[Show abstract][Hide abstract] ABSTRACT: Networks often represent systems that do not have a long history of studies
in traditional fields of physics, albeit there are some notable exceptions such
as energy landscapes and quantum gravity. Here we consider networks that
naturally arise in cosmology. Nodes in these networks are stationary observers
uniformly distributed in an expanding open FLRW universe with any scale factor,
and two observers are connected if one can causally influence the other. We
show that these networks are growing Lorentz-invariant graphs with power-law
distributions of node degrees. New links in these networks not only connect new
nodes to existing ones, but also appear at a certain rate between existing
nodes, as they do in many complex networks.
New Journal of Physics 10/2013; 16(9). · 4.06 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In statistical physics any given system can be either at an equilibrium or away from it. Networks are not an exception. Most network models can be classified as either equilibrium or growing. Here we show that under certain conditions there exists an equilibrium formulation for any growing network model, and vice versa. The equivalence between the equilibrium and nonequilibrium formulations is exact not only asymptotically, but even for any finite system size. The required conditions are satisfied in random geometric graphs in general and causal sets in particular, and to a large extent in some real networks.
Physical Review E 08/2013; 88(2):022808. · 2.31 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: As the Internet AS-level topology grows over time, some of its structural
properties remain unchanged. Such time- invariant properties are generally
interesting, because they tend to reflect some fundamental processes or
constraints behind Internet growth. As has been shown before, the
time-invariant structural properties of the Internet include some most basic
ones, such as the degree distribution or clustering. Here we add to this
time-invariant list a non-trivial property - k-dense decomposition. This
property is derived from a recursive form of edge multiplicity, defined as the
number of triangles that share a given edge. We show that after proper
normalization, the k- dense decomposition of the Internet has remained stable
over the last decade, even though the Internet size has approximately doubled,
and so has the k-density of its k-densest core. This core consists mostly of
content providers peering at Internet eXchange Points, and it only loosely
overlaps with the high-degree or high-rank AS core, consisting mostly of tier-1
transit providers. We thus show that high degrees and high k-densities reflect
two different Internet-specific properties of ASes (transit versus content
providers). As a consequence, even though degrees and k-densities of nodes are
correlated, the relative fluctuations are strong, and related to that, random
graphs with the same degree distribution or even degree correlations as in the
Internet, do not reproduce its k-dense decomposition. Therefore an interesting
open question is what Internet topology models or generators can fully explain
or at least reproduce the k-dense properties of the Internet.
[Show abstract][Hide abstract] ABSTRACT: Parallel Discrete Event Simulation (PDES) is based on the partitioning of the simulation model into distinct Logical Processes (LPs), each one modeling a portion of the entire system, which are allowed to execute simulation events concurrently. This ...
[Show abstract][Hide abstract] ABSTRACT: The principle that 'popularity is attractive' underlies preferential attachment, which is a common explanation for the emergence of scaling in growing networks. If new connections are made preferentially to more popular nodes, then the resulting distribution of the number of connections possessed by nodes follows power laws, as observed in many real networks. Preferential attachment has been directly validated for some real networks (including the Internet), and can be a consequence of different underlying processes based on node fitness, ranking, optimization, random walks or duplication. Here we show that popularity is just one dimension of attractiveness; another dimension is similarity. We develop a framework in which new connections optimize certain trade-offs between popularity and similarity, instead of simply preferring popular nodes. The framework has a geometric interpretation in which popularity preference emerges from local optimization. As opposed to preferential attachment, our optimization framework accurately describes the large-scale evolution of technological (the Internet), social (trust relationships between people) and biological (Escherichia coli metabolic) networks, predicting the probability of new links with high precision. The framework that we have developed can thus be used for predicting new links in evolving networks, and provides a different perspective on preferential attachment as an emergent phenomenon.
