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One of the most important tools available to limit the spread and impact of infectious diseases is vaccination. It is therefore important to understand what factors determine people's vaccination decisions. To this end, previous behavioural research made use of, (i) controlled but often abstract or hypothetical studies (e.g., vignettes) or, (ii) realistic but typically less flexible studies that make it difficult to understand individual decision processes (e.g., clinical trials). Combining the best of these approaches, we propose integrating real-world Bluetooth contacts via smartphones in several rounds of a game scenario, as a novel methodology to study vaccination decisions and disease spread. In our 12-week proof-of-concept study conducted with $N$ = 494 students, we found that participants strongly responded to some of the information provided to them during or after each decision round, particularly those related to their individual health outcomes. In contrast, information related to others' decisions and outcomes (e.g., the number of vaccinated or infected individuals) appeared to be less important. We discuss the potential of this novel method and point to fruitful areas for future research.

Disease propagation between countries strongly depends on their effective distance, a measure derived from the world air transportation network (WAN). It reduces the complex spreading patterns of a pandemic to a wave-like propagation from the outbreak country, i.e. a linear relationship to the arrival time of the unmitigated spread of a disease. However, in the early stage of an outbreak, what matters to countries' decision makers is knowledge about the relative risk of arrival of active cases, i.e. how likely it is that an active case that boarded at the outbreak location will arrive in their country. As accurate mechanistic models to estimate such risks are still lacking, we propose here the ``import risk'' model that defines an import probability by means of the effective-distance framework. The model assumes that airline passengers are distributed along the shortest path tree that starts at the outbreak's origin. In combination with a random walk, we account for all possible paths, thus inferring predominant connecting flights. Our model outperforms other mobility models, such as the radiation and gravity model with varying distance types, and it improves further if additional geographic information is included. The import risk model's precision increases for countries that are more connected within the WAN, and recovers a geographic distance-dependence that suggests a pull- rather than a push- dynamic of the distribution process., Comment: 11 pages, 7 figures

Vaccines are among the most powerful tools used to combat the COVID-19 pandemic. They are highly effective against infection and substantially reduce the risk of severe disease, hospitalization, ICU admission, and death. However, their potential for attenuating long-term effects of a SARS-CoV-2 infection, commonly denoted as Long COVID, remains elusive and is still subject of debate. Such long-term effects can be effectively monitored at the individual level by analyzing physiological data collected by consumer-grade wearable sensors. Here, we investigate changes in resting heart rate, daily physical activity, and sleep duration in response to a SARS-CoV-2 infection stratified by vaccination status. Data was collected over a period of two years in the context of the German Corona Data Donation Project with currently around 190,000 monthly active donors. Compared to their unvaccinated counterparts, we find that vaccinated individuals on average experience smaller changes in their vital data that also return to normal levels more quickly. Likewise, extreme changes in vitals during the acute phase of the disease occur less frequently in vaccinated individuals. Our results solidify evidence that vaccines can mitigate long-term detrimental effects of SARS-CoV-2 infections both in terms of duration and magnitude. Furthermore, they demonstrate the value of large scale, high-resolution wearable sensor data in public health research.

Large-scale human mobility data is a key resource in data-driven policy making and across many scientific fields. Most recently, mobility data was extensively used during the COVID-19 pandemic to study the effects of governmental policies and to inform epidemic models. Large-scale mobility is often measured using digital tools such as mobile phones. However, it remains an open question how truthfully these digital proxies represent the actual travel behavior of the general population. Here, we examine mobility datasets from multiple countries and identify two fundamentally different types of bias caused by unequal access to, and unequal usage of mobile phones. We introduce the concept of data generation bias, a previously overlooked type of bias, which is present when the amount of data that an individual produces influences their representation in the dataset. We find evidence for data generation bias in all examined datasets in that high-wealth individuals are overrepresented, with the richest 20% contributing over 50% of all recorded trips, substantially skewing the datasets. This inequality is consequential, as we find mobility patterns of different wealth groups to be structurally different, where the mobility networks of high-wealth users are denser and contain more long-range connections. To mitigate the skew, we present a framework to debias data and show how simple techniques can be used to increase representativeness. Using our approach we show how biases can severely impact outcomes of dynamic processes such as epidemic simulations, where biased data incorrectly estimates the severity and speed of disease transmission. Overall, we show that a failure to account for biases can have detrimental effects on the results of studies and urge researchers and practitioners to account for data-fairness in all future studies of human mobility.

