Abstract: The spontaneous ability of spatially extended systems to self-organize in space and time is proverbial and has been raised to paradigm in modern science. Collective behaviors are widespread in nature and mirror, at the macroscopic level, the microscopic interactions at play among elementary constituents. Convection instabilities in fluid dynamics, rhythms production and insect swarms are representative examples that emblematize the remarkable capacity of physical and biological systems to yield coherent dynamics. Instabilities triggered by random fluctuations are often patterns precursors. The imposed perturbation shakes e.g. an homogeneous equilibrium, seeding a resonant amplification mechanism that eventually materializes in magnificent patchy motifs, characterized by a vast gallery of shapes and geometries. Exploring possible routes to pattern formation, and unraveling novel avenues to symmetry breaking instability, is hence a challenge of both fundamental and applied importance. In the so-called modulational instability deviations from a periodic waveform are reinforced by nonlinearity, leading to spectral-sidebands and the breakup of the waveform into a train of pulses. The phenomenon was first conceptualized for periodic surface gravity waves on deep water by Benjamin and Feir, and for this reason it is customarily referred to as the Benjamin-Feir instability. The Benjamin-Feir instability has been later on discussed in the context of the Complex Ginzburg-Landau equation, a quintessential model for non linear physics. Here we revisit the Benjamin-Feir instability in the framework of the Complex Ginzburg-Landau equation, modified with the inclusion of a drift term. This latter is rigorously derived from a stochastic description of the microscopic coupling between adjacent oscillators. Generalized Benjamin-Feir instabilities occur, stimulated by the drift, outside the region of parameters for which the classical Benjamin-Feir instability is manifested. This observation, grounded on a detailed mathematical theory, contributes to considerably enrich the landscape of known instabilities, along a direction of investigation that can be experimentally substantiated.

Abstract: The N-Intertwined Mean Field Approximation (NIMFA) exponentially reduces the number of differential equations that need to be calculated in the Markovian susceptible-infected-susceptible (SIS) model in networks [1]. The non-zero steady state of NIMFA has been proved globally asymptotically stable when the number of initially infected nodes is larger than 0 and the effective spreading rate is above the NIMFA epidemic threshold [2,3]. Moreover, an accuracy criterion in the steady state has been derived in [4]. However, the accuracy of NIMFA is also determined by initial conditions of the epidemic process, which escaped the attention so far. We find that the virus die-out probability at an arbitrary time, which is determined both by the initial conditions and the network topology, impacts the accuracy of NIMFA. New results will be presented to show how the virus die-out probability influences the accuracy of time-dependent NIMFA prevalence in networks and a novel correction function for NIMFA has been found. Furthermore, the virus die-out probability is equivalent to the gambler’s ruin problem [5, page 231] in complete graphs and can also be solved numerically as a birth and death process in complete graphs. References: [1] P. Van Mieghem, J. Omic, and R. Kooij, IEEE/ACM Trans. on Networking, vol. 17, no. 1, pp. 1–14, Feb. 2009. [2] A. Khanafer, T. Başar, and B. Gharesifard, in 2014 American Control Conference, 2014, pp. 3579–3584. [3] S. Bonaccorsi, S. Ottaviano, D. Mugnolo, and F. Pellegrini, SIAM J. Appl. Math., vol. 75, no. 6, pp. 2421–2443, Jan. 2015. [4] P. Van Mieghem and R. van de Bovenkamp, Phys. Rev. E, vol. 91, no. 3, p. 032812, Mar. 2015. [5] P. Van Mieghem, Performance analysis of complex networks and systems. Cambridge: Cambridge University Press, 2014.

Abstract: Many methods have been developed to estimate the inter-dependence (called also coupling, information flow or Granger causality) between interacting variables of a dynamical system. Typically, this problem regards complex systems observed from multivariate time series, such as financial markets, climatic phenomena, brain activity and earthquakes. The methods tested so far are found to estimate the true underlying complex network with varying success. Here, a difficult but realistic setting of non-observed important variables (or subsystems) of the complex network is considered. In particular, the unobserved variables are assumed to be hidden sources of the complex network. The performance of different connectivity (Granger causality) measures on settings of unobserved source is evaluated with simulations on linear and nonlinear stochastic processes and dynamical systems of many variables. The results show that though some connectivity measures (of those tested) identify correctly the true complex network from the multivariate time series when no source is involved, all connectivity measures fail to estimate the true connections when a driving, but unobserved, hub is present in the true complex network. An example from finance is added to illustrate the problem.

Abstract: We will sketch an algorithmic take, i.e. message-passing algorithms, on networks and its relevance to some questions and insights in complex systems. Recently, message-passing algorithms have been shown to be an efficient, scalable approach to solve hard computational problems ranging from detecting community structures in networks to simulating probabilisitic epidemic dynamics on networks. The objective of the talk is two-fold. On on hand, we will discuss how the non-backtracking nature of message-passing avoids an “echo-chamber effects” of signal flow and thus makes a good tool to consider for problems in networks. On the other hand, we will also argue why insights gained from algorithms are equally important when exploring questions at the boundaries of scientific studies, such as networks and complex systems.

Abstract: Hypergraphs generalise graphs and networks by allowing edges to contain many vertices. An oriented hypergraph edge is a ‘simplex’. Simplices generalize oriented network edges: is a 1-dimensional edge, is a 2-dimensional triangle, is 3-dimensional tetrahedron, and … with (p+1) vertices being a p-dimensional polyhedron. Multiplex networks allow many relations to hold between sets of vertices. The ‘explicit relation’ notation can represent this: != . This generalizes to ‘hypersimplices’ with explicit relations on many vertices, != . Multiplex hypernetworks are formed from hypersimplices defined by many n-ary relations. Hypernetworks are foundational in a new theory of multilevel systems. The network family is integrated as follows (sorry if diagram doesn’t work): __________________orientation _____________________________many relations hypergraphs --------------> simplicial complexes ---------------> multiplex hypernetworks ^ . . . . . . . . . . . . . .__________. . . . . . . . ^ . . . . . . . . . . . . . __________. . . . . . . . . .^ | > 2 vertices. . . __________ . . . . . . . | > 2 vertices . . . . __________ . . . . . . . . . | > 2 vertices |. . . . . . . . . . . . . .__________ . . . . . . . . |. . . . . . . . . . . . . . .__________. . . . . . . . . | graphs -------------------------> networks -----------------------> multiplex networks __________________orientation _____________________________many relations It will be briefly explained how these structures contribute to understanding complexity.

Abstract: The Game of Life (LIFE), one of the two dimensional cellular automata, exhibits 1/f noise in the evolution starting from random configuration. 1/f noise has the power inversely proportional to frequency, that is, the exponent of the power spectrum is about -1. On the other hand, LIFE is capable of supporting universal computation. Considering these results together we can guess that there is a relationship between computational universality and 1/f noise in cellular automata. However the power spectrum of the computation process in which Turing machine is emulated on the array of LIFE has the exponent -0.3, that is not considered to be 1/f noise. In this research we conduct spectral analysis of the behavior of another computation model called universal register machine (URM) on the array of LIFE. Register machine is an abstract model of computer with a set of registers. Each register is of infinite extent and can hold any single non-negative integer. Universal register machine (URM) is a register machine that can imitate the computation process of any register machines. The first embodiment of URM on LIFE was accomplished by Chapman although we use the other one developed by Rendell (http://rendell-attic.org/gol/UCM/CMappNotes.html). The pattern of the URM is about 19,000 * 3,900 cells on the array and it takes about 32,586,000 time steps to halt. We calculate the power spectrum of the upper part of the pattern of URM with the array size of height * width = 3000 * 3800 = 11,400,000 cells over 65,536 time steps. As a result, the exponent of the power in the range between f = 1 .. 100 is about -1.5. This result shows that the power spectra of the computation process in LIFE vary according to its model and it contrasts clearly with the process starting from random configuration.

