13,839 research outputs found
Crystal frameworks, symmetry and affinely periodic flexes
Symmetry equations are obtained for the rigidity matrices associated with
various forms of infinitesimal flexibility for an idealised bond-node crystal
framework \C in \bR^d. These equations are used to derive symmetry-adapted
Maxwell-Calladine counting formulae for periodic self-stresses and affinely
periodic infinitesimal mechanisms. The symmetry equations also lead to general
Fowler-Guest formulae connecting the character lists of subrepresentations of
the crystallographic space and point groups which are associated with bonds,
nodes, stresses, flexes and rigid motions. A new derivation is also given for
the Borcea-Streinu rigidity matrix and the correspondence between its nullspace
and the space of affinely periodic infinitesimal flexes.Comment: This preprint has some new diagrams and clarifications. A final
version will appear in the New York Journal of Mathematic
Collective decision-making on triadic graphs
Many real-world networks exhibit community structures and non-trivial clustering associated with the occurrence of a considerable number of triangular subgraphs known as triadic motifs. Triads are a set of distinct triangles that do not share an edge with any other triangle in the network. Network motifs are subgraphs that occur significantly more often compared to random topologies. Two prominent examples, the feedforward loop and the feedback loop, occur in various real-world networks such as gene-regulatory networks, food webs or neuronal networks. However, as triangular connections are also prevalent in communication topologies of complex collective systems, it is worthwhile investigating the influence of triadic motifs on the collective decision-making dynamics. To this end, we generate networks called Triadic Graphs (TGs) exclusively from distinct triadic motifs. We then apply TGs as underlying topologies of systems with collective dynamics inspired from locust marching bands. We demonstrate that the motif type constituting the networks can have a paramount influence on group decision-making that cannot be explained solely in terms of the degree distribution. We find that, in contrast to the feedback loop, when the feedforward loop is the dominant subgraph, the resulting network is hierarchical and inhibits coherent behavior
Topological properties and fractal analysis of recurrence network constructed from fractional Brownian motions
Many studies have shown that we can gain additional information on time
series by investigating their accompanying complex networks. In this work, we
investigate the fundamental topological and fractal properties of recurrence
networks constructed from fractional Brownian motions (FBMs). First, our
results indicate that the constructed recurrence networks have exponential
degree distributions; the relationship between and of recurrence networks decreases with the Hurst
index of the associated FBMs, and their dependence approximately satisfies
the linear formula . Moreover, our numerical results of
multifractal analysis show that the multifractality exists in these recurrence
networks, and the multifractality of these networks becomes stronger at first
and then weaker when the Hurst index of the associated time series becomes
larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst
index possess the strongest multifractality. In addition, the
dependence relationships of the average information dimension on the Hurst index can also be
fitted well with linear functions. Our results strongly suggest that the
recurrence network inherits the basic characteristic and the fractal nature of
the associated FBM series.Comment: 25 pages, 1 table, 15 figures. accepted by Phys. Rev.
Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus Hurst index
Nonlinear time series analysis aims at understanding the dynamics of
stochastic or chaotic processes. In recent years, quite a few methods have been
proposed to transform a single time series to a complex network so that the
dynamics of the process can be understood by investigating the topological
properties of the network. We study the topological properties of horizontal
visibility graphs constructed from fractional Brownian motions with different
Hurst index . Special attention has been paid to the impact of Hurst
index on the topological properties. It is found that the clustering
coefficient decreases when increases. We also found that the mean
length of the shortest paths increases exponentially with for fixed
length of the original time series. In addition, increases linearly
with respect to when is close to 1 and in a logarithmic form when
is close to 0. Although the occurrence of different motifs changes with ,
the motif rank pattern remains unchanged for different . Adopting the
node-covering box-counting method, the horizontal visibility graphs are found
to be fractals and the fractal dimension decreases with . Furthermore,
the Pearson coefficients of the networks are positive and the degree-degree
correlations increase with the degree, which indicate that the horizontal
visibility graphs are assortative. With the increase of , the Pearson
coefficient decreases first and then increases, in which the turning point is
around . The presence of both fractality and assortativity in the
horizontal visibility graphs converted from fractional Brownian motions is
different from many cases where fractal networks are usually disassortative.Comment: 12 pages, 8 figure
Recurrence-based time series analysis by means of complex network methods
Complex networks are an important paradigm of modern complex systems sciences
which allows quantitatively assessing the structural properties of systems
composed of different interacting entities. During the last years, intensive
efforts have been spent on applying network-based concepts also for the
analysis of dynamically relevant higher-order statistical properties of time
series. Notably, many corresponding approaches are closely related with the
concept of recurrence in phase space. In this paper, we review recent
methodological advances in time series analysis based on complex networks, with
a special emphasis on methods founded on recurrence plots. The potentials and
limitations of the individual methods are discussed and illustrated for
paradigmatic examples of dynamical systems as well as for real-world time
series. Complex network measures are shown to provide information about
structural features of dynamical systems that are complementary to those
characterized by other methods of time series analysis and, hence,
substantially enrich the knowledge gathered from other existing (linear as well
as nonlinear) approaches.Comment: To be published in International Journal of Bifurcation and Chaos
(2011
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