Fragkiskos Papadopoulos

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 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.

Sustaining the Internet with hyperbolic mapping

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.

Greedy Forwarding in Dynamic Scale-Free Networks Embedded in Hyperbolic Metric Spaces

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.

Curvature and temperature of complex networks.

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.

Network mapping by replaying hyperbolic growth

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 networks 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 Autonomous Systems (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 tradeoffs between simplicity, accuracy, and computational complexity.

Hidden geometric correlations in real multiplex networks

Kaj-Kolja Kleineberg, ∗ Marián Boguñá, † M. Ángeles Serrano, 1, ‡ and Fragkiskos Papadopoulos § Departament de F́ısica Fonamental, Universitat de Barcelona, Mart́ı i Franquès 1, 08028 Barcelona, Spain Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Llúıs Companys 23, E-08010 Barcelona, Spain Department of Electrical Engineering, Computer Engineering and Informatics, Cyprus University of Technology, 33 Saripolou Street, 3036 Limassol, Cyprus (Dated: January 19, 2016)

Hamiltonian Dynamics of Preferential Attachment

Prediction and control of network dynamics are grand-challenge problems in network science. The lack of understanding of fundamental laws driving the dynamics of networks is among the reasons why many practical problems of great significance remain unsolved for decades. Here we study the dynamics of networks evolving according to preferential attachment, known to approximate well the large-scale growth dynamics of a variety of real networks. We show that this dynamics is Hamiltonian, thus casting the study of complex networks dynamics to the powerful canonical formalism, in which the time evolution of a dynamical system is described by Hamiltons equations. We derive the explicit form of the Hamiltonian that governs network growth in preferential attachment. This Hamiltonian turns out to be nearly identical to graph energy in the configuration model, which shows that the ensemble of random graphs generated by preferential attachment is nearly identical to the ensemble of random graphs with scale-free degree distributions. In other words, preferential attachment generates nothing but random graphs with power-law degree distribution. The extension of the developed canonical formalism for network analysis to richer geometric network models with non-degenerate groups of symmetries may eventually lead to a system of equations describing network dynamics at small scales.

Exploiting path diversity in datacenters using MPTCP-aware SDN

Recently, Multipath TCP (MPTCP) has been proposed as an alternative transport approach for datacenter networks. MPTCP provides the ability to split a flow into multiple paths thus providing better performance and resilience to failures. Usually, MPTCP is combined with flow-based Equal-Cost Multi-Path Routing (ECMP), which uses random hashing to split the MPTCP subflows over different paths. However, random hashing can be suboptimal as distinct subflows may end up using the same paths, while other available paths remain unutilized. In this paper, we explore an MPTCP-aware SDN controller that facilitates an alternative routing mechanism for the MPTCP subflows. The controller uses packet inspection to provide deterministic subflow assignment to paths. Using the controller, we show that MPTCP can deliver significantly improved performance when connections are not limited by the access links of hosts. To lessen the effect of throughput limitation due to access links, we also investigate the usage of multiple interfaces at the hosts. We demonstrate, using our modification of the MPTCP Linux Kernel, that using multiple subflows per pair of IP addresses can yield improved performance in multi-interface settings.

Modeling BitTorrent-like systems with many classes of users

BitTorrent is one of the most successful peer-to-peer systems. Researchers have studied a number of aspects of the system, including its scalability, performance, efficiency and fairness. However, the complexity of the system has forced most prior analytical work to make a number of simplifying assumptions, for example, user homogeneity, or even ignore some central aspects of the protocol altogether, for example, the rate-based Tit-for-Tat (TFT) unchoking scheme, in order to keep the analysis tractable. Motivated by this, in this article we propose two analytical models that accurately predict the performance of the system while considering the central details of the BitTorrent protocol. Our first model is a steady-state one, in the sense that it is valid during periods of time where the number of users remains fixed. Freed by the complications of user time-dynamics, we account for many of the central details of the BitTorrent protocol and accurately predict a number of performance metrics. Our second model combines prior work on fluid models with our first model to capture the transient behavior as new users join or old users leave, while modelling many major aspects of BitTorrent. To the best of our knowledge, this is the first model that attempts to capture the transient behavior of many classes of heterogeneous users. Finally, we use our analytical methodology to introduce and study the performance of a flexible token-based scheme for BitTorrent, show how this scheme can be used to block freeriders and tradeoff between higher-bandwidth and lower-bandwidth users performance, and evaluate the schemes parameters that achieve a target operational point.

Latent geometry of bipartite networks

Despite the abundance of bipartite networked systems, their organizing principles are less studied compared to unipartite networks. Bipartite networks are often analyzed after projecting them onto one of the two sets of nodes. As a result of the projection, nodes of the same set are linked together if they have at least one neighbor in common in the bipartite network. Even though these projections allow one to study bipartite networks using tools developed for unipartite networks, one-mode projections lead to significant loss of information and artificial inflation of the projected network with fully connected subgraphs. Here we pursue a different approach for analyzing bipartite systems that is based on the observation that such systems have a latent metric structure: network nodes are points in a latent metric space, while connections are more likely to form between nodes separated by shorter distances. This approach has been developed for unipartite networks, and relatively little is known about its applicability to bipartite systems. Here, we fully analyze a simple latent-geometric model of bipartite networks and show that this model explains the peculiar structural properties of many real bipartite systems, including the distributions of common neighbors and bipartite clustering. We also analyze the geometric information loss in one-mode projections in this model and propose an efficient method to infer the latent pairwise distances between nodes. Uncovering the latent geometry underlying real bipartite networks can find applications in diverse domains, ranging from constructing efficient recommender systems to understanding cell metabolism.