Thomas

Many real-world networks exhibit degree-assortativity, with nodes of similar degree more likely to link to one another. Particularly in social networks, the contribution to the total assortativity varies with degree, featuring a distinctive peak slightly past the average degree. The way traditional models imprint assortativity on top of pre-defined topologies is via degree-preserving link permutations, which however destroy the particular graph's hierarchical traits of clustering. Here, we propose the first generative model which creates heterogeneous networks with scale-free-like properties and tunable realistic assortativity. In our approach, two distinct populations of nodes are added to an initial network seed: one (the followers) that abides by usual preferential rules, and one (the potential leaders) connecting via anti-preferential attachments, i.e. selecting lower degree nodes for their initial links. The latter nodes come to develop a higher average degree, and convert eventually into the final hubs. Examining the evolution of links in Facebook, we present empirical validation for the connection between the initial anti-preferential attachment and long term high degree. Thus, our work sheds new light on the structure and evolution of social networks.

Donate to arXiv

Network reconstruction is the first step towards understanding, diagnosing and controlling the dynamics of complex networked systems. It allows us to infer properties of the interaction matrix, which characterizes how nodes in a system directly interact with each other. Despite a decade of extensive studies, network reconstruction remains an outstanding challenge. The fundamental limitations governing which properties of the interaction matrix (e.g., adjacency pattern, sign pattern and degree sequence) can be inferred from given temporal data of individual nodes remain unknown. Here we rigorously derive necessary conditions to reconstruct any property of the interaction matrix. These conditions characterize how uncertain can we be about the coupling functions that characterize the interactions between nodes, and how informative does the measured temporal data need to be; rendering two classes of fundamental limitations of network reconstruction. Counterintuitively, we find that reconstructing any property of the interaction matrix is generically as difficult as reconstructing the interaction matrix itself, requiring equally informative temporal data. Revealing these fundamental limitations shed light on the design of better network reconstruction algorithms, which offer practical improvements over existing methods.

Donate to arXiv

Author(s): Hyunsuk Hong, Hugues Chaté, Lei-Han Tang, and Hyunggyu Park

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population …

[Phys. Rev. E 92, 022122] Published Fri Aug 14, 2015

Author(s): Celso Freitas, Elbert Macau, and Ricardo Luiz Viana

Does the assignment order of a fixed collection of slightly distinct subsystems into given communication channels influence the overall ensemble behavior? We discuss this question in the context of complex networks of non-identical interacting oscillators. Three types of connection configurations ar…

[Phys. Rev. E] Published Wed Aug 12, 2015

Author(s): Tiago Pereira and Lai-Sang Young

We study isolation as a means to control epidemic outbreaks in complex networks, focusing on the consequences of delays in isolating infected nodes. Our analysis uncovers a tipping point: if infected nodes are isolated before a critical day d_c, the disease is effectively controlled, whereas for lon…

[Phys. Rev. E] Published Thu Aug 13, 2015

We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erd?s-R?nyi networks, based on recursion equations for the shells around a reference node and for the paths originating from it. The results are in agreement with numerical simulations for a broad range of network sizes and connectivities. The average and standard deviation of the distribution are also obtained. In the case in which the mean degree scales as ##IMG## [http://ej.iop.org/images/0295-5075/111/2/26006/epl17245ieqn1.gif] {$N^{\alpha}$} with the network size, the distribution becomes extremely narrow in the asymptotic limit, namely almost all pairs of nodes are equidistant, at distance ##IMG## [http://ej.iop.org/images/0295-5075/111/2/26006/epl17245ieqn2.gif] {$d=\lfloor1/\alpha\rfloor$} from each other. The distribution of shortest path lengths between nodes of degree m and the rest of the network is calcu...

With the rise of social media as an important channel for the debate and discussion of public affairs, online social networks such as Twitter have become important platforms for public information and engagement by policy makers. To communicate effectively through Twitter, policy makers need to understand how influence and interest propagate within its network of users. In this chapter we use graph-theoretic methods to analyse the Twitter debate surrounding NHS England's controversial care.data scheme. Directionality is a crucial feature of the Twitter social graph - information flows from the followed to the followers - but is often ignored in social network analyses; our methods are based on the behaviour of dynamic processes on the network and can be applied naturally to directed networks. We uncover robust communities of users and show that these communities reflect how information flows through the Twitter network. We are also able to classify users by their differing roles in directing the flow of information through the network. Our methods and results will be useful to policy makers who would like to use Twitter effectively as a communication medium.

