Contact patterns among high school students

Dynamics of collective cooperation under personalised strategy updates

Collective cooperation is essential for many social and biological systems, yet understanding how it evolves remains a challenge. Previous investigations report that the ubiquitous heterogeneous individual connections hinder cooperation by assuming individuals update strategies at identical rates. Here we develop a general framework by allowing individuals to update strategies at personalised rates, and provide the precise mathematical condition under which universal cooperation is favoured. Combining analytical and numerical calculations on synthetic and empirical networks, we find that when individuals' update rates vary inversely with their number of connections, heterogeneous connections actually outperform homogeneous ones in promoting cooperation. This surprising property undercuts the conventional wisdom that heterogeneous structure is generally antagonistic to cooperation and, further helps develop an efficient algorithm OptUpRat to optimise collective cooperation by designing individuals' update rates in any population structure. Our findings provide a unifying framework to understand the interplay between structural heterogeneity, behavioural rhythms, and cooperation.

Quality assessment and community detection methods for anonymized mobility data in the Italian Covid context

We discuss how to assess the reliability of partial, anonymized mobility data and compare two different methods to identify spatial communities based on movements: Greedy Modularity Clustering (GMC) and the novel Critical Variable Selection (CVS). These capture different aspects of mobility: direct population fluxes (GMC) and the probability for individuals to move between two nodes (CVS). As a test case, we consider movements of Italians before and during the SARS-Cov2 pandemic, using Facebook users' data and publicly available information from the Italian National Institute of Statistics (Istat) to construct daily mobility networks at the interprovincial level. Using the Perron-Frobenius (PF) theorem, we show how the mean stochastic network has a stationary population density state comparable with data from Istat, and how this ceases to be the case if even a moderate amount of pruning is applied to the network. We then identify the first two national lockdowns through temporal clustering of the mobility networks, define two representative graphs for the lockdown and non-lockdown conditions and perform optimal spatial community identification on both graphs using the GMC and CVS approaches. Despite the fundamental differences in the methods, the variation of information (VI) between them assesses that they return similar partitions of the Italian provincial networks in both situations. The information provided can be used to inform policy, for example, to define an optimal scale for lockdown measures. Our approach is general and can be applied to other countries or geographical scales.

Seasonal variations in social contact patterns in a rural population in north India: Implications for pandemic control

Social contact mixing patterns are critical to model the transmission of communicable diseases, and have been employed to model disease outbreaks including COVID-19. Nonetheless, there is a paucity of studies on contact mixing in low and middle-income countries such as India. Furthermore, mathematical models of disease outbreaks do not account for the temporal nature of social contacts. We conducted a longitudinal study of social contacts in rural north India across three seasons and analysed the temporal differences in contact patterns. A contact diary survey was performed across three seasons from October 2015-16, in which participants were queried on the number, duration, and characteristics of contacts that occurred on the previous day. A total of 8,421 responses from 3,052 respondents (49% females) recorded characteristics of 180,073 contacts. Respondents reported a significantly higher number and duration of contacts in the winter, followed by the summer and the monsoon season (Nemenyi post-hoc, p<0.001). Participants aged 0-9 years and 10-19 years of age reported the highest median number of contacts (16 (IQR 12-21), 17 (IQR 13-24) respectively) and were found to have the highest node centrality in the social network of the region (pageranks = 0.20, 0.17). A large proportion (>80%) of contacts that were reported in schools or on public transport involved physical contact. To the best of our knowledge, our study is the first from India to show that contact mixing patterns vary by the time of the year and provides useful implications for pandemic control. We compared the differences in the number, duration and location of contacts by age-group and gender, and studied the impact of the season, age-group, employment and day of the week on the number and duration of contacts using multivariate negative binomial regression. We created a social network to further understand the age and gender-specific contact patterns, and used the contact matrices in each season to parameterise a nine-compartment agent-based model for simulating a COVID-19 epidemic in each season. Our results can be used to parameterize more accurate mathematical models for prediction of epidemiological trends of infections in rural India.

Compressing the Chronology of a Temporal Network with Graph Commutators

Studies of dynamics on temporal networks often represent the network as a series of "snapshots," static networks active for short durations of time. We argue that successive snapshots can be aggregated if doing so has little effect on the overlying dynamics. We propose a method to compress network chronologies by progressively combining pairs of snapshots whose matrix commutators have the smallest dynamical effect. We apply this method to epidemic modeling on real contact tracing data and find that it allows for significant compression while remaining faithful to the epidemic dynamics.

On the inadequacy of nominal assortativity for assessing homophily in networks

Nominal assortativity (or discrete assortativity) is widely used to characterize group mixing patterns and homophily in networks, enabling researchers to analyze how groups interact with one another. Here we demonstrate that the measure presents severe shortcomings when applied to networks with unequal group sizes and asymmetric mixing. We characterize these shortcomings analytically and use synthetic and empirical networks to show that nominal assortativity fails to account for group imbalance and asymmetric group interactions, thereby producing an inaccurate characterization of mixing patterns. We propose the adjusted nominal assortativity and show that this adjustment recovers the expected assortativity in networks with various level of mixing. Furthermore, we propose an analytical method to assess asymmetric mixing by estimating the tendency of inter- and intra-group connectivities. Finally, we discuss how this approach enables uncovering hidden mixing patterns in real-world networks.

Heterogeneous message passing for heterogeneous networks

Message passing (MP) is a computational technique used to find approximate solutions to a variety of problems defined on networks. MP approximations are generally accurate in locally treelike networks but require corrections to maintain their accuracy level in networks rich with short cycles. However, MP may already be computationally challenging on very large networks and additional costs incurred by correcting for cycles could be prohibitive. We show how the issue can be addressed. By allowing each node in the network to have its own level of approximation, one can focus on improving the accuracy of MP approaches in a targeted manner. We perform a systematic analysis of 109 real-world networks and show that our node-based MP approximation is able to increase both the accuracy and speed of traditional MP approaches. We find that, compared to conventional MP, a heterogeneous approach based on a simple heuristic is more accurate in 81% of tested networks, faster in 64% of cases, and both more accurate and faster in 49% of cases.

