100 research outputs found
Epidemics with containment measures
(14 pages, 6 figures)(14 pages, 6 figures
Tensors in modelling multi-particle interactions
In this work we present recent results on application of low-rank tensor
decompositions to modelling of aggregation kinetics taking into account
multi-particle collisions (for three and more particles). Such kinetics can be
described by system of nonlinear differential equations with right-hand side
requiring operations for its straight-forward evaluation, where is
number of particles size classes and is number of particles colliding
simultaneously. Such a complexity can be significantly reduced by application
low rank tensor decompositions (either Tensor Train or Canonical Polyadic) to
acceleration of evaluation of sums and convolutions from right-hand side.
Basing on this drastic reduction of complexity for evaluation of right-hand
side we further utilize standard second order Runge-Kutta time integration
scheme and demonstrate that our approach allows to obtain numerical solutions
of studied equations with very high accuracy in modest times. We also show
preliminary results on parallel scalability of novel approach and conclude that
it can be efficiently utilized with use of supercomputers.Comment: LaTEX, 8 pages, 3 figures, submitted to proceedings of LSSC'19
conference, Sozopol, Bulgari
Evolutionary game of coalition building under external pressure
We study the fragmentation-coagulation (or merging and splitting)
evolutionary control model as introduced recently by one of the authors, where
small players can form coalitions to resist to the pressure exerted by the
principal. It is a Markov chain in continuous time and the players have a
common reward to optimize. We study the behavior as grows and show that the
problem converges to a (one player) deterministic optimization problem in
continuous time, in the infinite dimensional state space
Emergence of scale-free close-knit friendship structure in online social networks
Despite the structural properties of online social networks have attracted
much attention, the properties of the close-knit friendship structures remain
an important question. Here, we mainly focus on how these mesoscale structures
are affected by the local and global structural properties. Analyzing the data
of four large-scale online social networks reveals several common structural
properties. It is found that not only the local structures given by the
indegree, outdegree, and reciprocal degree distributions follow a similar
scaling behavior, the mesoscale structures represented by the distributions of
close-knit friendship structures also exhibit a similar scaling law. The degree
correlation is very weak over a wide range of the degrees. We propose a simple
directed network model that captures the observed properties. The model
incorporates two mechanisms: reciprocation and preferential attachment. Through
rate equation analysis of our model, the local-scale and mesoscale structural
properties are derived. In the local-scale, the same scaling behavior of
indegree and outdegree distributions stems from indegree and outdegree of nodes
both growing as the same function of the introduction time, and the reciprocal
degree distribution also shows the same power-law due to the linear
relationship between the reciprocal degree and in/outdegree of nodes. In the
mesoscale, the distributions of four closed triples representing close-knit
friendship structures are found to exhibit identical power-laws, a behavior
attributed to the negligible degree correlations. Intriguingly, all the
power-law exponents of the distributions in the local-scale and mesoscale
depend only on one global parameter -- the mean in/outdegree, while both the
mean in/outdegree and the reciprocity together determine the ratio of the
reciprocal degree of a node to its in/outdegree.Comment: 48 pages, 34 figure
A topological classification of convex bodies
The shape of homogeneous, generic, smooth convex bodies as described by the
Euclidean distance with nondegenerate critical points, measured from the center
of mass represents a rather restricted class M_C of Morse-Smale functions on
S^2. Here we show that even M_C exhibits the complexity known for general
Morse-Smale functions on S^2 by exhausting all combinatorial possibilities:
every 2-colored quadrangulation of the sphere is isomorphic to a suitably
represented Morse-Smale complex associated with a function in M_C (and vice
versa). We prove our claim by an inductive algorithm, starting from the path
graph P_2 and generating convex bodies corresponding to quadrangulations with
increasing number of vertices by performing each combinatorially possible
vertex splitting by a convexity-preserving local manipulation of the surface.
Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist,
this algorithm not only proves our claim but also generalizes the known
classification scheme in [36]. Our expansion algorithm is essentially the dual
procedure to the algorithm presented by Edelsbrunner et al. in [21], producing
a hierarchy of increasingly coarse Morse-Smale complexes. We point out
applications to pebble shapes.Comment: 25 pages, 10 figure
Popularity versus Similarity in Growing Networks
Popularity is attractive -- this is the formula underlying preferential
attachment, a popular explanation for the emergence of scaling in growing
networks. If new connections are made preferentially to more popular nodes,
then the resulting distribution of the number of connections that nodes have
follows power laws observed in many real networks. Preferential attachment has
been directly validated for some real networks, including the Internet.
Preferential attachment can also be a consequence of different underlying
processes based on node fitness, ranking, optimization, random walks, or
duplication. Here we show that popularity is just one dimension of
attractiveness. Another dimension is similarity. We develop a framework where
new connections, instead of preferring popular nodes, optimize certain
trade-offs between popularity and similarity. The framework admits a geometric
interpretation, in which popularity preference emerges from local optimization.
As opposed to preferential attachment, the optimization framework accurately
describes large-scale evolution of technological (Internet), social (web of
trust), and biological (E.coli metabolic) networks, predicting the probability
of new links in them with a remarkable precision. The developed framework can
thus be used for predicting new links in evolving networks, and provides a
different perspective on preferential attachment as an emergent phenomenon
Transport coefficients for inelastic Maxwell mixtures
The Boltzmann equation for inelastic Maxwell models is used to determine the
Navier-Stokes transport coefficients of a granular binary mixture in
dimensions. The Chapman-Enskog method is applied to solve the Boltzmann
equation for states near the (local) homogeneous cooling state. The mass, heat,
and momentum fluxes are obtained to first order in the spatial gradients of the
hydrodynamic fields, and the corresponding transport coefficients are
identified. There are seven relevant transport coefficients: the mutual
diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity,
the Dufour coefficient, the pressure energy coefficient, and the thermal
conductivity. All these coefficients are {\em exactly} obtained in terms of the
coefficients of restitution and the ratios of mass, concentration, and particle
sizes. The results are compared with known transport coefficients of inelastic
hard spheres obtained analytically in the leading Sonine approximation and by
means of Monte Carlo simulations. The comparison shows a reasonably good
agreement between both interaction models for not too strong dissipation,
especially in the case of the transport coefficients associated with the mass
flux.Comment: 9 figures, to be published in J. Stat. Phy
Boolean Dynamics with Random Couplings
This paper reviews a class of generic dissipative dynamical systems called
N-K models. In these models, the dynamics of N elements, defined as Boolean
variables, develop step by step, clocked by a discrete time variable. Each of
the N Boolean elements at a given time is given a value which depends upon K
elements in the previous time step.
We review the work of many authors on the behavior of the models, looking
particularly at the structure and lengths of their cycles, the sizes of their
basins of attraction, and the flow of information through the systems. In the
limit of infinite N, there is a phase transition between a chaotic and an
ordered phase, with a critical phase in between.
We argue that the behavior of this system depends significantly on the
topology of the network connections. If the elements are placed upon a lattice
with dimension d, the system shows correlations related to the standard
percolation or directed percolation phase transition on such a lattice. On the
other hand, a very different behavior is seen in the Kauffman net in which all
spins are equally likely to be coupled to a given spin. In this situation,
coupling loops are mostly suppressed, and the behavior of the system is much
more like that of a mean field theory.
We also describe possible applications of the models to, for example, genetic
networks, cell differentiation, evolution, democracy in social systems and
neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical
Sciences Serie
Characterizing and modeling citation dynamics
Citation distributions are crucial for the analysis and modeling of the
activity of scientists. We investigated bibliometric data of papers published
in journals of the American Physical Society, searching for the type of
function which best describes the observed citation distributions. We used the
goodness of fit with Kolmogorov-Smirnov statistics for three classes of
functions: log-normal, simple power law and shifted power law. The shifted
power law turns out to be the most reliable hypothesis for all citation
networks we derived, which correspond to different time spans. We find that
citation dynamics is characterized by bursts, usually occurring within a few
years since publication of a paper, and the burst size spans several orders of
magnitude. We also investigated the microscopic mechanisms for the evolution of
citation networks, by proposing a linear preferential attachment with time
dependent initial attractiveness. The model successfully reproduces the
empirical citation distributions and accounts for the presence of citation
bursts as well.Comment: 8 pages, 5 figure
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