75 research outputs found
Algorithmic information and incompressibility of families of multidimensional networks
This article presents a theoretical investigation of string-based generalized
representations of families of finite networks in a multidimensional space.
First, we study the recursive labeling of networks with (finite) arbitrary node
dimensions (or aspects), such as time instants or layers. In particular, we
study these networks that are formalized in the form of multiaspect graphs. We
show that, unlike classical graphs, the algorithmic information of a
multidimensional network is not in general dominated by the algorithmic
information of the binary sequence that determines the presence or absence of
edges. This universal algorithmic approach sets limitations and conditions for
irreducible information content analysis in comparing networks with a large
number of dimensions, such as multilayer networks. Nevertheless, we show that
there are particular cases of infinite nesting families of finite
multidimensional networks with a unified recursive labeling such that each
member of these families is incompressible. From these results, we study
network topological properties and equivalences in irreducible information
content of multidimensional networks in comparison to their isomorphic
classical graph.Comment: Extended preprint version of the pape
An algorithmically random family of MultiAspect Graphs and its topological properties
This article presents a theoretical investigation of incompressibility and randomness in generalized representations of graphs along with its implications on network topological properties. We extend previous studies on plain algorithmically random classical graphs to plain and prefix algorithmically random MultiAspect Graphs (MAGs). First, we show that there is an infinite recursively labeled infinite family of nested MAGs (or, as a particular case, of nested classical graphs) that behaves like (and is determined by) an algorithmically random real number. Then, we study some of their important topological properties, in particular, vertex degree, connectivity, diameter, and rigidity
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
MultiAspect Graphs: Algebraic representation and algorithms
We present the algebraic representation and basic algorithms for MultiAspect
Graphs (MAGs). A MAG is a structure capable of representing multilayer and
time-varying networks, as well as higher-order networks, while also having the
property of being isomorphic to a directed graph. In particular, we show that,
as a consequence of the properties associated with the MAG structure, a MAG can
be represented in matrix form. Moreover, we also show that any possible MAG
function (algorithm) can be obtained from this matrix-based representation.
This is an important theoretical result since it paves the way for adapting
well-known graph algorithms for application in MAGs. We present a set of basic
MAG algorithms, constructed from well-known graph algorithms, such as degree
computing, Breadth First Search (BFS), and Depth First Search (DFS). These
algorithms adapted to the MAG context can be used as primitives for building
other more sophisticated MAG algorithms. Therefore, such examples can be seen
as guidelines on how to properly derive MAG algorithms from basic algorithms on
directed graph. We also make available Python implementations of all the
algorithms presented in this paper.Comment: 59 pages, 6 figure
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