Spatial immunization to abate disease spreading in transportation hubs

Proximity social interactions are crucial for infectious diseases transmission. Crowded agglomerations pose serious risk of triggering superspreading events. Locations like transportation hubs (airports and stations) are designed to optimize logistic efficiency, not to reduce crowding, and are characterized by a constant in and out flow of people. Here, we analyze the paradigmatic example of London Heathrow, one of the busiest European airports. Thanks to a dataset of anonymized individuals' trajectories, we can model the spreading of different diseases to localize the contagion hotspots and to propose a spatial immunization policy targeting them to reduce disease spreading risk. We also detect the most vulnerable destinations to contagions produced at the airport and quantify the benefits of the spatial immunization technique to prevent regional and global disease diffusion. This method is immediately generalizable to train, metro and bus stations and to other facilities such as commercial or convention centers.

Impact of human contact patterns on epidemic spreading in time-varying networks

Human contact behaviors involve both dormant and active processes. The dormant (active) process goes from the disappearance (creation) to the creation (disappearance) of an edge. The dormant (active) time is the elapsed time since the edge became dormant (active). Many empirical studies have revealed that dormant and active times in human contact behaviors tend to show a long-tailed distribution. Previous researches focused on the impact of the dormant process on spreading dynamics. However, the epidemic spreading happens on the active process. This raises the question of how the active process affects epidemic spreading in complex networks. Here, we propose a novel time-varying network model in which the distributions of both the dormant time and active time of edges are adjustable. We develop a pairwise approximation method to describe the spreading dynamical processes in the time-varying networks. Through extensive numerical simulations, we find that the epidemic threshold is proportional to the mean dormant time and inversely proportional to the mean active time. The attack rate decreases with the increase of mean dormant time and increases with the increase of mean active time. It is worth noting that the epidemic threshold and the attack rate (e.g., the infected density in the steady state) are independent of the heterogeneities of the dormant time distribution and the active time distribution. Increasing the heterogeneity of the dormant time distribution accelerates epidemic spreading while increasing the heterogeneity of the active time distribution slows it down.

Characterization of interactions' persistence in time-varying networks

Many complex networked systems exhibit volatile dynamic interactions among their vertices, whose order and persistence reverberate on the outcome of dynamical processes taking place on them. To quantify and characterize the similarity of the snapshots of a time-varying network-a proxy for the persistence,-we present a study on the persistence of the interactions based on a descriptor named temporality. We use the average value of the temporality, [Formula: see text], to assess how "special" is a given time-varying network within the configuration space of ordered sequences of snapshots. We analyse the temporality of several empirical networks and find that empirical sequences are much more similar than their randomized counterparts. We study also the effects on [Formula: see text] induced by the (time) resolution at which interactions take place.

Longitudinal data collection to follow social network and language development dynamics at preschool

DyLNet is a large-scale longitudinal social experiment designed to observe the relations between child socialisation and oral language learning at preschool. During three years, a complete preschool in France was followed to record proximity interactions of about 200 children and adults every 5 seconds using autonomous Radio Frequency Identification Wireless Proximity Sensors. Data was collected monthly with one week-long deployments. In parallel, survey campaigns were carried out to record the socio-demographic and language background of children and their families, and to monitor the linguistic skills of the pupils at regular intervals. From data we inferred real social interactions and distinguished inter- and intra-class interactions in different settings. We share ten weeks of cleaned, pre-processed and reconstructed interaction data recorded over a complete school year, together with two sets of survey data providing details about the pupils' socio-demographic profile and language development level at the beginning and end of this period. Our dataset may stimulate researchers from several fields to study the simultaneous development of language and social interactions of children.

Dynamics of cold random hyperbolic graphs with link persistence

We consider and analyze a dynamic model of random hyperbolic graphs with link persistence. In the model, both connections and disconnections can be propagated from the current to the next snapshot with probability ω∈[0,1). Otherwise, with probability 1-ω, connections are reestablished according to the random hyperbolic graphs model. We show that while the persistence probability ω affects the averages of the contact and intercontact distributions, it does not affect the tails of these distributions, which decay as power laws with exponents that do not depend on ω. We also consider examples of real temporal networks, and we show that the considered model can adequately reproduce several of their dynamical properties. Our results advance our understanding of the realistic modeling of temporal networks and of the effects of link persistence on temporal network properties.

Cooperative evolution with opinion formation in a complex social environment

Cooperation is a significant aspect in the daily activities of human or animal populations that involves the process of agents making decisions based in part on the ideas or beliefs of cooperators. The importance of communication in the evolutionary game cannot be overstated. In this paper, we offer a co-evolutionary game model on a communication network, including expressed and private opinions. Then, we present our theoretical analysis of scenarios characterized by different parameters, elucidating the relationship between the agents' opinion formation process and the decision-making process. Finally, we run simulations with our proposed model. Surprisingly, we discover that whereas communication does not increase cooperation on Erdös-Rényi random networks, it does so on the Barabási-Albert scale-free network. Meanwhile, we discover that in the simulation results, the average of private opinions (simplified as values within [0,1]) changes in the same direction of the percentage of cooperators. Furthermore, we discover that deceivers who conceal their true private opinions may, to some extent, foster the formation of collaboration in the human population, which goes against our common sense.

Effective submodularity of influence maximization on temporal networks

We study influence maximization on temporal networks. This is a special setting where the influence function is not submodular, and there is no optimality guarantee for solutions achieved via greedy optimization. We perform an exhaustive analysis on both real and synthetic networks. We show that the influence function of randomly sampled sets of seeds often violates the necessary conditions for submodularity. However, when sets of seeds are selected according to the greedy optimization strategy, the influence function behaves effectively as a submodular function. Specifically, violations of the necessary conditions for submodularity are never observed in real networks, and only rarely in synthetic ones. The direct comparison with exact solutions obtained via brute-force search indicates that the greedy strategy provides approximate solutions that are well within the optimality gap guaranteed for strictly submodular functions. Greedy optimization appears, therefore, to be an effective strategy for the maximization of influence on temporal networks.

Full reconstruction of simplicial complexes from binary contagion and Ising data

Previous efforts on data-based reconstruction focused on complex networks with pairwise or two-body interactions. There is a growing interest in networks with higher-order or many-body interactions, raising the need to reconstruct such networks based on observational data. We develop a general framework combining statistical inference and expectation maximization to fully reconstruct 2-simplicial complexes with two- and three-body interactions based on binary time-series data from two types of discrete-state dynamics. We further articulate a two-step scheme to improve the reconstruction accuracy while significantly reducing the computational load. Through synthetic and real-world 2-simplicial complexes, we validate the framework by showing that all the connections can be faithfully identified and the full topology of the 2-simplicial complexes can be inferred. The effects of noisy data or stochastic disturbance are studied, demonstrating the robustness of the proposed framework.

