2,785 research outputs found
Forman-Ricci flow for change detection in large dynamic data sets
We present a viable solution to the challenging question of change detection
in complex networks inferred from large dynamic data sets. Building on Forman's
discretization of the classical notion of Ricci curvature, we introduce a novel
geometric method to characterize different types of real-world networks with an
emphasis on peer-to-peer networks. Furthermore we adapt the classical Ricci
flow that already proved to be a powerful tool in image processing and
graphics, to the case of undirected and weighted networks. The application of
the proposed method on peer-to-peer networks yields insights into topological
properties and the structure of their underlying data.Comment: Conference paper, accepted at ICICS 2016. (Updated version
Comparative analysis of two discretizations of Ricci curvature for complex networks
We have performed an empirical comparison of two distinct notions of discrete
Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and
Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci
curvature were developed based on different properties of the classical smooth
notion, and thus, the two notions shed light on different aspects of network
structure and behavior. Nevertheless, our extensive computational analysis in a
wide range of both model and real-world networks shows that the two
discretizations of Ricci curvature are highly correlated in many networks.
Moreover, we show that if one considers the augmented Forman-Ricci curvature
which also accounts for the two-dimensional simplicial complexes arising in
graphs, the observed correlation between the two discretizations is even
higher, especially, in real networks. Besides the potential theoretical
implications of these observations, the close relationship between the two
discretizations has practical implications whereby Forman-Ricci curvature can
be employed in place of Ollivier-Ricci curvature for faster computation in
larger real-world networks whenever coarse analysis suffices.Comment: Published version. New results added in this version. Supplementary
tables can be freely downloaded from the publisher websit
The Large Scale Curvature of Networks
Understanding key structural properties of large scale networks are crucial
for analyzing and optimizing their performance, and improving their reliability
and security. Here we show that these networks possess a previously unnoticed
feature, global curvature, which we argue has a major impact on core
congestion: the load at the core of a network with N nodes scales as N^2 as
compared to N^1.5 for a flat network. We substantiate this claim through
analysis of a collection of real data networks across the globe as measured and
documented by previous researchers.Comment: 4 pages, 5 figure
Hidden geometries in networks arising from cooperative self-assembly
Multilevel self-assembly involving small structured groups of nano-particles
provides new routes to development of functional materials with a sophisticated
architecture. Apart from the inter-particle forces, the geometrical shapes and
compatibility of the building blocks are decisive factors in each phase of
growth. Therefore, a comprehensive understanding of these processes is
essential for the design of large assemblies of desired properties. Here, we
introduce a computational model for cooperative self-assembly with simultaneous
attachment of structured groups of particles, which can be described by
simplexes (connected pairs, triangles, tetrahedrons and higher order cliques)
to a growing network, starting from a small seed. The model incorporates
geometric rules that provide suitable nesting spaces for the new group and the
chemical affinity of the system to accepting an excess number of
particles. For varying chemical affinity, we grow different classes of
assemblies by binding the cliques of distributed sizes. Furthermore, to
characterise the emergent large-scale structures, we use the metrics of graph
theory and algebraic topology of graphs, and 4-point test for the intrinsic
hyperbolicity of the networks. Our results show that higher Q-connectedness of
the appearing simplicial complexes can arise due to only geometrical factors,
i.e., for , and that it can be effectively modulated by changing the
chemical potential and the polydispersity of the size of binding simplexes. For
certain parameters in the model we obtain networks of mono-dispersed clicks,
triangles and tetrahedrons, which represent the geometrical descriptors that
are relevant in quantum physics and frequently occurring chemical clusters.Comment: 9 pages, 8 figure
Emergent Complex Network Geometry
Networks are mathematical structures that are universally used to describe a
large variety of complex systems such as the brain or the Internet.
Characterizing the geometrical properties of these networks has become
increasingly relevant for routing problems, inference and data mining. In real
growing networks, topological, structural and geometrical properties emerge
spontaneously from their dynamical rules. Nevertheless we still miss a model in
which networks develop an emergent complex geometry. Here we show that a single
two parameter network model, the growing geometrical network, can generate
complex network geometries with non-trivial distribution of curvatures,
combining exponential growth and small-world properties with finite spectral
dimensionality. In one limit, the non-equilibrium dynamical rules of these
networks can generate scale-free networks with clustering and communities, in
another limit planar random geometries with non-trivial modularity. Finally we
find that these properties of the geometrical growing networks are present in a
large set of real networks describing biological, social and technological
systems.Comment: (24 pages, 7 figures, 1 table
Deterministic Dynamics and Chaos: Epistemology and Interdisciplinary Methodology
We analyze, from a theoretical viewpoint, the bidirectional interdisciplinary
relation between mathematics and psychology, focused on the mathematical theory
of deterministic dynamical systems, and in particular, on the theory of chaos.
On one hand, there is the direct classic relation: the application of
mathematics to psychology. On the other hand, we propose the converse relation
which consists in the formulation of new abstract mathematical problems
appearing from processes and structures under research of psychology. The
bidirectional multidisciplinary relation from-to pure mathematics, largely
holds with the "hard" sciences, typically physics and astronomy. But it is
rather new, from the social and human sciences, towards pure mathematics
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