63,202 research outputs found
Average Distance in a General Class of Scale-Free Networks with Underlying Geometry
In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution (for which we fix an exponent ), and two vertices form an edge independently with probability proportional to the product of their weights. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry, which is typically Euclidean, such as the unit square, circle, or torus. The edge probability of two vertices then depends, say, inversely polynomial on their distance. We show that specific choices, such as the underlying geometry being Euclidean or the dependence on the distance being inversely polynomial, do not significantly influence the average distance, by studying a generic augmented version of Chung-Lu random graphs. Specifically, we analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs, i.e., proportional to the product of their weights. The resulting class contains Chung-Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases. Our main result is that this general model has the same average distance as Chung-Lu random graphs, up to a factor . The proof also yields that our model has a giant component and polylogarithmic diameter with high probability
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
Transport on complex networks: Flow, jamming and optimization
Many transport processes on networks depend crucially on the underlying network geometry, although the exact relationship between the structure of the network and the properties of transport processes remain elusive. In this paper we address this question by using numerical models in which both structure and dynamics are controlled systematically. We consider the traffic of information packets that include driving, searching and queuing. We present the results of extensive simulations on two classes of networks; a correlated cyclic scale-free network and an uncorrelated homogeneous weakly clustered network. By measuring different dynamical variables in the free flow regime we show how the global statistical properties of the transport are related to the temporal fluctuations at individual nodes (the traffic noise) and the links (the traffic flow). We then demonstrate that these two network classes appear as representative topologies for optimal traffic flow in the regimes of low density and high density traffic, respectively. We also determine statistical indicators of the pre-jamming regime on different network geometries and discuss the role of queuing and dynamical betweenness for the traffic congestion. The transition to the jammed traffic regime at a critical posting rate on different network topologies is studied as a phase transition with an appropriate order parameter. We also address several open theoretical problems related to the network dynamics
Hyperbolic Geometry of Complex Networks
We develop a geometric framework to study the structure and function of
complex networks. We assume that hyperbolic geometry underlies these networks,
and we show that with this assumption, heterogeneous degree distributions and
strong clustering in complex networks emerge naturally as simple reflections of
the negative curvature and metric property of the underlying hyperbolic
geometry. Conversely, we show that if a network has some metric structure, and
if the network degree distribution is heterogeneous, then the network has an
effective hyperbolic geometry underneath. We then establish a mapping between
our geometric framework and statistical mechanics of complex networks. This
mapping interprets edges in a network as non-interacting fermions whose
energies are hyperbolic distances between nodes, while the auxiliary fields
coupled to edges are linear functions of these energies or distances. The
geometric network ensemble subsumes the standard configuration model and
classical random graphs as two limiting cases with degenerate geometric
structures. Finally, we show that targeted transport processes without global
topology knowledge, made possible by our geometric framework, are maximally
efficient, according to all efficiency measures, in networks with strongest
heterogeneity and clustering, and that this efficiency is remarkably robust
with respect to even catastrophic disturbances and damages to the network
structure
Metric clusters in evolutionary games on scale-free networks
The evolution of cooperation in social dilemmas in structured populations has
been studied extensively in recent years. Whereas many theoretical studies have
found that a heterogeneous network of contacts favors cooperation, the impact
of spatial effects in scale-free networks is still not well understood. In
addition to being heterogeneous, real contact networks exhibit a high mean
local clustering coefficient, which implies the existence of an underlying
metric space. Here, we show that evolutionary dynamics in scale-free networks
self-organize into spatial patterns in the underlying metric space. The
resulting metric clusters of cooperators are able to survive in social dilemmas
as their spatial organization shields them from surrounding defectors, similar
to spatial selection in Euclidean space. We show that under certain conditions
these metric clusters are more efficient than the most connected nodes at
sustaining cooperation and that heterogeneity does not always favor--but can
even hinder--cooperation in social dilemmas. Our findings provide a new
perspective to understand the emergence of cooperation in evolutionary games in
realistic structured populations
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