Fundamental changes in brain structure and function during adolescence are well-characterized, but the extent to which experience modulates adolescent neurodevelopment is not. Musical experience provides an ideal case for examining this question because the influence of music training begun early in life is well-known. We investigated the effects of in-school...

Effects of hidden nodes on inference quality of observed network structure are explored based on a disordered Ising model with hidden nodes. We first study analytically small systems consisting of a few nodes, and find that the magnitude of the effective coupling grows as the coupling strength from the hidden common input nodes increases, while the field strength of the input node has opposite effects. Insights gained from analytic results of small systems are confirmed in numerical simulations of large systems. We also find that the inference quality deteriorates as the number of hidden nodes increases. Furthermore, increasing field variance of hidden nodes improves the inference quality of the effective couplings, but worsens the quality for the effective fields. In addition, attenuated coupling strengths involved in at least one hidden node lead to high quality of coupling inference.

Renewables will soon dominate energy production in our electric power system. And yet, how to integrate renewable energy into the grid and the market is still a subject of major debate. Decentral Smart Grid Control (DSGC) was recently proposed as a robust and decentralized approach to balance supply and demand and to guarantee a grid operation that is both economically and dynamically feasible. Here, we analyze the impact of network topology by assessing the stability of essential network motifs using both linear stability analysis and basin volume for delay systems. Our results indicate that if frequency measurements are averaged over sufficiently large time intervals, DSGC enhances the stability of extended power grid systems. We further investigate whether DSGC supports centralized and/or decentralized power production and fi?nd it to be applicable to both. However, our results on cycle-like systems suggest that DSGC favors systems with decentralized production. Here, lower line capacities and lower averaging times are required compared to those with centralized production.

Donate to arXiv

Several expressions for the $j$-th component $\left( x_{k}\right)_{j}$ of the $k$-th eigenvector $x_{k}$ of a symmetric matrix $A$ belonging to eigenvalue $\lambda_{k}$ and normalized as $x_{k}^{T}x_{k}=1$ are presented. In particular, the expression \[ \left( x_{k}\right)_{j}^{2}=-\frac{1}{c_{A}^{\prime}\left( \lambda_{k}\right) }\det\left( A_{\backslash\left\{ j\right\} }-\lambda_{k}I\right) \] where $c_{A}\left( \lambda\right) =\det\left( A-\lambda I\right) $ is the characteristic polynomial of $A$, $c_{A}^{\prime}\left( \lambda\right) =\frac{dc_{A}\left( \lambda\right) }{d\lambda}$ and $A_{\backslash\left\{ j\right\} }$ is obtained from $A$ by removal of row $j$ and column $j$, suggests us to consider the square eigenvector component as a graph centrality metric for node $j$ that reflects the impact of the removal of node $j$ from the graph at an eigenfrequency/eigenvalue $\lambda_{k}$ of a graph related matrix (such as the adjacency or Laplacian matrix). Removal of nodes in a graph relates to the robustness of a graph. The set of such nodal centrality metrics, the squared eigenvector components $\left( x_{k}\right)_{j}^{2}$ of the adjacency matrix over all eigenvalue $\lambda_{k}$ for each node $j$, is 'ideal' in the sense of being complete, \emph{almost} uncorrelated and mathematically precisely defined and computable. Fundamental weights (column sum of $X$) and dual fundamental weights (row sum of $X$) are introduced as spectral metrics that condense information embedded in the orthogonal eigenvector matrix $X$, with elements $X_{ij}=\left( x_{j}\right)_{i}$.

In addition to the criterion {\em If the algebraic connectivity is positive, then the graph is connected}, we found an alternative condition: {\em If $\min_{1\leq k\leq N}\left( \lambda_{k}^{2}(A)\right) =d_{\min}$, then the graph is disconnected.}

Donate to arXiv

Authors: Petter Holme, Leo Speidel, Taro Takaguchi and Naoki Masuda.
The European Physical Journal B Vol. 88 , page 203
Published online: 10/8/2015

Author(s): Pim van der Hoorn and Nelly Litvak

Analysis of degree-degree dependencies in complex networks, and their impact on processes on networks requires null models, i.e., models that generate uncorrelated scale-free networks. Most models to date, however, show structural negative dependencies, caused by finite size effects. We analyze the …

