Thomas

Spreading processes are ubiquitous in natural and artificial systems. They can be studied via a plethora of models, depending on the specific details of the phenomena under study. Disease contagion and rumor spreading are among the most important of these processes due to their practical relevance. However, despite the similarities between them, current models address both spreading dynamics separately. In this paper, we propose a general information spreading model that is based on discrete time Markov chains. The model includes all the transitions that are plausible for both a disease contagion process and rumor propagation. We show that our model not only covers the traditional spreading schemes, but that it also contains some features relevant in social dynamics, such as apathy, forgetting, and lost/recovering of interest. The model is evaluated analytically to obtain the spreading thresholds and the early time dynamical behavior for the contact and reactive processes in several scenarios. Comparison with Monte Carlo simulations shows that the Markov chain formalism is highly accurate while it excels in computational efficiency. We round off our work by showing how the proposed framework can be applied to the study of spreading processes occurring on social networks.

The increasing power of computer technology does not dispense with the need to extract meaningful in- formation out of data sets of ever growing size, and indeed typically exacerbates the complexity of this task. To tackle this general problem, two methods have emerged, at chronologically different times, that are now commonly used in the scientific community: data mining and complex network theory. Not only do complex network analysis and data mining share the same general goal, that of extracting information from complex systems to ultimately create a new compact quantifiable representation, but they also often address similar problems too. In the face of that, a surprisingly low number of researchers turn out to resort to both methodologies. One may then be tempted to conclude that these two fields are either largely redundant or totally antithetic. The starting point of this review is that this state of affairs should be put down to contingent rather than conceptual differences, and that these two fields can in fact advantageously be used in a synergistic manner. An overview of both fields is first provided, some fundamental concepts of which are illustrated. A variety of contexts in which complex network theory and data mining have been used in a synergistic manner are then presented. Contexts in which the appropriate integration of complex network metrics can lead to improved classification rates with respect to classical data mining algorithms and, conversely, contexts in which data mining can be used to tackle important issues in complex network theory applications are illustrated. Finally, ways to achieve a tighter integration between complex networks and data mining, and open lines of research are discussed.

Author(s): Marc Wiedermann, Jonathan F. Donges, Jürgen Kurths, and Reik V. Donner

Networks with nodes embedded in a metric space have gained increasing interest in recent years. The effects of spatial embedding on the networks' structural characteristics, however, are rarely taken into account when studying their macroscopic properties. Here, we propose a hierarchy of null models…

[Phys. Rev. E 93, 042308] Published Tue Apr 12, 2016

Author(s): Reimer Kühn

We describe a method for disentangling giant component and finite cluster contributions to sparse random matrix spectra, using sparse symmetric random matrices defined on Erdős-Rényi graphs as an example and test bed. Our methods apply to sparse matrices defined in terms of arbitrary graphs in the c…

[Phys. Rev. E 93, 042110] Published Tue Apr 12, 2016

One of the most celebrated findings in complex systems in the last decade is that different indexes y (e.g., patents) scale nonlinearly with the population~x of the cities in which they appear, i.e., $y\sim x^\beta, \beta \neq 1$. More recently, the generality of this finding has been questioned in studies using new databases and different definitions of city boundaries. In this paper we investigate the existence of nonlinear scaling using a probabilistic framework in which fluctuations are accounted explicitly. In particular, we show that this allows not only to (a) estimate $\beta$ and confidence intervals, but also to (b) quantify the evidence in favor of $\beta \neq 1$ and (c) test the hypothesis that the observations are compatible with the nonlinear scaling. We employ this framework to compare $5$ different models to $15$ different datasets and we find that the answers to points (a)-(c) crucially depend on the fluctuations contained in the data, on how they are modeled, and on the fact that the city sizes are heavy-tailed distributed.

Author(s): Justus A. Kromer, Lutz Schimansky-Geier, and Alexander B. Neiman

We study the emergence and coherence of stochastic oscillations in star networks of excitable elements in which peripheral nodes receive independent random inputs. A biophysical model of a distal branch of sensory neuron in which peripheral nodes of Ranvier are coupled to a central node by myelinate…

[Phys. Rev. E 93, 042406] Published Fri Apr 08, 2016

We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, $N$, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite $N$, the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite $N$ have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when $N \gg 1$. The leading correction to the infinite-$N$ result scales like either $N^{-3/2}$ or $N^{-1}$, depending on whether the frequencies are evenly spaced according to a midpoint rule or an endpoint rule. These scaling laws agree with numerical results obtained by Paz\'{o} [Phys. Rev. E 72, 046211 (2005)]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.

