Communicating sentiment and outlook reverses inaction against collective risks.

Collective risks permeate society, triggering social dilemmas in which working toward a common goal is impeded by selfish interests. One such dilemma is mitigating runaway climate change. To study the social aspects of climate-change mitigation, we organized an experimental game and asked volunteer groups of three different sizes to invest toward a common mitigation goal. If investments reached a preset target, volunteers would avoid all consequences and convert their remaining capital into monetary payouts. In the opposite case, however, volunteers would lose all their capital with 50% probability. The dilemma was, therefore, whether to invest one's own capital or wait for others to step in. We find that communicating sentiment and outlook helps to resolve the dilemma by a fundamental shift in investment patterns. Groups in which communication is allowed invest persistently and hardly ever give up, even when their current investment deficits are substantial. The improved investment patterns are robust to group size, although larger groups are harder to coordinate, as evidenced by their overall lower success frequencies. A clustering algorithm reveals three behavioral types and shows that communication reduces the abundance of the free-riding type. Climate-change mitigation, however, is achieved mainly by cooperator and altruist types stepping up and increasing contributions as the failure looms. Meanwhile, contributions from free riders remain flat throughout the game. This reveals that the mechanisms behind avoiding collective risks depend on an interaction between behavioral type, communication, and timing.

Distances in Higher-Order Networks and the Metric Structure of Hypergraphs.

We explore the metric structure of networks with higher-order interactions and introduce a novel definition of distance for hypergraphs that extends the classic methods reported in the literature. The new metric incorporates two critical factors: (1) the inter-node distance within each hyperedge, and (2) the distance between hyperedges in the network. As such, it involves the computation of distances in a weighted line graph of the hypergraph. The approach is illustrated with several ad hoc synthetic hypergraphs, where the structural information unveiled by the novel metric is highlighted. Moreover, the method's performance and effectiveness are shown through computations on large real-world hypergraphs, which indeed reveal new insights into the structural features of networks beyond pairwise interactions. Namely, using the new distance measure, we generalize the definitions of efficiency, closeness and betweenness centrality for the case of hypergraphs. Comparing the values of these generalized measures with their analogs calculated for the hypergraph clique projections, we show that our measures provide significantly different assessments on the characteristics (and roles) of the nodes from the information-transferability point of view. The difference is brighter for hypergraphs in which hyperedges of large sizes are frequent, and nodes relating to these hyperedges are rarely connected by other hyperedges of smaller sizes.

Universal behavior of cascading failures in interdependent networks.

Catastrophic and major disasters in real-world systems, such as blackouts in power grids or global failures in critical infrastructures, are often triggered by minor events which originate a cascading failure in interdependent graphs. We present here a self-consistent theory enabling the systematic analysis of cascading failures in such networks and encompassing a broad range of dynamical systems, from epidemic spreading, to birth-death processes, to biochemical and regulatory dynamics. We offer testable predictions on breakdown scenarios, and, in particular, we unveil the conditions under which the percolation transition is of the first-order or the second-order type, as well as prove that accounting for dynamics in the nodes always accelerates the cascading process. Besides applying directly to relevant real-world situations, our results give practical hints on how to engineer more robust networked systems.

Punishment diminishes the benefits of network reciprocity in social dilemma experiments.

Network reciprocity has been widely advertised in theoretical studies as one of the basic cooperation-promoting mechanisms, but experimental evidence favoring this type of reciprocity was published only recently. When organized in an unchanging network of social contacts, human subjects cooperate provided the following strict condition is satisfied: The benefit of cooperation must outweigh the total cost of cooperating with all neighbors. In an attempt to relax this condition, we perform social dilemma experiments wherein network reciprocity is aided with another theoretically hypothesized cooperation-promoting mechanism-costly punishment. The results reveal how networks promote and stabilize cooperation. This stabilizing effect is stronger in a smaller-size neighborhood, as expected from theory and experiments. Contrary to expectations, punishment diminishes the benefits of network reciprocity by lowering assortment, payoff per round, and award for cooperative behavior. This diminishing effect is stronger in a larger-size neighborhood. An immediate implication is that the psychological effects of enduring punishment override the rational response anticipated in quantitative models of cooperation in networks.