[Phys. Rev. E 92, 022803] Published Fri Aug 07, 2015

Author(s): Jan F. Totz, Razan Snari, Desmond Yengi, Mark R. Tinsley, Harald Engel, and Kenneth Showalter

Chemical oscillators with a broad frequency distribution are photochemically coupled in network topologies. Experiments and simulations show that the network synchronization occurs by phase-lag synchronization of clusters of oscillators with zero- or nearly zero-lag synchronization. Symmetry also pl…

[Phys. Rev. E] Published Thu Aug 06, 2015

Author(s): Kevin P. O’Keeffe, P. L. Krapivsky, and Steven H. Strogatz

An analysis of a model that might describe fireflies and neurons shows the path to synchronization occurring in groups of charge-and-fire oscillators.

[Phys. Rev. Lett. 115, 064101] Published Thu Aug 06, 2015

Many infrastructure networks have a modular structure and are also interdependent. While significant research has explored the resilience of interdependent networks, there has been no analysis of the effects of modularity. Here we develop a theoretical framework for attacks on interdependent modular networks and support our results by simulations. We focus on the case where each network has the same number of communities and the dependency links are restricted to be between pairs of communities of different networks. This is very realistic for infrastructure across cities. Each city has its own infrastructures and different infrastructures are dependent within the city. However, each infrastructure is connected within and between cities. For example, a power grid will connect many cities as will a communication network, yet a power station and communication tower that are interdependent will likely be in the same city. It has been shown that single networks are very susceptible to the failure of the interconnected nodes (between communities) Shai et al. and that attacks on these nodes are more crippling than attacks based on betweenness da Cunha et al. In our example of cities these nodes have long range links which are more likely to fail. For both treelike and looplike interdependent modular networks we find distinct regimes depending on the number of modules, $m$. (i) In the case where there are fewer modules with strong intraconnections, the system first separates into modules in an abrupt first-order transition and then each module undergoes a second percolation transition. (ii) When there are more modules with many interconnections between them, the system undergoes a single transition. Overall, we find that modular structure can influence the type of transitions observed in interdependent networks and should be considered in attempts to make interdependent networks more resilient.

The power of any kind of network approach lies in the ability to simplify a complex system so that one can better understand its function as a whole. Sometimes it is beneficial, however, to include more information than in a simple graph of only nodes and links. Adding information about times of interactions can make predictions and mechanistic understanding more accurate. The drawback, however, is that there are not so many methods available, partly because temporal networks is a relatively young field, partly because it more difficult to develop such methods compared to for static networks. In this colloquium, we review the methods to analyze and model temporal networks and processes taking place on them, focusing mainly on the last three years. This includes the spreading of infectious disease, opinions, rumors, in social networks; information packets in computer networks; various types of signaling in biology, and more. We also discuss future directions.

Author(s): Michele Borassi, Alessandro Chessa, and Guido Caldarelli

In this work, we analyze the hyperbolicity of real-world networks, a geometric quantity that measures if a space is negatively curved. We provide two improvements in our understanding of this quantity: first of all, in our interpretation, a hyperbolic network is ``aristocratic'', since few elements …

[Phys. Rev. E] Published Tue Aug 04, 2015

Temporal networks come with a wide variety of heterogeneities, from burstiness of event sequences to correlations between timings of node and link activations. In this paper, we set to explore the latter by using greedy walks as probes of temporal network structure. Given a temporal network (a sequence of contacts), greedy walks proceed from node to node by always following the first available contact. Because of this, their structure is particularly sensitive to temporal-topological patterns involving repeated contacts between sets of nodes. This becomes evident in their small coverage per step as compared to a temporal reference model -- in empirical temporal networks, greedy walks often get stuck within small sets of nodes because of correlated contact patterns. While this may also happen in static networks that have pronounced community structure, the use of the temporal reference model takes the underlying static network structure out of the equation and indicates that there is a purely temporal reason for the observations. Further analysis of the structure of greedy walks indicates that burst trains, sequences of repeated contacts between node pairs, are the dominant factor. However, there are larger patterns too, as shown with non-backtracking greedy walks. We proceed further to study the entropy rates of greedy walks, and show that the sequences of visited nodes are more structured and predictable in original data as compared to temporally uncorrelated references. Taken together, these results indicate a richness of correlated temporal-topological patterns in temporal networks.