Abstract: Missing data is an issue ubiquitous in applied science. In the context of temporal networks, loss of data may correspond to missing contacts or missing node activities. In some cases, other data sources containing related information which can be used for data recovery are available. However, combining information from different sources is challenging. Here, we propose a technique based on joint factorization (Evrim et al., 2011) to recover missing activity in temporal networks. Each data source is encoded in a tensor. Assuming that there exists some correlation between the data from the multiple sources, a joint factorization of the tensors allows to infer missing information in one tensor from the others. We illustrate this method on an empirical contact network in which we simulated loss of data for the 10% of nodes by removing their activity for half the time span of the dataset. The resulting network was represented as a tensor where each slice gives the contacts of the nodes at a given time. In addition to the contact network, we have access to the location of the nodes with time. We built a second tensor where each slice gives the position of the nodes at a given time. We jointly factorized the two tensors, with a method handling missing values. Regularization terms were added to control the level of correlation between the contact network and the location network. We evaluated the correlation between the temporal activity of the nodes in the original network and in the recovered one over the missing period and found a good agreement. This suggests that our approach is an efficient method for recovering missing data for temporal networks. Moreover, it generalizes easily to more than two data sources which could all contain missing data.

Abstract: The introduction of multilayer neteorks poses the theoretical question of whether critical phenomena will behave differently on such networks with respect to traditional networks. So far theoretical studies have pointed out that such differences in the critical behaviours indeed exists. It has been showed that a multiplex network can exists in different structural phases, the transition among them being abrupt under some conditions. Three different topological scales can be naturally identified in a multiplex: that of the individual layers, that of the network of layers, and that of the aggregate network. The notion of quotient graph gives the connection between those scales in therms of spectral properties of the parent multiplex network an its aggregate representation. Much attention was paid to the coupling parameter p that weights the coupling between different representatives of the same node in different layers. It exists a value p∗ for which the derivative of the algebraic connectivity (the first non-zero eigenvalue of the supra-Laplacian) shows a discontinuity. At this point the Fiedler vector also changes abruptly and the Fiedler cut, that before p∗ only intercepts inter-layer edges, starts to intercepts intra-layer edges. We show that the same transition can be triggered by a completely different mechanism that we call layer degradation. Just acting on a single layer at a fixed p by removing intra-layer edges or tuning the weight of these, the multiplex undergoes to the same transition. Working on an approximation of the algebraic connectivity, we give an analytical description of the transition and its dependency on the value of p and on the value of the algebraic connectivities of isolated layers. We also study the transition numerically and the behaviour of the Shannon entropy of the Fiedler vector during the process, since one can interpret (a function of) this as an order parameter.

Abstract: It is intuitive to assume that differences in growth among diverse parts change the constitution of the population as a whole. The larger the differences, the faster the change. Fisher’s fundamental theorem of natural selection formalizes this idea. Counterintuitively, this article shows that large fitness differentials make it likely more beneficial to stop evolutionary selection by stabilizing population structure through informed intervention (i.e. bet-hedging). The larger the differences of fitness among types, the more likely it is that it pays of to hold the shares of type constant by counteracting natural selection. This possibility of such interventions depends on the availability of information about the environmental pattern. With information growth can be optimized through resource redistribution. Information is the key ingredient in order to be able to hedge ones bets and actively manage a portfolio in an uncertain environment. We treat information formally in terms of information theory, including Shannon’s absolute entropy and Kullback-Leibler’s relative entropy. The policy mechanism to convert information into growth is structural adjustment. An empirical analysis over nine levels of an international trade taxonomy shows that is more likely beneficial to intervene on more fine-grained levels, while blind natural selection is likely to optimize growth on more aggregate levels.

Abstract: The concept of Computational Irreducibility is formally defined in a rigorous way inside the computational model of Turing machines supplemented with a new kind of machines, the E-Turing machines (for enumerating Turing machine). This helps understanding the behavior of some cellular automata which seem apparently impredictible. Computational Irreducibilty is also mobilized to give an explanation to the concept of Emergence.

Abstract: Geometric representations of data provide useful tools for visualisation, classification and prediction. In particular, geometric approaches to network analysis have grown in popularity recently as the effectiveness of simple geometric models to generate complex networks with interesting properties becomes clearer. Techniques such as Multidimensional Scaling (MDS) provide a method of embedding high dimensional data, or networks into lower-dimensional spaces. However, classical MDS forces a representation in Euclidean space, which may not necessarily be appropriate. In this talk we use ideas from the causal set approach to quantum gravity to find a general method for embedding networks in spaces with any metric signature, not just a Riemannian one (with all eigenvalues of the metric positive). We demonstrate this on the most common case found in physics, a Lorentzian manifold (with one negative eigenvalue of the metric). These manifolds represent spacetime as used in relativity. We show that for networks which naturally form Directed Acyclic Graphs (DAG), such as citation networks, spacetime embeddings are appropriate. This is because the constraint of having no cycles in a DAG corresponds to the causal structure of events in spacetime: that if A caused B to happen, it cannot also be the case that B caused A. As an illustration of these techniques we embed well known citation networks from papers on the arXiv, and from cases of the US Supreme Court in flat spacetime and see that this embedding accurately predicts edges in the networks.

Abstract: Self-organization is defined as the spontaneous emergence of order arising out of local interactions in a complex system. The central to the idea of self-organization is the interaction between the agents, the particles, the elements that constitute the system. In biological systems, particle-particle interactions or particle-field interactions are often mediated by chemical trails (chemotaxis), or swarm behaviour that optimizes system efficiency. In non-biological systems, particle-field interaction plays the crucial role, as these interactions modify the surrounding field or often the topology of the energy landscape. Interactions in a system, or a system and its surroundings, are governed by energetic and entropic exchanges, either in terms of forces or in terms of statistical information. Since, energy and time, the two properties of matter and space we look into the Principle of Least Action to search for answers as it involves both time and energy into its formulation. Since, the Action Principle minimizes action and directs the system elements along least action trajectories on the energy landscape (surrounding field), it is imperative that a one to one correspondence exists between the Second Law of Thermodynamics and the Action Principle. In a system in equilibrium, the system particles can occupy all possible microstates whereas in a self-organizing, out-of-equilibrium system only certain microstates will be accessible to the system particles. In these systems, in order to organize efficiently the system particles interact locally and coordinate globally, in a way that lets swarms of agents to uniformly follow least action trajectories and simultaneously degrade their free-energies in order to maintain the organizational structure of the system at the expense of entropy export along the least action paths. The question we ask is how the symmetry breaking is mediated in a physical system?

Abstract: We present universal framework for generation, analysis and control of non-trivial states/patterns in the complex systems like kinetic hierarchies describing general set-up for non-equilibrium dynamics and their important reductions. We start from the proper underlying functional spaces and their internal hidden symmetries which generate all dynamical effects. The key ingredients are orbits of these symmetries, their representations, and Local Nonlinear Harmonic Analysis on these orbits. All that provides the possibility to consider the maximally localized fundamental generic modes, non-linear (in case of the non-abelian underlying symmetry) and non-gaussian, which are not so smooth as gaussians and as a consequence allowing to consider fractal-like images and possible scenarios for generation chaotic/stochastic dynamics on the level of representation theory only. As a generic example we consider the modeling of fusion dynamics in plasma physics. http://math.ipme.ru/zeitlin.html, http://mp.ipme.ru/zeitlin.html

Abstract: Complex networks have been conjectured to be embedded in hidden metric spaces, in which distances between nodes encode their similarity and, thus, their likelihood of being connected. This hypothesis, combined with a suitable underlying space, has offered a geometric interpretation of the complex topologies observed in real networks, including scale-free degree distributions, the small-world effect, strong clustering as a reflection of the triangle inequality, and self-similarity. It has also explained their efficient inter-node communication without a knowledge of the complete structure. Moreover, it has been shown that for networks whose degree distribution is scale-free, the natural geometry of their underlying metric space is hyperbolic. These results have led to geometric models for real growing networks that reproduce their evolution and in which preferential attachment emerges from local optimization principles. Using real networks from very different domains, we present empirical evidence that the magnitude of the connections, the weights, also have a geometric nature. To quantify the level of coupling between the topology, the weights, and the underlying metric space in real networks, we introduce the most general and versatile class of weighted networks embedded in hidden metric spaces. This framework allows us to independently regulate the coupling between the topology and the geometry and between the weights and the geometry. We show that such couplings can be significantly different in real networks, which supports the hypothesis that the formation of connections and the assignment of their magnitude is ruled by different processes. Our empirical findings, combined with our new class of models, open the path towards the uncovering of the natural geometry of real weighted complex networks.