Vaccines are the most powerful pharmaceutical tool to combat the COVID-19 pandemic. While the majority (about 65%) of the German population were fully vaccinated, incidence started growing exponentially in October 2021 with about 41% of recorded new cases aged twelve or above being symptomatic breakthrough infections, presumably also contributing to the dynamics. At the time, it (i) remains elusive how significant this contribution is and (ii) whether targeted non-pharmaceutical interventions (NPIs) may stop the amplification of the ongoing crisis. Here, we estimate that about 67%-76% of all new infections are caused by unvaccinated individuals, implying that only 24%-33% are caused by the vaccinated. Furthermore, we estimate 38%-51% of new infections to be caused by unvaccinated individuals infecting other unvaccinated individuals. In total, unvaccinated individuals are expected to be involved in 8-9 of 10 new infections. We further show that decreasing the transmissibility of the unvaccinated by, e.g. targeted NPIs, causes a steeper decrease in the effective reproduction number $\mathcal{R}$ than decreasing the transmissibility of vaccinated individuals, potentially leading to temporary epidemic control. Furthermore, reducing contacts between vaccinated and unvaccinated individuals serves to decrease $\mathcal R$ in a similar manner as increasing vaccine uptake. Taken together, our results contribute to the public discourse regarding policy changes in pandemic response and highlight the importance of combined measures, such as vaccination campaigns and contact reduction, to achieve epidemic control and preventing an overload of public health systems., Comment: 21 pages, 3 figures

Vaccination against COVID-19 with the recently approved mRNA vaccines BNT162b2 (BioNTech/Pfizer) and mRNA-1273 (Moderna) is currently underway in a large number of countries. However, high incidence rates and rapidly spreading SARS-CoV-2 variants are concerning. In combination with acute supply deficits in Europe in early 2021, the question arises of whether stretching the vaccine, for instance by delaying the second dose, can make a significant contribution to preventing deaths, despite associated risks such as lower vaccine efficacy, the potential emergence of escape mutants, enhancement, waning immunity, reduced social acceptance of off-label vaccination, and liability shifts. A quantitative epidemiological assessment of risks and benefits of non-standard vaccination protocols remains elusive. To clarify the situation and to provide a quantitative epidemiological foundation we develop a stochastic epidemiological model that integrates specific vaccine rollout protocols into a risk-group structured infectious disease dynamical model. Using the situation and conditions in Germany as a reference system, we show that delaying the second vaccine dose is expected to prevent deaths in the four to five digit range, should the incidence resurge. We show that this considerable public health benefit relies on the fact that both mRNA vaccines provide substantial protection against severe COVID-19 and death beginning 12 to 14 days after the first dose. The benefits of protocol change are attenuated should vaccine compliance decrease substantially. To quantify the impact of protocol change on vaccination adherence we performed a large-scale online survey. We find that, in Germany, changing vaccination protocols may lead to small reductions in vaccination intention. In sum, we therefore expect the benefits of a strategy change to remain substantial and stable., Comment: 22 pages, 9 figures, 10 tables

In the wake of the COVID-19 pandemic many countries implemented containment measures to reduce disease transmission. Studies using digital data sources show that the mobility of individuals was effectively reduced in multiple countries. However, it remains unclear whether these reductions caused deeper structural changes in mobility networks, and how such changes may affect dynamic processes on the network. Here we use movement data of mobile phone users to show that mobility in Germany has not only been reduced considerably: Lockdown measures caused substantial and long-lasting structural changes in the mobility network. We find that long-distance travel was reduced disproportionately strongly. The trimming of long-range network connectivity leads to a more local, clustered network and a moderation of the "small-world" effect. We demonstrate that these structural changes have a considerable effect on epidemic spreading processes by "flattening" the epidemic curve and delaying the spread to geographically distant regions., Comment: 20 pages, 8 figures