Data-driven recovery of topology is challenging for networks beyond pairwise interactions. The authors propose a framework to reconstruct complex networks with higher-order interactions from time series, focusing on networks with 2-simplexes where social contagion and Ising dynamics generate binary data.

Dynamics of hot random hyperbolic graphs

We derive the most basic dynamical properties of random hyperbolic graphs (the distributions of contact and intercontact durations) in the hot regime (network temperature T>1). We show that for sufficiently large networks the contact distribution decays as a power law with exponent 2+T>3 for durations t>T, while for t<T it exhibits exponential-like decays. This result holds irrespective of the expected degree distribution, as long as it has a finite Tth moment. Otherwise, the contact distribution depends on the expected degree distribution and we show that if the latter is a power law with exponent γ∈(2,T+1], then the former decays as a power law with exponent γ+1>3. However, the intercontact distribution exhibits power-law decays with exponent 2-T∈(0,1) for T∈(1,2), while for T>2 it displays linear decays with a slope that depends on the observation interval. This result holds irrespective of the expected degree distribution as long as it has a finite Tth moment if T∈(1,2), or a finite second moment if T>2. Otherwise, the intercontact distribution depends on the expected degree distribution and if the latter is a power law with exponent γ∈(2,3), then the former decays as a power law with exponent 3-γ∈(0,1). Thus, hot random hyperbolic graphs can give rise to contact and intercontact distributions that both decay as power laws. These power laws, however, are unrealistic for the case of the intercontact distribution, as their exponent is always less than one. These results mean that hot random hyperbolic graphs are not adequate for modeling real temporal networks, in stark contrast to cold random hyperbolic graphs (T<1). Since the configuration model emerges at T→∞, these results also suggest that this is not an adequate null temporal network model.

Mitigate SIR epidemic spreading via contact blocking in temporal networks

Progress has been made in how to suppress epidemic spreading on temporal networks via blocking all contacts of targeted nodes or node pairs. In this work, we develop contact blocking strategies that remove a fraction of contacts from a temporal (time evolving) human contact network to mitigate the spread of a Susceptible-Infected-Recovered epidemic. We define the probability that a contact c(i, j, t) is removed as a function of a given centrality metric of the corresponding link l(i, j) in the aggregated network and the time t of the contact. The aggregated network captures the number of contacts between each node pair. A set of 12 link centrality metrics have been proposed and each centrality metric leads to a unique contact removal strategy. These strategies together with a baseline strategy (random removal) are evaluated in empirical contact networks via the average prevalence, the peak prevalence and the time to reach the peak prevalence. We find that the epidemic spreading can be mitigated the best when contacts between node pairs that have fewer contacts and early contacts are more likely to be removed. A strategy tends to perform better when the average number contacts removed from each node pair varies less. The aggregated pruned network resulted from the best contact removal strategy tends to have a large largest eigenvalue, a large modularity and probably a small largest connected component size.

Using social contact data to improve the overall effect estimate of a cluster-randomized influenza vaccination program in Senegal

This study estimates the overall effect of two influenza vaccination programs consecutively administered in a cluster-randomized trial in western Senegal over the course of two influenza seasons from 2009-2011. We apply cutting-edge methodology combining social contact data with infection data to reduce bias in estimation arising from contamination between clusters. Our time-varying estimates reveal a reduction in seasonal influenza from the intervention and a nonsignificant increase in H1N1 pandemic influenza. We estimate an additive change in overall cumulative incidence (which was 6.13% in the control arm) of -0.68 percentage points during Year 1 of the study (95% CI: -2.53, 1.18). When H1N1 pandemic infections were excluded from analysis, the estimated change was -1.45 percentage points and was significant (95% CI, -2.81, -0.08). Because cross-cluster contamination was low (0-3% of contacts for most villages), an estimator assuming no contamination was only slightly attenuated (-0.65 percentage points). These findings are encouraging for studies carefully designed to minimize spillover. Further work is needed to estimate contamination - and its effect on estimation - in a variety of settings.

Systematic comparison of graph embedding methods in practical tasks

Network embedding techniques aim to represent structural properties of graphs in geometric space. Those representations are considered useful in downstream tasks such as link prediction and clustering. However, the number of graph embedding methods available on the market is large, and practitioners face the nontrivial choice of selecting the proper approach for a given application. The present work attempts to close this gap of knowledge through a systematic comparison of 11 different methods for graph embedding. We consider methods for embedding networks in the hyperbolic and Euclidean metric spaces, as well as nonmetric community-based embedding methods. We apply these methods to embed more than 100 real-world and synthetic networks. Three common downstream tasks - mapping accuracy, greedy routing, and link prediction - are considered to evaluate the quality of the various embedding methods. Our results show that some Euclidean embedding methods excel in greedy routing. As for link prediction, community-based and hyperbolic embedding methods yield an overall performance that is superior to that of Euclidean-space-based approaches. We compare the running time for different methods and further analyze the impact of different network characteristics such as degree distribution, modularity, and clustering coefficients on the quality of the embedding results. We release our evaluation framework to provide a standardized benchmark for arbitrary embedding methods.

Within and between classroom transmission patterns of seasonal influenza among primary school students in Matsumoto city, Japan

Schools play a central role in the transmission of many respiratory infections. Heterogeneous social contact patterns associated with the social structures of schools (i.e., classes/grades) are likely to influence the within-school transmission dynamics, but data-driven evidence on fine-scale transmission patterns between students has been limited. Using a mathematical model, we analyzed a large-scale dataset of seasonal influenza outbreaks in Matsumoto city, Japan, to infer social interactions within and between classes/grades from observed transmission patterns. While the relative contribution of within-class and within-grade transmissions to the reproduction number varied with the number of classes per grade, the overall within-school reproduction number, which determines the initial growth of cases and the risk of sustained transmission, was only minimally associated with class sizes and the number of classes per grade. This finding suggests that interventions that change the size and number of classes, e.g., splitting classes and staggered attendance, may have a limited effect on the control of school outbreaks. We also found that vaccination and mask-wearing of students were associated with reduced susceptibility (vaccination and mask-wearing) and infectiousness (mask-wearing), and hand washing was associated with increased susceptibility. Our results show how analysis of fine-grained transmission patterns between students can improve understanding of within-school disease dynamics and provide insights into the relative impact of different approaches to outbreak control.

