31 research outputs found
Nonextensive statistical mechanics and complex scale-free networks
One explanation for the impressive recent boom in network theory might be
that it provides a promising tool for an understanding of complex systems.
Network theory is mainly focusing on discrete large-scale topological
structures rather than on microscopic details of interactions of its elements.
This viewpoint allows to naturally treat collective phenomena which are often
an integral part of complex systems, such as biological or socio-economical
phenomena. Much of the attraction of network theory arises from the discovery
that many networks, natural or man-made, seem to exhibit some sort of
universality, meaning that most of them belong to one of three classes: {\it
random}, {\it scale-free} and {\it small-world} networks. Maybe most important
however for the physics community is, that due to its conceptually intuitive
nature, network theory seems to be within reach of a full and coherent
understanding from first principles ..
Randomness and Complexity in Networks
I start by reviewing some basic properties of random graphs. I then consider
the role of random walks in complex networks and show how they may be used to
explain why so many long tailed distributions are found in real data sets. The
key idea is that in many cases the process involves copying of properties of
near neighbours in the network and this is a type of short random walk which in
turn produce a natural preferential attachment mechanism. Applying this to
networks of fixed size I show that copying and innovation are processes with
special mathematical properties which include the ability to solve a simple
model exactly for any parameter values and at any time. I finish by looking at
variations of this basic model.Comment: Survey paper based on talk given at the workshop on ``Stochastic
Networks and Internet Technology'', Centro di Ricerca Matematica Ennio De
Giorgi, Matematica nelle Scienze Naturali e Sociali, Pisa, 17th - 21st
September 2007. To appear in proceeding
Analysis of scale-free networks based on a threshold graph with intrinsic vertex weights
Many real networks are complex and have power-law vertex degree distribution,
short diameter, and high clustering. We analyze the network model based on
thresholding of the summed vertex weights, which belongs to the class of
networks proposed by Caldarelli et al. (2002). Power-law degree distributions,
particularly with the dynamically stable scaling exponent 2, realistic
clustering, and short path lengths are produced for many types of weight
distributions. Thresholding mechanisms can underlie a family of real complex
networks that is characterized by cooperativeness and the baseline scaling
exponent 2. It contrasts with the class of growth models with preferential
attachment, which is marked by competitiveness and baseline scaling exponent 3.Comment: 5 figure
Preferential attachment without vertex growth: emergence of the giant component
We study the following preferential attachment variant of the classical
Erdos-Renyi random graph process. Starting with an empty graph on n vertices,
new edges are added one-by-one, and each time an edge is chosen with
probability roughly proportional to the product of the current degrees of its
endpoints (note that the vertex set is fixed). We determine the asymptotic size
of the giant component in the supercritical phase, confirming a conjecture of
Pittel from 2010. Our proof uses a simple method: we condition on the vertex
degrees (of a multigraph variant), and use known results for the configuration
model.Comment: 20 page
Enhancing the spectral gap of networks by node removal
Dynamics on networks are often characterized by the second smallest
eigenvalue of the Laplacian matrix of the network, which is called the spectral
gap. Examples include the threshold coupling strength for synchronization and
the relaxation time of a random walk. A large spectral gap is usually
associated with high network performance, such as facilitated synchronization
and rapid convergence. In this study, we seek to enhance the spectral gap of
undirected and unweighted networks by removing nodes because, practically, the
removal of nodes often costs less than the addition of nodes, addition of
links, and rewiring of links. In particular, we develop a perturbative method
to achieve this goal. The proposed method realizes better performance than
other heuristic methods on various model and real networks. The spectral gap
increases as we remove up to half the nodes in most of these networks.Comment: 5 figure
Rigorous results on the threshold network model
We analyze the threshold network model in which a pair of vertices with
random weights are connected by an edge when the summation of the weights
exceeds a threshold. We prove some convergence theorems and central limit
theorems on the vertex degree, degree correlation, and the number of prescribed
subgraphs. We also generalize some results in the spatially extended cases.Comment: 21 pages, Journal of Physics A, in pres
Randomly Evolving Idiotypic Networks: Structural Properties and Architecture
We consider a minimalistic dynamic model of the idiotypic network of
B-lymphocytes. A network node represents a population of B-lymphocytes of the
same specificity (idiotype), which is encoded by a bitstring. The links of the
network connect nodes with complementary and nearly complementary bitstrings,
allowing for a few mismatches. A node is occupied if a lymphocyte clone of the
corresponding idiotype exists, otherwise it is empty. There is a continuous
influx of new B-lymphocytes of random idiotype from the bone marrow.
B-lymphocytes are stimulated by cross-linking their receptors with
complementary structures. If there are too many complementary structures,
steric hindrance prevents cross-linking. Stimulated cells proliferate and
secrete antibodies of the same idiotype as their receptors, unstimulated
lymphocytes die.
Depending on few parameters, the autonomous system evolves randomly towards
patterns of highly organized architecture, where the nodes can be classified
into groups according to their statistical properties. We observe and describe
analytically the building principles of these patterns, which allow to
calculate number and size of the node groups and the number of links between
them. The architecture of all patterns observed so far in simulations can be
explained this way. A tool for real-time pattern identification is proposed.Comment: 19 pages, 15 figures, 4 table