241 research outputs found
Complexity of Road Coloring with Prescribed Reset Words
By the Road Coloring Theorem (Trahtman, 2008), the edges of any aperiodic
directed multigraph with a constant out-degree can be colored such that the
resulting automaton admits a reset word. There may also be a need for a
particular reset word to be admitted. For certain words it is NP-complete to
decide whether there is a suitable coloring of a given multigraph. We present a
classification of all words over the binary alphabet that separates such words
from those that make the problem solvable in polynomial time. We show that the
classification becomes different if we consider only strongly connected
multigraphs. In this restricted setting the classification remains incomplete.Comment: To be presented at LATA 201
Enhancing the Guidance of the Intentional Model "MAP": Graph Theory Application
The MAP model was introduced in information system engineering in order to
model processes on a flexible way. The intentional level of this model helps an
engineer to execute a process with a strong relationship to the situation of
the project at hand. In the literature, attempts for having a practical use of
maps are not numerous. Our aim is to enhance the guidance mechanisms of the
process execution by reusing graph algorithms. After clarifying the existing
relationship between graphs and maps, we improve the MAP model by adding
qualitative criteria. We then offer a way to express maps with graphs and
propose to use Graph theory algorithms to offer an automatic guidance of the
map. We illustrate our proposal by an example and discuss its limitations.Comment: 9 page
Multilayer Networks
In most natural and engineered systems, a set of entities interact with each
other in complicated patterns that can encompass multiple types of
relationships, change in time, and include other types of complications. Such
systems include multiple subsystems and layers of connectivity, and it is
important to take such "multilayer" features into account to try to improve our
understanding of complex systems. Consequently, it is necessary to generalize
"traditional" network theory by developing (and validating) a framework and
associated tools to study multilayer systems in a comprehensive fashion. The
origins of such efforts date back several decades and arose in multiple
disciplines, and now the study of multilayer networks has become one of the
most important directions in network science. In this paper, we discuss the
history of multilayer networks (and related concepts) and review the exploding
body of work on such networks. To unify the disparate terminology in the large
body of recent work, we discuss a general framework for multilayer networks,
construct a dictionary of terminology to relate the numerous existing concepts
to each other, and provide a thorough discussion that compares, contrasts, and
translates between related notions such as multilayer networks, multiplex
networks, interdependent networks, networks of networks, and many others. We
also survey and discuss existing data sets that can be represented as
multilayer networks. We review attempts to generalize single-layer-network
diagnostics to multilayer networks. We also discuss the rapidly expanding
research on multilayer-network models and notions like community structure,
connected components, tensor decompositions, and various types of dynamical
processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure
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