression under least absolute deviation. Then, we characterize a class
of matrix embeddings that can be used to efficiently update SVD of a dynamic
graph. For adjacency matrix and Laplacian matrix, we study those graph update
operations for which SVD (and low rank approximation) can be updated
efficiently
Large Scale Spectral Clustering Using Approximate Commute Time Embedding
Spectral clustering is a novel clustering method which can detect complex
shapes of data clusters. However, it requires the eigen decomposition of the
graph Laplacian matrix, which is proportion to and thus is not
suitable for large scale systems. Recently, many methods have been proposed to
accelerate the computational time of spectral clustering. These approximate
methods usually involve sampling techniques by which a lot information of the
original data may be lost. In this work, we propose a fast and accurate
spectral clustering approach using an approximate commute time embedding, which
is similar to the spectral embedding. The method does not require using any
sampling technique and computing any eigenvector at all. Instead it uses random
projection and a linear time solver to find the approximate embedding. The
experiments in several synthetic and real datasets show that the proposed
approach has better clustering quality and is faster than the state-of-the-art
approximate spectral clustering methods
Stability of graph communities across time scales
The complexity of biological, social and engineering networks makes it
desirable to find natural partitions into communities that can act as
simplified descriptions and provide insight into the structure and function of
the overall system. Although community detection methods abound, there is a
lack of consensus on how to quantify and rank the quality of partitions. We
show here that the quality of a partition can be measured in terms of its
stability, defined in terms of the clustered autocovariance of a Markov process
taking place on the graph. Because the stability has an intrinsic dependence on
time scales of the graph, it allows us to compare and rank partitions at each
time and also to establish the time spans over which partitions are optimal.
Hence the Markov time acts effectively as an intrinsic resolution parameter
that establishes a hierarchy of increasingly coarser clusterings. Within our
framework we can then provide a unifying view of several standard partitioning
measures: modularity and normalized cut size can be interpreted as one-step
time measures, whereas Fiedler's spectral clustering emerges at long times. We
apply our method to characterize the relevance and persistence of partitions
over time for constructive and real networks, including hierarchical graphs and
social networks. We also obtain reduced descriptions for atomic level protein
structures over different time scales.Comment: submitted; updated bibliography from v
Edge Label Inference in Generalized Stochastic Block Models: from Spectral Theory to Impossibility Results
The classical setting of community detection consists of networks exhibiting
a clustered structure. To more accurately model real systems we consider a
class of networks (i) whose edges may carry labels and (ii) which may lack a
clustered structure. Specifically we assume that nodes possess latent
attributes drawn from a general compact space and edges between two nodes are
randomly generated and labeled according to some unknown distribution as a
function of their latent attributes. Our goal is then to infer the edge label
distributions from a partially observed network. We propose a computationally
efficient spectral algorithm and show it allows for asymptotically correct
inference when the average node degree could be as low as logarithmic in the
total number of nodes. Conversely, if the average node degree is below a
specific constant threshold, we show that no algorithm can achieve better
inference than guessing without using the observations. As a byproduct of our
analysis, we show that our model provides a general procedure to construct
random graph models with a spectrum asymptotic to a pre-specified eigenvalue
distribution such as a power-law distribution.Comment: 17 page
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