An efficient procedure for mining egocentric temporal motifs

Temporal graphs are structures which model relational data between entities that change over time. Due to the complex structure of data, mining statistically significant temporal subgraphs, also known as temporal motifs, is a challenging task. In this work, we present an efficient technique for extracting temporal motifs in temporal networks. Our method is based on the novel notion of egocentric temporal neighborhoods, namely multi-layer structures centered on an ego node. Each temporal layer of the structure consists of the first-order neighborhood of the ego node, and corresponding nodes in sequential layers are connected by an edge. The strength of this approach lies in the possibility of encoding these structures into a unique bit vector, thus bypassing the problem of graph isomorphism in searching for temporal motifs. This allows our algorithm to mine substantially larger motifs with respect to alternative approaches. Furthermore, by bringing the focus on the temporal dynamics of the interactions of a specific node, our model allows to mine temporal motifs which are visibly interpretable. Experiments on a number of complex networks of social interactions confirm the advantage of the proposed approach over alternative non-egocentric solutions. The egocentric procedure is indeed more efficient in revealing similarities and discrepancies among different social environments, independently of the different technologies used to collect data, which instead affect standard non-egocentric measures.

From temporal network data to the dynamics of social relationships

Networks are well-established representations of social systems, and temporal networks are widely used to study their dynamics. However, going from temporal network data (i.e. a stream of interactions between individuals) to a representation of the social group's evolution remains a challenge. Indeed, the temporal network at any specific time contains only the interactions taking place at that time and aggregating on successive time-windows also has important limitations. Here, we present a new framework to study the dynamic evolution of social networks based on the idea that social relationships are interdependent: as the time we can invest in social relationships is limited, reinforcing a relationship with someone is done at the expense of our relationships with others. We implement this interdependence in a parsimonious two-parameter model and apply it to several human and non-human primates' datasets to demonstrate that this model detects even small and short perturbations of the networks that cannot be detected using the standard technique of successive aggregated networks. Our model solves a long-standing problem by providing a simple and natural way to describe the dynamic evolution of social networks, with far-reaching consequences for the study of social networks and social evolution.

Avoiding cross-sex friendships: The separability of people with and without cross-sex friends

Prior studies on individual differences in the preference for cross-sex friendships found that this preference was not normally distributed but was instead bimodal. In one group of people, the preferences for higher or lower proportions of cross-sex friendships appear to be normally distributed, whereas in a second and unexpectedly large group of people, the preference for cross-sex friends is exactly zero. If the people in the second group with no cross-sex friends at all actively avoid forming cross-sex friendships, then these individuals may be expected to differ systematically and meaningfully from individuals who report having at least one cross-sex friend. The present study tests this hypothesis. The Big Five, homophobia, physical attraction to the opposite sex, and demographic variables from a data set of 491 adult participants were used as potential predictors of group membership. Results showed that most predictors except the Big Five contributed to supporting the separability of the two groups. Findings are discussed with regard to the differentiation between close and general friends and the potential influence of cultural factors.

Mitigating COVID-19 outbreaks in workplaces and schools by hybrid telecommuting

The COVID-19 epidemic has forced most countries to impose contact-limiting restrictions at workplaces, universities, schools, and more broadly in our societies. Yet, the effectiveness of these unprecedented interventions in containing the virus spread remain largely unquantified. Here, we develop a simulation study to analyze COVID-19 outbreaks on three real-life contact networks stemming from a workplace, a primary school and a high school in France. Our study provides a fine-grained analysis of the impact of contact-limiting strategies at workplaces, schools and high schools, including: (1) Rotating strategies, in which workers are evenly split into two shifts that alternate on a daily or weekly basis; and (2) On-Off strategies, where the whole group alternates periods of normal work interactions with complete telecommuting. We model epidemics spread in these different setups using a stochastic discrete-time agent-based transmission model that includes the coronavirus most salient features: super-spreaders, infectious asymptomatic individuals, and pre-symptomatic infectious periods. Our study yields clear results: the ranking of the strategies, based on their ability to mitigate epidemic propagation in the network from a first index case, is the same for all network topologies (workplace, primary school and high school). Namely, from best to worst: Rotating week-by-week, Rotating day-by-day, On-Off week-by-week, and On-Off day-by-day. Moreover, our results show that below a certain threshold for the original local reproduction number [Formula: see text] within the network (< 1.52 for primary schools, < 1.30 for the workplace, < 1.38 for the high school, and < 1.55 for the random graph), all four strategies efficiently control outbreak by decreasing effective local reproduction number to [Formula: see text] < 1. These results can provide guidance for public health decisions related to telecommuting.

Modeling of network based digital contact tracing and testing strategies, including the pre-exposure notification system, for the COVID-19 pandemic

With more than 1.7 million COVID-19 deaths, identifying effective measures to prevent COVID-19 is a top priority. We developed a mathematical model to simulate the COVID-19 pandemic with digital contact tracing and testing strategies. The model uses a real-world social network generated from a high-resolution contact data set of 180 students. This model incorporates infectivity variations, test sensitivities, incubation period, and asymptomatic cases. We present a method to extend the weighted temporal social network and present simulations on a network of 5000 students. The purpose of this work is to investigate optimal quarantine rules and testing strategies with digital contact tracing. The results show that the traditional strategy of quarantining direct contacts reduces infections by less than 20% without sufficient testing. Periodic testing every 2 weeks without contact tracing reduces infections by less than 3%. A variety of strategies are discussed including testing second and third degree contacts and the pre-exposure notification system, which acts as a social radar warning users how far they are from COVID-19. The most effective strategy discussed in this work was combining the pre-exposure notification system with testing second and third degree contacts. This strategy reduces infections by 18.3% when 30% of the population uses the app, 45.2% when 50% of the population uses the app, 72.1% when 70% of the population uses the app, and 86.8% when 95% of the population uses the app. When simulating the model on an extended network of 5000 students, the results are similar with the contact tracing app reducing infections by up to 79%.

