1,463 research outputs found
k-core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects
We develop the theory of the k-core (bootstrap) percolation on uncorrelated
random networks with arbitrary degree distributions. We show that the k-core
percolation is an unusual, hybrid phase transition with a jump emergence of the
k-core as at a first order phase transition but also with a critical
singularity as at a continuous transition. We describe the properties of the
k-core, explain the meaning of the order parameter for the k-core percolation,
and reveal the origin of the specific critical phenomena. We demonstrate that a
so-called ``corona'' of the k-core plays a crucial role (corona is a subset of
vertices in the k-core which have exactly k neighbors in the k-core). It turns
out that the k-core percolation threshold is at the same time the percolation
threshold of finite corona clusters. The mean separation of vertices in corona
clusters plays the role of the correlation length and diverges at the critical
point. We show that a random removal of even one vertex from the k-core may
result in the collapse of a vast region of the k-core around the removed
vertex. The mean size of this region diverges at the critical point. We find an
exact mapping of the k-core percolation to a model of cooperative relaxation.
This model undergoes critical relaxation with a divergent rate at some critical
moment.Comment: 11 pages, 8 figure
Degree-degree correlations in random graphs with heavy-tailed degrees
Mixing patterns in large self-organizing networks, such as the Internet, the
World Wide Web, social and biological networks are often characterized by
degree-degree {dependencies} between neighbouring nodes. One of the problems
with the commonly used Pearson's correlation coefficient (termed as the
assortativity coefficient) is that {in disassortative networks its magnitude
decreases} with the network size. This makes it impossible to compare mixing
patterns, for example, in two web crawls of different size.
We start with a simple model of two heavy-tailed highly correlated random
variable and , and show that the sample correlation coefficient
converges in distribution either to a proper random variable on , or to
zero, and if then the limit is non-negative. We next show that it is
non-negative in the large graph limit when the degree distribution has an
infinite third moment. We consider the alternative degree-degree dependency
measure, based on the Spearman's rho, and prove that it converges to an
appropriate limit under very general conditions. We verify that these
conditions hold in common network models, such as configuration model and
Preferential Attachment model. We conclude that rank correlations provide a
suitable and informative method for uncovering network mixing patterns
Random acyclic networks
Directed acyclic graphs are a fundamental class of networks that includes
citation networks, food webs, and family trees, among others. Here we define a
random graph model for directed acyclic graphs and give solutions for a number
of the model's properties, including connection probabilities and component
sizes, as well as a fast algorithm for simulating the model on a computer. We
compare the predictions of the model to a real-world network of citations
between physics papers and find surprisingly good agreement, suggesting that
the structure of the real network may be quite well described by the random
graph.Comment: 4 pages, 2 figure
The percolation critical polynomial as a graph invariant
Every lattice for which the bond percolation critical probability can be
found exactly possesses a critical polynomial, with the root in [0,1] providing
the threshold. Recent work has demonstrated that this polynomial may be
generalized through a definition that can be applied on any periodic lattice.
The polynomial depends on the lattice and on its decomposition into identical
finite subgraphs, but once these are specified, the polynomial is essentially
unique. On lattices for which the exact percolation threshold is unknown, the
polynomials provide approximations for the critical probability with the
estimates appearing to converge to the exact answer with increasing subgraph
size. In this paper, I show how this generalized critical polynomial can be
viewed as a graph invariant, similar to the Tutte polynomial. In particular,
the critical polynomial is computed on a finite graph and may be found using
the recursive deletion-contraction algorithm. This allows calculation on a
computer, and I present such results for the kagome lattice using subgraphs of
up to 36 bonds. For one of these, I find the prediction p_c=0.52440572...,
which differs from the numerical value, p_c=0.52440503(5), by only 6.9 x
10^{-7}
Diameters in preferential attachment models
In this paper, we investigate the diameter in preferential attachment (PA-)
models, thus quantifying the statement that these models are small worlds. The
models studied here are such that edges are attached to older vertices
proportional to the degree plus a constant, i.e., we consider affine PA-models.
