12,856 research outputs found
Theory of Interface: Category Theory, Directed Networks and Evolution of Biological Networks
Biological networks have two modes. The first mode is static: a network is a
passage on which something flows. The second mode is dynamic: a network is a
pattern constructed by gluing functions of entities constituting the network.
In this paper, first we discuss that these two modes can be associated with the
category theoretic duality (adjunction) and derive a natural network structure
(a path notion) for each mode by appealing to the category theoretic
universality. The path notion corresponding to the static mode is just the
usual directed path. The path notion for the dynamic mode is called lateral
path which is the alternating path considered on the set of arcs. Their general
functionalities in a network are transport and coherence, respectively. Second,
we introduce a betweenness centrality of arcs for each mode and see how the two
modes are embedded in various real biological network data. We find that there
is a trade-off relationship between the two centralities: if the value of one
is large then the value of the other is small. This can be seen as a kind of
division of labor in a network into transport on the network and coherence of
the network. Finally, we propose an optimization model of networks based on a
quality function involving intensities of the two modes in order to see how
networks with the above trade-off relationship can emerge through evolution. We
show that the trade-off relationship can be observed in the evolved networks
only when the dynamic mode is dominant in the quality function by numerical
simulations. We also show that the evolved networks have features qualitatively
similar to real biological networks by standard complex network analysis.Comment: 59 pages, minor corrections from v
Network Models
Networks can be combined in various ways, such as overlaying one on top of
another or setting two side by side. We introduce "network models" to encode
these ways of combining networks. Different network models describe different
kinds of networks. We show that each network model gives rise to an operad,
whose operations are ways of assembling a network of the given kind from
smaller parts. Such operads, and their algebras, can serve as tools for
designing networks. Technically, a network model is a lax symmetric monoidal
functor from the free symmetric monoidal category on some set to
, and the construction of the corresponding operad proceeds via a
symmetric monoidal version of the Grothendieck construction.Comment: 46 page
Towards a navigational logic for graphical structures
One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for reasoning about graphs, is its generality: this logic can be used in the framework of many classes of graphs and graphical structures. It is enough that the category of these structures satisfies certain basic conditions.
In a previous paper [14], we extended this logic to be able to deal with graph properties including paths, but this extension was only defined for the category of untyped directed graphs. In addition it seemed difficult to talk about paths abstractly, that is, independently of the given category of graphical structures. In this paper we approach this problem. In particular, given an arbitrary category of graphical structures, we assume that for every object of this category there is an associated edge relation that can be used to define a path relation. Moreover, we consider that edges have some kind of labels and paths can be specified by associating them to a set of label sequences. Then, after the presentation of that general framework, we show how it can be applied to several classes of graphs. Moreover, we present a set of sound inference rules for reasoning in the logic.Peer ReviewedPostprint (author's final draft
RCFT with defects: Factorization and fundamental world sheets
It is known that for any full rational conformal field theory, the
correlation functions that are obtained by the TFT construction satisfy all
locality, modular invariance and factorization conditions, and that there is a
small set of fundamental correlators to which all others are related via
factorization - provided that the world sheets considered do not contain any
non-trivial defect lines. In this paper we generalize both results to oriented
world sheets with an arbitrary network of topological defect lines.Comment: 46 pages, several pictures. v2: typos correcte
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