5,301 research outputs found
A JOURNEY DOWN THE RABBIT HOLE: PONDERING PREFERENTIAL ATTACHMENT MODELS WITH LOCATION
We investigate the use of stochastic approximation as a method of identifying conditions necessary to facilitate condensation and coexistence. We did this for a variety of preferential attachment models which are growing by way of some predetermined selection criteria.
The main results presented in this thesis concern the choice of r model. This growth method uses preferential attachment to select r vertices from a graph at time n. These r vertices are subsequently ranked according to fixed location assigned at each of their creations and used as an extra level of comparison between vertices. A new vertex is then attached to one of these r selected vertices according to a predetermined vector of probabilities corresponding to this ranking. We have shown that condensation can occur for any of these vectors, if we can find at least two stable fixed points to the corresponding set of stochastic approximation equations.
Following this we investigate the degree distribution and complexity associated to the introduction of a higher dimensional location coefficient.
Our concluding chapter investigates the coexistence between vertices in preferential attachment networks where vertices posses different types and locations. Using similar methods as in the choice of r model we have shown that coexistence can occur in location type models with phase transitions helping to classify different cases
Multiple Scale-Free Structures in Complex Ad-Hoc Networks
This paper develops a framework for analyzing and designing dynamic networks
comprising different classes of nodes that coexist and interact in one shared
environment. We consider {\em ad hoc} (i.e., nodes can leave the network
unannounced, and no node has any global knowledge about the class identities of
other nodes) {\em preferentially grown networks}, where different classes of
nodes are characterized by different sets of local parameters used in the
stochastic dynamics that all nodes in the network execute. We show that
multiple scale-free structures, one within each class of nodes, and with
tunable power-law exponents (as determined by the sets of parameters
characterizing each class) emerge naturally in our model. Moreover, the
coexistence of the scale-free structures of the different classes of nodes can
be captured by succinct phase diagrams, which show a rich set of structures,
including stable regions where different classes coexist in heavy-tailed and
light-tailed states, and sharp phase transitions. Finally, we show how the
dynamics formulated in this paper will serve as an essential part of {\em
ad-hoc networking protocols}, which can lead to the formation of robust and
efficiently searchable networks (including, the well-known Peer-To-Peer (P2P)
networks) even under very dynamic conditions
A Complex Network Approach to Topographical Connections
The neuronal networks in the mammals cortex are characterized by the
coexistence of hierarchy, modularity, short and long range interactions,
spatial correlations, and topographical connections. Particularly interesting,
the latter type of organization implies special demands on the evolutionary and
ontogenetic systems in order to achieve precise maps preserving spatial
adjacencies, even at the expense of isometry. Although object of intensive
biological research, the elucidation of the main anatomic-functional purposes
of the ubiquitous topographical connections in the mammals brain remains an
elusive issue. The present work reports on how recent results from complex
network formalism can be used to quantify and model the effect of topographical
connections between neuronal cells over a number of relevant network properties
such as connectivity, adjacency, and information broadcasting. While the
topographical mapping between two cortical modules are achieved by connecting
nearest cells from each module, three kinds of network models are adopted for
implementing intracortical connections (ICC), including random,
preferential-attachment, and short-range networks. It is shown that, though
spatially uniform and simple, topographical connections between modules can
lead to major changes in the network properties, fostering more effective
intercommunication between the involved neuronal cells and modules. The
possible implications of such effects on cortical operation are discussed.Comment: 5 pages, 5 figure
Structure of Business Firm Networks and Scale-Free Models
We study the structure of business firm networks and scale-free models with
degree distribution using the method of
-shell decomposition.We find that the Life Sciences industry network consist
of three components: a ``nucleus,'' which is a small well connected subgraph,
``tendrils,'' which are small subgraphs consisting of small degree nodes
connected exclusively to the nucleus, and a ``bulk body'' which consists of the
majority of nodes. At the same time we do not observe the above structure in
the Information and Communication Technology sector of industry. We also
conduct a systematic study of these three components in random scale-free
networks. Our results suggest that the sizes of the nucleus and the tendrils
decrease as increases and disappear for . We compare
the -shell structure of random scale-free model networks with two real world
business firm networks in the Life Sciences and in the Information and
Communication Technology sectors. Our results suggest that the observed
behavior of the -shell structure in the two industries is consistent with a
recently proposed growth model that assumes the coexistence of both
preferential and random agreements in the evolution of industrial networks
The Fractional Preferential Attachment Scale-Free Network Model
Many networks generated by nature have two generic properties: they are
formed in the process of {preferential attachment} and they are scale-free.
Considering these features, by interfering with mechanism of the {preferential
attachment}, we propose a generalisation of the Barab\'asi--Albert model---the
'Fractional Preferential Attachment' (FPA) scale-free network model---that
generates networks with time-independent degree distributions with degree exponent (where corresponds
to the typical value of the BA model). In the FPA model, the element
controlling the network properties is the parameter, where . Depending on the different values of parameter, we study the
statistical properties of the numerically generated networks. We investigate
the topological properties of FPA networks such as degree distribution, degree
correlation (network assortativity), clustering coefficient, average node
degree, network diameter, average shortest path length and features of
fractality. We compare the obtained values with the results for various
synthetic and real-world networks. It is found that, depending on , the FPA
model generates networks with parameters similar to the real-world networks.
Furthermore, it is shown that parameter has a significant impact on, among
others, degree distribution and degree correlation of generated networks.
Therefore, the FPA scale-free network model can be an interesting alternative
to existing network models. In addition, it turns out that, regardless of the
value of , FPA networks are not fractal.Comment: 16 pages, 6 figure
Structure of Business Firm Networks and Scale-Free Models.
We study the structure of business firm networks in the Life Sciences (LS) and the Information and Communication Technology (ICT) sectors. We analyze business firm networks and scale-free models with degree distribution P(q) proportional to (q + c)^-λ using the method of k-shell decomposition. We find that the LS network consists of three components: a "nucleus", which is a small well connected subgraph, "tendrils", which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus, and a "bulk body" which consists of the majority of nodes. At the same time we do not observe the above structure in the ICT network. Our results suggest that the sizes of the nucleus and the tendrils decrease as λ increases and disappear for λ greater or equal to 3. We compare the k-shell structure of random scale-free model networks with the real world business firm networks. The observed behavior of the k-shell structure in the two industries is consistent with a recently proposed growth model that assumes the coexistence of both preferential and random regimes in the evolution of industry networks.
Modeling the IPv6 Internet AS-level Topology
To measure the IPv6 internet AS-level topology, a network topology discovery
system, called Dolphin, was developed. By comparing the measurement result of
Dolphin with that of CAIDA's Scamper, it was found that the IPv6 Internet at AS
level, similar to other complex networks, is also scale-free but the exponent
of its degree distribution is 1.2, which is much smaller than that of the IPv4
Internet and most other scale-free networks. In order to explain this feature
of IPv6 Internet we argue that the degree exponent is a measure of uniformity
of the degree distribution. Then, for the purpose on modeling the networks, we
propose a new model based on the two major factors affecting the exponent of
the EBA model. It breaks the lower bound of degree exponent which is 2 for most
models. To verify the validity of this model, both theoretical and experimental
analyses have been carried out. Finally, we demonstrate how this model can be
successfully used to reproduce the topology of the IPv6 Internet.Comment: 15 pages, 5 figure
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