Abstract: Dynamical processes on networks are currently being considered in different domains of cross-disciplinary interest. Reaction-diffusion systems hosted on directed graphs are in particular relevant for their widespread applications, from neuroscience, to computer networks and traffic systems. We shall here consider as a paradigmatic example the complex Ginzburg-Landau equation defined on a asymmetric lattice, with next-nearest-neighbors couplings. The peculiar spectrum of the discrete Laplacian operator yields an extended class of modulational instabilities which can be analytically predicted. Specifically, the synchronized regime can turn unstable to external perturbation because of the imposed degree of spatial asymmetry, and for a choice of the paramaters for which the instability cannot develop on undirected graphs. The excited modes can then stabilize in traveling wave solutions. Alternatively, the modulus of the complex quantity that obeys Ginzburg-Landau equation can display a complicated mosaic of patterned structures in the non linear regime of evolution. The bifurcation between the different regimes is predicted by the theory and ultimately relates to the long range nature of the spatial coupling imposed. The case of an heterogeneous generic directed network will be also discussed.

Abstract: The interplay between network evolution and the dynamics of a spreading process impacts the conditions for large-scale propagation, by altering the epidemic threshold, i.e., the critical transmissibility value above which the disease turns epidemic. In order to study these processes many works have resorted to discrete-time approximations of both network and spreading dynamics. While this is practical from both the numerical and the theoretical point of view, it is known to bias results in some circumstances. Here we analytically derive the epidemic threshold for a generic time varying network, in both continuous and discrete time. We focus on the susceptible-infected-susceptible spreading process, within the quenched mean field framework. In discrete time, the epidemic threshold of a generic network can be computed from the spectral radius of the infection propagator, a matrix encoding a multi-layer representation of both network and spreading dynamics [Valdano et al (2015) Phys Rev X, Valdano et al (2015) Eur Phys J B]. We build a new theoretical framework that allows us to perform the continuous time limit of the infection propagator, and we find it correctly describes the linearized form of the continuous-time Markov equations of the process. This yields a solution for the epidemic threshold in continuous time that admits an explicit form in specific cases. The mathematical formalism proposed here allows addressing the effect of discretization and temporal resolution on the epidemic threshold accounting for different network properties and parameter regimes.

Abstract: Multiple sinks competition is investigated for a walker diffusing on directed complex networks. The asymmetry of the imposed spatial support makes the system non transitive. As a consequence, it is always possible to identify a suitable location for the second absorbing sink that screens at most the flux of agents directed against the first trap, whose position has been preliminarily assigned. The degree of mutual competition between pairs of nodes is analytically quantified through apt indicators that build on the topological characteristics of the hosting graph. Moreover, the positioning of the second trap can be chosen so as to minimize, at the same time the probability of being in turn shaded by a thirdly added trap. Supervised placing of absorbing traps on a asymmetric disordered and complex graph is hence possible, as follows a robust optimization protocol. This latter is here discussed and successfully tested against synthetic data.

Abstract: Most of the real world complex systems consist of multiple subsystems that interact with each other, which can be represented as multilayer networks. Implications of such multilayer topologies have been studied for system robustness, cascading failures, reaction-diffusion dynamics, etc., but little research has been conducted on the dynamical stability of multilayer networks. Here we study the stability of multilayer networks and its relationships with the stabilities of networks within individual layers. Specifically, we generated two random networks following the method of May's random ecological network model, and then coupled them to create a random multilayer network. Two parameters were systematically varied: the strength of within-layer connections, alpha, and the strength of inter-layer coupling, kappa. Stabilities of those networks were evaluated by calculating eigenvalues of their (supra-) adjacency matrices. Numerical analysis showed that, with alpha above a certain threshold, individual networks become unstable, as is well known from May's work. We also found, however, that the whole multilayer network can be unstable with alpha below the threshold, if the inter-layer coupling kappa is above another threshold. This form of instability was previously not known, and it indicates that a system made of individually stable layers may get destabilized because of too strong inter-layer coupling. We also investigated the effects of the network size in each layer. As the number of nodes increases, the thresholds of alpha and kappa both decrease, further manifesting the network instability induced by inter-layer coupling. Our study illustrates that the strength of inter-layer coupling has a critical role in the stability of the whole network.

Abstract: We would like to draw the network science community's attention to a scenario in which the graphlet correlation distance (GCD) method, introduced by Yaveroglu et al. [1], can give misleading results. The aim of the talk is to introduce attendees to the method and offer an appreciation of its subtleties. Graphlet correlation distance (GCD) has been proposed as an alignment-free method for the comparison of networks in systems biology and other domains. In particular, GCD has been shown to outperform other alignment-free methods, both in terms of accuracy and computational efficiency [1, 2]. The method correlates counts of graphlets, that is, induced subgraphs of a network that are of order 5 or less, for individual nodes of a network. A set of networks can then be clustered based upon a distance between their graphlet correlation matrices. In this talk we argue that GCD is very sensitive to low counts of graphlets; which, in turn, can bias results. Thus, when a set of networks contains negligible amounts of some graphlets, correlations between these underrepresented graphlets dominates the resulting partition. We illustrate this problem by constructing artificial networks which lack some graphlets in their makeup. We then add a statistically insignificant amount of edges to individual networks such that they now contain an instance of the missing graphlet. We conclude by proposing a modification to the method and then applying it to a dataset of street networks. [1] Yaveroğlu, Ömer Nebil, et al. "Revealing the hidden language of complex networks." Scientific reports 4 (2014). [2] Yaveroğlu, Ömer Nebil, Tijana Milenković, and Nataša Pržulj. "Proper evaluation of alignment-free network comparison methods." Bioinformatics 31.16 (2015): 2697-2704.

Abstract: Bipartite networks are currently regarded as providing a major insight into the organization of real-world systems, unveiling the mechanisms shaping the interactions occurring between separate groups of nodes. One of the major problems encountered when dealing with bipartite networks is obtaining a (monopartite) projection over the layer of interest, preserving as much as possible the information encoded into the original bipartite structure. In the present paper we propose an algorithm able to obtain statistically-validated projections of bipartite networks: our method com- pares the number of shared neighbors between two nodes on the same layer with the expectation from a recently null-models for bipartite networks, [1,2]. The output of our method is a p-value per couple of nodes, encoding the statistical significance of the observation: if the p-value is small, the observation is not explained by the null-model and it is statistically relevant. A validated projection can thus be obtained by choosing a threshold and linking the nodes passing the test. We also show an alternative approach, intended to retain all the information provided by the whole matrix of p-values: upon doing so, we are able to detect the (potentially) hierarchical relationships occurring between different subsets of elements. We test our procedure on the bipartite projection of the trade network (i.e. usual Economic Complexity framework) and on the bipartite social network of MovieLens100k between users and rated films. The resulting similarities between countries have interesting explanations in terms of the economic development of different nations; on the social side, our methods is able to cluster films respect to some non trivial features (cast, directors, sub-genres..). [1] Saracco F., et al. Randomizing bipartite networks: the case of the World Trade Web. Sci. Rep. 5(10595) [2] Saracco F., et al. Detecting the bipartite world trade web evolution across 2007: a motifs-based analysis, arXiv:1508.03533.

