Nucleation phenomena and extreme vulnerability of spatial k-core systems.

K-core percolation is a fundamental dynamical process in complex networks with applications that span numerous real-world systems. Earlier studies focus primarily on random networks without spatial constraints and reveal intriguing mixed-order transitions. However, real-world systems, ranging from transportation and communication networks to complex brain networks, are not random but are spatially embedded. Here, we study k-core percolation on two-dimensional spatially embedded networks and show that, in contrast to regular percolation, the length of connections can control the transition type, leading to four different types of phase transitions associated with interesting phenomena and a rich phase diagram. A key finding is the existence of a metastable phase where microscopic localized damage, independent of system size, can cause a macroscopic phase transition, a result which cannot be achieved in traditional percolation. In this case, local failures spontaneously propagate the damage radially until the system collapses, a phenomenon analogous to the nucleation process.

A blueprint for a new commercial driving epidemiology: An emerging paradigm grounded in integrative exposome and network epistemologies.

Excess health and safety risks of commercial drivers are largely determined by, embedded in, or operate as complex, dynamic, and randomly determined systems with interacting parts. Yet, prevailing epidemiology is entrenched in narrow, deterministic, and static exposure-response frameworks along with ensuing inadequate data and limiting methods, thereby perpetuating an incomplete understanding of commercial drivers' health and safety risks. This paper is grounded in our ongoing research that conceptualizes health and safety challenges of working people as multilayered "wholes" of interacting work and nonwork factors, exemplified by complex-systems epistemologies. Building upon and expanding these assumptions, herein we: (a) discuss how insights from integrative exposome and network-science-based frameworks can enhance our understanding of commercial drivers' chronic disease and injury burden; (b) introduce the "working life exposome of commercial driving" (WLE-CD)-an array of multifactorial and interdependent work and nonwork exposures and associated biological responses that concurrently or sequentially impact commercial drivers' health and safety during and beyond their work tenure; (c) conceptualize commercial drivers' health and safety risks as multilayered networks centered on the WLE-CD and network relational patterns and topological properties-that is, arrangement, connections, and relationships among network components-that largely govern risk dynamics; and (d) elucidate how integrative exposome and network-science-based innovations can contribute to a more comprehensive understanding of commercial drivers' chronic disease and injury risk dynamics. Development, validation, and proliferation of this emerging discourse can move commercial driving epidemiology to the frontier of science with implications for policy, action, other working populations, and population health at large.

The indispensable whole of work and population health: How the working life exposome can advance empirical research, policy, and action.

OBJECTIVES: The thesis of this paper is that health and safety challenges of working people can only be fully understood by examining them as wholes with interacting parts. This paper unravels this indispensable whole by introducing the working life exposome and elucidating how associated epistemologies and methodologies can enhance empirical research. METHODS: Network and population health scientists have initiated an ongoing discourse on the state of empirical work-health-safety-well-being research. RESULTS: Empirical research has not fully captured the totality and complexity of multiple and interacting work and nonwork factors defining the health of working people over their life course. We challenge the prevailing paradigm by proposing to expand it from narrow work-related exposures and associated monocausal frameworks to the holistic study of work and population health grounded in complexity and exposome sciences. Health challenges of working people are determined by, embedded in, and/or operate as complex systems comprised of multilayered and interdependent components. One can identify many potentially causal factors as sufficient and component causes where removal of one or more of these can impact disease progression. We, therefore, cannot effectively study them by an a priori determination of a set of components and/or properties to be examined separately and then recombine partial approaches, attempting to form a picture of the whole. Instead, we must examine these challenges as wholes from the start, with an emphasis on interactions among their multifactorial components and their emergent properties. Despite various challenges, working-life-exposome-grounded frameworks and associated innovations have the potential to accomplish that. CONCLUSIONS: This emerging paradigm shift can move empirical work-health-safety-well-being research to cutting-edge science and enable more impactful policies and actions.

Effect of cities and distance on COVID-19 spreading in the United States.

