306 research outputs found
Numerical evaluation of the upper critical dimension of percolation in scale-free networks
We propose a numerical method to evaluate the upper critical dimension
of random percolation clusters in Erd\H{o}s-R\'{e}nyi networks and in
scale-free networks with degree distribution ,
where is the degree of a node and is the broadness of the degree
distribution. Our results report the theoretical prediction, for scale-free networks with and
for Erd\H{o}s-R\'{e}nyi networks and scale-free networks with .
When the removal of nodes is not random but targeted on removing the highest
degree nodes we obtain for all . Our method also yields
a better numerical evaluation of the critical percolation threshold, , for
scale-free networks. Our results suggest that the finite size effects increases
when approaches 3 from above.Comment: 10 pages, 5 figure
Percolation theory applied to measures of fragmentation in social networks
We apply percolation theory to a recently proposed measure of fragmentation
for social networks. The measure is defined as the ratio between the
number of pairs of nodes that are not connected in the fragmented network after
removing a fraction of nodes and the total number of pairs in the original
fully connected network. We compare with the traditional measure used in
percolation theory, , the fraction of nodes in the largest cluster
relative to the total number of nodes. Using both analytical and numerical
methods from percolation, we study Erd\H{o}s-R\'{e}nyi (ER) and scale-free (SF)
networks under various types of node removal strategies. The removal strategies
are: random removal, high degree removal and high betweenness centrality
removal. We find that for a network obtained after removal (all strategies) of
a fraction of nodes above percolation threshold, . For fixed and close to percolation threshold
(), we show that better reflects the actual fragmentation. Close
to , for a given , has a broad distribution and it is
thus possible to improve the fragmentation of the network. We also study and
compare the fragmentation measure and the percolation measure
for a real social network of workplaces linked by the households of the
employees and find similar results.Comment: submitted to PR
Robustness of onion-like correlated networks against targeted attacks
Recently, it was found by Schneider et al. [Proc. Natl. Acad. Sci. USA, 108,
3838 (2011)], using simulations, that scale-free networks with "onion
structure" are very robust against targeted high degree attacks. The onion
structure is a network where nodes with almost the same degree are connected.
Motivated by this work, we propose and analyze, based on analytical
considerations, an onion-like candidate for a nearly optimal structure against
simultaneous random and targeted high degree node attacks. The nearly optimal
structure can be viewed as a hierarchically interconnected random regular
graphs, the degrees and populations of which are specified by the degree
distribution. This network structure exhibits an extremely assortative
degree-degree correlation and has a close relationship to the "onion
structure." After deriving a set of exact expressions that enable us to
calculate the critical percolation threshold and the giant component of a
correlated network for an arbitrary type of node removal, we apply the theory
to the cases of random scale-free networks that are highly vulnerable against
targeted high degree node removal. Our results show that this vulnerability can
be significantly reduced by implementing this onion-like type of degree-degree
correlation without much undermining the almost complete robustness against
random node removal. We also investigate in detail the robustness enhancement
due to assortative degree-degree correlation by introducing a joint
degree-degree probability matrix that interpolates between an uncorrelated
network structure and the onion-like structure proposed here by tuning a single
control parameter. The optimal values of the control parameter that maximize
the robustness against simultaneous random and targeted attacks are also
determined. Our analytical calculations are supported by numerical simulations.Comment: 12 pages, 8 figure
Optimal Paths in Complex Networks with Correlated Weights: The World-wide Airport Network
We study complex networks with weights, , associated with each link
connecting node and . The weights are chosen to be correlated with the
network topology in the form found in two real world examples, (a) the
world-wide airport network, and (b) the {\it E. Coli} metabolic network. Here
, where and are the degrees of
nodes and , is a random number and represents the
strength of the correlations. The case represents correlation
between weights and degree, while represents anti-correlation and
the case reduces to the case of no correlations. We study the
scaling of the lengths of the optimal paths, , with the system
size in strong disorder for scale-free networks for different . We
calculate the robustness of correlated scale-free networks with different
, and find the networks with to be the most robust
networks when compared to the other values of . We propose an
analytical method to study percolation phenomena on networks with this kind of
correlation. We compare our simulation results with the real world-wide airport
network, and we find good agreement
Structure of shells in complex networks
In a network, we define shell as the set of nodes at distance
with respect to a given node and define as the fraction of nodes
outside shell . In a transport process, information or disease usually
diffuses from a random node and reach nodes shell after shell. Thus,
understanding the shell structure is crucial for the study of the transport
property of networks. For a randomly connected network with given degree
distribution, we derive analytically the degree distribution and average degree
of the nodes residing outside shell as a function of . Further,
we find that follows an iterative functional form
, where is expressed in terms of the generating
function of the original degree distribution of the network. Our results can
explain the power-law distribution of the number of nodes found in
shells with larger than the network diameter , which is the average
distance between all pairs of nodes. For real world networks the theoretical
prediction of deviates from the empirical . We introduce a
network correlation function to
characterize the correlations in the network, where is the
empirical value and is the theoretical prediction.