Abstract: Bubbly flows are frequently observed in nature or industrial applications. Is some cases, inception of such flows are undesirable, like boiling of coolant liquid which prevents effective heat loss. On other hand, there are processes, like bubble chemical reactors, where formation of bubbly flows are necessary but should be carried out in specific fashion. Prediction of such regimes are crucial for robust equipment design. It is not always possible to apply experimental methods for investigation of bubbly flows, thus modern numerical methods are of high importance and can be used to predict flow patterns, bubble formation and evolution for complex systems and set-ups. Numerical simulation of three-dimensional bubbly column will be presented in the scope of that work as a representative case of polydisperse bubbly flow. Numerical algorithm is based on the mathematical model which utilizes Euler-Euler approach for description of dispersed phases, supplemented by k-omega SST turbulence model for bubbly flows and extended for polydisperse flow description by multi-class approach. Main problem arises from coupling between carrier phase and multiple dispersed classes, which is overcome by specific numerical realization of the mathematical model. In-house computer code was developed on the basis of the proposed numerical algorithm. Main feature of that code is unstructured meshes, pseudo-time solution algorithm, modified phase-coupled SIMPLEC method, TVD-based interpolation schemes. Velocity and gas concentration distributions presented for bubbly flow column and compared with the existed experimental data (Sokolichin at al 1999). Column is represented as parallelepiped with gas injector (sparger) installed on the bottom surface with the offset towards one vertical wall. It was shown, that numerical simulation predicts well entire jet structure produced by buoyant bubbles, but also provides detailed information of flow near bubble inlet and close to vertical walls, with formation of thin regions with local maximum of bubble concentration.

Abstract: It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. We find that, while graphs with a given number of links are ensemble equivalent, graphs with a given degree sequence are not. This result holds irrespective of whether the energy is nonadditive (as in unipartite graphs) or additive (as in bipartite graphs). In contrast with previous expectations, our results show that (1) physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by longrange interactions or nonadditivity, (2) mathematically, nonequivalence is determined by a different large-deviation behavior of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in general.

Abstract: Sports and games can be described as complex systems. We describe these systems through a mathematical theory to find a universal behavior in the dynamics of different sports and games related to the players performance. This description can be done studying the ranking dynamics. We employed data from tennis, chess, golf, poker and football. Players are ordered by each official association. Each player or team has a score that varies with time, this results in a time dependent ranking. We tested five models to describe the distribution of scores for a given time. The rank diversity is introduced to quantify evolution of rankings; it is defined as the number of distinct elements of the complex set which have a given rank k within a given period of time. In other words, we analyze the time dependence of the ranks and observed a global behavior, despite the differences in distributions. Rank diversity in six cases follows closely the cumulative of a Gaussian. We introduce a simple model to reproduce the observed behavior. A member with rank k(t), at the discrete time t, is converted to rank k(t+1), tanking k(t) we add a Gaussian random number generator of the standard deviation corresponding to the original rank diversity getting s(t+1), once we obtained this for all members they are ordered according to their magnitude, a new data set is gotten. Good agreement is obtained.

Abstract: Social interactions shape the patterns of spreading processes in a population. Techniques such as diaries or proximity sensors allow to collect data about encounters and to build networks of contacts between individuals. The contact networks obtained from these different techniques are however quantitatively different. Here, we first show how these discrepancies affect the prediction of the epidemic risk when these data are fed to numerical models of epidemic spread: low participation rate, under-reporting of contacts and overestimation of contact durations in contact diaries with respect to sensor data determine indeed important differences in the outcomes of the corresponding simulations. Most importantly, we investigate if and how information gathered from contact diaries can be used in such simulations in order to yield an accurate description of the epidemic risk, assuming that data from sensors represent the ground truth. The contact networks built from contact sensors and diaries present indeed several structural similarities: this suggests the possibility to construct, using only the contact diary network information, a surrogate contact network such that simulations using this surrogate network give the same estimation of the epidemic risk as simulations using the contact sensor network. We present and evaluate several methods to build such surrogate data and show that it is indeed possible to obtain a good agreement between the outcomes of simulations using surrogate and sensor data, as long as the contact diary information is complemented by publicly available data describing the heterogeneity of the durations of human contacts.

Abstract: The pervasiveness of the Internet and smartphones enables individuals to participate in multiple networks such as communications networks and the physical contact network. The feedback between these networks opens new possibilities to mitigate epidemic spreading. For instance, the spread of a disease in a physical contact network may trigger the propagation of the information related to this disease in a communications network, which in turn may increase the alertness of some individuals resulting in their avoidance of contact with their infected neighbours in the physical contact network, possibly protecting the population from the infection [1,2]. In this work, we aim to understand how the time scale of the information propagation (speed at which information is spreaded and forgotten) in the communications network relative to that of the epidemic spread in the physical contract network influences such mitigation using awareness information. We firstly propose a model of the interaction between information propagation and epidemic spread, taking into account their relative time scale. We analytically derived the average fraction of infected nodes in the meta-stable state for this model (i) by developing an individual based Mean-Field Approximation method and (ii) by extending the Microscopic Markov Chain Approach, firstly introduced in [1,2]. Furthermore, we find that the optimal mitigation, can be achieved at a relative time scale, not necessarily zero nor infinity depending on the rate at which an infected individual gets aware. Contrary to our intuition, fast information spread in the communications network could even reduce the mitigation effect. Finally, we find that the mitigation tends to perform better when more node pairs are connected both in the physical contact and communications network. We explain these findings using both theoretical analysis and physical interpretations. [1] Granell C, Gomez S, Arenas A, Phys.Rev.E. 2014;90(1):012808. [2] Granell C, Gomez S, Arenas A, Phys.Rev.L. 2013;111(12):128701.

Abstract: Across many scientific domains, there is common need to automatically extract a simplified view or a coarse-graining of how a complex system's components interact. This general task is called community detection in networks and is analogous to searching for clusters in vector data. It is common to evaluate the performance of community detection algorithms by their ability to find so-called "ground truth" communities. This works well in synthetic networks with planted communities because such networks' links are formed explicitly based on the planted communities. In real world networks there are no planted communities. Instead, it is standard practice to treat some observed discrete-valued node attributes, or metadata, as ground truth. However, failure to find a good division that correlates with our metadata is a highly confounded outcome, arising for any of several reasons: (i) these particular metadata are irrelevant to the structure of the network, (ii) the detected communities and the metadata capture different aspects of the network's structure, (iii) the network contains no communities, or (iv) the community detection algorithm performed poorly. Most work on community detection assumes that failure to find communities that correlate with metadata implies case (iv). Here, we show that metadata are not the same as ground truth, and that treating them as such induces severe theoretical and practical problems, and can lead to incorrect scientific conclusions even in relatively benign circumstances. However, node metadata still have value and a careful exploration of their relationship with network structure can yield insights of genuine worth. We illustrate this point by introducing two statistical techniques that can quantify the relationship between metadata and community structure for a broad class of community structure models. We demonstrates both techniques using sets of synthetic and real-world networks, featuring multiple types of metadata and community structure.

