48,819 research outputs found
A Unifying Model for Representing Time-Varying Graphs
Graph-based models form a fundamental aspect of data representation in Data
Sciences and play a key role in modeling complex networked systems. In
particular, recently there is an ever-increasing interest in modeling dynamic
complex networks, i.e. networks in which the topological structure (nodes and
edges) may vary over time. In this context, we propose a novel model for
representing finite discrete Time-Varying Graphs (TVGs), which are typically
used to model dynamic complex networked systems. We analyze the data structures
built from our proposed model and demonstrate that, for most practical cases,
the asymptotic memory complexity of our model is in the order of the
cardinality of the set of edges. Further, we show that our proposal is an
unifying model that can represent several previous (classes of) models for
dynamic networks found in the recent literature, which in general are unable to
represent each other. In contrast to previous models, our proposal is also able
to intrinsically model cyclic (i.e. periodic) behavior in dynamic networks.
These representation capabilities attest the expressive power of our proposed
unifying model for TVGs. We thus believe our unifying model for TVGs is a step
forward in the theoretical foundations for data analysis of complex networked
systems.Comment: Also appears in the Proc. of the IEEE International Conference on
Data Science and Advanced Analytics (IEEE DSAA'2015
Simultaneous Representation of Proper and Unit Interval Graphs
In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs - the simultaneous version of arguably one of the most well-studied graph classes - is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more "rigid" and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G=(V,E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary
SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels
We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant
of deep neural networks for irregular structured and geometric input, e.g.,
graphs or meshes. Our main contribution is a novel convolution operator based
on B-splines, that makes the computation time independent from the kernel size
due to the local support property of the B-spline basis functions. As a result,
we obtain a generalization of the traditional CNN convolution operator by using
continuous kernel functions parametrized by a fixed number of trainable
weights. In contrast to related approaches that filter in the spectral domain,
the proposed method aggregates features purely in the spatial domain. In
addition, SplineCNN allows entire end-to-end training of deep architectures,
using only the geometric structure as input, instead of handcrafted feature
descriptors. For validation, we apply our method on tasks from the fields of
image graph classification, shape correspondence and graph node classification,
and show that it outperforms or pars state-of-the-art approaches while being
significantly faster and having favorable properties like domain-independence.Comment: Presented at CVPR 201
Graph Metrics for Temporal Networks
Temporal networks, i.e., networks in which the interactions among a set of
elementary units change over time, can be modelled in terms of time-varying
graphs, which are time-ordered sequences of graphs over a set of nodes. In such
graphs, the concepts of node adjacency and reachability crucially depend on the
exact temporal ordering of the links. Consequently, all the concepts and
metrics proposed and used for the characterisation of static complex networks
have to be redefined or appropriately extended to time-varying graphs, in order
to take into account the effects of time ordering on causality. In this chapter
we discuss how to represent temporal networks and we review the definitions of
walks, paths, connectedness and connected components valid for graphs in which
the links fluctuate over time. We then focus on temporal node-node distance,
and we discuss how to characterise link persistence and the temporal
small-world behaviour in this class of networks. Finally, we discuss the
extension of classic centrality measures, including closeness, betweenness and
spectral centrality, to the case of time-varying graphs, and we review the work
on temporal motifs analysis and the definition of modularity for temporal
graphs.Comment: 26 pages, 5 figures, Chapter in Temporal Networks (Petter Holme and
Jari Saram\"aki editors). Springer. Berlin, Heidelberg 201
Model validation of simple-graph representations of metabolism
The large-scale properties of chemical reaction systems, such as the
metabolism, can be studied with graph-based methods. To do this, one needs to
reduce the information -- lists of chemical reactions -- available in
databases. Even for the simplest type of graph representation, this reduction
can be done in several ways. We investigate different simple network
representations by testing how well they encode information about one
biologically important network structure -- network modularity (the propensity
for edges to be cluster into dense groups that are sparsely connected between
each other). To reach this goal, we design a model of reaction-systems where
network modularity can be controlled and measure how well the reduction to
simple graphs capture the modular structure of the model reaction system. We
find that the network types that best capture the modular structure of the
reaction system are substrate-product networks (where substrates are linked to
products of a reaction) and substance networks (with edges between all
substances participating in a reaction). Furthermore, we argue that the
proposed model for reaction systems with tunable clustering is a general
framework for studies of how reaction-systems are affected by modularity. To
this end, we investigate statistical properties of the model and find, among
other things, that it recreate correlations between degree and mass of the
molecules.Comment: to appear in J. Roy. Soc. Intefac
- …