# Evolving Nature of Human Contact Networks with Its Impact on Epidemic Processes

Authors: Cong Li; Jing Li; Xiang Li
Journal: Complexity
 (2021)

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## Abstract

Human contact networks constitute a multitude of individuals and pairwise contacts among them. However, the dynamic nature, which generates the evolution of human contact networks, of contact patterns is unknown yet. Here, we analyse three empirical datasets and identify two crucial mechanisms of the evolution of temporal human contact networks, i.e., the activity state transition laws for an individual to be socially active and the contact establishment mechanism that active individuals adopt. We consider both of the two mechanisms to propose a temporal network model, the so-called memory-driven (MD) model, of human contact networks. Then, we study the susceptible-infected (SI) spreading processes on empirical human contact networks as well as four corresponding temporal network models and compare the full prevalence time of SI processes with various infection rates on the networks. The full prevalence time of SI processes in the MD model is the same as that in real-world human contact networks. Moreover, we find that the individual activity state transition promotes the spreading process, while the contact establishment of active individuals suppresses the prevalence. Besides, we observe that individuals who establish social ties with a small exploring rate are still able to induce an endemic which prevails in the networks. The study offers new insights to predict and control the diffusion processes on networks.


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## Body

## 1. Introduction

The evolution nature of human interactions creates diverse temporal properties, which fundamentally influence the epidemic spreading. Interactions may be characterized and modeled by the human interaction networks [1–5], where a node represents a person, and the interaction between two persons is a link. In 2004, Pentland et al. from the MIT Media Laboratory took the lead in tracking and recording human interactions with personal mobile phones and proposed a stochastic process-based model to capture the coevolution of social relationships and individual behaviors [6, 7]. Many previous studies of interaction networks assumed that the interactions are fixed, or the spreading processes are much faster than the evolution of networks. Nowadays, with the rapid technological advances, high quality and time-resolved datasets of human interactive behaviors can be easily obtained [8–13] and have provided an unprecedented opportunity to better understand human society dynamics with temporal human interaction networks [14–16]. However, it remains an important challenge to unveil the mechanisms driving the evolution of human interaction networks and to further capture the effects of such mechanisms on dynamical processes on networks [17–20].In human interaction networks, in general, each agent at timet is either active, contacting with other agents, or inactive, isolated. The activity state, i.e., activeness or inactiveness of an agent and the contacts between pairwise agents alter with time. Rocha and Blondel [21] introduced a temporal network model, where the time interval between two activity states of each agent follows a stochastic process with a certain distribution, and the inactive agents keep isolated from other active ones in the network. Nevertheless, more research efforts assumed that the nodes in the inactive state are not totally isolated and can receive contacts from other active nodes. A typical example is the framework of activity-driven networks [22–25]. Perra et al. [22] proposed the original activity-driven model of temporal networks, in which an agent is either active with probability p or inactive with probability 1−p at each unit time. Since the active probability of an agent is decided by its activity potential which follows a given distribution, regardless of its activity state at the last time step, the activity-driven model generates a sequence of memoryless networks. Besides, the social contacts between active nodes and other nodes are randomly established. An extension of this model reduced the randomness of link establishments by introducing a mutual selection mechanism, where the destination of a contact depends mutually on the activity potential of the agents on both ends of the connection [23]. Similarly, some studies added a time-invariant quantity, namely, attractiveness for each agent, accounting for the fact that some agents are more likely to be connected by other agents when building a contact [24, 25].A wealth of empirical observations has shown that the mechanisms which govern the evolution of human contact networks are far from random [14–16, 18, 26–29]. Individuals often remember the agents whom they previously interacted with and thus form their own social circles [18]. Besides, due to the continuity of social interactions, the activity state of an agent is unlikely to be altered in a sudden [13], especially when the agent is actively engaging in a social event with a lot of peers. Valdano et al. [28] proposed a homogeneous assumption that the transition probability between the active state and inactive state of each agent is identically assigned with the same value. Although the assumption partially captures the evolution of the activity state, an explanation of the transition mechanism is still missing. When it comes to the evolution of contacts between agents, the effect of memory, i.e., an agent preserving their pervious contacts with a certain probability, should be considered within the temporal network model. The model introduced in [19] incorporates four distinct memory mechanisms to describe the rates of creation and disappearance of contacts in empirical networks, where the rates follow a power-law decaying functional forms, akin to the preferential attachment mechanism [30]. Recently, more realistic mechanisms based on the activity-driven framework have been explored [14, 15, 26, 27]. They considered a non-Markovian reinforcement process, in which agents are more inclined to allocate their contacts towards already existing social ties rather than create new relationships. In particular, the probability of an agent exploring a new social relationship is determined by its cumulative degree, i.e., the number of distinct agents contacted during observation period; however, the transition of the activity states of each agent is not taken into account.The aforementioned studies have investigated various topological and temporal features of real social systems from different perspectives. However, a better understanding of the mechanisms driving the evolution of the temporal social contact networks and the effects of network evolution on dynamic processes, such as epidemic spreading [31–34], information propagation [20, 35], and innovation diffusion [36, 37], is still urgently needed.Exploring several datasets of human contact networks, we report two generic mechanisms of the evolution of real networks: (i) the transition probability of activity states is a function of the degree of agents and (ii) the time interval between the reconnection of node pairs follows a power-law distribution. Based on the two mechanisms, we propose a temporal network model, namely, the memory-driven model, to characterize the evolution of social networks. The memory here refers that an agent inclines to concentrate towards recently contacted partners. With the new framework, we study the epidemic spreading processes in both synthetic and real temporal networks. The full prevalence time of susceptible-infected (SI) spreading in the memory-driven model is the same as that in real networks. Moreover, we find that the two mechanisms of the network evolution have distinct effects on the epidemic spreading on real networks; that is, the activity state transition promotes the spreading process; however, the contact memory hampers the spreading. Apart from this, we demonstrate that even a smaller fraction of interactions connecting to new social relationships is able to induce an explosive spreading on networks.This paper is organized as follows. In Section2, we describe three datasets of real-world human interactive activities and introduce the definitions of human contact networks. In Section 3, we analyse the empirical datasets to obtain statistic properties of human contact networks and propose a temporal network model. Section 4 focuses on the epidemic spreading processes on the synthetic networks and real networks. Finally, we conclude this work in Section 5.


