4,268 research outputs found
On the complexity of color-avoiding site and bond percolation
The mathematical analysis of robustness and error-tolerance of complex
networks has been in the center of research interest. On the other hand, little
work has been done when the attack-tolerance of the vertices or edges are not
independent but certain classes of vertices or edges share a mutual
vulnerability. In this study, we consider a graph and we assign colors to the
vertices or edges, where the color-classes correspond to the shared
vulnerabilities. An important problem is to find robustly connected vertex
sets: nodes that remain connected to each other by paths providing any type of
error (i.e. erasing any vertices or edges of the given color). This is also
known as color-avoiding percolation. In this paper, we study various possible
modeling approaches of shared vulnerabilities, we analyze the computational
complexity of finding the robustly (color-avoiding) connected components. We
find that the presented approaches differ significantly regarding their
complexity.Comment: 14 page
Path finding strategies in scale-free networks
We numerically investigate the scale-free network model of Barab{\'a}si and
Albert [A. L. Barab{\'a}si and R. Albert, Science {\bf 286}, 509 (1999)]
through the use of various path finding strategies. In real networks, global
network information is not accessible to each vertex, and the actual path
connecting two vertices can sometimes be much longer than the shortest one. A
generalized diameter depending on the actual path finding strategy is
introduced, and a simple strategy, which utilizes only local information on the
connectivity, is suggested and shown to yield small-world behavior: the
diameter of the network increases logarithmically with the network size
, the same as is found with global strategy. If paths are sought at random,
is found.Comment: 4 pages, final for
Efficiency of Scale-Free Networks: Error and Attack Tolerance
The concept of network efficiency, recently proposed to characterize the
properties of small-world networks, is here used to study the effects of errors
and attacks on scale-free networks. Two different kinds of scale-free networks,
i.e. networks with power law P(k), are considered: 1) scale-free networks with
no local clustering produced by the Barabasi-Albert model and 2) scale-free
networks with high clustering properties as in the model by Klemm and Eguiluz,
and their properties are compared to the properties of random graphs
(exponential graphs). By using as mathematical measures the global and the
local efficiency we investigate the effects of errors and attacks both on the
global and the local properties of the network. We show that the global
efficiency is a better measure than the characteristic path length to describe
the response of complex networks to external factors. We find that, at variance
with random graphs, scale-free networks display, both on a global and on a
local scale, a high degree of error tolerance and an extreme vulnerability to
attacks. In fact, the global and the local efficiency are unaffected by the
failure of some randomly chosen nodes, though they are extremely sensititive to
the removal of the few nodes which play a crucial role in maintaining the
network's connectivity.Comment: 23 pages, 10 figure
- …