Self-organized interdependence among populations promotes cooperation by means of coevolution.

We show that self-organized interdependence promotes the evolution of cooperation in interdependent networks. The evolution of connections between networks occurs according to the following rule: if a player often wins against its opponent (regardless of its strategy), it is allowed to form an external link with the corresponding partner in another network to obtain additional benefit; otherwise, the opportunity to increase its benefit is lost. Through numerical simulation, it is unveiled that cooperation can be significantly promoted due to interdependent network reciprocity. Interestingly, the synchronization of evolutionary processes emerges on both networks, and individuals can take advantage of interdependent network reciprocity when both the strategies and the coevolving times in the two networks are synchronous.

Interplay of delay and multiplexing: Impact on cluster synchronization.

Communication delays and multiplexing are ubiquitous features of real-world network systems. We here introduce a simple model where these two features are simultaneously present and report the rich phenomenology which is actually due to their interplay on cluster synchronization. A delay in one layer has non trivial impacts on the collective dynamics of the other layers, enhancing or suppressing synchronization. At the same time, multiplexing may also enhance cluster synchronization of delayed layers. We elucidate several nontrivial (and anti-intuitive) scenarios, which are of interest and potential application in various real-world systems, where the introduction of a delay may render synchronization of a layer robust against changes in the properties of the other layers.

Introduction to Focus Issue: Complex Dynamics in Networks, Multilayered Structures and Systems.

In the last years, network scientists have directed their interest to the multi-layer character of real-world systems, and explicitly considered the structural and dynamical organization of graphs made of diverse layers between its constituents. Most complex systems include multiple subsystems and layers of connectivity and, in many cases, the interdependent components of systems interact through many different channels. Such a new perspective is indeed found to be the adequate representation for a wealth of features exhibited by networked systems in the real world. The contributions presented in this Focus Issue cover, from different points of view, the many achievements and still open questions in the field of multi-layer networks, such as: new frameworks and structures to represent and analyze heterogeneous complex systems, different aspects related to synchronization and centrality of complex networks, interplay between layers, and applications to logistic, biological, social, and technological fields.

Explosive synchronization coexists with classical synchronization in the Kuramoto model.

Explosive synchronization has recently been reported in a system of adaptively coupled Kuramoto oscillators, without any conditions on the frequency or degree of the nodes. Here, we find that, in fact, the explosive phase coexists with the standard phase of the Kuramoto oscillators. We determine this by extending the mean-field theory of adaptively coupled oscillators with full coupling to the case with partial coupling of a fraction f. This analysis shows that a metastable region exists for all finite values of f > 0, and therefore explosive synchronization is expected for any perturbation of adaptively coupling added to the standard Kuramoto model. We verify this theory with GPU-accelerated simulations on very large networks (N ∼ 10(6)) and find that, in fact, an explosive transition with hysteresis is observed for all finite couplings. By demonstrating that explosive transitions coexist with standard transitions in the limit of f → 0, we show that this behavior is far more likely to occur naturally than was previously believed.

Graph-based unsupervised segmentation algorithm for cultured neuronal networks' structure characterization and modeling.

Large scale phase-contrast images taken at high resolution through the life of a cultured neuronal network are analyzed by a graph-based unsupervised segmentation algorithm with a very low computational cost, scaling linearly with the image size. The processing automatically retrieves the whole network structure, an object whose mathematical representation is a matrix in which nodes are identified neurons or neurons' clusters, and links are the reconstructed connections between them. The algorithm is also able to extract any other relevant morphological information characterizing neurons and neurites. More importantly, and at variance with other segmentation methods that require fluorescence imaging from immunocytochemistry techniques, our non invasive measures entitle us to perform a longitudinal analysis during the maturation of a single culture. Such an analysis furnishes the way of individuating the main physical processes underlying the self-organization of the neurons' ensemble into a complex network, and drives the formulation of a phenomenological model yet able to describe qualitatively the overall scenario observed during the culture growth.

Explosive synchronization in adaptive and multilayer networks.