The COVID-19 pandemic has evolved over time through multiple spatial and temporal dynamics. The varying extent of interactions among different geographical areas can result in a complex pattern of spreading so that influences between these areas can be hard to discern. Here, we use cross-correlation analysis to detect synchronous evolution and potential interinfluences in the time evolution of new COVID-19 cases at the county level in the United States. Our analysis identified two main time periods with distinguishable features in the behavior of correlations. In the first phase, there were few strong correlations that only emerged between urban areas. In the second phase of the epidemic, strong correlations became widespread and there was a clear directionality of influence from urban-to-rural areas. In general, the effect of distance between two counties was much weaker than that of the counties' population. Such analysis can provide possible clues on the evolution of the disease and may identify parts of the country where intervention may be more efficient in limiting the disease spread.

Spatial correlations in geographical spreading of COVID-19 in the United States.

The global spread of the COVID-19 pandemic has followed complex pathways, largely attributed to the high virus infectivity, human travel patterns, and the implementation of multiple mitigation measures. The resulting geographic patterns describe the evolution of the epidemic and can indicate areas that are at risk of an outbreak. Here, we analyze the spatial correlations of new active cases in the USA at the county level and characterize the extent of these correlations at different times. We show that the epidemic did not progress uniformly and we identify various stages which are distinguished by significant differences in the correlation length. Our results indicate that the correlation length may be large even during periods when the number of cases declines. We find that correlations between urban centers were much more significant than between rural areas and this finding indicates that long-range spreading was mainly facilitated by travel between cities, especially at the first months of the epidemic. We also show the existence of a percolation transition in November 2020, when the largest part of the country was connected to a spanning cluster, and a smaller-scale transition in January 2021, with both times corresponding to the peak of the epidemic in the country.

A generic arboviral model framework for exploring trade-offs between vector control and environmental concerns.

Effective public health measures must balance potentially conflicting demands from populations they serve. In the case of infectious disease risks from mosquito-borne infections, such as Zika virus, public concern about the pathogen may be counterbalanced by public concern about environmental contamination from chemical agents used for vector control. Here we introduce a generic framework for modeling how the spread of an infectious pathogen might lead to varying public perceptions, and therefore tolerance, of both disease risk and pesticide use. We consider how these dynamics might impact the spread of a vector-borne disease. We tailor and parameterize our model for direct application to Zika virus as spread by Aedes aegypti mosquitoes, though the framework itself has broad applicability to any arboviral infection. We demonstrate how public risk perception of both disease and pesticides may drastically impact the spread of a mosquito-borne disease in a susceptible population. We conclude that models hoping to inform public health decision making about how best to mitigate arboviral disease risks should explicitly consider the potential public demand for, or rejection of, chemical control of mosquito populations.

Propinquity drives the emergence of network structure and density.

The lack of large-scale, continuously evolving empirical data usually limits the study of networks to the analysis of snapshots in time. This approach has been used for verification of network evolution mechanisms, such as preferential attachment. However, these studies are mostly restricted to the analysis of the first links established by a new node in the network and typically ignore connections made after each node's initial introduction. Here, we show that the subsequent actions of individuals, such as their second network link, are not random and can be decoupled from the mechanism behind the first network link. We show that this feature has strong influence on the network topology. Moreover, snapshots in time can now provide information on the mechanism used to establish the second connection. We interpret these empirical results by introducing the "propinquity model," in which we control and vary the distance of the second link established by a new node and find that this can lead to networks with tunable density scaling, as found in real networks. Our work shows that sociologically meaningful mechanisms are influencing network evolution and provides indications of the importance of measuring the distance between successive connections.

Simple and efficient self-healing strategy for damaged complex networks.

The process of destroying a complex network through node removal has been the subject of extensive interest and research. Node loss typically leaves the network disintegrated into many small and isolated clusters. Here we show that these clusters typically remain close to each other and we suggest a simple algorithm that is able to reverse the inflicted damage by restoring the network's functionality. After damage, each node decides independently whether to create a new link depending on the fraction of neighbors it has lost. In addition to relying only on local information, where nodes do not need knowledge of the global network status, we impose the additional constraint that new links should be as short as possible (i.e., that the new edge completes a shortest possible new cycle). We demonstrate that this self-healing method operates very efficiently, both in model and real networks. For example, after removing the most connected airports in the USA, the self-healing algorithm rejoined almost 90% of the surviving airports.

The Effect of Disease-Induced Mortality on Structural Network Properties.