Author(s): Lucia Valentina Gambuzza, Mattia Frasca, Luigi Fortuna, and Stefano Boccaletti

Relay synchronization is a collective state, originally found in chains of interacting oscillators, in which uncoupled dynamical units synchronize through the action of mismatched inner nodes that relay the information but do not synchronize with them. It is demonstrated herein that relay synchroniz…

[Phys. Rev. E 93, 042203] Published Thu Apr 07, 2016

Extended numerical simulations of threshold models have been performed on a human brain network with N=836733 connected nodes available from the Open Connectome project. While in case of simple threshold models a sharp discontinuous phase transition without any critical dynamics arises, variable thresholds models exhibit extended power-law scaling regions. This is attributed to fact that Griffiths effects, stemming from the topological/interaction heterogeneity of the network, can become relevant if the input sensitivity of nodes is equalized. I have studied the effects effects of link directness, as well as the consequence of inhibitory connections. Non-universal power-law avalanche size and time distributions have been found with exponents agreeing with the values obtained in electrode experiments of the human brain. The dynamical critical region occurs in an extended control parameter space without the assumption of self organized criticality.

The collective dynamics in populations of magnetic spin torque oscillators (STO) is an intensely studied topic in modern magnetism. Here, we show that arrays of STO coupled via dipolar fields can be modeled using a variant of the Kuramoto model, a well-known mathematical model in non-linear dynamics. By investigating the collective dynamics in arrays of STO we find that the synchronization in such systems is a finite size effect and show that the critical coupling-for a complete synchronized state-scales with the number of oscillators. Using realistic values of the dipolar coupling strength between STO we show that this imposes an upper limit for the maximum number of oscillators that can be synchronized. Further, we show that the lack of long range order is associated with the formation of topological defects in the phase field similar to the two-dimensional XY model of ferromagnetism. Our results shed new light on the synchronization of STO, where controlling the mutual synchronization of several oscillators is considered crucial for applications.

In recent years, many variants of percolation have been used to study network structure and the behavior of processes spreading on networks. These include bond percolation, site percolation, $k$-core percolation, bootstrap percolation, the generalized epidemic process, and the Watts Threshold Model (WTM). We show that --- except for bond percolation --- each of these processes arises as a special case of the WTM and bond percolation arises from a small modification. In fact "heterogeneous $k$-core percolation", a corresponding "heterogeneous bootstrap percolation" model, and the generalized epidemic process are completely equivalent to one another and the WTM. We further show that a natural generalization of the WTM in which individuals "transmit" or "send a message" to their neighbors with some probability less than $1$ can be reformulated in terms of the WTM, and so this apparent generalization is in fact not more general. Finally, we show that in bond percolation, finding the set of nodes in the component containing a given node is equivalent to finding the set of nodes activated if that node is initially activated and the node thresholds are chosen from the appropriate distribution. A consequence of these results is that mathematical techniques developed for the WTM apply to these other models as well, and techniques that were developed for some particular case may in fact apply much more generally.

Network models with latent geometry have been used successfully in many applications in network science and other disciplines, yet it is usually impossible to tell if a given real network is geometric, meaning if it is a typical element in an ensemble of random geometric graphs. Here we identify structural properties of networks that guarantee that random graphs having these properties are geometric. Specifically we show that random graphs in which expected degree and clustering of every node are fixed to some constants are equivalent to random geometric graphs on the real line, if clustering is sufficiently strong. Large numbers of triangles, homogeneously distributed across all nodes as in real networks, are thus a consequence of network geometricity. The methods we use to prove this are quite general and applicable to other network ensembles, geometric or not, and to certain problems in quantum gravity.

This paper aims at providing an introduction to the behavior of authors submitting a paper to a scientific journal. Dates of electronic submission of papers to the Journal of the Serbian Chemical Society have been recorded from the 1st January 2013 till the 31st December 2014, thus over 2 years.

There is no Monday or Friday effect like in financial markets, but rather a Tuesday-Wednesday effect occurs: papers are more often submitted on Wednesday; however, the relative number of going to be accepted papers is larger if these are submitted on Tuesday. On the other hand, weekend days (Saturday and Sunday) are not the best days to finalize and submit manuscripts. An interpretation based on the type of submitted work ("experimental chemistry") and on the influence of (senior) coauthors is presented. A thermodynamic connection is proposed within an entropy context. A (new) entropic distance is defined in order to measure the "opaqueness" = disorder) of the submission process.

