54,928 research outputs found
Statistical mechanics of topological phase transitions in networks
We provide a phenomenological theory for topological transitions in
restructuring networks. In this statistical mechanical approach energy is
assigned to the different network topologies and temperature is used as a
quantity referring to the level of noise during the rewiring of the edges. The
associated microscopic dynamics satisfies the detailed balance condition and is
equivalent to a lattice gas model on the edge-dual graph of a fully connected
network. In our studies -- based on an exact enumeration method, Monte-Carlo
simulations, and theoretical considerations -- we find a rich variety of
topological phase transitions when the temperature is varied. These transitions
signal singular changes in the essential features of the global structure of
the network. Depending on the energy function chosen, the observed transitions
can be best monitored using the order parameters Phi_s=s_{max}/M, i.e., the
size of the largest connected component divided by the number of edges, or
Phi_k=k_{max}/M, the largest degree in the network divided by the number of
edges. If, for example the energy is chosen to be E=-s_{max}, the observed
transition is analogous to the percolation phase transition of random graphs.
For this choice of the energy, the phase-diagram in the [,T] plane is
constructed. Single vertex energies of the form
E=sum_i f(k_i), where k_i is the degree of vertex i, are also studied.
Depending on the form of f(k_i), first order and continuous phase transitions
can be observed. In case of f(k_i)=-(k_i+c)ln(k_i), the transition is
continuous, and at the critical temperature scale-free graphs can be recovered.Comment: 12 pages, 12 figures, minor changes, added a new refernce, to appear
in PR
StructMatrix: large-scale visualization of graphs by means of structure detection and dense matrices
Given a large-scale graph with millions of nodes and edges, how to reveal
macro patterns of interest, like cliques, bi-partite cores, stars, and chains?
Furthermore, how to visualize such patterns altogether getting insights from
the graph to support wise decision-making? Although there are many algorithmic
and visual techniques to analyze graphs, none of the existing approaches is
able to present the structural information of graphs at large-scale. Hence,
this paper describes StructMatrix, a methodology aimed at high-scalable visual
inspection of graph structures with the goal of revealing macro patterns of
interest. StructMatrix combines algorithmic structure detection and adjacency
matrix visualization to present cardinality, distribution, and relationship
features of the structures found in a given graph. We performed experiments in
real, large-scale graphs with up to one million nodes and millions of edges.
StructMatrix revealed that graphs of high relevance (e.g., Web, Wikipedia and
DBLP) have characterizations that reflect the nature of their corresponding
domains; our findings have not been seen in the literature so far. We expect
that our technique will bring deeper insights into large graph mining,
leveraging their use for decision making.Comment: To appear: 8 pages, paper to be published at the Fifth IEEE ICDM
Workshop on Data Mining in Networks, 2015 as Hugo Gualdron, Robson Cordeiro,
Jose Rodrigues (2015) StructMatrix: Large-scale visualization of graphs by
means of structure detection and dense matrices In: The Fifth IEEE ICDM
Workshop on Data Mining in Networks 1--8, IEE
High transitivity for more groups acting on trees
We establish a new sharp sufficient condition for groups acting on trees to
be highly transitive. This give new examples of highly transitive groups,
including icc non-solvable Baumslag-Solitar groups, thus answering a question
of Hull and Osin.Comment: Version 2 : correction of a mistake in an example from section 8.3
and general improvement of section 8.
The failure tolerance of mechatronic software systems to random and targeted attacks
This paper describes a complex networks approach to study the failure
tolerance of mechatronic software systems under various types of hardware
and/or software failures. We produce synthetic system architectures based on
evidence of modular and hierarchical modular product architectures and known
motifs for the interconnection of physical components to software. The system
architectures are then subject to various forms of attack. The attacks simulate
failure of critical hardware or software. Four types of attack are
investigated: degree centrality, betweenness centrality, closeness centrality
and random attack. Failure tolerance of the system is measured by a 'robustness
coefficient', a topological 'size' metric of the connectedness of the attacked
network. We find that the betweenness centrality attack results in the most
significant reduction in the robustness coefficient, confirming betweenness
centrality, rather than the number of connections (i.e. degree), as the most
conservative metric of component importance. A counter-intuitive finding is
that "designed" system architectures, including a bus, ring, and star
architecture, are not significantly more failure-tolerant than interconnections
with no prescribed architecture, that is, a random architecture. Our research
provides a data-driven approach to engineer the architecture of mechatronic
software systems for failure tolerance.Comment: Proceedings of the 2013 ASME International Design Engineering
Technical Conferences & Computers and Information in Engineering Conference
IDETC/CIE 2013 August 4-7, 2013, Portland, Oregon, USA (In Print
Eccentric connectivity index
The eccentric connectivity index is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as \,, where and
denote the vertex degree and eccentricity of \,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.Comment: 25 pages, 5 figure
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