The presence of burstiness in temporal social networks, revealed by a power-law form of the waiting time distribution of consecutive interactions, is expected to produce aging effects in the corresponding time-integrated network. Here, we propose an analytically tractable model, in which interactions among the agents are ruled by a renewal process, that is able to reproduce this aging behavior. We develop an analytic solution for the topological properties of the integrated network produced by the model, finding that the time translation invariance of the degree distribution is broken. We validate our predictions against numerical simulations, and we check for the presence of aging effects in a empirical temporal network, ruled by bursty social interactions. DOI: 10.1103/PhysRevLett.114.108701 PACS numbers: 89.75.Hc Our understanding of the structure and properties of social interactions has experienced a boost in recent years due to the new availability of large amounts of digital empirical data [1,2]. This endeavor has found the necessary theoretical grounding in network science [3,4]. A first round of studies focused on a static network representation [5,6], in which nodes (standing for individuals) and edges (indicating social interactions) are constant and never change in time. From such a static representation, a wealth of complex topological information was obtained, concerning, e.g., the presence of scale-free, power-law degree distributions PðkÞ ∼ k −γ , large clustering, positive degree correlations, or a distinct community structure [7]. More recently, this framework has been challenged by the empirical observation of a temporal dimension in networks, particularly evident in social systems, due to the fact that social relationships are continuously created and terminated. From these temporal networks [8], a static representation is obtained by means of an integration of the instantaneous interactions over a time window of width t, and its associated topological properties, such as the degree distribution P t ðkÞ, are thus to be understood to depend on the integration time [9]. The empirical study of the temporal aspects of social networks has unveiled an additional level of complexity, embodied in many statistical properties showing heavy-tailed distributions. Remarkable among them are the distribution ψðτÞ of interevent or waiting times between two consecutive social interactions, revealing the bursty nature of human dynamics, and the distribution FðaÞ of social activity, measuring the probability per unit time of establishing a new social relation. Both distributions approximately obey power-law decays of the form ψðτÞ ∼ τ −ð1þαÞ and FðaÞ ∼ a −δ , respectively [10][11][12][13][14]. Noteworthy, the bursty nature of social interactions represents a common feature of human dynamics that can be related to the prioritized behavior of human beings [11]. This twofold nature of social interactions naturally raises the issue of the relation between the temporal correlation properties of time-varying netw...
Many progresses in the understanding of epidemic spreading models have been obtained thanks to numerous modeling efforts and analytical and numerical studies, considering host populations with very different structures and properties, including complex and temporal interaction networks. Moreover, a number of recent studies have started to go beyond the assumption of an absence of coupling between the spread of a disease and the structure of the contacts on which it unfolds. Models including awareness of the spread have been proposed, to mimic possible precautionary measures taken by individuals that decrease their risk of infection, but have mostly considered static networks. Here, we adapt such a framework to the more realistic case of temporal networks of interactions between individuals. We study the resulting model by analytical and numerical means on both simple models of temporal networks and empirical time-resolved contact data. Analytical results show that the epidemic threshold is not affected by the awareness but that the prevalence can be significantly decreased. Numerical studies on synthetic temporal networks highlight, however, the presence of very strong finite-size effects, resulting in a significant shift of the effective epidemic threshold in the presence of risk awareness. For empirical contact networks, the awareness mechanism leads as well to a shift in the effective threshold and to a strong reduction of the epidemic prevalence.
We present an exhaustive mathematical analysis of the recently proposed Non-Poissonian Activity Driven (NoPAD) model [Moinet et al. Phys. Rev. Lett., 114 (2015)], a temporal network model incorporating the empirically observed bursty nature of social interactions. We focus on the aging effects emerging from the Non-Poissonian dynamics of link activation, and on their effects on the topological properties of time-integrated networks, such as the degree distribution. Analytic expressions for the degree distribution of integrated networks as a function of time are derived, exploring both limits of vanishing and strong aging. We also address the percolation process occurring on these temporal networks, by computing the threshold for the emergence of a giant connected component, highlighting the aging dependence. Our analytic predictions are checked by means of extensive numerical simulations of the NoPAD model.
We study the behavior of a generalized consensus dynamics on a temporal network of interactions, the activity-driven network with attractiveness. In this temporal network model, agents are endowed with an intrinsic activity a, ruling the rate at which they generate connections, and an intrinsic attractiveness b, modulating the rate at which they receive connections. The consensus dynamics considered is a mixed voter and Moran dynamics. Each agent, either in state 0 or 1, modifies his or her state when connecting with a peer. Thus, an active agent copies his or her state from the peer (with probability p) or imposes his or her state to him or her (with the complementary probability 1-p). Applying a heterogeneous mean-field approach, we derive a differential equation for the average density of voters with activity a and attractiveness b in state 1, which we use to evaluate the average time to reach consensus and the exit probability, defined as the probability that a single agent with activity a and attractiveness b eventually imposes his or her state to a pool of initially unanimous population in the opposite state. We study a number of particular cases, finding an excellent agreement with numerical simulations of the model. Interestingly, we observe a symmetry between voter and Moran dynamics in pure activity-driven networks and their static integrated counterparts that exemplifies the strong differences that a time-varying network can impose on dynamical processes.