Impact of environmental changes on the dynamics of temporal networks

Dynamics of complex social systems has often been described in the framework of temporal networks, where links are considered to exist only at the moment of interaction between nodes. Such interaction patterns are not only driven by internal interaction mechanisms, but also affected by environmental changes. To investigate the impact of the environmental changes on the dynamics of temporal networks, we analyze several face-to-face interaction datasets using the multiscale entropy (MSE) method to find that the observed temporal correlations can be categorized according to the environmental similarity of datasets such as classes and break times in schools. By devising and studying a temporal network model considering a periodically changing environment as well as a preferential activation mechanism, we numerically show that our model could successfully reproduce various empirical results by the MSE method in terms of multiscale temporal correlations. Our results demonstrate that the environmental changes can play an important role in shaping the dynamics of temporal networks when the interactions between nodes are influenced by the environment of the systems.

A Visual Analytics Framework for Reviewing Multivariate Time-Series Data with Dimensionality Reduction

Data-driven problem solving in many real-world applications involves analysis of time-dependent multivariate data, for which dimensionality reduction (DR) methods are often used to uncover the intrinsic structure and features of the data. However, DR is usually applied to a subset of data that is either single-time-point multivariate or univariate time-series, resulting in the need to manually examine and correlate the DR results out of different data subsets. When the number of dimensions is large either in terms of the number of time points or attributes, this manual task becomes too tedious and infeasible. In this paper, we present MulTiDR, a new DR framework that enables processing of time-dependent multivariate data as a whole to provide a comprehensive overview of the data. With the framework, we employ DR in two steps. When treating the instances, time points, and attributes of the data as a 3D array, the first DR step reduces the three axes of the array to two, and the second DR step visualizes the data in a lower-dimensional space. In addition, by coupling with a contrastive learning method and interactive visualizations, our framework enhances analysts' ability to interpret DR results. We demonstrate the effectiveness of our framework with four case studies using real-world datasets.

C, Galloway D, Schweizer ML, Guclu H, Cousins J, Lingle C, Yearwood GMH, Li K, Calderone P, Noble E, Gao H, Rainey J, Uzicanin A, Read JM. Age-specific social mixing of school-aged children in a US setting using proximity detecting sensors and contact surveys

Comparisons of the utility and accuracy of methods for measuring social interactions relevant to disease transmission are rare. To increase the evidence base supporting specific methods to measure social interaction, we compared data from self-reported contact surveys and wearable proximity sensors from a cohort of schoolchildren in the Pittsburgh metropolitan area. Although the number and type of contacts recorded by each participant differed between the two methods, we found good correspondence between the two methods in aggregate measures of age-specific interactions. Fewer, but longer, contacts were reported in surveys, relative to the generally short proximal interactions captured by wearable sensors. When adjusted for expectations of proportionate mixing, though, the two methods produced highly similar, assortative age-mixing matrices. These aggregate mixing matrices, when used in simulation, resulted in similar estimates of risk of infection by age. While proximity sensors and survey methods may not be interchangeable for capturing individual contacts, they can generate highly correlated data on age-specific mixing patterns relevant to the dynamics of respiratory virus transmission.

Advances in the agent-based modeling of economic and social behavior

In this review we discuss advances in the agent-based modeling of economic and social systems. We show the state of the art of the heuristic design of agents and how behavioral economics and laboratory experiments have improved the modeling of agent behavior. We further discuss how economic networks and social systems can be modeled and we discuss novel methodology and data sources. Lastly, we present an overview of estimation techniques to calibrate and validate agent-based models and show avenues for future research.

Individual variations lead to universal and cross-species patterns of social behavior

The duration of interaction events in a society is a fundamental measure of its collective nature and potentially reflects variability in individual behavior. Here we performed a high-throughput measurement of trophallaxis and face-to-face event durations experienced by a colony of honeybees over their entire lifetimes. The interaction time distribution is heavy-tailed, as previously reported for human face-to-face interactions. We developed a theory of pair interactions that takes into account individual variability and predicts the scaling behavior for both bee and extant human datasets. The individual variability of worker honeybees was nonzero but less than that of humans, possibly reflecting their greater genetic relatedness. Our work shows how individual differences can lead to universal patterns of behavior that transcend species and specific mechanisms for social interactions.

Influence maximization on temporal networks

We consider the optimization problem of seeding a spreading process on a temporal network so that the expected size of the resulting outbreak is maximized. We frame the problem for a spreading process following the rules of the susceptible-infected-recovered model with temporal scale equal to the one characterizing the evolution of the network topology. We perform a systematic analysis based on a corpus of 12 real-world temporal networks and quantify the performance of solutions to the influence maximization problem obtained using different level of information about network topology and dynamics. We find that having perfect knowledge of the network topology but in a static and/or aggregated form is not helpful in solving the influence maximization problem effectively. Knowledge, even if partial, of the early stages of the network dynamics appears instead essential for the identification of quasioptimal sets of influential spreaders.

Generative models of simultaneously heavy-tailed distributions of interevent times on nodes and edges

Intervals between discrete events representing human activities, as well as other types of events, often obey heavy-tailed distributions, and their impacts on collective dynamics on networks such as contagion processes have been intensively studied. The literature supports that such heavy-tailed distributions are present for interevent times associated with both individual nodes and individual edges in networks. However, the simultaneous presence of heavy-tailed distributions of interevent times for nodes and edges is a nontrivial phenomenon, and its origin has been elusive. In the present study, we propose a generative model and its variants to explain this phenomenon. We assume that each node independently transits between a high-activity and low-activity state according to a continuous-time two-state Markov process and that, for the main model, events on an edge occur at a high rate if and only if both end nodes of the edge are in the high-activity state. In other words, two nodes interact frequently only when both nodes prefer to interact with others. The model produces distributions of interevent times for both individual nodes and edges that resemble heavy-tailed distributions across some scales. It also produces positive correlation in consecutive interevent times, which is another stylized observation for empirical data of human activity. We expect that our modeling framework provides a useful benchmark for investigating dynamics on temporal networks driven by non-Poissonian event sequences.