At this time, explosive synchronization (ES) of networked oscillators is thought of as being rooted in the setting of specific microscopic correlation features between the natural frequencies of the oscillators and their effective coupling strengths. We show that ES is, in fact, far more general and can occur in adaptive and multilayer networks in the absence of such correlation properties. We first report evidence of ES for single-layer networks where a fraction f of the nodes have links adaptively controlled by a local order parameter, and we then extend the study to a variety of two-layer networks with a fraction f of their nodes coupled with each other by means of dependency links. In the latter case, we give evidence of ES regardless of the differences in the frequency distribution, in the topology of connections between the layers, or both. Finally, we provide a rigorous, analytical treatment to properly ground all of the observed scenarios and to advance the understanding of the actual mechanisms at the basis of ES in real-world systems.

The transition to synchronization of networked systems.

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network's nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems.

Eigenvector centrality of nodes in multiplex networks.

We extend the concept of eigenvector centrality to multiplex networks, and introduce several alternative parameters that quantify the importance of nodes in a multi-layered networked system, including the definition of vectorial-type centralities. In addition, we rigorously show that, under reasonable conditions, such centrality measures exist and are unique. Computer experiments and simulations demonstrate that the proposed measures provide substantially different results when applied to the same multiplex structure, and highlight the non-trivial relationships between the different measures of centrality introduced.

Graphical notation reveals topological stability criteria for collective dynamics in complex networks.

We propose a graphical notation by which certain spectral properties of complex systems can be rewritten concisely and interpreted topologically. Applying this notation to analyze the stability of a class of networks of coupled dynamical units, we reveal stability criteria on all scales. In particular, we show that in systems such as the Kuramoto model the Coates graph of the Jacobian matrix must contain a spanning tree of positive elements for the system to be locally stable.

Topological synchronization of chaotic systems.

A chaotic dynamics is typically characterized by the emergence of strange attractors with their fractal or multifractal structure. On the other hand, chaotic synchronization is a unique emergent self-organization phenomenon in nature. Classically, synchronization was characterized in terms of macroscopic parameters, such as the spectrum of Lyapunov exponents. Recently, however, we attempted a microscopic description of synchronization, called topological synchronization, and showed that chaotic synchronization is, in fact, a continuous process that starts in low-density areas of the attractor. Here we analyze the relation between the two emergent phenomena by shifting the descriptive level of topological synchronization to account for the multifractal nature of the visited attractors. Namely, we measure the generalized dimension of the system and monitor how it changes while increasing the coupling strength. We show that during the gradual process of topological adjustment in phase space, the multifractal structures of each strange attractor of the two coupled oscillators continuously converge, taking a similar form, until complete topological synchronization ensues. According to our results, chaotic synchronization has a specific trait in various systems, from continuous systems and discrete maps to high dimensional systems: synchronization initiates from the sparse areas of the attractor, and it creates what we termed as the 'zipper effect', a distinctive pattern in the multifractal structure of the system that reveals the microscopic buildup of the synchronization process. Topological synchronization offers, therefore, a more detailed microscopic description of chaotic synchronization and reveals new information about the process even in cases of high mismatch parameters.

Epidemic spreading under mutually independent intra- and inter-host pathogen evolution.

The dynamics of epidemic spreading is often reduced to the single control parameter R0 (reproduction-rate), whose value, above or below unity, determines the state of the contagion. If, however, the pathogen evolves as it spreads, R0 may change over time, potentially leading to a mutation-driven spread, in which an initially sub-pandemic pathogen undergoes a breakthrough mutation. To predict the boundaries of this pandemic phase, we introduce here a modeling framework to couple the inter-host network spreading patterns with the intra-host evolutionary dynamics. We find that even in the extreme case when these two process are driven by mutually independent selection forces, mutations can still fundamentally alter the pandemic phase-diagram. The pandemic transitions, we show, are now shaped, not just by R0, but also by the balance between the epidemic and the evolutionary timescales. If mutations are too slow, the pathogen prevalence decays prior to the appearance of a critical mutation. On the other hand, if mutations are too rapid, the pathogen evolution becomes volatile and, once again, it fails to spread. Between these two extremes, however, we identify a broad range of conditions in which an initially sub-pandemic pathogen can breakthrough to gain widespread prevalence.

Principles of recovery from traumatic brain injury: reorganization of functional networks.

