1,437 research outputs found
Giant strongly connected component of directed networks
We describe how to calculate the sizes of all giant connected components of a
directed graph, including the {\em strongly} connected one. Just to the class
of directed networks, in particular, belongs the World Wide Web. The results
are obtained for graphs with statistically uncorrelated vertices and an
arbitrary joint in,out-degree distribution . We show that if
does not factorize, the relative size of the giant strongly
connected component deviates from the product of the relative sizes of the
giant in- and out-components. The calculations of the relative sizes of all the
giant components are demonstrated using the simplest examples. We explain that
the giant strongly connected component may be less resilient to random damage
than the giant weakly connected one.Comment: 4 pages revtex, 4 figure
Diluted antiferromagnet in a ferromagnetic enviroment
The question of robustness of a network under random ``attacks'' is treated
in the framework of critical phenomena. The persistence of spontaneous
magnetization of a ferromagnetic system to the random inclusion of
antiferromagnetic interactions is investigated. After examing the static
properties of the quenched version (in respect to the random antiferromagnetic
interactions) of the model, the persistence of the magnetization is analysed
also in the annealed approximation, and the difference in the results are
discussed
Percolation in invariant Poisson graphs with i.i.d. degrees
Let each point of a homogeneous Poisson process in R^d independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution mu on the positive integers. We consider
translation-invariant schemes for perfectly matching the stubs to obtain a
simple graph with degree distribution mu. Leaving aside degenerate cases, we
prove that for any mu there exist schemes that give only finite components as
well as schemes that give infinite components. For a particular matching scheme
that is a natural extension of Gale-Shapley stable marriage, we give sufficient
conditions on mu for the absence and presence of infinite components
Network robustness and fragility: Percolation on random graphs
Recent work on the internet, social networks, and the power grid has
addressed the resilience of these networks to either random or targeted
deletion of network nodes. Such deletions include, for example, the failure of
internet routers or power transmission lines. Percolation models on random
graphs provide a simple representation of this process, but have typically been
limited to graphs with Poisson degree distribution at their vertices. Such
graphs are quite unlike real world networks, which often possess power-law or
other highly skewed degree distributions. In this paper we study percolation on
graphs with completely general degree distribution, giving exact solutions for
a variety of cases, including site percolation, bond percolation, and models in
which occupation probabilities depend on vertex degree. We discuss the
application of our theory to the understanding of network resilience.Comment: 4 pages, 2 figure
Vertex similarity in networks
We consider methods for quantifying the similarity of vertices in networks.
We propose a measure of similarity based on the concept that two vertices are
similar if their immediate neighbors in the network are themselves similar.
This leads to a self-consistent matrix formulation of similarity that can be
evaluated iteratively using only a knowledge of the adjacency matrix of the
network. We test our similarity measure on computer-generated networks for
which the expected results are known, and on a number of real-world networks
Transport Properties of Random Walks on Scale-Free/Regular-Lattice Hybrid Networks
We study numerically the mean access times for random walks on hybrid
disordered structures formed by embedding scale-free networks into regular
lattices, considering different transition rates for steps across lattice bonds
() and across network shortcuts (). For fast shortcuts () and
low shortcut densities, traversal time data collapse onto an universal curve,
while a crossover behavior that can be related to the percolation threshold of
the scale-free network component is identified at higher shortcut densities, in
analogy to similar observations reported recently in Newman-Watts small-world
networks. Furthermore, we observe that random walk traversal times are larger
for networks with a higher degree of inhomogeneity in their shortcut
distribution, and we discuss access time distributions as functions of the
initial and final node degrees. These findings are relevant, in particular,
when considering the optimization of existing information networks by the
addition of a small number of fast shortcut connections.Comment: 8 pages, 6 figures; expanded discussions, added figures and
references. To appear in J Stat Phy
Kinetic growth walks on complex networks
Kinetically grown self-avoiding walks on various types of generalized random
networks have been studied. Networks with short- and long-tailed degree
distributions were considered (, degree or connectivity), including
scale-free networks with . The long-range behaviour of
self-avoiding walks on random networks is found to be determined by finite-size
effects. The mean self-intersection length of non-reversal random walks, ,
scales as a power of the system size $N$: $ \sim N^{\beta}$, with an
exponent $\beta = 0.5$ for short-tailed degree distributions and $\beta < 0.5$
for scale-free networks with $\gamma < 3$. The mean attrition length of kinetic
growth walks, , scales as , with an exponent
which depends on the lowest degree in the network. Results of
approximate probabilistic calculations are supported by those derived from
simulations of various kinds of networks. The efficiency of kinetic growth
walks to explore networks is largely reduced by inhomogeneity in the degree
distribution, as happens for scale-free networks.Comment: 10 pages, 8 figure
Random graphs with arbitrary degree distributions and their applications
Recent work on the structure of social networks and the internet has focussed
attention on graphs with distributions of vertex degree that are significantly
different from the Poisson degree distributions that have been widely studied
in the past. In this paper we develop in detail the theory of random graphs
with arbitrary degree distributions. In addition to simple undirected,
unipartite graphs, we examine the properties of directed and bipartite graphs.
Among other results, we derive exact expressions for the position of the phase
transition at which a giant component first forms, the mean component size, the
size of the giant component if there is one, the mean number of vertices a
certain distance away from a randomly chosen vertex, and the average
vertex-vertex distance within a graph. We apply our theory to some real-world
graphs, including the world-wide web and collaboration graphs of scientists and
Fortune 1000 company directors. We demonstrate that in some cases random graphs
with appropriate distributions of vertex degree predict with surprising
accuracy the behavior of the real world, while in others there is a measurable
discrepancy between theory and reality, perhaps indicating the presence of
additional social structure in the network that is not captured by the random
graph.Comment: 19 pages, 11 figures, some new material added in this version along
with minor updates and correction
Evolutionary Events in a Mathematical Sciences Research Collaboration Network
This study examines long-term trends and shifting behavior in the
collaboration network of mathematics literature, using a subset of data from
Mathematical Reviews spanning 1985-2009. Rather than modeling the network
cumulatively, this study traces the evolution of the "here and now" using
fixed-duration sliding windows. The analysis uses a suite of common network
diagnostics, including the distributions of degrees, distances, and clustering,
to track network structure. Several random models that call these diagnostics
as parameters help tease them apart as factors from the values of others. Some
behaviors are consistent over the entire interval, but most diagnostics
indicate that the network's structural evolution is dominated by occasional
dramatic shifts in otherwise steady trends. These behaviors are not distributed
evenly across the network; stark differences in evolution can be observed
between two major subnetworks, loosely thought of as "pure" and "applied",
which approximately partition the aggregate. The paper characterizes two major
events along the mathematics network trajectory and discusses possible
explanatory factors.Comment: 30 pages, 14 figures, 1 table; supporting information: 5 pages, 5
figures; published in Scientometric
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