Abstract: Epidemic processes describe several real-world dynamic activities on networks, where we are concerned about how fast the virus will overspread the network. We use the average pervasion time to measure the spreading velocity in a SIS process, where the pervasion time is defined as the minimum time when every node in the network is infected at least once. We show that the pervasion time resembles a lognormal distribution, and the average pervasion time for the same spreaders exponentially decreases with the effective infection rate. Increasing the number of initially infected nodes can accelerate the spreading. The simulation results show the different effect of multiple spreaders on the spreading velocity for the different underlying topology, which provide some clues whether more spreaders are worthy to be invested. The investment utility is defined as the normalized decrement of the average pervasion time after adding a new spreader, which measures the effect of this spreader. We show that the investment utility decreases steadily with the number of initial spreaders in the homogeneous graphs like complete graph and lattice; However, in the network with power-law degree distribution, the investment utility of the newest spreader becomes suddenly low when the number of spreaders is over a value. After that, the investment utility of the lowest degree node dramatically becomes the highest among the rest of nodes. In addition, we reveal that the total investment utility for the same number of spreaders increases with the effective infection rate in a network, which implies the higher investment value to deploy multiple spreaders for the process with a higher effective infection rate.

Abstract: The dynamics of random walks and thermal diffusion on network is interlinked with the spectral properties of the associated graph representing the network. For example, the eigenvalues of the normalized Laplacian L of the graph characterize the time scales for the convergence to the equilibrium probability distribution over the nodes of the graph. The normalized Laplacian also describes the thermal heat conduction on networks via the Fourier heat transport equation. The heat conduction on networks has been investigated experimentally using a 2 dimensional network cut from strongly thermal conducting pyrolytic graphite films and observing the diffusion of heat with a thermal camera. The effect of network topology, length and width of the connections in the network is analyzed using the general concepts of the spectral theory of graphs. Cospectral graphs have identical eigenspectrum for the matrices associated with the graph such as the adjacency matrix A, the Laplacian matrix L=D-A, with D is the degree matrix for the vertices, and the normalized Laplacian matrix. The dynamics of a random walk, -hence also the heat conduction -, on these graphs can be found by iterating the probability transfer matrix P= D^(-1) A and is strongly related to the spectra of the normalized Laplacian L = D^(-1/2) AD^(-1/2). In the past decades methods have been devised to systematically generate sets of graphs with cospectral properties. For example L cospectral graphs can be generated systematically for inflated star graphs by modifying the arm composition (swapping). The graphs have identical eigenvalues, but different topology. Furthermore, some of these graphs have multiple zero eigenvalues, making the final distribution dependent on the initial starting conditions. The effect of cospectral behavior and multiple zero eigenvalue degeneracy on the dynamics of thermal conduction is demonstrated using the experiments and by numerical and analytical modeling .

Abstract: Evolutionary game theory has been studied intensely as a mathematical framework for understudying Darwinian evolution. Despite the fact that evolutionary dynamics, whether in biological ecosystems or, say in human environments like the economy, takes place in the form of co-evolving webs of interactions; game theory is usually studied in the limit of very few strategies. Here, we study replicator dynamics with extinctions and mutations in a large game with many strategies. We perform numerical simulations of a game where each player can play one of a large number of strategies. The game is defined with a payoff matrix whose elements are random numbers which can be positive or negative, with majority being zero. At the beginning of the simulation we choose randomly a small number of strategies to be played. Reproduction of players is done according to the replicator equation in which a small amount of mutations is introduced. In this way new strategies can appear in the system. Additionally we introduce an extinction threshold; strategies with a frequency less than this threshold are removed from the system. The resulting behaviour shows complex intermittent time dependence similar to those of the Tangled Nature model. The dynamics has two types of phases: a quasi stable phase in which the number of active strategies is more or less constant and hectic phases during which creation and extinct of strategies happens at a high rate. We see that the complex behaviour of the Tangled Nature model, which is in good agreement with observations on ecosystems, also arises from the game theoretic basis of the replicator dynamics. Finally we investigate various lifetime distributions and find fat tail distributions similar to those often observed for real systems and show that these distributions can be explained using supestatistics.

Abstract: Many real-world complex systems exhibit a community structure, where groups of nodes share a high internal connection density or strength but have fewer or weaker connections with nodes belonging to different groups. Identifying these groups of nodes is a key challenge in network theory and much attention has been devoted to this specific task. Amongst the methods proposed so far, those based on the modularity function are the most popular and widely used. Modularity is a quality function that assigns a score to each partition of the network, with higher scores corresponding to better partitions. However, despite its wide use, modularity has also been shown to suffer from severe limitations, such as a resolution limit or finding communities on random graphs. We present the properties of a new objective function, modularity density, which was introduced in recent years by Szymanski and collaborators to address some of the limitations mentioned above. Modularity density presents interesting properties as a quality function, and we will derive the analytical dependence of modularity density on connection density in random graphs of any given size. We will also show that modularity density allows for a comparison of the community structure between networks with different number of nodes. Finally, we will present a detailed community detection algorithm based on modularity density, discussing its computational complexity. This presentation will allow me to further disseminate my work to an audience with broad interests. It will help me establish a personal network of connections in the complex systems scientific community. This may enable collaborations, which would be of great benefit in my career as a young researchers. This talk is aimed at researchers with an interest in network science and methods to analyse networked data.

Abstract: How much information does a complex networks integrate as a whole over the sum of its parts? Can the complexity of such networks be quantified in an information-theoretic way and be meaningfully coupled to its function? Recently, measures of dynamical complexity such as integrated information have been proposed. However, problems related to the normalization and Bell number of partitions associated to these measures make these approaches computationally infeasible for large-scale networks. Our goal in this work is to address this problem. Our formulation of network integrated information is based on the Kullback-Leibler divergence between the multivariate distribution on the set of network states versus the corresponding factorized distribution over its parts. We find that implementing the maximum information partition optimizes computations. These methods are well-suited for large networks with linear stochastic dynamics. As an application of our formalism, we compute the information complexity of the human brain’s connectome network. Compared to a randomly re-wired network, we find that the specific topology of the brain generates greater information complexity.

Abstract: One of the main objectives to study on complex systems and complex networks is to understand how to efficiently control them, or alternatively, to make them difficult to be controlled (and therefore become robust under certain circumstances). While extensive studies have been carried out on controllability of complex networks, which typically aim to finding the minimum set of driver nodes to be connected to external controllers for ensuring that the system can be driven to any state, little has been done for an arguably even more important problem: the optimal control of complex networks, which aims to driving a system to a certain state with the minimum energy cost. A complex system with perfect controllability may not be actually controllable in the real life, if the cost for driving it to the target state is too high. To start, we consider the “simplest” optimal control problem, trying to find the solution for driving a static complex network at the minimum energy cost with a given number of controllers. A projected gradient method has been proposed, which works efficiently in both synthetic and real-life networks. The study is then extended to the case when each controller can only be connected to a single network node for having the lowest connection complexity. Interesting insights reveal that such connections basically avoid high-degree nodes of the network, which is in resonance with recent observations on controllability of complex networks. Such results open the technical path to enabling minimum cost control of complex networks, and contribute new insights into locating the key nodes from a minimum cost control perspective. We shall also discuss on the possible approaches for enhancing the scalability of the proposed method and provide some preliminary testing results on some of these approaches

Abstract: Statistical significance of network clustering has been an unresolved problem since it was observed that community detection algorithms produce false positives even in random graphs. After a phase transition between undetectable and detectable cluster structures was discovered [1,2], the spectra of adjacency matrices and detectability limits were shown to be connected, and they were calculated for a wide range of networks with arbitrary degree distributions and community structure [3]. In practice, given a real-world network neither the hypothetical network model nor its full eigenspectrum is known, and whether a given network has any communities within detectability regime cannot be easily established. Based on the global clustering coefficient (GCC) we construct a criterion telling whether in an undirected, unweighted network there is some/no detectable community structure, or if the network is in a transient regime. To that end we use approximations of GCC and the spectra of adjacency matrices, together with the empirical observations showing that: a) for graphs with community structure GCC reaches the random graph value before its connectivity is fully random, and b) this saturation of GCC coincides with the theoretically predicted limit of detectability. We compare the criterion against existing methods of assessing significance of network partitioning. We compute it on various benchmark graphs, as well as on real-world networks from SNAP database, and compare it with the results of state-of-the-art community detection methods. The method is simple and faster than methods involving bootstrapping; it is robust also on sparse networks. Analogous criteria are plausible also for directed graphs. [1] J. Reichardt and M. Leone, Phys. Rev. Lett. 101, 078701 (2008). [2] A. Decelle, F. Krzakala, C. Moore, and L. Zdeborova, Phys. Rev. Lett. 107, 065701 (2011). [3] X. Zhang, R. R. Nadakuditi, and M. E. J. Newman, Phys. Rev. E 89, 042816 (2014).