Sensing social interactions through BLE beacons and commercial mobile devices

Wearable sensing devices can provide high-resolution data useful to characterise and identify complex human behaviours. Sensing human social interactions through wearable devices represents one of the emerging field in mobile social sensing, considering their impact on different user categories and on different social contexts. However, it is important to limit the collection and use of sensitive information characterising individual users and their social interactions in order to maintain the user compliance. For this reason, we decided to focus mainly on physical proximity and, specifically, on the analysis of BLE wireless signals commonly used by commercial mobile devices. In this work, we present the SocializeME framework designed to collect proximity information and to detect social interactions through heterogeneous personal mobile devices. We also present the results of an experimental data collection campaign conducted with real users, highlighting technical limitations and performances in terms of quality of RSS, packet loss, and channel symmetry, and how they are influenced by different configurations of the user's body and the position of the personal device. Specifically, we obtained a dataset with more than 820.000 Bluetooth signals (BLE beacons) collected, with a total monitoring of over 11 h. The dataset collected reproduces 4 different configurations by mixing two user posture's layouts (standing and sitting) and different positions of the receiver device (in hand, in the front pocket and in the back pocket). The large number of experiments in those different configurations, well cover the common way of holding a mobile device, and the layout of a dyad involved in a social interaction. We also present the results obtained by SME-D algorithm, designed to automatically detect social interactions based on the collected wireless signals, which obtained an overall accuracy of 81.56% and F-score 84.7%. The collected and labelled dataset is also released to the mobile social sensing community in order to evaluate and compare new algorithms.

Understanding the complexities of Bluetooth for representing real-life social networks: A methodology for inferring and validating Bluetooth-based social network graphs

Bluetooth (BT) data has been extensively used for recognizing social patterns and inferring social networks, as BT is widely present in everyday technological devices. However, even though collecting BT data is subject to random noise and may result in substantial measurement errors, there is an absence of rigorous procedures for validating the quality of the inferred BT social networks. This paper presents a methodology for inferring and validating BT-based social networks based on parameter optimization algorithm and social network analysis (SNA). The algorithm performs edge inference in a brute-force search over a given BT data set, for deriving optimal BT social networks by validating them with predefined ground truth (GT) networks. The algorithm seeks to optimize a set of parameters, predefined considering some reliability challenges associated to the BT technology itself. The outcomes show that optimizing the parameters can reduce the number of BT data false positives or generate BT networks with the minimum amount of BT data observations. The subsequent SNA shows that the inferred BT social networks are unable to reproduce some network characteristics present in the corresponding GT networks. Finally, the generalizability of the proposed methodology is demonstrated by applying the algorithm on external BT data sets, while obtaining comparable results.

Age-specific social mixing of school-aged children in a US setting using proximity detecting sensors and contact surveys

Comparisons of the utility and accuracy of methods for measuring social interactions relevant to disease transmission are rare. To increase the evidence base supporting specific methods to measure social interaction, we compared data from self-reported contact surveys and wearable proximity sensors from a cohort of schoolchildren in the Pittsburgh metropolitan area. Although the number and type of contacts recorded by each participant differed between the two methods, we found good correspondence between the two methods in aggregate measures of age-specific interactions. Fewer, but longer, contacts were reported in surveys, relative to the generally short proximal interactions captured by wearable sensors. When adjusted for expectations of proportionate mixing, though, the two methods produced highly similar, assortative age-mixing matrices. These aggregate mixing matrices, when used in simulation, resulted in similar estimates of risk of infection by age. While proximity sensors and survey methods may not be interchangeable for capturing individual contacts, they can generate highly correlated data on age-specific mixing patterns relevant to the dynamics of respiratory virus transmission.

Evolution of cooperation on temporal networks

Population structure is a key determinant in fostering cooperation among naturally self-interested individuals in microbial populations, social insect groups, and human societies. Traditional research has focused on static structures, and yet most real interactions are finite in duration and changing in time, forming a temporal network. This raises the question of whether cooperation can emerge and persist despite an intrinsically fragmented population structure. Here we develop a framework to study the evolution of cooperation on temporal networks. Surprisingly, we find that network temporality actually enhances the evolution of cooperation relative to comparable static networks, despite the fact that bursty interaction patterns generally impede cooperation. We resolve this tension by proposing a measure to quantify the amount of temporality in a network, revealing an intermediate level that maximally boosts cooperation. Our results open a new avenue for investigating the evolution of cooperation and other emergent behaviours in more realistic structured populations.

Population structure enables emergence of cooperation among individuals, but the impact of the dynamic nature of real interaction networks is not understood. Here, the authors study the evolution of cooperation on temporal networks and find that temporality enhances the evolution of cooperation.

Transmission routes of rare seasonal diseases: the case of norovirus infections

Norovirus (NoV) is the most commonly recognized cause of acute gastroenteritis, with over a million cases globally per year. While usually self-limiting, NoV poses a substantial economic burden because it is highly contagious and there are multiple transmission routes. Infection occurs through inhalation of vomitus; faecal-oral spread; and food, water and environmental contamination. While the incidence of the disease is predictably seasonal, much less is known about the relative contribution of the various exposure pathways in causing disease. Additionally, asymptomatic excretion and viral shedding make forecasting disease burden difficult. We develop a novel stochastic dynamic network model to investigate the contributions of different transmission pathways in multiple coupled social networks representing schools, hospitals, care-homes and family households in a community setting. We analyse how the networks impact on transmission. We used ward-level demographic data from Northumberland, UK to create a simulation cohort. We compared the results with extant data on NoV cases from the IID2 study. Connectivity across the simulated cohort was high. Cases of NoV showed marked seasonality, peaking in early winter and declining through the summer. For the first time, we show that fomites and food appear to be the most important exposure routes in determining the population burden of disease. This article is part of the theme issue 'Modelling infectious disease outbreaks in humans, animals and plants: epidemic forecasting and control'. This theme issue is linked with the earlier issue 'Modelling infectious disease outbreaks in humans, animals and plants: approaches and important themes'.

Measuring social networks in primates: wearable sensors versus direct observations

Network analysis represents a valuable and flexible framework to understand the structure of individual interactions at the population level in animal societies. The versatility of network representations is moreover suited to different types of datasets describing these interactions. However, depending on the data collection method, different pictures of the social bonds between individuals could a priori emerge. Understanding how the data collection method influences the description of the social structure of a group is thus essential to assess the reliability of social studies based on different types of data. This is however rarely feasible, especially for animal groups, where data collection is often challenging. Here, we address this issue by comparing datasets of interactions between primates collected through two different methods: behavioural observations and wearable proximity sensors. We show that, although many directly observed interactions are not detected by the sensors, the global pictures obtained when aggregating the data to build interaction networks turn out to be remarkably similar. Moreover, sensor data yield a reliable social network over short time scales and can be used for long-term studies, showing their important potential for detailed studies of the evolution of animal social groups.