The recent outbreak of COVID-19 in Mainland China is characterized by a distinctive algebraic, sub-exponential increase of confirmed cases during the early phase of the epidemic, contrasting an initial exponential growth expected for an unconstrained outbreak with sufficiently large reproduction rate. Although case counts vary significantly between affected provinces in Mainland China, the scaling law $t^{\mu}$ is surprisingly universal, with a range of exponents $\mu=2.1\pm0.3$. The universality of this behavior indicates that despite social, regional, demographical, geographical, and socio-economical heterogeneities of affected Chinese provinces, this outbreak is dominated by fundamental mechanisms that are not captured by standard epidemiological models. We show that the observed scaling law is a direct consequence of containment policies that effectively deplete the susceptible population. To this end we introduce a parsimonious model that captures both, quarantine of symptomatic infected individuals as well as population wide isolation in response to mitigation policies or behavioral changes. For a wide range of parameters, the model reproduces the observed scaling law in confirmed cases and explains the observed exponents. Quantitative fits to empirical data permit the identification of peak times in the number of asymptomatic or oligo-symptomatic, unidentified infected individuals, as well as estimates of local variations in the basic reproduction number. The model implies that the observed scaling law in confirmed cases is a direct signature of effective contaiment strategies and/or systematic behavioral changes that affect a substantial fraction of the susceptible population. These insights may aid the implementation of containment strategies in potential export induced COVID-19 secondary outbreaks elsewhere or similar future outbreaks of other emergent infectious diseases., Comment: 9 pages, 3 figures

Big Bayes is the computationally intensive co-application of big data and large, expressive Bayesian models for the analysis of complex phenomena in scientific inference and statistical learning. Standing as an example, Bayesian multidimensional scaling (MDS) can help scientists learn viral trajectories through space-time, but its computational burden prevents its wider use. Crucial MDS model calculations scale quadratically in the number of observations. We partially mitigate this limitation through massive parallelization using multi-core central processing units, instruction-level vectorization and graphics processing units (GPUs). Fitting the MDS model using Hamiltonian Monte Carlo, GPUs can deliver more than 100-fold speedups over serial calculations and thus extend Bayesian MDS to a big data setting. To illustrate, we employ Bayesian MDS to infer the rate at which different seasonal influenza virus subtypes use worldwide air traffic to spread around the globe. We examine 5392 viral sequences and their associated 14 million pairwise distances arising from the number of commercial airline seats per year between viral sampling locations. To adjust for shared evolutionary history of the viruses, we implement a phylogenetic extension to the MDS model and learn that subtype H3N2 spreads most effectively, consistent with its epidemic success relative to other seasonal influenza subtypes. Finally, we provide MassiveMDS, an open-source, stand-alone C++ library and rudimentary R package, and discuss program design and high-level implementation with an emphasis on important aspects of computing architecture that become relevant at scale.

Networks that are organized as a hierarchy of modules have been the subject of much research, mainly focusing on algorithms that can extract this community structure from data. The question of why modular hierarchical organizations are so ubiquitous in nature, however, has received less attention. One hypothesis is that modular hierarchical topologies may provide an optimal structure for certain dynamical processes. We revisit a modular hierarchical network model that interpolates, using a single parameter, between two known network topologies: from strong hierarchical modularity to an Erd\H{o}s-R\'enyi random connectivity structure. We show that this model displays a similar small-world effect as the Kleinberg model, where the connection probability between nodes decays algebraically with distance. We find that there is an optimal structure, in both models, for which the pair-averaged first passage time (FPT) and mean cover time of a discrete-time random walk are minimal, and provide a heuristic explanation for this effect. Finally, we show that analytic predictions for the pair-averaged FPT based on an effective medium approximation fail to reproduce these minima, which implies that their presence is due to a network structure effect., Comment: 19 pages, 10 figures

We present an analytical method for computing the mean cover time of a random walk process on arbitrary, complex networks. The cover time is defined as the time a random walker requires to visit every node in the network at least once. This quantity is particularly important for random search processes and target localization in network topologies. Based on the global mean first passage time of target nodes we derive an estimate for the cumulative distribution function of the cover time based on first passage time statistics. We show that our result can be applied to various model networks, including Erd\H{o}s-R\'enyi and Barab\'asi-Albert networks, as well as various real-world networks. Our results reveal an intimate link between first passage and cover time statistics in networks in which structurally induced temporal correlations decay quickly and offer a computationally efficient way for estimating cover times in network related applications.