There is a substantial amount of literature proving that, quite generally,
PA-graphs possess power-law degree sequences with a power-law exponent \tau>2.
We prove that the diameter of the PA-model is bounded above by a constant
times \log{t}, where t is the size of the graph. When the power-law exponent
\tau exceeds 3, then we prove that \log{t} is the right order, by proving a
lower bound of this order, both for the diameter as well as for the typical
distance. This shows that, for \tau>3, distances are of the order \log{t}. For
\tau\in (2,3), we improve the upper bound to a constant times \log\log{t}, and
prove a lower bound of the same order for the diameter. Unfortunately, this
proof does not extend to typical distances. These results do show that the
diameter is of order \log\log{t}.
These bounds partially prove predictions by physicists that the typical
distance in PA-graphs are similar to the ones in other scale-free random
graphs, such as the configuration model and various inhomogeneous random graph
models, where typical distances have been shown to be of order \log\log{t} when
\tau\in (2,3), and of order \log{t} when \tau>3
Statistical Self-Similar Properties of Complex Networks
It has been shown that many complex networks shared distinctive features,
which differ in many ways from the random and the regular networks. Although
these features capture important characteristics of complex networks, their
applicability depends on the type of networks. To unravel ubiquitous
characteristics that complex networks may have in common, we adopt the
clustering coefficient as the probability measure, and present a systematic
analysis of various types of complex networks from the perspective of
statistical self-similarity. We find that the probability distribution of the
clustering coefficient is best characterized by the multifractal; moreover, the
support of the measure had a fractal dimension. These two features enable us to
describe complex networks in a unified way; at the same time, offer unforeseen
possibilities to comprehend complex networks.Comment: 11 pages, 4 figure
Continuous Dynamical Decoupling with Bounded Controls
We develop a theory of continuous decoupling with bounded controls from a
geometric perspective. Continuous decoupling with bounded controls can
accomplish the same decoupling effect as the bang-bang control while using
realistic control resources and it is robust against systematic implementation
errors. We show that the decoupling condition within this framework is
equivalent to average out error vectors whose trajectories are determined by
the control Hamiltonian. The decoupling pulses can be intuitively designed once
the structure function of the corresponding SU(n) is known and is represented
from the geometric perspective. Several examples are given to illustrate the
basic idea. From the physical implementation point of view we argue that the
efficiency of the decoupling is determined not by the order of the decoupling
group but by the minimal time required to finish a decoupling cycle
Local majority dynamics on preferential attachment graphs
Suppose in a graph vertices can be either red or blue. Let be odd. At
each time step, each vertex in polls random neighbours and takes
the majority colour. If it doesn't have neighbours, it simply polls all of
them, or all less one if the degree of is even. We study this protocol on
the preferential attachment model of Albert and Barab\'asi, which gives rise to
a degree distribution that has roughly power-law ,
as well as generalisations which give exponents larger than . The setting is
as follows: Initially each vertex of is red independently with probability
, and is otherwise blue. We show that if is
sufficiently biased away from , then with high probability,
consensus is reached on the initial global majority within
steps. Here is the number of vertices and is the minimum of
and (or if is even), being the number of edges each new
vertex adds in the preferential attachment generative process. Additionally,
our analysis reduces the required bias of for graphs of a given degree
sequence studied by the first author (which includes, e.g., random regular
graphs)
Spectral Analysis of Protein-Protein Interactions in Drosophila melanogaster
Within a case study on the protein-protein interaction network (PIN) of
Drosophila melanogaster we investigate the relation between the network's
spectral properties and its structural features such as the prevalence of
specific subgraphs or duplicate nodes as a result of its evolutionary history.
The discrete part of the spectral density shows fingerprints of the PIN's
topological features including a preference for loop structures. Duplicate
nodes are another prominent feature of PINs and we discuss their representation
in the PIN's spectrum as well as their biological implications.Comment: 9 pages RevTeX including 8 figure
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