Abstract: Peer review is the most prevalent mechanism for validating the quality of published research. While scientists are not blind to many of the shortcomings of this system (e.g. bias), studies show that peer review is perceived as a necessary tool that does improve the quality of published papers. Surprisingly, despite the importance of this process and its place in the scientific world, peer review has not been extensively studied until recently. The goal of our work was twofold. Firstly, we wanted to deepen our understanding of the peer review process as perceived from the viewpoint of a journal editor. Secondly, through extensive numerical experiments based on real-world data, we wanted to improve the effectiveness of the process. In order to understand some of the dynamical properties of the peer review process, we analysed a dataset which contained information about 58 papers submitted to the Journal of the Serbian Chemical Society. The dataset consisted of 311 “review threads” - full descriptions of the review process initiated by sending an invitation to a potential reviewer. We were able to separate the process into distinct phases, recreate transition probabilities between phases as well as their durations. This allowed us to uncover many interesting properties of the process and to create a framework that can be used as a basis for simulations. We were particularly interested in how editorial workflows – sets of guidelines and rules that journal editors use to acquire reviews for submitted manuscripts – can be improved. To this end, we employed Cartesian Genetic Programming to evolve and optimise editorial strategies. We showed that these evolved strategies are better than typical strategies used by some editors and lead to shorter review times.

Abstract: 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 [1]. 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 non-parametric Bayesian framework [2] 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. 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. [1] J. Yang and J. Leskovec, "Structure and overlaps of ground-truth communities in networks", ACM Trans. Intell. Syst. Technol., vol. 5, pp. 26:1–26:35, Apr. 2014. [2] T. P. Peixoto, "Parsimonious Module Inference in Large Networks", Phys. Rev. Lett., vol. 110, p. 148701, Apr. 2013.

Abstract: Complex systems have large numbers of components with correspondingly large numbers of possible interactions. Even very large systems, however, can be simple in that they can be understood in terms of additive combinations of the contributions of their parts. Even the largest computer can be fully analysed this way. Furthermore, their behaviour for a given input is fully predictable for any length of time one chooses. It was long assumed that all systems were similarly analysable and predictable. Our only problem was to show this for more complex systems. This assumption was justified both by striking successes, but also on theoretical grounds. However, such systems are special among the range of all possible systems (Rosen 1991), and some systems with few components violate them by being both irreducible and noncomputable. We call such systems complexly organized. The complexity is not in the numbers, but in their behaviour. Such systems can be given a dynamical characterization so that for energy based systems and information based systems the key is the loss of available energy or information as entropy disposed outside the system, maintained by an external source of energy or information. This permits what is called self-organization, though not all complexly organized systems are self-organized. The unpredictability and irreducibility of complexly organized systems fits well with ideas of emergence developed in the 19th Century to explain living systems, and formalized more carefully by C.D. Broad in the 20th Century. I will argue that the dynamics of complex systems implies the logical conditions that these philosophers laid down. The advantage of having a dynamical account is that it gives us a way to interact with and test the systems for their properties. The logical conditions, on the other hand, were mostly attributed qualitatively (and sometimes erroneously), which is distressingly common even today.

Abstract: In their self-organization, complex systems increase the entropy in their surroundings and decrease their internal entropy, but the mechanisms, the reasons and the physical laws leading to this processes are still a question of debate. In our approach, energy gradients across complex systems lead to change in the structure of systems, decreasing their internal entropy to ensure the most efficient energy transport and therefore maximum entropy production in the surroundings. This approach stems from fundamental variational principles in physics, such as the principle of least action, which determine the motion of particles. We compare energy transport through a fluid cell which has random motion of its molecules, and a cell which can form convection cells. We examine the signs of change of entropy, and the action needed for the motion inside those systems. The system in which convective motion occurs, reduces the time for energy transmission, compared to random motion. For more complex systems, those convection cells form a network of transport channels, for the purpose of obeying the equations of motion in this geometry. Those transport networks are an essential feature of complex systems in biology, ecology, economy and society in general. This approach can help explain some of the features of those transport networks, and how they correlate with the level of organization of systems.

Abstract: Real networks often form interacting parts of larger and more complex systems. Examples can be found in different domains, ranging from the Internet to structural and functional brain networks. Here, we show that these multiplex systems are not random combinations of single network layers. Instead, they are organized in specific ways dictated by hidden geometric correlations interweaving the layers. We find that these correlations are significant in different real multiplexes, and form a key framework for answering many important questions. Specifically, we show that these geometric correlations facilitate: (i) the definition and detection of multidimensional communities, which are sets of nodes that are simultaneously similar in multiple layers; (ii) accurate trans-layer link prediction, where connections in one layer can be predicted by observing the hidden geometric space of another layer; and (iii) efficient targeted navigation in the multilayer system using only local knowledge, which outperforms navigation in the single layers only if the geometric correlations are sufficiently strong. Importantly, if optimal correlations are present, the fraction of failed deliveries is mitigated superlinerarly with the number of layers, suggesting that more layers with the right correlations quickly make multiplex systems almost perfectly navigable. Our findings uncover fundamental organizing principles behind real multiplexes and can have important applications in diverse domains, ranging from improving information transport and navigation or search in multilayer communication systems and decentralized data architectures, to understanding functional and structural brain networks and deciphering their precise relationship(s), to predicting links among nodes (e.g., terrorists) in a specific network by knowing their connectivity in some other network.

Abstract: Living organisms capitalize on their ability to predict their environment to maximize their available free energy, and invest this energy in turn to create new complex structures. For example, a lion metabolizes the structure of an antelope (destroying it in the process), and uses the energy released to build more lion. Is there a preferred method by which this manipulation of structure should be done? Our intuition is “simpler is better,” but this is only a guiding principal. By formalizing the manipulation of patterns – structured sequences of data – this intuitive preference for simplicity can be substantiated through physical reasoning based on thermodynamics. Using techniques from complexity science and information theory, we consider devices that can manipulate (i.e. create, change or destroy) patterns. In order to operate continually, such devices must utilize an internal memory in order to keep track of their current position within the pattern. However, the exact structure of this internal memory is not uniquely defined, and all choices are not equivalent when it comes to their thermal properties. Here, we present the fundamental bounds of the cost of pattern manipulation. When it comes to generating a pattern, we see indeed that the machine with the simplest memory capable of the task is indeed the best choice thermodynamically. Using the simplest internal memory for generation grants the advantage that less antelope needs to be consumed in order to produce the same amount of lion. However, contrary to intuition, when it comes to extracting work from a pattern, any device capable of making statistically accurate predictions can recover all available energy from the structure. This apparent paradox can be explained by careful consideration of nature of the information-processing tasks at hand: namely, one of logical irreversibility. [See also arXiv:1510.00010.]

Abstract: Recently Gao et al.[1] showed that classes of complex networks could be described in a universal way. In particular it was stated that the dynamics of a complex network consisting of many nodes and links is governed by a one-dimensional effective dynamical equation, which was obtained by averaging over all network configurations. In this paper we address the question how well the averaged effective equation describes classes of networks by numerical calculation of variances in dynamics. It appears that huge variances in the dynamics can arise. To examine the consequences of our work to practical situations, we apply our findings to specific networks occurring in transport and supply chains. References [1] Jianxi Gao1, Baruch Barzel, Albert-László Barabási, Universal resilience patterns in complex networks, Nature 530, 307 (2016).