We analyze the synchronization transition of a globally coupled network of N phase oscillators with inertia (rotators) whose natural frequencies are unimodally or bimodally distributed. In the unimodal case, the system exhibits a discontinuous hysteretic transition from an incoherent to a partially synchronized (PS) state. For sufficiently large inertia, the system reveals the coexistence of a PS state and of a standing wave (SW) solution. In the bimodal case, the hysteretic synchronization transition involves several states. Namely, the system becomes coherent passing through traveling waves (TWs), SWs and finally arriving to a PS regime. The transition to the PS state from the SW occurs always at the same coupling, independently of the system size, while its value increases linearly with the inertia. On the other hand the critical coupling required to observe TWs and SWs increases with N suggesting that in the thermodynamic limit the transition from incoherence to PS will occur without any intermediate states. Finally a linear stability analysis reveals that the system is hysteretic not only at the level of macroscopic indicators, but also microscopically as verified by measuring the maximal Lyapunov exponent.

We propose a method to associate a differentiable Riemannian manifold to a generic many-degrees-of-freedom discrete system which is not described by a Hamiltonian function. Then, in analogy with classical statistical mechanics, we introduce an entropy as the logarithm of the volume of the manifold. The geometric entropy so defined is able to detect a paradigmatic phase transition occurring in random graphs theory: the appearance of the ?giant component? according to the Erd?s-R?nyi theorem.

Author(s): Maxim Komarov and Arkady Pikovsky

We report on finite-sized-induced transitions to synchrony in a population of phase oscillators coupled via a nonlinear mean field, which microscopically is equivalent to a hypernetwork organization of interactions. Using a self-consistent approach and direct numerical simulations, we argue that a t…

[Phys. Rev. E 92, 020901(R)] Published Mon Aug 03, 2015

The chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. In this paper, the first evidence of three-dimensional chimera states is reported for the Kuramoto model of phase oscillators in 3D grid topology with periodic boundary conditions. Systematic analysis of the dependence of the spatiotemporal dynamics on the range and strength of coupling shows that there are two principal classes of the chimera patterns which exist in large domains of the parameter space: (I) oscillating and (II) spirally rotating. Characteristic examples from the first class include coherent as well as incoherent balls, tubes, crosses, and layers in incoherent or coherent surrounding; the second class includes scroll waves with incoherent, randomized rolls of different modality and dynamics. Numerical simulations started from various initial conditions indicate that the...

The statistical significance of network properties is conditioned on null models which satisfy spec- ified properties but that are otherwise random. Exponential random graph models are a principled theoretical framework to generate such constrained ensembles, but which often fail in practice, either due to model inconsistency, or due to the impossibility to sample networks from them. These problems affect the important case of networks with prescribed clustering coefficient or number of small connected subgraphs (motifs). In this paper we use the Wang-Landau method to obtain a multicanonical sampling that overcomes both these problems. We sample, in polynomial time, net- works with arbitrary degree sequences from ensembles with imposed motifs counts. Applying this method to social networks, we investigate the relation between transitivity and homophily, and we quantify the correlation between different types of motifs, finding that single motifs can explain up to 60% of the variation of motif profiles.

Author(s): M. E. J. Newman and Tiago P. Peixoto

A substantial volume of research has been devoted to studies of community structure in networks, but communities are not the only possible form of large-scale network structure. Here we describe a broad extension of community structure that encompasses traditional communities but includes a wide ran…

[Phys. Rev. Lett.] Published Tue Jul 28, 2015

Author(s): Hyunsuk Hong, Hugues Chaté, Lei-Han Tang, and Hyunggyu Park

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population …

[Phys. Rev. E] Published Wed Jul 29, 2015

Author(s): Malte Schröder, Manu Mannattil, Debabrata Dutta, Sagar Chakraborty, and Marc Timme

Finding conditions that support synchronization is a fertile and active area of research with applications across multiple disciplines. Here we present and analyze a scheme for synchronizing chaotic dynamical systems by transiently uncoupling them. Specifically, systems coupled only in a fraction of…

[Phys. Rev. Lett. 115, 054101] Published Wed Jul 29, 2015

Author(s): Ruud van de Bovenkamp and Piet Van Mieghem

The survival time T is the longest time that a virus, a meme or a failure can propagate in a network. Using the hitting time of the absorbing state in an uniformised embedded Markov chain of the continuous-time Susceptible-Infected-Susceptible (SIS) Markov process, we derive an exact expression for …

[Phys. Rev. E] Published Fri Jul 24, 2015