[Show abstract][Hide abstract] ABSTRACT: Recent years have shown a promising progress in understanding geometric
underpinnings behind the structure, function, and dynamics of many complex
networks in nature and society. However these promises cannot be readily
fulfilled and lead to important practical applications, without a simple,
reliable, and fast network mapping method to infer the latent geometric
coordinates of nodes in a real network. Here we present HyperMap, a simple
method to map a given real network to its hyperbolic space. The method utilizes
a recent geometric theory of complex networks modeled as random geometric
graphs in hyperbolic spaces. The method replays the network's geometric growth,
estimating at each time step the hyperbolic coordinates of new nodes in a
growing network by maximizing the likelihood of the network snapshot in the
model. We apply HyperMap to the AS Internet, and find that: 1) the method
produces meaningful results, identifying soft communities of ASs belonging to
the same geographic region; 2) the method has a remarkable predictive power:
using the resulting map, we can predict missing links in the Internet with high
precision, outperforming popular existing methods; and 3) the resulting map is
highly navigable, meaning that a vast majority of greedy geometric routing
paths are successful and low-stretch. Even though the method is not without
limitations, and is open for improvement, it occupies a unique attractive
position in the space of trade-offs between simplicity, accuracy, and
IEEE/ACM Transactions on Networking 05/2012; · 2.01 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Many social, biological and technological systems can be conveniently
represented as bipartite networks, consisting of two disjoint sets of
elements along with edges connecting only elements from different sets.
Many of such systems are characterized by high values of bipartite
clustering coefficient. We also find that pairs of elements in these
bipartite systems tend to have many common neighbors. We present a
natural interpretation of these observations. We suggest that elements
of the above bipartite systems exist in underlying metric spaces, such
that the observed high clustering is a topological reflection of the
triangle inequality, the key property of metric space. We propose a
simple stochastic mechanism of formation of bipartite networks embedded
in metric spaces. We prove that this mechanism is able to reproduce the
observed topological properties of bipartite networks. We also discuss
the possibility of constructive embedding of real bipartite systems into
metric spaces. In my talk I will overview the concept of hidden metric
spaces with respect to both unipartite and bipartite networks. I will
also discuss existing methods used to infer hidden metric spaces in real
networks and possible applications for bipartite networks.
[Show abstract][Hide abstract] ABSTRACT: Prediction and control of the dynamics of complex networks is a central problem in network science. Structural and dynamical similarities of different real networks suggest that some universal laws might accurately describe the dynamics of these networks, albeit the nature and common origin of such laws remain elusive. Here we show that the causal network representing the large-scale structure of spacetime in our accelerating universe is a power-law graph with strong clustering, similar to many complex networks such as the Internet, social, or biological networks. We prove that this structural similarity is a consequence of the asymptotic equivalence between the large-scale growth dynamics of complex networks and causal networks. This equivalence suggests that unexpectedly similar laws govern the dynamics of complex networks and spacetime in the universe, with implications to network science and cosmology.
[Show abstract][Hide abstract] ABSTRACT: We introduce and study random bipartite networks with hidden variables. Nodes in these networks are characterized by hidden variables that control the appearance of links between node pairs. We derive analytic expressions for the degree distribution, degree correlations, the distribution of the number of common neighbors, and the bipartite clustering coefficient in these networks. We also establish the relationship between degrees of nodes in original bipartite networks and in their unipartite projections. We further demonstrate how hidden variable formalism can be applied to analyze topological properties of networks in certain bipartite network models, and verify our analytical results in numerical simulations.
[Show abstract][Hide abstract] ABSTRACT: We provide a simple proof that graphs in a general class of self-similar networks have zero percolation threshold. The considered self-similar networks include random scale-free graphs with given expected node degrees and zero clustering, scale-free graphs with finite clustering and metric structure, growing scale-free networks, and many real networks. The proof and the derivation of the giant component size do not require the assumption that networks are treelike. Our results rely only on the observation that self-similar networks possess a hierarchy of nested subgraphs whose average degree grows with their depth in the hierarchy. We conjecture that this property is pivotal for percolation in networks.