Online petitions are an important avenue for direct political action, yet the dynamics that determine when a petition will be successful are not well understood. Here we analyze the temporal characteristics of online-petition signing behavior in order to identify systematic differences between popular petitions, which receive a high volume of signatures, and unpopular ones. We find that, in line with other temporal characterizations of human activity, the signing process is typically non-Poissonian and non-homogeneous in time. However, this process exhibits anomalously high memory for human activity, possibly indicating that synchronized external influence or contagion play and important role. More interestingly, we find clear differences in the characteristics of the inter-event time distributions depending on the total number of signatures that petitions receive, independently of the total duration of the petitions. Specifically, popular petitions that attract a large volume of signatures exhibit more variance in the distribution of inter-event times than unpopular petitions with only a few signatures, which could be considered an indication that the former are more bursty. However, petitions with large signature volume are less bursty according to measures that consider the time ordering of inter-event times. Our results, therefore, emphasize the importance of accounting for time ordering to characterize human activity., Comment: 16 pages, 8 figures

We show that the recently introduced logarithmic metrics used to predict disease arrival times on complex networks are approximations of more general network-based measures derived from random walks theory. Using the daily air-traffic transportation data we perform numerical experiments to compare the infection arrival time with this alternative metric that is obtained by accounting for multiple walks instead of only the most probable path. The comparison with direct simulations of arrival times reveals a higher correlation compared to the shortest path approach used previously. In addition our method allows to connect fundamental observables in epidemic spreading with the cumulant generating function of the hitting time for a Markov chain. Our results provides a general and computationally efficient approach to the problem using only algebraic methods., Comment: F.I. and A.K. contributed equally to this work. Skripts can be found on GitHub: https://github.com/andreaskoher/effective_distance

We investigate the effects of cooperativity between contagion processes that spread and persist in a host population. We propose and analyze a dynamical model in which individuals that are affected by one transmissible agent $A$ exhibit a higher than baseline propensity of being affected by a second agent $B$ and vice versa. The model is a natural extension of the traditional SIS (Susceptible-Infected-Susceptible) model used for modeling single contagion processes. We show that cooperativity changes the dynamics of the system considerably when cooperativity is strong. The system exhibits discontinuous phase transitions not observed in single agent contagion, multi-stability, a separation of the traditional epidemic threshold into different thresholds for inception and extinction as well as hysteresis. These properties are robust and are corroborated by stochastic simulations on lattices and generic network topologies. Finally, we investigate wave propagation and transients in a spatially extended version of the model and show that especially for intermediate values of baseline reproduction ratios the system is characterized by various types of wave-front speeds. The system can exhibit spatially heterogeneous stationary states for some parameters and negative front speeds (receding wave fronts). The two agent model can be employed as a starting point for more complex contagion processes, involving several interacting agents, a model framework particularly suitable for modeling the spread and dynamics of microbiological ecosystems in host populations., Comment: 10 pages, 7 figures

We discuss models and data of crowd disasters, crime, terrorism, war and disease spreading to show that conventional recipes, such as deterrence strategies, are often not effective and sufficient to contain them. Many common approaches do not provide a good picture of the actual system behavior, because they neglect feedback loops, instabilities and cascade effects. The complex and often counter-intuitive behavior of social systems and their macro-level collective dynamics can be better understood by means of complexity science. We highlight that a suitable system design and management can help to stop undesirable cascade effects and to enable favorable kinds of self-organization in the system. In such a way, complexity science can help to save human lives., Comment: 67 pages, 25 figures; accepted for publication in Journal of Statistical Physics [for related work see http://www.futurict.eu/]

Real world network datasets often contain a wealth of complex topological information. In the face of these data, researchers often employ methods to extract reduced networks containing the most important structures or pathways, sometimes known as `skeletons' or `backbones'. Numerous such methods have been developed. Yet data are often noisy or incomplete, with unknown numbers of missing or spurious links. Relatively little effort has gone into understanding how salient network extraction methods perform in the face of noisy or incomplete networks. We study this problem by comparing how the salient features extracted by two popular methods change when networks are perturbed, either by deleting nodes or links, or by randomly rewiring links. Our results indicate that simple, global statistics for skeletons can be accurately inferred even for noisy and incomplete network data, but it is crucial to have complete, reliable data to use the exact topologies of skeletons or backbones. These results also help us understand how skeletons respond to damage to the network itself, as in an attack scenario.