A strong reduction in diversity around a specific locus is often interpreted as a recent rapid fixation of a positively selected allele, a phenomenon called a selective sweep. Rapid fixation of neutral variants can however lead to similar reduction in local diversity, especially when the population experiences changes in population size, e.g., bottlenecks or range expansions. The fact that demographic processes can lead to signals of nucleotide diversity very similar to signals of selective sweeps is at the core of an ongoing discussion about the roles of demography and natural selection in shaping patterns of neutral variation. Here we quantitatively investigate the shape of such neutral valleys of diversity under a simple model of a single population size change, and we compare it to signals of a selective sweep. We analytically describe the expected shape of such “neutral sweeps” and show that selective sweep valleys of diversity are, for the same fixation time, wider than neutral valleys. On the other hand, it is always possible to parametrize our model to find a neutral valley that has the same width as a given selected valley. Our findings provide further insight in how simple demographic models can create valleys of genetic diversity similar to those attributed to positive selection.
The interest in non-Markovian dynamics within the complex systems community has recently blossomed, due to a new wealth of time-resolved data pointing out the bursty dynamics of many natural and human interactions, manifested in an inter-event time between consecutive interactions showing a heavy-tailed distribution. In particular, empirical data has shown that the bursty dynamics of temporal networks can have deep consequences on the behavior of the dynamical processes running on top of them. Here, we study the case of random walks, as a paradigm of diffusive processes, unfolding on temporal networks generated by a non-Poissonian activity driven dynamics. We derive analytic expressions for the steady state occupation probability and first passage time distribution in the infinite network size and strong aging limits, showing that the random walk dynamics on non-Markovian networks are fundamentally different from what is observed in Markovian networks. We found a particularly surprising behavior in the limit of diverging average inter-event time, in which the random walker feels the network as homogeneous, even though the activation probability of nodes is heterogeneously distributed. Our results are supported by extensive numerical simulations. We anticipate that our findings may be of interest among the researchers studying non-Markovian dynamics on time-evolving complex topologies. IntroductionTemporal networks [1, 2] constitute a recent new description of complex systems, that, moving apart from the classical static paradigm of network science [3], in which nodes and edges do not change in time, consider dynamic connections that can be created, destroyed or rewired at different time scales. Within this framework, a first round of studies proposed temporal network models ruled by homogeneous Markovian dynamics [4]. A prominent example is represented by the activity-driven model [5] (see also [6]), in which nodes are characterized by a different degree of activity, i.e. the constant rate at which an agent sends links to other peers, following a Poissonian process. The memoryless property implied by the Markovian dynamics greatly simplifies the mathematical treatment of these models, regarding both the topological properties of the time-integrated network representation [7], and the description of the dynamical processes unfolding on activity-driven networks [8][9][10][11][12][13].However, the Markovian assumption in temporal network modeling has been challenged by the increasing availability of time-resolved data on different kinds of interactions, ranging from phone communications [14] and face-to-face interactions [15], to natural phenomena [16,17], biological processes [18] and physiological systems [19][20][21]. These empirical observations have uncovered a rich variety of dynamical properties, in particular that the inter-event times t between two successive interactions (either the creation of the same edge or two consecutive creations of an edge by the same node), ψ(t), follows heavy-tailed distributions...
The distribution of genetic diversity over geographical space has long been investigated in population genetics and serves as a useful tool to understand evolution and history of populations. Within some species or across regions of contact between two species, there are instances where there is no apparent ecological determinant of sharp changes in allele frequencies or divergence. To further understand these patterns of spatial genetic structure and potential species divergence, we model the establishment of clines that occur due to the surfing of underdominant alleles during range expansions. We provide analytical approximations for the fixation probability of underdominant alleles at expansion fronts and demonstrate that gene surfing can lead to clines in one-dimensional range expansions. We extend these results to multiple loci via a mixture of analytical theory and individual-based simulations. We study the interaction between the strength of selection against heterozygotes, migration rates, and local recombination rates on the formation of stable hybrid zones. Clines created by surfing at different loci can attract each other and align after expansion, if they are sufficiently close in space and in terms of recombination distance. Our findings suggest that range expansions can set the stage for parapatric speciation due to the alignment of multiple selective clines, even in the absence of ecologically divergent selection. This article is part of the theme issue ‘Species' ranges in the face of changing environments (part I)’.
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