98 research outputs found
Minimal energy point systems on the unit circle and the real line
In this paper, we investigate discrete logarithmic energy problems in the
unit circle. We study the equilibrium configuration of electrons and
pairs of external protons of charge . It is shown that all the critical
points of the discrete logarithmic energy are global minima, and they are the
solutions of certain equations involving Blaschke products. As a nontrivial
application, we refine a recent result of Simanek, namely, we prove that any
configuration of electrons in the unit circle is in stable equilibrium
(that is, they are not just critical points but are of minimal energy) with
respect to an external field generated by pairs of protons.Comment: Revised and extended versio
Towards a Better Understanding of the Characteristics of Fractal Networks
The fractal nature of complex networks has received a great deal of research
interest in the last two decades. Similarly to geometric fractals, the
fractality of networks can also be defined with the so-called box-covering
method. A network is called fractal if the minimum number of boxes needed to
cover the entire network follows a power-law relation with the size of the
boxes. The fractality of networks has been associated with various network
properties throughout the years, for example, disassortativity, repulsion
between hubs, long-range-repulsive correlation, and small edge betweenness
centralities. However, these assertions are usually based on tailor-made
network models and on a small number of real networks, hence their ubiquity is
often disputed.
Since fractal networks have been shown to have important properties, such as
robustness against intentional attacks, it is in dire need to uncover the
underlying mechanisms causing fractality. Hence, the main goal of this work is
to get a better understanding of the origins of fractality in complex networks.
To this end, we systematically review the previous results on the relationship
between various network characteristics and fractality. Moreover, we perform a
comprehensive analysis of these relations on five network models and a large
number of real-world networks originating from six domains. We clarify which
characteristics are universally present in fractal networks and which features
are just artifacts or coincidences
Investigating the Origins of Fractality Based on Two Novel Fractal Network Models
Numerous network models have been investigated to gain insights into the
origins of fractality. In this work, we introduce two novel network models, to
better understand the growing mechanism and structural characteristics of
fractal networks. The Repulsion Based Fractal Model (RBFM) is built on the
well-known Song-Havlin-Makse (SHM) model, but in RBFM repulsion is always
present among a specific group of nodes. The model resolves the contradiction
between the SHM model and the Hub Attraction Dynamical Growth model, by showing
that repulsion is the characteristic that induces fractality. The Lattice
Small-world Transition Model (LSwTM) was motivated by the fact that repulsion
directly influences the node distances. Through LSwTM we study the
fractal-small-world transition. The model illustrates the transition on a fixed
number of nodes and edges using a preferential-attachment-based edge rewiring
process. It shows that a small average distance works against fractal scaling,
and also demonstrates that fractality is not a dichotomous property, continuous
transition can be observed between the pure fractal and non-fractal
characteristics.Comment: 12 pages, 5 figures, to appear in: 978-3-031-17657-9, Pacheco et al
(eds.): Complex Networks XII
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