Herein we provide a closed form perturbative solution to a general $M$-node network SIS model using the transport rates between nodes as a perturbation parameter. We separate the dynamics into a short-time regime and a medium/long-time regime. We solve the short-time dynamics of the system and provide a limit before which our explicit, analytical result of the first-order perturbation for the medium/long-time regime is to be employed. These stitched calculations provide an approximation to the full temporal dynamics for rather general initial conditions. To further corroborate our results, we solve the mean-field equations numerically for an infectious SIS outbreak in New Zealand (NZ, \emph{Aotearoa}) recomposed into 23 subpopulations where the virus is spread to different subpopulations via (documented) air traffic data, and the country is internationally quarantined. We demonstrate that our analytical predictions compare well to the numerical solution., Comment: Revised. 11 pages, 4 figures, 2 Tables

Global mobility flow data are at the heart of spatial epidemiological models used to predict infectious disease behavior but this wealth of data on human mobility has been largely neglected by reconstructions of pathogen evolutionary dynamics using viral genetic data. Although stochastic models of viral evolution may potentially be informed by such data, a major challenge lies in deciding which mobility processes are critical and to what extent they contribute to shaping contemporaneous distributions of pathogen diversity. Here, we develop a framework to integrate predictors of viral diffusion with phylogeographic inference and estimate human influenza H3N2 migration history while simultaneously testing and quantifying the factors that underly it. We provide evidence for air travel governing the global dynamics of human influenza whereas other processes act at a more local scale., Comment: 16 pages, 4 figures, 3 tables

Network models with preferential attachment, where new nodes are injected into the network and form links with existing nodes proportional to their current connectivity, have been well studied for some time. Extensions have been introduced where nodes attach proportionally to arbitrary fitness functions. However, in these models, attaching to a node always increases the ability of that node to gain more links in the future. We study network growth where nodes attach proportionally to the clustering coefficients, or local densities of triangles, of existing nodes. Attaching to a node typically lowers its clustering coefficient, in contrast to preferential attachment or rich-get-richer models. This simple modification naturally leads to a variety of rich phenomena, including aging, non-Poissonian bursty dynamics, and community formation. This theoretical model shows that complex network structure can be generated without artificially imposing multiple dynamical mechanisms and may reveal potentially overlooked mechanisms present in complex systems., Comment: 6 pages, 4 figures

One of the key challenges in modeling the dynamics of contagion phenomena is to understand how the structure of social interactions shapes the time course of a disease. Complex network theory has provided significant advances in this context. However, awareness of an epidemic in a population typically yields behavioral changes that correspond to changes in the network structure on which the disease evolves. This feedback mechanism has not been investigated in depth. For example, one would intuitively expect susceptible individuals to avoid other infecteds. However, doctors treating patients or parents tending sick children may also increase the amount of contact made with an infecteds, in an effort to speed up recovery but also exposing themselves to higher risks of infection. We study the role of these caretaker links in an adaptive network models where individuals react to a disease by increasing or decreasing the amount of contact they make with infected individuals. We find that pure avoidance, with only few caretaker links, is the best strategy for curtailing an SIS disease in networks that possess a large topological variability. In more homogeneous networks, disease prevalence is decreased for low concentrations of caretakers whereas a high prevalence emerges if caretaker concentration passes a well defined critical value., Comment: 8 pages, 9 figures

Complex networks in natural, social, and technological systems generically exhibit an abundance of rich information. Extracting meaningful structural features from data is one of the most challenging tasks in network theory. Many methods and concepts have been proposed to address this problem such as centrality statistics, motifs, community clusters, and backbones, but such schemes typically rely on external and arbitrary parameters. It is unknown whether generic networks permit the classification of elements without external intervention. Here we show that link salience is a robust approach to classifying network elements based on a consensus estimate of all nodes. A wide range of empirical networks exhibit a natural, network-implicit classification of links into qualitatively distinct groups, and the salient skeletons have generic statistical properties. Salience also predicts essential features of contagion phenomena on networks, and points towards a better understanding of universal features in empirical networks that are masked by their complexity., Comment: Nature Communications 3 (2012)

Human mobility is a key factor in spatial disease dynamics and related phenomena. In computational models host mobility is typically modelled by diffusion in space or on metapolulation networks. Alternatively, an effective force of infection across distance has been introduced to capture spatial dispersal implicitly. Both approaches do not account for important aspects of natural human mobility, diffusion does not capture the high degree of predictability in natural human mobility patters, e.g. the high percentage of return movements to individuals' base location, the effective force of infection approach assumes immediate equilibrium with respect to dispersal. These conditions are typically not met in natural scenarios. We investigate an epidemiological model that explicitly captures natural individual mobility patterns. We systematically investigate generic dynamical features of the model on regular lattices as well as metapopulation networks and show that generally the model exhibits significant dynamical differences in comparison to ordinary diffusion and effective force of infection models. For instance, the natural human mobility model exhibits a saturation of wave front speeds and a novel type of invasion threshold that is a function of the return rate in mobility patterns. In the light of these new findings and with the availability of precise and pervasive data on human mobility our approach provides a framework for a more sophisticated modeling of spatial disease dynamics.

