35,345 research outputs found
Degree Correlations in Random Geometric Graphs
Spatially embedded networks are important in several disciplines. The
prototypical spatial net- work we assume is the Random Geometric Graph of which
many properties are known. Here we present new results for the two-point degree
correlation function in terms of the clustering coefficient of the graphs for
two-dimensional space in particular, with extensions to arbitrary finite
dimension
Spectral statistics of random geometric graphs
We use random matrix theory to study the spectrum of random geometric graphs,
a fundamental model of spatial networks. Considering ensembles of random
geometric graphs we look at short range correlations in the level spacings of
the spectrum via the nearest neighbour and next nearest neighbour spacing
distribution and long range correlations via the spectral rigidity Delta_3
statistic. These correlations in the level spacings give information about
localisation of eigenvectors, level of community structure and the level of
randomness within the networks. We find a parameter dependent transition
between Poisson and Gaussian orthogonal ensemble statistics. That is the
spectral statistics of spatial random geometric graphs fits the universality of
random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert
and Watts-Strogatz random graph.Comment: 19 pages, 6 figures. Substantially updated from previous versio
Algorithm and Complexity for a Network Assortativity Measure
We show that finding a graph realization with the minimum Randi\'c index for
a given degree sequence is solvable in polynomial time by formulating the
problem as a minimum weight perfect b-matching problem. However, the
realization found via this reduction is not guaranteed to be connected.
Approximating the minimum weight b-matching problem subject to a connectivity
constraint is shown to be NP-Hard. For instances in which the optimal solution
to the minimum Randi\'c index problem is not connected, we describe a heuristic
to connect the graph using pairwise edge exchanges that preserves the degree
sequence. In our computational experiments, the heuristic performs well and the
Randi\'c index of the realization after our heuristic is within 3% of the
unconstrained optimal value on average. Although we focus on minimizing the
Randi\'c index, our results extend to maximizing the Randi\'c index as well.
Applications of the Randi\'c index to synchronization of neuronal networks
controlling respiration in mammals and to normalizing cortical thickness
networks in diagnosing individuals with dementia are provided.Comment: Added additional section on application
Random Graph Models with Hidden Color
We demonstrate how to generalize two of the most well-known random graph
models, the classic random graph, and random graphs with a given degree
distribution, by the introduction of hidden variables in the form of extra
degrees of freedom, color, applied to vertices or stubs (half-edges). The color
is assumed unobservable, but is allowed to affect edge probabilities. This
serves as a convenient method to define very general classes of models within a
common unifying formalism, and allowing for a non-trivial edge correlation
structure.Comment: 17 pages, 2 figures; contrib. to the Workshop on Random Geometry in
Krakow, May 200
Riemannian-geometric entropy for measuring network complexity
A central issue of the science of complex systems is the quantitative
characterization of complexity. In the present work we address this issue by
resorting to information geometry. Actually we propose a constructive way to
associate to a - in principle any - network a differentiable object (a
Riemannian manifold) whose volume is used to define an entropy. The
effectiveness of the latter to measure networks complexity is successfully
proved through its capability of detecting a classical phase transition
occurring in both random graphs and scale--free networks, as well as of
characterizing small Exponential random graphs, Configuration Models and real
networks.Comment: 15 pages, 3 figure
Cascading failures in spatially-embedded random networks
Cascading failures constitute an important vulnerability of interconnected
systems. Here we focus on the study of such failures on networks in which the
connectivity of nodes is constrained by geographical distance. Specifically, we
use random geometric graphs as representative examples of such spatial
networks, and study the properties of cascading failures on them in the
presence of distributed flow. The key finding of this study is that the process
of cascading failures is non-self-averaging on spatial networks, and thus,
aggregate inferences made from analyzing an ensemble of such networks lead to
incorrect conclusions when applied to a single network, no matter how large the
network is. We demonstrate that this lack of self-averaging disappears with the
introduction of a small fraction of long-range links into the network. We
simulate the well studied preemptive node removal strategy for cascade
mitigation and show that it is largely ineffective in the case of spatial
networks. We introduce an altruistic strategy designed to limit the loss of
network nodes in the event of a cascade triggering failure and show that it
performs better than the preemptive strategy. Finally, we consider a real-world
spatial network viz. a European power transmission network and validate that
our findings from the study of random geometric graphs are also borne out by
simulations of cascading failures on the empirical network.Comment: 13 pages, 15 figure
Worm Epidemics in Wireless Adhoc Networks
A dramatic increase in the number of computing devices with wireless
communication capability has resulted in the emergence of a new class of
computer worms which specifically target such devices. The most striking
feature of these worms is that they do not require Internet connectivity for
their propagation but can spread directly from device to device using a
short-range radio communication technology, such as WiFi or Bluetooth. In this
paper, we develop a new model for epidemic spreading of these worms and
investigate their spreading in wireless ad hoc networks via extensive Monte
Carlo simulations. Our studies show that the threshold behaviour and dynamics
of worm epidemics in these networks are greatly affected by a combination of
spatial and temporal correlations which characterize these networks, and are
significantly different from the previously studied epidemics in the Internet
Clique topology reveals intrinsic geometric structure in neural correlations
Detecting meaningful structure in neural activity and connectivity data is
challenging in the presence of hidden nonlinearities, where traditional
eigenvalue-based methods may be misleading. We introduce a novel approach to
matrix analysis, called clique topology, that extracts features of the data
invariant under nonlinear monotone transformations. These features can be used
to detect both random and geometric structure, and depend only on the relative
ordering of matrix entries. We then analyzed the activity of pyramidal neurons
in rat hippocampus, recorded while the animal was exploring a two-dimensional
environment, and confirmed that our method is able to detect geometric
organization using only the intrinsic pattern of neural correlations.
Remarkably, we found similar results during non-spatial behaviors such as wheel
running and REM sleep. This suggests that the geometric structure of
correlations is shaped by the underlying hippocampal circuits, and is not
merely a consequence of position coding. We propose that clique topology is a
powerful new tool for matrix analysis in biological settings, where the
relationship of observed quantities to more meaningful variables is often
nonlinear and unknown.Comment: 29 pages, 4 figures, 13 supplementary figures (last two authors
contributed equally
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