Aging and percolation dynamics in a Non-Poissonian temporal network model

We present an exhaustive mathematical analysis of the recently proposed Non-Poissonian Ac- tivity 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, ex- ploring 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

Effect of risk perception on epidemic spreading in temporal networks

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 highlight however the presence of very strong finite-size effects, in particular for the more realistic synthetic temporal networks, 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

Generalized Voter-like model on activity driven networks with attractiveness

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/Moran dynamics. Each agent, either in state $0$ or $1$, modifies his/her state when connecting with a peer. Thus, an active agent copies his/her state from the peer (with probability $p$) or imposes his/her state to him/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 $1$, that 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/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

Burstiness and aging in social temporal networks

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, and 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

Random walks in non-Poissoinan activity driven temporal networks

International audienceThe 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 of time-evolving complex topologies

Processus dynamiques sur réseaux temporels non-Markoviens

A complex system is formed by a large number of interacting elementary components, and presenting a non-trivial and adaptive architecture. The understanding of these systems has inflated through the recent development of network science, which is rooted in graph theory. This approach allows to highlight a complex function-structure relation, and to simplify the theoretical study of many dynamic processes possibly taking place in a network. The temporal nature of these networks, i.e. the fact that the links representing elementary interactions appear and disappear in time, has recently drawn a lot of attention. In the particular case of social networks, the patterns of interactions exhibit non-Markovian dynamics, i.e. memory effects are at play. This thesis dives into these issues, expanding the understanding of networks and in particular evaluating the impact of a non-Markovian dynamics on dynamical processes running on top of such systems.We build an extension of an existing stochastic agent based model of social dynamics, generalizing to the study of arbitrary distributions of the waiting time between two interactions of an agent. Assumed to be exponential in the original model, this distribution, measurable in real social networks, is in fact power law, revealing the non-Markovian character of human social dynamics. Taking this property into account in our model, we provide a detailed investigation of the topological properties of the aggregated network, formed by the union of all the links that have appeared at least once in a given temporal window.Les systèmes complexes résultent de l’interaction d’un grand nombre de constituants élémentaires, et présentent une architecture non triviale et adaptative. La compréhension de ces sytèmes s’est accrue considérablement grâce au récent développement de la science des réseaux, qui s’appuie sur la théorie des graphes. Cette approche permet de mettre en lumière une relation fonction-structure complexe, et simplifie l’étude théorique de nombreux processus dynamiques dont ils peuvent être le siège. On s’intéresse depuis peu à la dynamique instantanée de ces réseaux. Dans le cas particulier des réseaux sociaux, ces dynamiques non-markoviennes, c'est-à-dire présentant des effets de mémoire. Cette thèse a pour but d’approfondir notre compréhension des réseaux sociaux dynamiques, et en particulier de déterminer l'impact d’une topologie à dynamique non-markovienne sur les processus collectifs, de diffusion ou de propagation susceptibles d’apparaître au sein de tels systèmes. Nous appuyant sur un modèle stochastique existant de dynamique sociale à base d'agents actifs, nous généralisons à l'étude d'une distribution quelconque du temps d'attente entre deux interacions d'un agent. Supposée exponentielle dans le modèle original, cette distribution, mesurable dans les réseaux sociaux réels, suit en réalité une loi de puissance, révélant le caractère non-markovien de la dynamique sociale chez les êtres humains. Incluant cette propriété dans notre modèle, nous donnons une étude théorique détaillée des propriétés topologiques du réseau agrégé, union de tous les liens apparus au moins une fois dans une fenêtre temporelle donnée

Strong neutral sweeps occurring during a population contraction

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. We apply our framework to the case of a putative selective sweep signal around the gene Quetzalcoatl in D. melanogaster and show that the valley of diversity in the vicinity of this gene is compatible with a short bottleneck scenario without selection. Our findings provide further insight in how simple demographic models can create valleys of genetic diversity that may falsely be attributed to positive selection

Strong neutral sweeps occurring during a population contraction.

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

Burstiness and aging in social temporal networks

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