Territorial subdivisions and geographic borders are essential for understanding phenomena in sociology, political science, history, and economics. They influence the interregional flow of information and cross-border trade and affect the diffusion of innovation and technology. However, most existing administrative borders were determined by a variety of historic and political circumstances along with some degree of arbitrariness. Societies have changed drastically, and it is doubtful that currently existing borders reflect the most logical divisions. Fortunately, at this point in history we are in a position to actually measure some aspects of the geographic structure of society through human mobility. Large-scale transportation systems such as trains and airlines provide data about the number of people traveling between geographic locations, and many promising human mobility proxies are being discovered, such as cell phones, bank notes, and various online social networks. In this chapter we apply two optimization techniques to a human mobility proxy (bank note circulation) to investigate the effective geographic borders that emerge from a direct analysis of human mobility.

We investigate a model for spatial epidemics explicitly taking into account bi-directional movements between base and destination locations on individual mobility networks. We provide a systematic analysis of generic dynamical features of the model on regular and complex metapopulation network topologies and show that significant dynamical differences exist to ordinary reaction-diffusion and effective force of infection models. On a lattice we calculate an expression for the velocity of the propagating epidemic front and find that in contrast to the diffusive systems, our model predicts a saturation of the velocity with increasing traveling rate. Furthermore, we show that a fully stochastic system exhibits a novel threshold for attack ratio of an outbreak absent in diffusion and force of infection models. These insights not only capture natural features of human mobility relevant for the geographical epidemic spread, they may serve as a starting point for modeling important dynamical processes in human and animal epidemiology, population ecology, biology and evolution.

We present a comparative network theoretic analysis of the two largest global transportation networks: The worldwide air-transportation network (WAN) and the global cargoship network (GCSN). We show that both networks exhibit striking statistical similarities despite significant differences in topology and connectivity. Both networks exhibit a discontinuity in node and link betweenness distributions which implies that these networks naturally segragate in two different classes of nodes and links. We introduce a technique based on effective distances, shortest paths and shortest-path trees for strongly weighted symmetric networks and show that in a shortest-path-tree representation the most significant features of both networks can be readily seen. We show that effective shortest-path distance, unlike conventional geographic distance measures, strongly correlates with node centrality measures. Using the new technique we show that network resilience can be investigated more precisely than with contemporary techniques that are based on percolation theory. We extract a functional relationship between node characteristics and resilience to network disruption. Finally we discuss the results, their implications and conclude that dynamic processes that evolve on both networks are expected to share universal dynamic characteristics.

We investigate the dynamic impact of heterogeneous environments on superdiffusive random walks known as L\'evy flights. We devote particular attention to the relative weight of source and target locations on the rates for spatial displacements of the random walk. Unlike ordinary random walks which are slowed down for all values of the relative weight of source and target, non-local superdiffusive processes show distinct regimes of attenuation and acceleration for increased source and target weight, respectively. Consequently, spatial inhomogeneities can facilitate the spread of superdiffusive processes, in contrast to common belief that external disorder generally slows down stochastic processes. Our results are based on a novel type of fractional Fokker-Planck equation which we investigate numerically and by perturbation theory for weak disorder., Comment: 8 pages, 5 figures

We consider different generalizations of the Fokker-Planck-equation devised to describe Levy processes in potential force fields. We show that such generalizations can proceed along different lines. On one hand, Levy statistics can emerge from the fractal temporal nature of the underlying process, i.e. a high variability in the rate of microscopic events. On the other hand, they may be a direct consequence of the scale-free spatial structure on which the process evolves. Although both forms considered lead to Boltzmann equilibrium, the relaxation patterns are quite different. As an example, generalized diffusion in a double-well potential is considered., Comment: 21 pages, 5 figures