indicates perfect agreement between empirical results and theory. We apply
to several model and real world networks. We find that the networks
fall into two distinct classes: (i) a class of {\it poorly-connected} networks
with , which have larger average distances compared with randomly
connected networks with the same degree distributions; and (ii) a class of {\it
well-connected} networks with
Transport in weighted networks: Partition into superhighways and roads
Transport in weighted networks is dominated by the minimum spanning tree
(MST), the tree connecting all nodes with the minimum total weight. We find
that the MST can be partitioned into two distinct components, having
significantly different transport properties, characterized by centrality --
number of times a node (or link) is used by transport paths. One component, the
{\it superhighways}, is the infinite incipient percolation cluster; for which
we find that nodes (or links) with high centrality dominate. For the other
component, {\it roads}, which includes the remaining nodes, low centrality
nodes dominate. We find also that the distribution of the centrality for the
infinite incipient percolation cluster satisfies a power law, with an exponent
smaller than that for the entire MST. The significance of this finding is that
one can improve significantly the global transport by improving a tiny fraction
of the network, the superhighways.Comment: 12 pages, 5 figure
Percolation of Partially Interdependent Scale-free Networks
We study the percolation behavior of two interdependent scale-free (SF)
networks under random failure of 1- fraction of nodes. Our results are based
on numerical solutions of analytical expressions and simulations. We find that
as the coupling strength between the two networks reduces from 1 (fully
coupled) to 0 (no coupling), there exist two critical coupling strengths
and , which separate three different regions with different behavior of
the giant component as a function of . (i) For , an abrupt
collapse transition occurs at . (ii) For , the giant
component has a hybrid transition combined of both, abrupt decrease at a
certain followed by a smooth decrease to zero for as decreases to zero. (iii) For , the giant
component has a continuous second-order transition (at ). We find that
for , ; and for ,
decreases with increasing . In the hybrid transition, at the
region, the mutual giant component jumps
discontinuously at to a very small but non-zero value, and when
reducing , continuously approaches to 0 at for
for . Thus, the known theoretical
for a single network with is expected to be valid
also for strictly partial interdependent networks.Comment: 20 pages, 17 figure
Robustness of a Network of Networks
Almost all network research has been focused on the properties of a single
network that does not interact and depends on other networks. In reality, many
real-world networks interact with other networks. Here we develop an analytical
framework for studying interacting networks and present an exact percolation
law for a network of interdependent networks. In particular, we find that
for Erd\H{o}s-R\'{e}nyi networks each of average degree , the giant
component, , is given by
where is the initial fraction of removed nodes. Our general result
coincides for with the known Erd\H{o}s-R\'{e}nyi second-order phase
transition for a single network. For any cascading failures occur
and the transition becomes a first-order percolation transition. The new law
for shows that percolation theory that is extensively studied in
physics and mathematics is a limiting case () of a more general general
and different percolation law for interdependent networks.Comment: 7 pages, 3 figure
Truncation of power law behavior in "scale-free" network models due to information filtering
We formulate a general model for the growth of scale-free networks under
filtering information conditions--that is, when the nodes can process
information about only a subset of the existing nodes in the network. We find
that the distribution of the number of incoming links to a node follows a
universal scaling form, i.e., that it decays as a power law with an exponential
truncation controlled not only by the system size but also by a feature not
previously considered, the subset of the network ``accessible'' to the node. We
test our model with empirical data for the World Wide Web and find agreement.Comment: LaTeX2e and RevTeX4, 4 pages, 4 figures. Accepted for publication in
Physical Review Letter
Primary accumulation in the Soviet transition
The Soviet background to the idea of primary socialist accumulation is presented. The mobilisation of labour power and of products into public sector investment from outside are shown to have been the two original forms of the concept. In Soviet primary accumulation the mobilisation of labour power was apparently more decisive than the mobilisation of products. The primary accumulation process had both intended and unintended results. Intended results included bringing most of the economy into the public sector, and industrialisation of the economy as a whole. Unintended results included substantial economic losses, and the proliferation of coercive institutions damaging to attainment of the ultimate goal - the building of a communist society
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