Dmitri Krioukov

Northeastern University, Boston, Massachusetts, United States

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Publications (62)181.17 Total impact

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    ABSTRACT: Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the dk-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks-the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain-and find that many important local and global structural properties of these networks are closely reproduced by dk-random graphs whose degree distributions, degree correlations and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network randomness, and release software to generate dk-random graphs.
    Full-text · Article · Oct 2015 · Nature Communications
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    Maksim Kitsak · Ahmed Elmokashfi · Shlomo Havlin · Dmitri Krioukov
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    ABSTRACT: Data transfer is one of the main functions of the Internet. The Internet consists of a large number of interconnected subnetworks or domains, known as Autonomous Systems. Due to privacy and other reasons the information about what route to use to reach devices within other Autonomous Systems is not readily available to any given Autonomous System. The Border Gateway Protocol is responsible for discovering and distributing this reachability information to all Autonomous Systems. Since the topology of the Internet is highly dynamic, all Autonomous Systems constantly exchange and update this reachability information in small chunks, known as routing control packets or Border Gateway Protocol updates. Motivated by scalability and predictability issues with the dynamics of these updates in the quickly growing Internet, we conduct a systematic time series analysis of Border Gateway Protocol update rates. We find that Border Gateway Protocol update time series are extremely volatile, exhibit long-term correlations and memory effects, similar to seismic time series, or temperature and stock market price fluctuations. The presented statistical characterization of Border Gateway Protocol update dynamics could serve as a ground truth for validation of existing and developing better models of Internet interdomain routing.
    Full-text · Article · Jul 2015 · PLoS ONE
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    ABSTRACT: Common sense suggests that networks are not random mazes of purposeless connections, but that these connections are organized so that networks can perform their functions well. One function common to many networks is targeted transport or navigation. Here, using game theory, we show that minimalistic networks designed to maximize the navigation efficiency at minimal cost share basic structural properties with real networks. These idealistic networks are Nash equilibria of a network construction game whose purpose is to find an optimal trade-off between the network cost and navigability. We show that these skeletons are present in the Internet, metabolic, English word, US airport, Hungarian road networks, and in a structural network of the human brain. The knowledge of these skeletons allows one to identify the minimal number of edges, by altering which one can efficiently improve or paralyse navigation in the network.
    Full-text · Article · Jul 2015 · Nature Communications
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    ABSTRACT: Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the $dk$-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks---the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain---and find that many important local and global structural properties of these networks are closely reproduced by $dk$-random graphs whose degree distributions, degree correlations, and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network randomness.
    Full-text · Article · May 2015
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    ABSTRACT: 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 Hamilton's 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.
    Preview · Article · Apr 2015
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    Rodrigo Aldecoa · Chiara Orsini · Dmitri Krioukov
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    ABSTRACT: Networks representing many complex systems in nature and society share some common structural properties like heterogeneous degree distributions and strong clustering. Recent research on network geometry has shown that those real networks can be adequately modeled as random geometric graphs in hyperbolic spaces. In this paper, we present a computer program to generate such graphs. Besides real-world-like networks, the program can generate random graphs from other well-known graph ensembles, such as the soft configuration model, random geometric graphs on a circle, or Erd\H{o}s-R\'enyi random graphs. The simulations show a good match between the expected values of different network structural properties and the corresponding empirical values measured in generated graphs, confirming the accurate behavior of the program.
    Full-text · Article · Mar 2015 · Computer Physics Communications
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    Fragkiskos Papadopoulos · Dmitri Krioukov
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    ABSTRACT: We introduce and explore a new method for inferring hidden geometric coordinates of nodes in complex networks based on the number of common neighbors between the nodes. We compare this approach to the one in [1], which is based on the connections (and disconnections) between the nodes, i.e., on the links that the nodes have (or do not have). We find that for high degree nodes the common-neighbors approach yields a more accurate inference than the link-based method, unless heuristic periodic adjustments (or "correction steps") are used in the latter. The common-neighbors approach is computationally intensive, requiring $O(t^4)$ running time to map a network of $t$ nodes, versus $O(t^3)$ in the link-based method. But we also develop a hybrid method with $O(t^3)$ running time, which combines the common-neighbors and link-based approaches, and explore a heuristic that reduces its running time further to $O(t^2)$, without significant reduction in the mapping accuracy. We apply this method to the Autonomous Systems (AS) Internet, and reveal how soft communities of ASes evolve over time in the similarity space. We further demonstrate the method's predictive power by forecasting future links between ASes. Taken altogether, our results advance our understanding of how to efficiently and accurately map real networks to their latent geometric spaces, which is an important necessary step towards understanding the laws that govern the dynamics of nodes in these spaces, and the fine-grained dynamics of network connections.
    Preview · Article · Feb 2015 · Physical Review E
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    Konstantin Zuev · Or Eisenberg · Dmitri Krioukov
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    ABSTRACT: Exponential random graph models have attracted significant research attention over the past decades. These models are maximum-entropy ensembles under the constraints that the expected values of a set of graph observables are equal to given values. Here we extend these maximum-entropy ensembles to random simplicial complexes, which are more adequate and versatile constructions to model complex systems in many applications. We show that many random simplicial complex models considered in the literature can be casted as maximum-entropy ensembles under certain constraints. We introduce and analyze the most general random simplicial complex ensemble $\mathbf{\Delta}$ with statistically independent simplices. Our analysis is simplified by the observation that any distribution $\mathbb{P}(O)$ on any collection of objects $\mathcal{O}=\{O\}$, including graphs and simplicial complexes, is maximum-entropy under the constraint that the expected value of $-\ln \mathbb{P}(O)$ is equal to the entropy of the distribution. With the help of this observation, we prove that ensemble $\mathbf{\Delta}$ is maximum-entropy under two types of constraints that fix the expected numbers of simplices and their boundaries.
    Preview · Article · Feb 2015 · Journal of Physics A Mathematical and Theoretical
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    ABSTRACT: All real networks are different, but many have some structural properties in common. There seems to be no consensus on what the most common properties are, but scale-free degree distributions, strong clustering, and community structure are frequently mentioned without question. Surprisingly, there exists no simple generative mechanism explaining all the three properties at once in growing networks. Here we show how latent network geometry coupled with preferential attachment of nodes to this geometry fills this gap. We call this mechanism geometric preferential attachment (GPA), and validate it against the Internet. GPA gives rise to soft communities that provide a different perspective on the community structure in networks. The connections between GPA and cosmological models, including inflation, are also discussed.
    Full-text · Article · Jan 2015 · Scientific Reports
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    ABSTRACT: The common sense suggests that networks are not random mazes of purposeless connections, but that these connections are organised so that networks can perform their functions. One common function that many networks perform is targeted transport or navigation. Here with the help of game theory we show that minimalistic networks designed to maximise the navigation efficiency at minimal cost share basic structural properties of real networks. These idealistic networks are Nash equilibria of a network construction game whose purpose is to find an optimal trade-off between the network cost and navigability. They are navigation skeletons that we show are present in the Internet, {\it E. coli} metabolic network, English word network, US airport network, and the Hungarian road network. The knowledge of these skeletons allows one to identify the minimal number of edges by altering which one can dramatically improve or paralyse the navigation in the network.
    Full-text · Article · Dec 2014
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    Dmitri Krioukov