Efficient sentinel surveillance strategies for preventing epidemics on networks

Surveillance plays a crucial role in preventing emerging infectious diseases from becoming epidemic. In circumstances where it is possible to monitor the infection status of certain people, transport hubs, or hospitals, early detection of the disease allows interventions to be implemented before most of the damage can occur, or at least its impact can be mitigated. This paper addresses the question of which nodes we should select in a network of individuals susceptible to some infectious disease in order to minimize the number of casualties. By simulating disease outbreaks on a collection of empirical and synthetic networks we show that the best strategy depends on topological characteristics of the network. For highly modular or spatially embedded networks it is better to place the sentinels on nodes distributed across different regions. However, if the degree heterogeneity is high, then a strategy that targets network hubs is preferred. We further consider the consequences of having an incomplete sample of the network and demonstrate that the value of new information diminishes as more data is collected. Finally we find further marginal improvements using two heuristics informed by known results in graph theory that exploit the fragmented structure of sparse network data.

In a network of individuals susceptible to some infectious disease, what are the best locations to monitor in order to detect the infection before most damage can be done? In this paper we address this question by considering various heuristic strategies for sentinel placement that can potentially be implemented in real-world situations without requiring excessive amounts of computation, or even having perfect data about the structure of the network. We find that strategies that attempt to distribute sentinels over different regions of the network perform best in highly modular or spatially embedded networks, whereas the strategy of targeting the most well connected individuals works best when there is a considerable amount of contact heterogeneity between individuals. Our results may be used as a guideline to help decide when certain strategies should, or should not, be implemented.

Latent geometry and dynamics of proximity networks

Proximity networks are time-varying graphs representing the closeness among humans moving in a physical space. Their properties have been extensively studied in the past decade as they critically affect the behavior of spreading phenomena and the performance of routing algorithms. Yet the mechanisms responsible for their observed characteristics remain elusive. Here we show that many of the observed properties of proximity networks emerge naturally and simultaneously in a simple latent space network model, called dynamic-S^{1}. The dynamic-S^{1} does not model node mobility directly but captures the connectivity in each snapshot-each snapshot in the model is a realization of the S^{1} model of traditional complex networks, which is isomorphic to hyperbolic geometric graphs. By forgoing the motion component the model facilitates mathematical analysis, allowing us to prove the contact, intercontact, and weight distributions. We show that these distributions are power laws in the thermodynamic limit with exponents lying within the ranges observed in real systems. Interestingly, we find that network temperature plays a central role in network dynamics, dictating the exponents of these distributions, the time-aggregated agent degrees, and the formation of unique and recurrent components. Further, we show that paradigmatic epidemic and rumor-spreading processes perform similarly in real and modeled networks. The dynamic-S^{1} or extensions of it may apply to other types of time-varying networks and constitute the basis of maximum likelihood estimation methods that infer the node coordinates and their evolution in the latent spaces of real systems.

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.

Backbone reconstruction in temporal networks from epidemic data

Many complex systems are characterized by time-varying patterns of interactions. These interactions comprise strong ties, driven by dyadic relationships, and weak ties, based on node-specific attributes. The interplay between strong and weak ties plays an important role on dynamical processes that could unfold on complex systems. However, seldom do we have access to precise information about the time-varying topology of interaction patterns. A particularly elusive question is to distinguish strong from weak ties, on the basis of the sole node dynamics. Building upon analytical results, we propose a statistically-principled algorithm to reconstruct the backbone of strong ties from data of a spreading process, consisting of the time series of individuals' states. Our method is numerically validated over a range of synthetic datasets, encapsulating salient features of real-world systems. Motivated by compelling evidence, we propose the integration of our algorithm in a targeted immunization strategy that prioritizes influential nodes in the inferred backbone. Through Monte Carlo simulations on synthetic networks and a real-world case study, we demonstrate the viability of our approach.

Lifetime distributions for adjacency relationships in a Vicsek model

We investigate the statistical properties of adjacency relationships in a two-dimensional Vicsek model. We define adjacent edges for all particles at every time step by (a) Delaunay triangulation and (b) Euclidean distance, and obtain cumulative distributions P(τ) of lifetime τ of the edges. We find that the shape of P(τ) changes from an exponential to a power law depending on the interaction radius, which is a parameter of the Vicsek model. We discuss the emergence of the power-law distribution from the viewpoint of first passage time problem for a fractional Brownian motion.

Study design and protocol for investigating social network patterns in rural and urban schools and households in a coastal setting in Kenya using wearable proximity sensors

Background: Social contact patterns shape the transmission of respiratory infections spread via close interactions. There is a paucity of observational data from schools and households, particularly in developing countries. Portable wireless sensors can record unbiased proximity events between individuals facing each other, shedding light on pathways of infection transmission.

Design and methods: The aim is to characterize face-to-face contact patterns that may shape the transmission of respiratory infections in schools and households in Kilifi, Kenya. Two schools, one each from a rural and urban area, will be purposively selected. From each school, 350 students will be randomly selected proportional to class size and gender to participate. Nine index students from each school will be randomly selected and followed-up to their households. All index household residents will be recruited into the study. A further 3-5 neighbouring households will also be recruited to give a maximum of 350 participants per household setting. The sample size per site is limited by the number of sensors available for data collection. Each participant will wear a wireless proximity sensor lying on their chest area for 7 consecutive days. Data on proximal dyadic interactions will be collected automatically by the sensors only for participants who are face-to-face. Key characteristics of interest include the distribution of degree and the frequency and duration of contacts and their variation in rural and urban areas. These will be stratified by age, gender, role, and day of the week.

Expected results: Resultant data will inform on social contact patterns in rural and urban areas of a previously unstudied population. Ensuing data will be used to parameterize mathematical simulation models of transmission of a range of respiratory viruses, including respiratory syncytial virus, and used to explore the impact of intervention measures such as vaccination and social distancing.