Recovery after brain injury is an excellent platform to study the mechanism underlying brain plasticity, the reorganization of networks. Do complex network measures capture the physiological and cognitive alterations that occurred after a traumatic brain injury and its recovery? Patients as well as control subjects underwent resting-state MEG recording following injury and after neurorehabilitation. Next, network measures such as network strength, path length, efficiency, clustering and energetic cost were calculated. We show that these parameters restore, in many cases, to control ones after recovery, specifically in delta and alpha bands, and we design a model that gives some hints about how the functional networks modify their weights in the recovery process. Positive correlations between complex network measures and some of the general index of the WAIS-III test were found: changes in delta-based path-length and those in Performance IQ score, and alpha-based normalized global efficiency and Perceptual Organization Index. These results indicate that: 1) the principle of recovery depends on the spectral band, 2) the structure of the functional networks evolves in parallel to brain recovery with correlations with neuropsychological scales, and 3) energetic cost reveals an optimal principle of recovery.

Analyses of antigen dependency networks unveil immune system reorganization between birth and adulthood.

Much effort has been devoted to assess the importance of nodes in complex biological networks (such as gene transcriptional regulatory networks, protein interaction networks, and neural networks). Examples of commonly used measures of node importance include node degree, node centrality, and node vulnerability score (the effect of the node deletion on the network efficiency). Here, we present a new approach to compute and investigate the mutual dependencies between network nodes from the matrices of node-node correlations. To this end, we first define the dependency of node i on node j (or the influence of node j on node i), D(i, j) as the average over all nodes k of the difference between the i - k correlation and the partial correlations between these nodes with respect to node j. Note that the dependencies, D(i, j) define a directed weighted matrix, since, in general, D(i, j) differs from D( j, i). For this reason, many of the commonly used measures of node importance, such as node centrality, cannot be used. Hence, to assess the node importance of the dependency networks, we define the system level influence (SLI) of antigen j, SLI( j) as the sum of the influence of j on all other antigens i. Next, we define the system level influence or the influence score of antigen j, SLI( j) as the sum of D(i, j) over all nodes i. We introduce the new approach and demonstrate that it can unveil important biological information in the context of the immune system. More specifically, we investigated antigen dependency networks computed from antigen microarray data of autoantibody reactivity of IgM and IgG isotypes present in the sera of ten mothers and their newborns. We found that the analysis was able to unveil that there is only a subset of antigens that have high influence scores (SLI) common both to the mothers and newborns. Networks comparison in terms of modularity (using the Newman's algorithm) and of topology (measured by the divergence rate) revealed that, at birth, the IgG networks exhibit a more profound global reorganization while the IgM networks exhibit a more profound local reorganization. During immune system development, the modularity of the IgG network increases and becomes comparable to that of the IgM networks at adulthood. We also found the existence of several conserved IgG and IgM network motifs between the maternal and newborns networks, which might retain network information as our immune system develops. If correct, these findings provide a convincing demonstration of the effectiveness of the new approach to unveil most significant biological information. Whereas we have introduced the new approach within the context of the immune system, it is expected to be effective in the studies of other complex biological social, financial, and manmade networks.

Steering complex networks toward desired dynamics.

We consider networks of dynamical units that evolve in time according to different laws, and are coupled to each other in highly irregular ways. Studying how to steer the dynamics of such systems towards a desired evolution is of great practical interest in many areas of science, as well as providing insight into the interplay between network structure and dynamical behavior. We propose a pinning protocol for imposing specific dynamic evolutions compatible with the equations of motion on a networked system. The method does not impose any restrictions on the local dynamics, which may vary from node to node, nor on the interactions between nodes, which may adopt in principle any nonlinear mathematical form and be represented by weighted, directed or undirected links. We first explore our method on small synthetic networks of chaotic oscillators, which allows us to unveil a correlation between the ordered sequence of pinned nodes and their topological influence in the network. We then consider a 12-species trophic web network, which is a model of a mammalian food web. By pinning a relatively small number of species, one can make the system abandon its spontaneous evolution from its (typically uncontrolled) initial state towards a target dynamics, or periodically control it so as to make the populations evolve within stipulated bounds. The relevance of these findings for environment management and conservation is discussed.