As the understanding of the importance of social contact networks in the spread of infectious diseases has increased, so has the interest in understanding the feedback process of the disease altering the social network. While many studies have explored the influence of individual epidemiological parameters and/or underlying network topologies on the resulting disease dynamics, we here provide a systematic overview of the interactions between these two influences on population-level disease outcomes. We show that the sensitivity of the population-level disease outcomes to the combination of epidemiological parameters that describe the disease are critically dependent on the topological structure of the population's contact network. We introduce a new metric for assessing disease-driven structural damage to a network as a population-level outcome. Lastly, we discuss how the expected individual-level disease burden is influenced by the complete suite of epidemiological characteristics for the circulating disease and the ongoing process of network compromise. Our results have broad implications for prediction and mitigation of outbreaks in both natural and human populations.

IMDB network revisited: unveiling fractal and modular properties from a typical small-world network.

We study a subset of the movie collaboration network, http://www.imdb.com, where only adult movies are included. We show that there are many benefits in using such a network, which can serve as a prototype for studying social interactions. We find that the strength of links, i.e., how many times two actors have collaborated with each other, is an important factor that can significantly influence the network topology. We see that when we link all actors in the same movie with each other, the network becomes small-world, lacking a proper modular structure. On the other hand, by imposing a threshold on the minimum number of links two actors should have to be in our studied subset, the network topology becomes naturally fractal. This occurs due to a large number of meaningless links, namely, links connecting actors that did not actually interact. We focus our analysis on the fractal and modular properties of this resulting network, and show that the renormalization group analysis can characterize the self-similar structure of these networks.

Collective behavior in the spatial spreading of obesity.

Obesity prevalence is increasing in many countries at alarming levels. A difficulty in the conception of policies to reverse these trends is the identification of the drivers behind the obesity epidemics. Here, we implement a spatial spreading analysis to investigate whether obesity shows spatial correlations, revealing the effect of collective and global factors acting above individual choices. We find a regularity in the spatial fluctuations of their prevalence revealed by a pattern of scale-free long-range correlations. The fluctuations are anomalous, deviating in a fundamental way from the weaker correlations found in the underlying population distribution indicating the presence of collective behavior, i.e., individual habits may have negligible influence in shaping the patterns of spreading. Interestingly, we find the same scale-free correlations in economic activities associated with food production. These results motivate future interventions to investigate the causality of this relation providing guidance for the implementation of preventive health policies.

The conundrum of functional brain networks: small-world efficiency or fractal modularity.

The human brain has been studied at multiple scales, from neurons, circuits, areas with well-defined anatomical and functional boundaries, to large-scale functional networks which mediate coherent cognition. In a recent work, we addressed the problem of the hierarchical organization in the brain through network analysis. Our analysis identified functional brain modules of fractal structure that were inter-connected in a small-world topology. Here, we provide more details on the use of network science tools to elaborate on this behavior. We indicate the importance of using percolation theory to highlight the modular character of the functional brain network. These modules present a fractal, self-similar topology, identified through fractal network methods. When we lower the threshold of correlations to include weaker ties, the network as a whole assumes a small-world character. These weak ties are organized precisely as predicted by theory maximizing information transfer with minimal wiring costs.

A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks.

The human brain is organized in functional modules. Such an organization presents a basic conundrum: Modules ought to be sufficiently independent to guarantee functional specialization and sufficiently connected to bind multiple processors for efficient information transfer. It is commonly accepted that small-world architecture of short paths and large local clustering may solve this problem. However, there is intrinsic tension between shortcuts generating small worlds and the persistence of modularity, a global property unrelated to local clustering. Here, we present a possible solution to this puzzle. We first show that a modified percolation theory can define a set of hierarchically organized modules made of strong links in functional brain networks. These modules are "large-world" self-similar structures and, therefore, are far from being small-world. However, incorporating weaker ties to the network converts it into a small world preserving an underlying backbone of well-defined modules. Remarkably, weak ties are precisely organized as predicted by theory maximizing information transfer with minimal wiring cost. This trade-off architecture is reminiscent of the "strength of weak ties" crucial concept of social networks. Such a design suggests a natural solution to the paradox of efficient information flow in the highly modular structure of the brain.