[Show abstract][Hide abstract] ABSTRACT: 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 noninteracting 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.
[Show abstract][Hide abstract] ABSTRACT: We show that complex (scale-free) network topologies naturally emerge from hyperbolic metric spaces. Hyperbolic geometry facilitates maximally efficient greedy forwarding in these networks. Greedy forwarding is topology-oblivious. Nevertheless, greedy packets find their destinations with 100% probability following almost optimal shortest paths. This remarkable efficiency sustains even in highly dynamic networks. Our findings suggest that forwarding information through complex networks, such as the Internet, is possible without the overhead of existing routing protocols, and may also find practical applications in overlay networks for tasks such as application-level routing, information sharing, and data distribution.
[Show abstract][Hide abstract] ABSTRACT: The Internet infrastructure is severely stressed. Rapidly growing overheads associated with the primary function of the Internet-routing information packets between any two computers in the world-cause concerns among Internet experts that the existing Internet routing architecture may not sustain even another decade. In this paper, we present a method to map the Internet to a hyperbolic space. Guided by a constructed map, which we release with this paper, Internet routing exhibits scaling properties that are theoretically close to the best possible, thus resolving serious scaling limitations that the Internet faces today. Besides this immediate practical viability, our network mapping method can provide a different perspective on the community structure in complex networks.
[Show abstract][Hide abstract] ABSTRACT: We show that heterogeneous degree distributions in observed scale-free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space. Fermi-Dirac statistics provides a physical interpretation of hyperbolic distances as energies of links. The hidden space curvature affects the heterogeneity of the degree distribution, while clustering is a function of temperature. We embed the internet into the hyperbolic plane and find a remarkable congruency between the embedding and our hyperbolic model. Besides proving our model realistic, this embedding may be used for routing with only local information, which holds significant promise for improving the performance of internet routing.
[Show abstract][Hide abstract] ABSTRACT: Network motifs are small building blocks of complex networks, such as gene regulatory networks. The frequent appearance of a motif may be an indication of some network-specific utility for that motif, such as speeding up the response times of gene circuits. However, the precise nature of the connection between motifs and the global structure and function of networks remains unclear. Here we show that the global structure of some real networks is statistically determined by the distributions of local motifs of size at most 3, once we augment motifs to include node degree information. That is, remarkably, the global properties of these networks are fixed by the probability of the presence of links between node triples, once this probability accounts for the degree of the individual nodes. We consider a social web of trust, protein interactions, scientific collaborations, air transportation, the Internet, and a power grid. In all cases except the power grid, random networks that maintain the degree-enriched connectivity profiles for node triples in the original network reproduce all its local and global properties. This finding provides an alternative statistical explanation for motif significance. It also impacts research on network topology modeling and generation. Such models and generators are guaranteed to reproduce essential local and global network properties as soon as they reproduce their 3-node connectivity statistics.
[Show abstract][Hide abstract] ABSTRACT: Random scale-free networks are ultrasmall worlds. The average length of the shortest paths in networks of size N scales as lnlnN. Here we show that these ultrasmall worlds can be navigated in ultrashort time. Greedy routing on scale-free networks embedded in metric spaces finds paths with the average length scaling also as lnlnN. Greedy routing uses only local information to navigate a network. Nevertheless, it finds asymptotically the shortest paths, a direct computation of which requires global topology knowledge. Our findings imply that the peculiar structure of complex networks ensures that the lack of global topological awareness has asymptotically no impact on the length of communication paths. These results have important consequences for communication systems such as the Internet, where maintaining knowledge of current topology is a major scalability bottleneck.
[Show abstract][Hide abstract] ABSTRACT: We show that heterogeneous degree distributions in observed scale-free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space. Fermi-Dirac statistics provides a physical interpretation of hyperbolic distances as energies of links. The hidden space curvature aects the heterogeneity of the degree distribution, while clustering is