Abstract: Assortative mixing (or homophily) in networks is the tendency for nodes with the same attributes, or metadata to link to each other. For instance in social networks we may observe more interactions between people with the same age, race, or political belief. Quantifying the level of assortativity or disassortativity (the preference of linking to nodes with different attributes) can shed light on the factors involved in the formation of links in complex networks. It is common practice to measure the level of assortativity according to the assortativity coefficient, or modularity in the case of discrete-valued metadata. This global value is an average behaviour across the network and may not be a representative statistic when mixing patterns are heterogeneous. For example, a social network that spans the globe may exhibit local differences in mixing patterns as a consequence of differences in cultural norms. Here, we present a new approach to localise these global measures so that we can describe the assortativity at the node level. Consequently we are able to capture and qualitatively evaluate the distribution of mixing patterns in the network. We develop a statistical hypothesis test with null models that preserve the global mixing pattern and degree distribution so that we may quantitatively determine the representativeness of the global assortativity. Using synthetic examples we describe cases of heterogeneous assortativity and demonstrate that for many real-world networks the global assortativity is not representative of the mixing patterns throughout the network.

Abstract: We will sketch an algorithmic take, i.e. message-passing algorithms, on networks and its relevance to some questions and insight in complex systems. Recently, message-passing algorithms have been shown to be an efficient, scalable approach to solve hard computational problems ranging from detecting community structures in networks to simulating probabilisitic epidemic dynamics on networks. The objective of the talk is two fold. On on hand, we will discuss how the non-backtracking nature of message-passing avoids an “echo-chamber effects” of signal flow and thus makes a good tool to consider for problems in networks. On the other hand, we will also argue why insight gained from algorithms are equally important when exploring questions at the boundaries of scientific studies, such as networks and complex systems.

Abstract: The determination of the most central agents in complex networks is important because they are responsible for a faster propagation of information, epidemics, failures and congestion, among others. A challenging problem is to identify them in networked systems characterized by different types of interactions, forming interconnected multilayer networks. Here we describe a mathematical framework that allows us to calculate centrality in such networks and rank nodes accordingly, finding the ones that play the most central roles in the cohesion of the whole structure, bridging together different types of relations. These nodes are the most versatile in the multilayer network. We then present two applications. First, we propose a method based on the analysis of bipartite interconnected multilayer networks of citations and disciplines, to assess scholars, institutions and countries interdisciplinary importance. Using data about physics publications and US patents, we show that our method allows to reward, using a quantitative approach, scholars and institutions that have carried out interdisciplinary work and have had an impact in different scientific areas. Second, we investigate the diffusion of microfinance within rural India villages accounting for the whole multilayer structure of the underlying social networks. We define a new measure of node centrality on multilayer networks, diffusion versatility, and show that this is a better predictor of microfinance participation rate than previously introduced measures defined on aggregated single-layer social networks. Moreover, we untangle the role played by each social dimension and find that the most prominent role is played by the nodes that are central on the layer representing medical help ties, shedding new light on the key triggers of the diffusion of microfinance.

Abstract: We propose a new analytical method to study stochastic, binary-state models on complex networks. Moving beyond the usual mean-field theories, this alternative approach is based on the introduction of an annealed approximation for uncorrelated networks, allowing to deal with the network structure as parametric heterogeneity. As an illustration, we study the noisy voter model, a modification of the original voter model including random changes of state. The proposed method is able to unfold the dependence of the model not only on the mean degree (the mean-field prediction) but also on more complex averages over the degree distribution. In particular, we find that the degree heterogeneity—variance of the underlying degree distribution—has a strong influence on the location of the critical point of a noise-induced, finite-size transition occurring in the model, on the local ordering of the system, and on the functional form of its temporal correlations. Finally, we show how this latter point opens the possibility of inferring the degree heterogeneity of the underlying network by observing only the aggregate behavior of the system as a whole, an issue of interest for systems where only macroscopic, population level variables can be measured.

Abstract: In this talk I discuss the definition(s) of complex systems from a mathematical perspective. The basic question since complex systems theory was established is about what properties we should expect, and then develop analytical tools specific for this set of systems. In phrases like ‘the sum (system) is more than its parts’ one implicitly assumes a system can be arranged into system components, and their interactions. On this level, a system can very well be described inside network theory, with components projected into vertices, and interactions illustrated by arrows between vertices. Typically, the interactions then should be highly nonlinear, in order to guarantee potential ‘surprising’ behaviour. This idea leads to the discussion of feedback loops, and how they can be analysed mathematically. Such existence or absence of feedback loops is essential in modern scientific theory, as an example we mention climate models. However, such considerations do not take into account the multi-scale structure of most systems. Mathematically, the description on the micro-scale is often stochastic in nature, whereas the macroscopic scales are often described deterministically, for example with the help of partial differential equations. In order to understand these relationship two operations need to be investigated, first discretisation (and resulting transition from deterministic to stochastic, and vice versa), and secondly up-scaling, typically done in the form of a continuum limit. This discussion is also relevant for data science, as data from different scales of the system are increasingly available. The data question will end the talk.

Abstract: Since the turn of the millennium, there is a shift towards big data. This makes the plotting and testing of empirical distributions more important. However, many continue to rely on visual cues (eg. classifying a seemingly straight line on a log-log plot as a power law). In 2009, Clauset and Newman published a series of statistical methods to test empirical data sets and classify them into various distributions. The paper have been cited over a thousand times in journals across various disciplines. However, in many of these papers, the exponential distribution always scores poorly. To understand why there is this apparent systematic bias against the exponential distribution, we sample data points from such a distribution before adding additive and multiplicative noise of different amplitudes. We then perform p-testing on these noisy exponentially-distributed data, to see how the confidence p decreases with increasing noise amplitude. We do the same for the power-law distribution. Based on these results, we discuss how to perform p-tests in an unbiased fashion across different distributions.

Abstract: We address the issue of the reducibility of the dynamics on a multilayer network to an equivalent process on an aggregated single-layer network. As a typical example of models for opinion formation in social networks, we implement the voter model on a two-layer multiplex network, and we study its dynamics as a function of two control parameters, namely the fraction of edges simultaneously existing in both layers of the network (edge overlap), and the fraction of nodes participating in both layers (interlayer connectivity or degree of multiplexity). We compute the asymptotic value of the number of active links (interface density) in the thermodynamic limit, and the time to reach an absorbing state for finite systems, and we compare the numerical results with the analytical predictions on equivalent single-layer networks obtained through various possible aggregation procedures. We find a large region of parameters where the interface density of large multiplexes gives systematic deviations from that of the aggregates. We show that neither of the standard unweighted aggregation procedures is able to capture the highly nonlinear increase in the lifetime of a finite size multiplex at small interlayer connectivity. These results indicate that multiplexity should be appropriately taken into account when studying voter model dynamics, and that, in general, single-layer approximations might be not accurate enough to properly understand processes occurring on multiplex networks, since they might flatten out relevant dynamical details.

