346,480 research outputs found
Degree correlations in scale-free null models
We study the average nearest neighbor degree of vertices with degree
. In many real-world networks with power-law degree distribution
falls off in , a property ascribed to the constraint that any two vertices
are connected by at most one edge. We show that indeed decays in in
three simple random graph null models with power-law degrees: the erased
configuration model, the rank-1 inhomogeneous random graph and the hyperbolic
random graph. We consider the large-network limit when the number of nodes
tends to infinity. We find for all three null models that starts to
decay beyond and then settles on a power law , with the degree exponent.Comment: 21 pages, 4 figure
Superdiffusion in a class of networks with marginal long-range connections
A class of cubic networks composed of a regular one-dimensional lattice and a
set of long-range links is introduced. Networks parametrized by a positive
integer k are constructed by starting from a one-dimensional lattice and
iteratively connecting each site of degree 2 with a th neighboring site of
degree 2. Specifying the way pairs of sites to be connected are selected,
various random and regular networks are defined, all of which have a power-law
edge-length distribution of the form with the marginal
exponent s=1. In all these networks, lengths of shortest paths grow as a power
of the distance and random walk is super-diffusive. Applying a renormalization
group method, the corresponding shortest-path dimensions and random-walk
dimensions are calculated exactly for k=1 networks and for k=2 regular
networks; in other cases, they are estimated by numerical methods. Although,
s=1 holds for all representatives of this class, the above quantities are found
to depend on the details of the structure of networks controlled by k and other
parameters.Comment: 10 pages, 9 figure
Biological network comparison using graphlet degree distribution
Analogous to biological sequence comparison, comparing cellular networks is
an important problem that could provide insight into biological understanding
and therapeutics. For technical reasons, comparing large networks is
computationally infeasible, and thus heuristics such as the degree distribution
have been sought. It is easy to demonstrate that two networks are different by
simply showing a short list of properties in which they differ. It is much
harder to show that two networks are similar, as it requires demonstrating
their similarity in all of their exponentially many properties. Clearly, it is
computationally prohibitive to analyze all network properties, but the larger
the number of constraints we impose in determining network similarity, the more
likely it is that the networks will truly be similar.
We introduce a new systematic measure of a network's local structure that
imposes a large number of similarity constraints on networks being compared. In
particular, we generalize the degree distribution, which measures the number of
nodes 'touching' k edges, into distributions measuring the number of nodes
'touching' k graphlets, where graphlets are small connected non-isomorphic
subgraphs of a large network. Our new measure of network local structure
consists of 73 graphlet degree distributions (GDDs) of graphlets with 2-5
nodes, but it is easily extendible to a greater number of constraints (i.e.
graphlets). Furthermore, we show a way to combine the 73 GDDs into a network
'agreement' measure. Based on this new network agreement measure, we show that
almost all of the 14 eukaryotic PPI networks, including human, are better
modeled by geometric random graphs than by Erdos-Reny, random scale-free, or
Barabasi-Albert scale-free networks.Comment: Proceedings of the 2006 European Conference on Computational Biology,
ECCB'06, Eilat, Israel, January 21-24, 200
Leaders of neuronal cultures in a quorum percolation model
We present a theoretical framework using quorum-percolation for describing
the initiation of activity in a neural culture. The cultures are modeled as
random graphs, whose nodes are excitatory neurons with kin inputs and kout
outputs, and whose input degrees kin = k obey given distribution functions pk.
We examine the firing activity of the population of neurons according to their
input degree (k) classes and calculate for each class its firing probability
\Phi_k(t) as a function of t. The probability of a node to fire is found to be
determined by its in-degree k, and the first-to-fire neurons are those that
have a high k. A small minority of high-k classes may be called "Leaders", as
they form an inter-connected subnetwork that consistently fires much before the
rest of the culture. Once initiated, the activity spreads from the Leaders to
the less connected majority of the culture. We then use the distribution of
in-degree of the Leaders to study the growth rate of the number of neurons
active in a burst, which was experimentally measured to be initially
exponential. We find that this kind of growth rate is best described by a
population that has an in-degree distribution that is a Gaussian centered
around k = 75 with width {\sigma} = 31 for the majority of the neurons, but
also has a power law tail with exponent -2 for ten percent of the population.
Neurons in the tail may have as many as k = 4, 700 inputs. We explore and
discuss the correspondence between the degree distribution and a dynamic
neuronal threshold, showing that from the functional point of view, structure
and elementary dynamics are interchangeable. We discuss possible geometric
origins of this distribution, and comment on the importance of size, or of
having a large number of neurons, in the culture.Comment: Keywords: Neuronal cultures, Graph theory, Activation dynamics,
Percolation, Statistical mechanics of networks, Leaders of activity, Quorum.
http://www.weizmann.ac.il/complex/tlusty/papers/FrontCompNeuro2010.pd
Degree distributions in AB random geometric graphs
In this paper, we provide degree distributions for random geometric
graphs, in which points of type connect to the closest points of type
. The motivating example to derive such degree distributions is in 5G
wireless networks with multi-connectivity, where users connect to their closest
base stations. It is important to know how many users a particular base
station serves, which gives the degree of that base station. To obtain these
degree distributions, we investigate the distribution of area sizes of the
th order Voronoi cells of -points. Assuming that the -points are
Poisson distributed, we investigate the amount of users connected to a certain
-point, which is equal to the degree of this point. In the simple case where
the -points are placed in an hexagonal grid, we show that all -th order
Voronoi areas are equal and thus all degrees follow a Poisson distribution.
However, this observation does not hold for Poisson distributed -points, for
which we show that the degree distribution follows a compound Poisson-Erlang
distribution in the 1-dimensional case. We then approximate the degree
distribution in the 2-dimensional case with a compound Poisson-Gamma degree
distribution and show that this one-parameter fit performs well for different
values of . Moreover, we show that for increasing , these degree
distributions become more concentrated around the mean. This means that
-connected random graphs balance the loads of -type nodes more
evenly as increases. Finally, we provide a case study on real data of base
stations. We show that with little shadowing in the distances between users and
base stations, the Poisson distribution does not capture the degree
distribution of these data, especially for . However, under strong
shadowing, our degree approximations perform quite good even for these
non-Poissonian location data.Comment: 23 pages, 13 figure
Inducing Effect on the Percolation Transition in Complex Networks
Percolation theory concerns the emergence of connected clusters that
percolate through a networked system. Previous studies ignored the effect that
a node outside the percolating cluster may actively induce its inside
neighbours to exit the percolating cluster. Here we study this inducing effect
on the classical site percolation and K-core percolation, showing that the
inducing effect always causes a discontinuous percolation transition. We
precisely predict the percolation threshold and core size for uncorrelated
random networks with arbitrary degree distributions. For low-dimensional
lattices the percolation threshold fluctuates considerably over realizations,
yet we can still predict the core size once the percolation occurs. The core
sizes of real-world networks can also be well predicted using degree
distribution as the only input. Our work therefore provides a theoretical
framework for quantitatively understanding discontinuous breakdown phenomena in
various complex systems.Comment: Main text and appendices. Title has been change
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