Academic research is driven by several factors causing different disciplines to act as "sources" or "sinks" of knowledge. However, how the flow of authors' research interests -- a proxy of human knowledge -- evolved across time is still poorly understood. Here, we build a comprehensive map of such flows across one century, revealing fundamental periods in the raise of interest in areas of human knowledge. We identify and quantify the most attractive topics over time, when a relatively significant number of researchers moved from their original area to another one, causing what we call a "diaspora of the knowledge" towards sinks of scientific interest, and we relate these points to crucial historical and political events. Noticeably, only a few areas -- like Medicine, Physics or Chemistry -- mainly act as sources of the diaspora, whereas areas like Material Science, Chemical Engineering, Neuroscience, Immunology and Microbiology or Environmental Science behave like sinks.

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Randomly coupled neural fields demonstrate chaotic variation of firing rates, if the coupling is strong enough, as has been shown by Sompolinsky et. al [Phys. Rev. Lett., v. 61, 259 (1988)]. We present a method for reconstruction of the coupling matrix from the observations of the chaotic firing rates. The approach is based on the particular property of the nonlinearity in the coupling, as the latter is determined by a sigmoidal gain function. We demonstrate that for a large enough data set, the method gives an accurate estimation of the coupling matrix and of other parameters of the system, including the gain function.

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The empirical validation of community detection methods is often based on available annotations on the nodes that serve as putative indicators of the large-scale network structure. Most often, the suitability of the annotations as topological descriptors itself is not assessed, and without this it is not possible to ultimately distinguish between actual shortcomings of the community detection algorithms on one hand, and the incompleteness, inaccuracy or structured nature of the data annotations themselves on the other. In this work we present a principled method to access both aspects simultaneously. We construct a joint generative model for the data and metadata, and a nonparametric Bayesian framework to infer its parameters from annotated datasets. We assess the quality of the metadata not according to its direct alignment with the network communities, but rather in its capacity to predict the placement of edges in the network. We also show how this feature can be used to predict the connections to missing nodes when only the metadata is available, as well as missing metadata. By investigating a wide range of datasets, we show that while there are seldom exact agreements between metadata tokens and the inferred data groups, the metadata is often informative of the network structure nevertheless, and can improve the prediction of missing nodes. This shows that the method uncovers meaningful patterns in both the data and metadata, without requiring or expecting a perfect agreement between the two.

Online social media have greatly affected the way in which we communicate with each other. However, little is known about what are the fundamental mechanisms driving dynamical information flow in online social systems. Here, we introduce a generative model for online sharing behavior that is analytically tractable and which can reproduce several characteristics of empirical micro-blogging data on hashtag usage, such as (time-dependent) heavy-tailed distributions of meme popularity. The presented framework constitutes a null model for social spreading phenomena which, in contrast to purely empirical studies or simulation-based models, clearly distinguishes the roles of two distinct factors affecting meme popularity: the memory time of users and the connectivity structure of the social network.

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Author(s): Toni Vallès-Català, Francesco A. Massucci, Roger Guimerà, and Marta Sales-Pardo

Multiple interaction layers are a fact of life in real-world networks. Scientists model how well networks can be represented using superpositions of layers assembled using either AND or OR logic.

[Phys. Rev. X 6, 011036] Published Thu Mar 31, 2016

Author(s): Tommaso Coletta and Philippe Jacquod

We investigate the influence that adding a new coupling has on the linear stability of the synchronous state in coupled-oscillator networks. Using a simple model, we show that, depending on its location, the new coupling can lead to enhanced or reduced stability. We extend these results to electric …

[Phys. Rev. E 93, 032222] Published Thu Mar 31, 2016

Author(s): J. Poulter

This pedagogical comment highlights three misconceptions concerning the usefulness of the concept of negative temperature, being derived from the usual, often termed Boltzmann, definition of entropy. First, both the Boltzmann and Gibbs entropies must obey the same thermodynamic consistency relation.…

[Phys. Rev. E 93, 032149] Published Wed Mar 30, 2016

Author(s): Dirk Witthaut, Martin Rohden, Xiaozhu Zhang, Sarah Hallerberg, and Marc Timme

Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet, it is still not well understood which links are particularly prone to inducing such outages. Here we analyze how the nature and location of each link impact the network’s capability to maintain a stabl…

[Phys. Rev. Lett. 116, 138701] Published Wed Mar 30, 2016

We study the network dismantling problem, which consists in determining a minimal set of vertices whose removal leaves the network broken into connected components of sub-extensive size. For a large class of random graphs, this problem is tightly connected to the decycling problem (the removal of vertices leaving the graph acyclic). Exploiting this connection and recent works on epidemic spreading we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed (light-tailed) degree distribution. Building on the statistical mechanics perspective we propose a three-stage Min-Sum algorithm for efficiently dismantling networks, including heavy-tailed ones for which the dismantling and decycling problems are not equivalent. We also provide further insights into the dismantling problem concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well performing nodes.

Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must be met by every graph, soft constraints must be met only on average, subject to maximal entropy. In Squartini et al. (2015) it was shown that ensembles of random graphs are non-equivalent when the degrees of the nodes are constrained, in the sense of a non-zero limiting specific relative entropy as the number of nodes diverges. In that paper, the nodes were placed either on a single layer (uni-partite graphs) or on two layers (bi-partite graphs). In the present paper we consider an arbitrary number of intra-connected and inter-connected layers, thus allowing for modular graphs with a multi-partite, multiplex, block-model or community structure. We give a full classification of ensemble equivalence, proving that breakdown occurs if and only if the number of local constraints (i.e., the number of constrained degrees) is extensive in the number of nodes, irrespective of the layer structure. In addition, we derive a formula for the specific relative entropy and provide an interpretation of this formula in terms of Poissonisation of the degrees.

We analyze a variant of a model proposed by Kuramoto, Shinomoto, and Sakaguchi for a large population of coupled oscillatory and excitable elements. Using the Ott-Antonsen ansatz, we reduce the behavior of the population to a two-dimensional dynamical system with three parameters. We present the stability diagram and calculate several of its bifurcation curves analytically, for both excitatory and inhibitory coupling. Our main result is that when the coupling function is broad, the system can display bistability between steady states of constant high and low activity, whereas when the coupling function is narrow and inhibitory, one of the states in the bistable regime can show persistent pulsations in activity.

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Author(s): Reimer Kühn

We describe a method for disentangling giant component and finite cluster contributions to sparse random matrix spectra, using sparse symmetric random matrices defined on Erdös-Renyi graphs as an example and test-bed. Our methods apply to sparse matrices defined in terms of arbitrary graphs in …

[Phys. Rev. E] Published Mon Mar 28, 2016

Multilayer networks have been the subject of intense research during the last few years, as they represent better the interdependent nature of many real world systems. Here, we address the question of describing the three different structural phases in which a multiplex network might exist. We show that each phase can be characterized by the presence of gaps in the spectrum of the supra-Laplacian of the multiplex network. We therefore unveil the existence of different topological scales in the system, whose relation characterizes each phase. Moreover, by capitalizing on the coarse-grained representation that is given in terms of quotient graphs, we explain the mechanisms that produce those gaps as well as their dynamical consequences.

Author(s): Wesley Cota, Silvio C. Ferreira, and Géza Ódor

We provide numerical evidence for slow dynamics of the susceptible-infected-susceptible model evolving on finite-size random networks with power-law degree distributions. Extensive simulations were done by averaging the activity density over many realizations of networks. We investigated the effects…

[Phys. Rev. E 93, 032322] Published Mon Mar 28, 2016

For a system of coupled identical phase oscillators with full permutation symmetry, any broken symmetries in dynamical behaviour must come from spontaneous symmetry breaking, i.e. from the nonlinear dynamics of the system. The dynamics of phase differences for such a system depends only on the coupling (phase interaction) function $g(\varphi)$ and the number of oscillators $N$. This paper briefly reviews some results for such systems in the case of general coupling $g$ before exploring two cases in detail: (a) general two harmonic form: $g(\varphi)=q\sin(\varphi-\alpha)+r\sin(2\varphi-\beta)$ and $N$ small (b) the coupling $g$ is odd or even. We extend previously published bifurcation analyses to the general two harmonic case, and show for even $g$ that the dynamics of phase differences has a number of time-reversal symmetries. For the case of even $g$ with one harmonic it is known the system has $N-2$ constants of the motion. This is true for $N=4$ and any $g$, while for $N=4$ and more than two harmonics in $g$, we show the system must have fewer independent constants of the motion.

Author(s): P. Van Mieghem

A general two-layer network (and similar for a general m-layer network) consists of two networks G1 and G2, whose interconnection pattern is specified by the interconnectivity matrix B. We deduce desirable properties of B from a dynamic process point of view. Many dynamic processes are described by …

[Phys. Rev. E] Published Mon Mar 21, 2016

Author(s): Adrian van Kan, Jannes Jegminat, Jonathan F. Donges, and Jürgen Kurths

Transient dynamics are of large interest in many areas of science. Here, a generalization of basin stability (BS) is presented: constrained basin stability (CBS) that is sensitive to various different types of transients arising from finite size perturbations. CBS is applied to the paradigmatic Lore…

[Phys. Rev. E] Published Tue Mar 22, 2016