Modularity map of the network of human cell differentiation.

Cell differentiation in multicellular organisms is a complex process whose mechanism can be understood by a reductionist approach, in which the individual processes that control the generation of different cell types are identified. Alternatively, a large-scale approach in search of different organizational features of the growth stages promises to reveal its modular global structure with the goal of discovering previously unknown relations between cell types. Here, we sort and analyze a large set of scattered data to construct the network of human cell differentiation (NHCD) based on cell types (nodes) and differentiation steps (links) from the fertilized egg to a developed human. We discover a dynamical law of critical branching that reveals a self-similar regularity in the modular organization of the network, and allows us to observe the network at different scales. The emerging picture clearly identifies clusters of cell types following a hierarchical organization, ranging from sub-modules to super-modules of specialized tissues and organs on varying scales. This discovery will allow one to treat the development of a particular cell function in the context of the complex network of human development as a whole. Our results point to an integrated large-scale view of the network of cell types systematically revealing ties between previously unrelated domains in organ functions.

Scaling of degree correlations and its influence on diffusion in scale-free networks.

Connectivity correlations play an important role in the structure of scale-free networks. While several empirical studies exist, there is no general theoretical analysis that can explain the largely varying behavior of real networks. Here, we use scaling theory to quantify the degree of correlations in the particular case of networks with a power-law degree distribution. These networks are classified in terms of their correlation properties, revealing additional information on their structure. For instance, the studied social networks and the Internet at the router level are clustered around the line of random networks, implying a strongly connected core of hubs. On the contrary, some biological networks and the WWW exhibit strong anticorrelations. The present approach can be used to study robustness or diffusion, where we find that anticorrelations tend to accelerate the diffusion process.

Scaling theory of transport in complex biological networks.

Transport is an important function in many network systems and understanding its behavior on biological, social, and technological networks is crucial for a wide range of applications. However, it is a property that is not well understood in these systems, probably because of the lack of a general theoretical framework. Here, based on the finding that renormalization can be applied to bionetworks, we develop a scaling theory of transport in self-similar networks. We demonstrate the networks invariance under length scale renormalization, and we show that the problem of transport can be characterized in terms of a set of critical exponents. The scaling theory allows us to determine the influence of the modular structure on transport in metabolic and protein-interaction networks. We also generalize our theory by presenting and verifying scaling arguments for the dependence of transport on microscopic features, such as the degree of the nodes and the distance between them. Using transport concepts such as diffusion and resistance, we exploit this invariance, and we are able to explain, based on the topology of the network, recent experimental results on the broad flow distribution in metabolic networks.

Improving immunization strategies.

We introduce an immunization method where the percentage of required vaccinations for immunity are close to the optimal value of a targeted immunization scheme of highest degree nodes. Our strategy retains the advantage of being purely local, without the need for knowledge on the global network structure or identification of the highest degree nodes. The method consists of selecting a random node and asking for a neighbor that has more links than himself or more than a given threshold and immunizing him. We compare this method to other efficient strategies on three real social networks and on a scale-free network model and find it to be significantly more effective.

Characteristics of reaction-diffusion on scale-free networks.

We examine some characteristic properties of reaction-diffusion processes of the A+A-->0 type on scale-free networks. Due to the inhomogeneity of the structure of the substrate, as compared to usual lattices, we focus on the characteristics of the nodes where the annihilations occur. We show that at early times the majority of these events take place on low-connectivity nodes, while as time advances the process moves towards the high-connectivity nodes, the so-called hubs. This pattern remarkably accelerates the annihilation of the particles, and it is in agreement with earlier predictions that the rates of reaction-diffusion processes on scale-free networks are much faster than the equivalent ones on lattice systems.

Reaction-diffusion processes on correlated and uncorrelated scale-free networks.

We compare reaction-diffusion processes of the A+A-->0 type on scale-free networks created with either the configuration model or the uncorrelated configuration model. We show via simulations that except for the difference in the behavior of the two models, different results are observed within the same model when the minimum number of connections for a node varies from k(min) =1 to k(min) =2 . This difference is attributed to the varying local properties of the two systems. In all cases, we are able to identify a power-law behavior of the density decay with time, with an exponent f>1 , considerably larger than its lattice counterpart.