Hypernetworks for reconstructing the dynamics of multilevel systems

Networks are fundamental for reconstructing the dynamics of many systems, but have the drawback that they are restricted to binary relations. Hypergraphs extend relational structure to multi-vertex edges, but are essentially set-theoretic and unable to represent essential structural properties. Hypernetworks are a natural multidimensional generalisation of networks, representing n-ary relations by simplices with n vertices. The assembly of vertices to make simplices is key for moving between levels in multilevel systems, and integrating dynamics between levels. It is argued that hypernetworks are necessary, if not sufficient, for reconstructing the dynamics of multilevel complex systems

Visualizing and Interacting with Concept Hierarchies

Concept Hierarchies and Formal Concept Analysis are theoretically well grounded and largely experimented methods. They rely on line diagrams called Galois lattices for visualizing and analysing object-attribute sets. Galois lattices are visually seducing and conceptually rich for experts. However they present important drawbacks due to their concept oriented overall structure: analysing what they show is difficult for non experts, navigation is cumbersome, interaction is poor, and scalability is a deep bottleneck for visual interpretation even for experts. In this paper we introduce semantic probes as a means to overcome many of these problems and extend usability and application possibilities of traditional FCA visualization methods. Semantic probes are visual user centred objects which extract and organize reduced Galois sub-hierarchies. They are simpler, clearer, and they provide a better navigation support through a rich set of interaction possibilities. Since probe driven sub-hierarchies are limited to users focus, scalability is under control and interpretation is facilitated. After some successful experiments, several applications are being developed with the remaining problem of finding a compromise between simplicity and conceptual expressivity

Structure Entropy, Self-Organization, and Power Laws in Urban Street Networks: Evidence for Alexander's Ideas

Easy and intuitive navigability is of central importance in cities. The actual scale-free networking of urban street networks in their topological space, where navigation information is encoded by mapping roads to nodes and junctions to links between nodes, has still no simple explanation. Emphasizing the road-junction hierarchy in a holistic and systematic way leads us to envisage urban street networks as evolving social systems subject to a Boltzmann-mesoscopic entropy conservation. This conservation, which we may interpret in terms of surprisal, ensures the passage from the road-junction hierarchy to a scale-free coherence. To wit, we recover the actual scale-free probability distribution for natural roads in self-organized cities. We obtain this passage by invoking Jaynes's Maximum Entropy principle (statistical physics), while we capitalize on modern ideas of quantification (information physics) and well known results on structuration (lattice theory) to measure the information network entropy. The emerging paradigm, which applies to systems with more intricate hierarchies as actual cities, appears to reflect well the influential ideas on cities of the urbanist Christopher Alexander

A Galois lattice for qualitative spatial reasoning and representation

Colloque avec actes et comité de lecture. internationale.International audienceThis paper presents an original approach to qualitative spatial representation and reasoning with topological relations based on the use of a Galois lattice of topological relations. This approach has been developed for qualitative spatial reasoning in the domain of agricultural landscape analysis. The paper describes first the general framework of topological relations, and then the design of a Galois lattice of topological relations. The elements of representation and reasoning based on this Galois lattice are discussed and examples are given, together with a brief description of the implementation of the Galois lattice within an object-based representation system

Mining Complex Hydrobiological Data with Galois Lattices

International audienceWe used Galois lattices for mining hydrobiological data about macrophytes, i.e. macroscopic plants living in water bodies. These plants are characterized by several biological traits, that are divided into several modalities. Our aim was to cluster the plants according to their common traits and modalities and to find out the relations between the traits. Galois lattices are efficient methods for such an aim, but apply to binary data. In this article, we detail a few of the approaches we used to turn complex hydrobiological data into binary data and compare the first results obtained thanks to Galois lattices

Meta-Modeling And Structural Paradigm For Strategic Alignment Of Information System

The information system is strongly sensitive to strategic evolutions of the enterprise: organisational change, change of objectives, modified variety, new objects and business processes, etc. With the objective to control strategic alignment of information systems, we propose an approach based on our extended enterprise meta-model ISO/DSI 19440. This extension is borrowed from the COBIT framework for IT processes. In order to better lead evolutions of the information system, this extension integrates necessary structures for developing systemic tools, based on a structural paradigm. In this work we propose to build an extension of the meta-model ISO 19440, so that we can explicitly bring the issue of alignment of various aspects of the information system. The strengths of the strategic alignment are interactions and couplings between different views of the meta-model: the interaction between activities and resources, the linkage between business processes and activities, the resource interdependence of entities and objects of the enterprise, the coupling between the capabilities and resources, etc. We propose to use the COBIT best practices for driving IT processes. Thus we add the abstract concept objective which will be specialized. We will also add a specialization of functional entity to model IT processes. In this work, we also offer a variety of algebraic structures to establish structural measures on the information system. For each class of structure we define its role and contribution to the governance of the information system

On the essential logical structure of inter-universal Teichmüller theory in terms of logical AND “∧”/logical OR “∨” relations: Report on the occasion of the publication of the four main papers on inter-universal Teichmüller theory

The main goal of the present paper is to give a detailed exposition of the essential logical structure of inter-universal Teichmüller theory from the point of view of the Boolean operators --such as the logical AND “∧”logical OR “∨” operators-- of propositional calculus. This essential logical structure of inter-universal Teichmüller theory may be summarized symbolically as follows: A ∧ B = A ∧ (B₁ ∨˙ B₂ ∨˙...) ⇒ A ∧ (B₁∨˙B₂∨˙...∨˙ B́₁ ∨˙ B́₂ ∨˙...) -- where · the “∨˙” denotes the Boolean operator exclusive-OR, i.e., “XOR”; · A, B, B₁, B₂, B́₁, B́₂, denote various propositions; · the logical AND “∧'s” correspond to the Θ-link of inter-universal Teichmüller theory and are closely related to the multiplicative structures of the rings that appear in the domain and codomain of the Θ-link; · the logical XOR “∨˙'s” correspond to various indeterminacies that arise mainly from the log-Kummer-correspondence, i.e., from sequences of iterates of the log-link of inter-universal Teichmüller theory, which may be thought of as a device for constructing additive log-shells. This sort of concatenation of logical AND “∧'s” and logical XOR “∨˙ 's” is reminiscent of the well-known description of the “carry-addition” operation on Teichmüller representatives of the truncated Witt ring ℤ/4ℤ in terms of Boolean addition “∨˙” and Boolean multiplication “∧” in the field F₂ and may be regarded as a sort of “Boolean intertwining” that mirrors, in a remarkable fashion, the “arithmetic intertwining” between addition and multiplication in number fields and local fields, which is, in some sense, the main object of study in inter-universal Teichmüller theory. One important topic in this exposition is the issue of “redundant copies”, i.e., the issue of how the arbitrary identification of copies of isomorphic mathematical objects that appear in the various constructions of inter-universal Teichmüller theory impacts-- and indeed invalidates-- the essential logical structure of inter-universal Teichmüller theory. This issue has been a focal point of fundamental misunderstandings and entirely unnecessary confusion concerning inter-universal Teichmüller theory in certain sectors of the mathematical community. The exposition of the topic of “redundant copies” makes use of many interesting elementary examples from the history of mathematics