5,577 research outputs found
Finding the Graph of Epidemic Cascades
We consider the problem of finding the graph on which an epidemic cascade
spreads, given only the times when each node gets infected. While this is a
problem of importance in several contexts -- offline and online social
networks, e-commerce, epidemiology, vulnerabilities in infrastructure networks
-- there has been very little work, analytical or empirical, on finding the
graph. Clearly, it is impossible to do so from just one cascade; our interest
is in learning the graph from a small number of cascades.
For the classic and popular "independent cascade" SIR epidemics, we
analytically establish the number of cascades required by both the global
maximum-likelihood (ML) estimator, and a natural greedy algorithm. Both results
are based on a key observation: the global graph learning problem decouples
into local problems -- one for each node. For a node of degree , we show
that its neighborhood can be reliably found once it has been infected times (for ML on general graphs) or times (for greedy on
trees). We also provide a corresponding information-theoretic lower bound of
; thus our bounds are essentially tight. Furthermore, if we
are given side-information in the form of a super-graph of the actual graph (as
is often the case), then the number of cascade samples required -- in all cases
-- becomes independent of the network size .
Finally, we show that for a very general SIR epidemic cascade model, the
Markov graph of infection times is obtained via the moralization of the network
graph.Comment: To appear in Proc. ACM SIGMETRICS/Performance 201
Learning loopy graphical models with latent variables: Efficient methods and guarantees
The problem of structure estimation in graphical models with latent variables
is considered. We characterize conditions for tractable graph estimation and
develop efficient methods with provable guarantees. We consider models where
the underlying Markov graph is locally tree-like, and the model is in the
regime of correlation decay. For the special case of the Ising model, the
number of samples required for structural consistency of our method scales
as , where p is the
number of variables, is the minimum edge potential, is
the depth (i.e., distance from a hidden node to the nearest observed nodes),
and is a parameter which depends on the bounds on node and edge
potentials in the Ising model. Necessary conditions for structural consistency
under any algorithm are derived and our method nearly matches the lower bound
on sample requirements. Further, the proposed method is practical to implement
and provides flexibility to control the number of latent variables and the
cycle lengths in the output graph.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1070 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
High-Dimensional Gaussian Graphical Model Selection: Walk Summability and Local Separation Criterion
We consider the problem of high-dimensional Gaussian graphical model
selection. We identify a set of graphs for which an efficient estimation
algorithm exists, and this algorithm is based on thresholding of empirical
conditional covariances. Under a set of transparent conditions, we establish
structural consistency (or sparsistency) for the proposed algorithm, when the
number of samples n=omega(J_{min}^{-2} log p), where p is the number of
variables and J_{min} is the minimum (absolute) edge potential of the graphical
model. The sufficient conditions for sparsistency are based on the notion of
walk-summability of the model and the presence of sparse local vertex
separators in the underlying graph. We also derive novel non-asymptotic
necessary conditions on the number of samples required for sparsistency
Graphs in machine learning: an introduction
Graphs are commonly used to characterise interactions between objects of
interest. Because they are based on a straightforward formalism, they are used
in many scientific fields from computer science to historical sciences. In this
paper, we give an introduction to some methods relying on graphs for learning.
This includes both unsupervised and supervised methods. Unsupervised learning
algorithms usually aim at visualising graphs in latent spaces and/or clustering
the nodes. Both focus on extracting knowledge from graph topologies. While most
existing techniques are only applicable to static graphs, where edges do not
evolve through time, recent developments have shown that they could be extended
to deal with evolving networks. In a supervised context, one generally aims at
inferring labels or numerical values attached to nodes using both the graph
and, when they are available, node characteristics. Balancing the two sources
of information can be challenging, especially as they can disagree locally or
globally. In both contexts, supervised and un-supervised, data can be
relational (augmented with one or several global graphs) as described above, or
graph valued. In this latter case, each object of interest is given as a full
graph (possibly completed by other characteristics). In this context, natural
tasks include graph clustering (as in producing clusters of graphs rather than
clusters of nodes in a single graph), graph classification, etc. 1 Real
networks One of the first practical studies on graphs can be dated back to the
original work of Moreno [51] in the 30s. Since then, there has been a growing
interest in graph analysis associated with strong developments in the modelling
and the processing of these data. Graphs are now used in many scientific
fields. In Biology [54, 2, 7], for instance, metabolic networks can describe
pathways of biochemical reactions [41], while in social sciences networks are
used to represent relation ties between actors [66, 56, 36, 34]. Other examples
include powergrids [71] and the web [75]. Recently, networks have also been
considered in other areas such as geography [22] and history [59, 39]. In
machine learning, networks are seen as powerful tools to model problems in
order to extract information from data and for prediction purposes. This is the
object of this paper. For more complete surveys, we refer to [28, 62, 49, 45].
In this section, we introduce notations and highlight properties shared by most
real networks. In Section 2, we then consider methods aiming at extracting
information from a unique network. We will particularly focus on clustering
methods where the goal is to find clusters of vertices. Finally, in Section 3,
techniques that take a series of networks into account, where each network i
Active Learning for Undirected Graphical Model Selection
This paper studies graphical model selection, i.e., the problem of estimating
a graph of statistical relationships among a collection of random variables.
Conventional graphical model selection algorithms are passive, i.e., they
require all the measurements to have been collected before processing begins.
We propose an active learning algorithm that uses junction tree representations
to adapt future measurements based on the information gathered from prior
measurements. We prove that, under certain conditions, our active learning
algorithm requires fewer scalar measurements than any passive algorithm to
reliably estimate a graph. A range of numerical results validate our theory and
demonstrates the benefits of active learning.Comment: AISTATS 201
Structure Selection from Streaming Relational Data
Statistical relational learning techniques have been successfully applied in
a wide range of relational domains. In most of these applications, the human
designers capitalized on their background knowledge by following a
trial-and-error trajectory, where relational features are manually defined by a
human engineer, parameters are learned for those features on the training data,
the resulting model is validated, and the cycle repeats as the engineer adjusts
the set of features. This paper seeks to streamline application development in
large relational domains by introducing a light-weight approach that
efficiently evaluates relational features on pieces of the relational graph
that are streamed to it one at a time. We evaluate our approach on two social
media tasks and demonstrate that it leads to more accurate models that are
learned faster
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