    Preview · Article · Oct 2014 · Frontiers in Computational Neuroscience
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    Marian Boguna · Maksim Kitsak · Dmitri Krioukov
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    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.
    Full-text · Article · Oct 2013 · New Journal of Physics
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    Dmitri Krioukov · Massimo Ostilli
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    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.
    Full-text · Article · Aug 2013 · Physical Review E
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    Chiara Orsini · Enrico Gregori · Luciano Lenzini · Dmitri Krioukov
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    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.
    Full-text · Article · Jan 2013 · IEEE/ACM Transactions on Networking
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    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 ...
    No preview · Article · Dec 2012 · ACM SIGMETRICS Performance Evaluation Review
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    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.
    Full-text · Article · Nov 2012 · Scientific Reports
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    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.
    Full-text · Article · Sep 2012 · Nature
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    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 computational complexity.
    Preview · Article · May 2012 · IEEE/ACM Transactions on Networking
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    Full-text · Article · May 2012
  • Maksim Kitsak · Fragkiskos Papadopoulos · Dmitri Krioukov
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    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.
    No preview · Article · Feb 2012

Publication Stats

2k Citations
181.17 Total Impact Points

Institutions

  • 2013-2015
    • Northeastern University
      • • Department of Mathematics
      • • Department of Physics
      Boston, Massachusetts, United States
  • 2007-2013
    • University of California, San Diego
      • Department of Computer Science and Engineering (CSE)
      San Diego, California, United States