Networks of face-to-face social contacts in Niakhar, Senegal

We present the first analysis of face-to-face contact network data from Niakhar, Senegal. Participants in a cluster-randomized influenza vaccine trial were interviewed about their contact patterns when they reported symptoms during their weekly household surveillance visit. We employ a negative binomial model to estimate effects of covariates on contact degree. We estimate the mean contact degree for asymptomatic Niakhar residents to be 16.5 (95% C.I. 14.3, 18.7) in the morning and 14.8 in the afternoon (95% C.I. 12.7, 16.9). We estimate that symptomatic people make 10% fewer contacts than asymptomatic people (95% C.I. 5%, 16%; p = 0.006), and those aged 0-5 make 33% fewer contacts than adults (95% C.I. 29%, 37%; p < 0.001). By explicitly modelling the partial rounding pattern observed in our data, we make inference for both the underlying (true) distribution of contacts as well as for the reported distribution. We created an estimator for homophily by compound (household) membership and estimate that 48% of contacts by symptomatic people are made to their own compound members in the morning (95% CI, 45%, 52%) and 60% in the afternoon/evening (95% CI, 56%, 64%). We did not find a significant effect of symptom status on compound homophily. We compare our findings to those from other countries and make design recommendations for future surveys.

Spreading dynamics in a cattle trade network: Size, speed, typical profile and consequences on epidemic control strategies

Infections can spread among livestock notably because infected animals can be brought to uncontaminated holdings, therefore exposing a new group of susceptible animals to the disease. As a consequence, the structure and dynamics of animal trade networks is a major focus of interest to control zoonosis. We investigate the impact of the chronology of animal trades on the dynamics of the process. Precisely, in the context of a basic SI model spreading, we measure on the French database of bovine transfers to what extent a snapshot-based analysis of the cattle trade networks overestimates the epidemic risks. We bring into light that an analysis taking into account the chronology of interactions would give a much more accurate assessment of both the size and speed of the process. For this purpose, we model data as a temporal network that we analyze using the link stream formalism in order to mix structural and temporal aspects. We also show that in this dataset, a basic SI spreading comes down in most cases to a simple two-phases scenario: a waiting period, with few contacts and low activity, followed by a linear growth of the number of infected holdings. Using this portrait of the spreading process, we identify efficient strategies to control a potential outbreak, based on the identification of specific elements of the link stream which have a higher probability to be involved in a spreading process.

Study design and protocol for investigating social network patterns in rural and urban schools and households in a coastal setting in Kenya using wearable proximity sensors

Background: Social contact patterns shape the transmission of respiratory infections spread via close interactions. There is a paucity of observational data from schools and households, particularly in developing countries. Portable wireless sensors can record unbiased proximity events between individuals facing each other, shedding light on pathways of infection transmission. Design and methods: The aim is to characterize face-to-face contact patterns that may shape the transmission of respiratory infections in schools and households in Kilifi, Kenya. Two schools, one each from a rural and urban area, will be purposively selected. From each school, 350 students will be randomly selected proportional to class size and gender to participate. Nine index students from each school will be randomly selected and followed-up to their households. All index household residents will be recruited into the study. A further 3-5 neighbouring households will also be recruited to give a maximum of 350 participants per household setting. The sample size per site is limited by the number of sensors available for data collection. Each participant will wear a wireless proximity sensor lying on their chest area for 7 consecutive days. Data on proximal dyadic interactions will be collected automatically by the sensors only for participants who are face-to-face. Key characteristics of interest include the distribution of degree and the frequency and duration of contacts and their variation in rural and urban areas. These will be stratified by age, gender, role, and day of the week. Expected results: Resultant data will inform on social contact patterns in rural and urban areas of a previously unstudied population. Ensuing data will be used to parameterize mathematical simulation models of transmission of a range of respiratory viruses, including respiratory syncytial virus, and used to explore the impact of intervention measures such as vaccination and social distancing.

Information diffusion backbones in temporal networks

Progress has been made in understanding how temporal network features affect the percentage of nodes reached by an information diffusion process. In this work, we explore further: which node pairs are likely to contribute to the actual diffusion of information, i.e., appear in a diffusion trajectory? How is this likelihood related to the local temporal connection features of the node pair? Such deep understanding of the role of node pairs is crucial to tackle challenging optimization problems such as which kind of node pairs or temporal contacts should be stimulated in order to maximize the prevalence of information spreading. We start by using Susceptible-Infected (SI) model, in which an infected (information possessing) node could spread the information to a susceptible node with a given infection probability β whenever a contact happens between the two nodes, as the information diffusion process. We consider a large number of real-world temporal networks. First, we propose the construction of an information diffusion backbone GB(β) for a SI spreading process with an infection probability β on a temporal network. The backbone is a weighted network where the weight of each node pair indicates how likely the node pair appears in a diffusion trajectory starting from an arbitrary node. Second, we investigate the relation between the backbones with different infection probabilities on a temporal network. We find that the backbone topology obtained for low and high infection probabilities approach the backbone GB(β → 0) and GB(β = 1), respectively. The backbone GB(β → 0) equals the integrated weighted network, where the weight of a node pair counts the total number of contacts in between. Finally, we explore node pairs with what local connection features tend to appear in GB(β = 1), thus actually contribute to the global information diffusion. We discover that a local connection feature among many other features we proposed, could well identify the (high-weight) links in GB(β = 1). This local feature encodes the time that each contact occurs, pointing out the importance of temporal features in determining the role of node pairs in a dynamic process.

Temporal information gathering process for node ranking in time-varying networks

Many systems are dynamic and time-varying in the real world. Discovering the vital nodes in temporal networks is more challenging than that in static networks. In this study, we proposed a temporal information gathering (TIG) process for temporal networks. The TIG-process, as a node's importance metric, can be used to do the node ranking. As a framework, the TIG-process can be applied to explore the impact of temporal information on the significance of the nodes. The key point of the TIG-process is that nodes' importance relies on the importance of its neighborhood. There are four variables: temporal information gathering depth n, temporal distance matrix D, initial information c, and weighting function f. We observed that the TIG-process can degenerate to classic metrics by a proper combination of these four variables. Furthermore, the fastest arrival distance based TIG-process ( fad-tig) is performed optimally in quantifying nodes' efficiency and nodes' spreading influence. Moreover, for the fad-tig process, we can find an optimal gathering depth n that makes the TIG-process perform optimally when n is small.