28,307 research outputs found
Average nearest neighbor degrees in scale-free networks
The average nearest neighbor degree (ANND) of a node of degree is widely
used to measure dependencies between degrees of neighbor nodes in a network. We
formally analyze ANND in undirected random graphs when the graph size tends to
infinity. The limiting behavior of ANND depends on the variance of the degree
distribution. When the variance is finite, the ANND has a deterministic limit.
When the variance is infinite, the ANND scales with the size of the graph, and
we prove a corresponding central limit theorem in the configuration model (CM,
a network with random connections). As ANND proved uninformative in the
infinite variance scenario, we propose an alternative measure, the average
nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic
function whenever the degree distribution has finite mean. We then consider the
erased configuration model (ECM), where self-loops and multiple edges are
removed, and investigate the well-known `structural negative correlations', or
`finite-size effects', that arise in simple graphs, such as ECM, because large
nodes can only have a limited number of large neighbors. Interestingly, we
prove that for any fixed , ANNR in ECM converges to the same limit as in CM.
However, numerical experiments show that finite-size effects occur when
scales with
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
Ising Model on Edge-Dual of Random Networks
We consider Ising model on edge-dual of uncorrelated random networks with
arbitrary degree distribution. These networks have a finite clustering in the
thermodynamic limit. High and low temperature expansions of Ising model on the
edge-dual of random networks are derived. A detailed comparison of the critical
behavior of Ising model on scale free random networks and their edge-dual is
presented.Comment: 23 pages, 4 figures, 1 tabl
Information Theoretic Operating Regimes of Large Wireless Networks
In analyzing the point-to-point wireless channel, insights about two
qualitatively different operating regimes--bandwidth- and power-limited--have
proven indispensable in the design of good communication schemes. In this
paper, we propose a new scaling law formulation for wireless networks that
allows us to develop a theory that is analogous to the point-to-point case. We
identify fundamental operating regimes of wireless networks and derive
architectural guidelines for the design of optimal schemes.
Our analysis shows that in a given wireless network with arbitrary size,
area, power, bandwidth, etc., there are three parameters of importance: the
short-distance SNR, the long-distance SNR, and the power path loss exponent of
the environment. Depending on these parameters we identify four qualitatively
different regimes. One of these regimes is especially interesting since it is
fundamentally a consequence of the heterogeneous nature of links in a network
and does not occur in the point-to-point case; the network capacity is {\em
both} power and bandwidth limited. This regime has thus far remained hidden due
to the limitations of the existing formulation. Existing schemes, either
multihop transmission or hierarchical cooperation, fail to achieve capacity in
this regime; we propose a new hybrid scheme that achieves capacity.Comment: 12 pages, 5 figures, to appear in IEEE Transactions on Information
Theor
Generation of uncorrelated random scale-free networks
Uncorrelated random scale-free networks are useful null models to check the
accuracy an the analytical solutions of dynamical processes defined on complex
networks. We propose and analyze a model capable to generate random
uncorrelated scale-free networks with no multiple and self-connections. The
model is based on the classical configuration model, with an additional
restriction on the maximum possible degree of the vertices. We check
numerically that the proposed model indeed generates scale-free networks with
no two and three vertex correlations, as measured by the average degree of the
nearest neighbors and the clustering coefficient of the vertices of degree ,
respectively
Optimizing transport efficiency on scale-free networks through assortative or dissortative topology
We find that transport on scale-free random networks depends strongly on
degree-correlated network topologies whereas transport on
Erds-Rnyi networks is insensitive to the degree
correlation. An approach for the tuning of scale-free network transport
efficiency through assortative or dissortative topology is proposed. We
elucidate that the unique transport behavior for scale-free networks results
from the heterogeneous distribution of degrees.Comment: 4 pages, 3 figure
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