106,691 research outputs found
Higher order clustering coefficients in Barabasi-Albert networks
Higher order clustering coefficients are introduced for random
networks. The coefficients express probabilities that the shortest distance
between any two nearest neighbours of a certain vertex equals , when one
neglects all paths crossing the node . Using we found that in the
Barab\'{a}si-Albert (BA) model the average shortest path length in a node's
neighbourhood is smaller than the equivalent quantity of the whole network and
the remainder depends only on the network parameter . Our results show that
small values of the standard clustering coefficient in large BA networks are
due to random character of the nearest neighbourhood of vertices in such
networks.Comment: 10 pages, 4 figure
Rich-club connectivity dominates assortativity and transitivity of complex networks
Rich-club, assortativity and clustering coefficients are frequently-used
measures to estimate topological properties of complex networks. Here we find
that the connectivity among a very small portion of the richest nodes can
dominate the assortativity and clustering coefficients of a large network,
which reveals that the rich-club connectivity is leveraged throughout the
network. Our study suggests that more attention should be payed to the
organization pattern of rich nodes, for the structure of a complex system as a
whole is determined by the associations between the most influential
individuals. Moreover, by manipulating the connectivity pattern in a very small
rich-club, it is sufficient to produce a network with desired assortativity or
transitivity. Conversely, our findings offer a simple explanation for the
observed assortativity and transitivity in many real world networks --- such
biases can be explained by the connectivities among the richest nodes.Comment: 5 pages, 2 figures, accepted by Phys. Rev.
Bayesian nonparametric sparse VAR models
High dimensional vector autoregressive (VAR) models require a large number of
parameters to be estimated and may suffer of inferential problems. We propose a
new Bayesian nonparametric (BNP) Lasso prior (BNP-Lasso) for high-dimensional
VAR models that can improve estimation efficiency and prediction accuracy. Our
hierarchical prior overcomes overparametrization and overfitting issues by
clustering the VAR coefficients into groups and by shrinking the coefficients
of each group toward a common location. Clustering and shrinking effects
induced by the BNP-Lasso prior are well suited for the extraction of causal
networks from time series, since they account for some stylized facts in
real-world networks, which are sparsity, communities structures and
heterogeneity in the edges intensity. In order to fully capture the richness of
the data and to achieve a better understanding of financial and macroeconomic
risk, it is therefore crucial that the model used to extract network accounts
for these stylized facts.Comment: Forthcoming in "Journal of Econometrics" ---- Revised Version of the
paper "Bayesian nonparametric Seemingly Unrelated Regression Models" ----
Supplementary Material available on reques
Spatial preferential attachment networks: Power laws and clustering coefficients
We define a class of growing networks in which new nodes are given a spatial
position and are connected to existing nodes with a probability mechanism
favoring short distances and high degrees. The competition of preferential
attachment and spatial clustering gives this model a range of interesting
properties. Empirical degree distributions converge to a limit law, which can
be a power law with any exponent . The average clustering coefficient
of the networks converges to a positive limit. Finally, a phase transition
occurs in the global clustering coefficients and empirical distribution of edge
lengths when the power-law exponent crosses the critical value . Our
main tool in the proof of these results is a general weak law of large numbers
in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Network dynamics of ongoing social relationships
Many recent large-scale studies of interaction networks have focused on
networks of accumulated contacts. In this paper we explore social networks of
ongoing relationships with an emphasis on dynamical aspects. We find a
distribution of response times (times between consecutive contacts of different
direction between two actors) that has a power-law shape over a large range. We
also argue that the distribution of relationship duration (the time between the
first and last contacts between actors) is exponentially decaying. Methods to
reanalyze the data to compensate for the finite sampling time are proposed. We
find that the degree distribution for networks of ongoing contacts fits better
to a power-law than the degree distribution of the network of accumulated
contacts do. We see that the clustering and assortative mixing coefficients are
of the same order for networks of ongoing and accumulated contacts, and that
the structural fluctuations of the former are rather large.Comment: to appear in Europhys. Let
Sampling Geometric Inhomogeneous Random Graphs in Linear Time
Real-world networks, like social networks or the internet infrastructure,
have structural properties such as large clustering coefficients that can best
be described in terms of an underlying geometry. This is why the focus of the
literature on theoretical models for real-world networks shifted from classic
models without geometry, such as Chung-Lu random graphs, to modern
geometry-based models, such as hyperbolic random graphs.
With this paper we contribute to the theoretical analysis of these modern,
more realistic random graph models. Instead of studying directly hyperbolic
random graphs, we use a generalization that we call geometric inhomogeneous
random graphs (GIRGs). Since we ignore constant factors in the edge
probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic
cosines), while preserving the qualitative behaviour of hyperbolic random
graphs, and we suggest to replace hyperbolic random graphs by this new model in
future theoretical studies.
We prove the following fundamental structural and algorithmic results on
GIRGs. (1) As our main contribution we provide a sampling algorithm that
generates a random graph from our model in expected linear time, improving the
best-known sampling algorithm for hyperbolic random graphs by a substantial
factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in
{\Omega}(1), (3) we prove that GIRGs have small separators, i.e., it suffices
to delete a sublinear number of edges to break the giant component into two
large pieces, and (4) we show how to compress GIRGs using an expected linear
number of bits.Comment: 25 page
Unified model for network dynamics exhibiting nonextensive statistics
We introduce a dynamical network model which unifies a number of network
families which are individually known to exhibit -exponential degree
distributions. The present model dynamics incorporates static (non-growing)
self-organizing networks, preferentially growing networks, and (preferentially)
rewiring networks. Further, it exhibits a natural random graph limit. The
proposed model generalizes network dynamics to rewiring and growth modes which
depend on internal topology as well as on a metric imposed by the space they
are embedded in. In all of the networks emerging from the presented model we
find q-exponential degree distributions over a large parameter space. We
comment on the parameter dependence of the corresponding entropic index q for
the degree distributions, and on the behavior of the clustering coefficients
and neighboring connectivity distributions.Comment: 11 pages 8 fig
Function and form in networks of interacting agents
The main problem we address in this paper is whether function determines form
when a society of agents organizes itself for some purpose or whether the
organizing method is more important than the functionality in determining the
structure of the ensemble. As an example, we use a neural network that learns
the same function by two different learning methods. For sufficiently large
networks, very different structures may indeed be obtained for the same
functionality. Clustering, characteristic path length and hierarchy are
structural differences, which in turn have implications on the robustness and
adaptability of the networks. In networks, as opposed to simple graphs, the
connections between the agents are not necessarily symmetric and may have
positive or negative signs. New characteristic coefficients are introduced to
characterize this richer connectivity structure.Comment: 27 pages Latex, 11 figure
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