Abstract: Network science makes a major contribution to understanding complexity but has focused on relations between two entities and hardly embraced the generality of n-ary relations for any value of n. A set of elements is n-ary related if, on removing one of them, the relation ceases to hold, e.g., the notes {C, E, G} forming the chord of C major (3-ary relation), four people playing bridge (4-ary relation), and the characters of the word {w,h,o,l,e} (5-ary relation). N-ary relations are ubiquitous in complex systems. Hypergraphs, with edges sets of vertices, provide a powerful first step. Generally vertices need to be ordered to avoid, e.g., {g, r, o, w, n} = {w, r, o, n, g}. An ordered set of vertices is a ‘simplex’, e.g. != . Simplices take network edges to new dimensions, is a 1-dimensional edge, is a 2-dimensional triangle, is 3-dimensional tetrahedron, and … with (p+1) vertices is a p-dimensional polyhedron. Polyhedra are q-connected through their q-dimensional faces, leading to higher dimensional q-connectivity. Generally the same set of vertices can support many relations, e.g. the right facing smiling face :-) and the left facing frowning face )-: are both assembled from the same three symbols. The ‘explicit relation’ notation of multiplex hypersimplices, != , allows different structures with same vertices to be discriminated. It will be shown that multilevel multiplex hypernetworks, formed from hypersimplices, are part of a natural family integrating graphs, networks and multiplex networks with their higher dimensional extensions to hypergraphs, simplicial complexes and multiplex hypernetworks. Illustrative analyses will be presented. Embracing higher dimensions can make network science even more powerful.

Abstract: Complex networks are special mathematical objects lacking an appropriate definition of distance between their units. In fact, the shortest path between two nodes is not metric because it does not satisfy the triangle inequality. One possible alternative is to introduce a metric distance based on dynamical process, more specifically random walks. This "diffusion distance" between two nodes, already used in recent machine learning and image processing applications, quantify how easily a walker can diffuse between them and it is a true metric. We show that the diffusion distance allows to identify a metric tensor g whose properties depends on the eigenvalue spectrum of the Laplacian matrix of the underlying network G. We introduce a Riemannian manifold M endowed with such a metric tensor and the Levi-Civita connection as the natural affine connection. By requiring that distances on G are preserved on M, this choice allows us to embed the network into a metric space that we call "diffusion manifold". The embedding of a complex network into a metric manifold provides several advantages for the analysis. For instance, it is possible to exploit the mathematical properties of the manifold to better understand the diffusion landscape of the network: we discuss the cases of several real-world networks, from physical ones (such as transportation networks, which are naturally embedded in an Euclidean space) to non-physical ones (such as online social networks). More intriguingly, we show that the diffusion manifold allows to naturally define a dynamical renormalization of the underlying complex network that can be exploited to better understand its structure and its critical properties.

Abstract: The graph wave equation arises naturally from conservation laws on a network; there, the usual continuum Laplacian is replaced by the graph Laplacian. We consider such a wave equation with a cubic defocusing non-linearity on a general network. The model is well-posed. It is close to the φ4 model in condensed matter physics. Using the normal modes of the graph Laplacian as a basis, we derive the amplitude equations and define resonance conditions that relate the graph structure to the dynamics. For cycles and chains, the spectrum of the Laplacian is known; for these we analyze the amplitude equations in detail. The results are validated by comparison to the numerical solutions. This study could help understand other dynamical systems on networks from biology, physics or engineering.

Abstract: Signs of hierarchy are prevalent in a wide range of systems in nature and society. One of the key problems is quantifying the importance of hierarchical organisation in the structure of the network representing the interactions or connections between the fundamental units of the studied system. Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical. Here we propose a hierarchy measure based on random walks on the network. The novelty of our approach is that directed trees corresponding to multi level pyramidal structures obtain higher hierarchy scores compared to directed chains and directed stars. Furthermore, in the thermodynamic limit the hierarchy measure of regular trees is converging to a well defined limit depending only on the branching number. When applied to real networks, our method is computationally very effective, as the result can be evaluated with arbitrary precision by subsequent multiplications of the transition matrix describing the random walk process. In addition, the tests on real world networks provided very intuitive results, e.g., the trophic levels obtained from our approach on a food web were highly consistent with former results from ecology.

Abstract: Several real-world spreading and diffusion processes take place on time-varying networks. In general, the process dynamics are affected by changes of the topological structure. To understand the impact of temporal connectivity patterns, several studies have focused on the analysis of random walks, as they represent a paradigm for all diffusion processes. It was shown that the temporal connectivity patterns introduce features not found in the equivalent processes in quenched or annealed networks. The case of random walks on activity driven networks, where each node engages in interaction according to a fixed activity rate, is particularly transparent since its stationary state was derived analytically. However, this model accounts only for the heterogeneous activity of nodes, irrespectively of their potential to attract other connections. Here, we focus on the role played by the distribution of nodes with respect to their attractiveness, clarifying its impact on the diffusion of random walk processes and on the transport dynamics. We introduce a time-varying network model where each node is characterized by a fixed activity rate, giving the probability of engaging in interaction per time unit, and a fixed attractiveness rate, specifying the probability to receive a connection from an active node. We derive analytically the stationary state of the process and study the interplay between the distributions of nodes’ activity and attractiveness. Interestingly, the introduction of heterogeneous attractiveness distributions alters substantially the dynamical properties of the process and creates a rich phenomenology. Models based on heterogeneous attractiveness were introduced to explain features of face-to-face interaction networks, where the attractiveness may account for individuals’ different degree of social appeal. By clarifying the impact of this characteristic network feature on the diffusion of random walks, our findings can potentially shed light on real-world dynamical processes such as opinion and epidemics spreading on social networks.

Abstract: A link stream L models interactions over time: a link (b, e, u, v) is in L if two entities (nodes) u and v have continuously interacted from time b to time e. Link streams, also called temporal networks or time-varying graphs, model many real-world interactions between individuals, email exchanges, or network traffic. Real-world link streams typically contain millions, or even billions of interactions, and handling them computationally is a challenge in itself. Hence, module decomposition based aggregation is a possible approach to construct a smaller and more observable image of a link stream. In our study, we generalize the definition of modules to link streams. In a graph, a module M is a subset of nodes that have the same neighborhood. Analogously, we define a module in a link stream as a couple (M, T) consisting of a set of nodes M and a time duration T (not necessarily an interval) such that all nodes in M share the same neighborhood over T. With the goal of achieving more meaningful (yet lossy) module decomposition of real world datasets, we reduce the stringency of module definition to introduce “relaxed” modules. Intuitively, a relaxed module (M, T) is such that the neighborhoods’ similarities of nodes in M over T are higher than a given threshold. We define a relevant measure of spatiotemporal neighborhood similarity. Beyond the theoretical scope of our work, we use modules in link streams as a mean of compression of large real-world datasets. In addition to reducing the amount of computational resources to process the stream, compression has applications in event and mesoscale structure detection, at the cost of optimizable loss in information. Hence, we finally investigate the reconstruction error induced by link stream compression, and discuss possible ways to minimize this error.

Abstract: To speed up information propagation or mitigate disease spreading calls for an investigation of the interplay between dynamical processes and temporal network. Temporal (e.g. burstiness) and topological (e.g. community structure) network characteristics have been indeed shown to influence such processes. Following these findings, techniques to influence the dynamics of a process have been built on using either temporal or structural informations, but designing such intervention strategy on their coupling is still challenging. Here we propose a generative model of temporal networks where we can control separately temporal and topological structures and combine them. A temporal network is built from subnetworks whose links have a correlated activity. The underlying assumption is that individuals are usually engaged in different social activities (e.g. work, spare time) involving different people at different times (e.g. week, week-end). Each subnetwork can be seen as representing one of these social activities. Moreover, the hypothesis of such modeling is supported by previous works using tensor factorization, that both shows the existence of correlated activity patterns in networks and enables to extract them. Using this model, we generated synthetic networks resulting from several subnetworks. Each subnetwork is built from a static network whose activity is modulated in time. We considered subnetworks with various clustering coefficients and activity profiles. We simulated a susceptible-infected process over each network and over the networks generated by the removal of one subnetwork at a time to measure the impact of each of them on the dynamic process. It led to the identification of subnetworks - with their respective temporal and topological structure - having an impact on the process. The clustering coefficient value was found to be the most decisive factor to predict the impact on the spreading process. The model architecture allows to extend the study to richer temporal patterns.