## 2. Empirical Datasets and Definitions

In this paper, we use three empirical datasets of offline human interactive activities in various scenes. The datasets include the offline sex interaction dataset (“Sex6yr,” [38, 39]) and the physical proximity interaction datasets (“MIT_RM,” [6] and “School,” [40]). The “Sex6yr” dataset collects the real offline sex trading activities among sellers and buyers with a forum-like Brazilian web community. If a buyer A posted a comment in a thread about seller B, a contact between A and B exists. The “MIT_RM” dataset records the call logs, Bluetooth devices in proximity, cell tower IDs, application usage, and so on among students and faculty in the MIT Media Laboratory via the Bluetooth device in mobile phones. We here studied the physical proximity interactions with the sampling data of physical proximity via Bluetooth devices in the “MIT_RM” dataset. The “School” dataset collects the face-to-face proximity interactions among teachers and students in a French school by the Radio-Frequency IDentification (RFID) device embedded in badges. The contact records of each dataset are in the same format i,j,t, which represents a contact between two individuals i and j started at time t. More details of the datasets are introduced in Appendix A and summarized in Table 1.Table 1
Description of the three datasets.
DatasetT△tNLCRSex6yr2,232 day1 day16,10238,99550,185MIT_RM232 day6 min962,53955,306School8.67 hour20 sec2365,90137,402T is the time span of contact sequences, △t is the resolution (time interval between two records in time order), N is the number of agents, LC is the number of distinct contacts, and R is the number of records.The human contact network is a kind of temporal network, which consists of the evolution of network topology with time. We segment the empirical data into adjacency time steps of lengthtw=w△t, where △t is the resolution and w is the number of resolutions in a time step. A human contact network is a sequence of networks in time order Gtw=G1,G2,…,Gt,…,GT, where Gt=Nt,ℒt is the network taking place at time step t over time window t−1tw,t⋅tw and T is the total number of time steps. An illustration of the human contact network is shown in Figure 1. In this work, we select tw as 1 month, 1 day, and 5 mins for “Sex6yr,” “MIT_RM,” and “School,” respectively (explanation for the selection is given in Appendix B). The aggregated network of Gtw is GT=∪t=1t=TGt=N,ℒ, where N=∪t=1t=TNt and ℒ=∪t=1t=Tℒt. We denote the number of agents and the number of contacts by N=N and LC=ℒ, respectively.Figure 1
Schematic illustration of the construction of temporal networks and the time slice of a contact sequence, wheretw is the length of a time step.


## 3. Analysis of Temporal Properties

Before we introduce the temporal network model, we first empirically analyse statistic properties of human contact networks, i.e., the activity state transition of individuals and the contact memory of node pairs, which are the fundamental mechanisms for the design of our temporal network model.


### 3.1. Individual Activity State Transition

We here assume that each node at time stept in the network is in one of two states: active state, connecting with other nodes, denoted by a and inactive state i. The sets of nodes in active state and in inactive state at time step t are denoted by Nat and Nit, respectively. We here study whether the activity state of an individual j at time step t+1 depends on the state at previous step t and how the degree k of individual j at time step t influences the activity state transition. We denote the number of nodes with degree k in active state at time step t by Nk,at=Nk,at, where N represents the number of nodes in a node set N. The transition probability of nodes with degree k transferring from active state a to inactive state i at one time step is(1)PAIk=∑t=1T−1Nk,at∩Nit+1∑t=1T−1Nk,at.The transition probability of nodes with degreek staying at active state a in one time step is PAAk=1−PAIk. Note that the degree of an individual might change over time since the individual might take part in or leave from the interactions of the system across time.The transition probabilities characterize the relation between the activity state of nodei with degree k at the current time step and that at next time step. Although it has been observed in empirical networks (e.g., Facebook) that the strong link between activity and degree [18], the detail about how they are related and how to apply it to network modeling is not fully discussed previously. Here, we study how the transition probability, i.e., PAIk or PAAk, varies with the degree k. Figures 2(a)–2(c) show the transition probabilities of remaining in the active status or switching to the inactive status calculated on three empirical networks. Remarkably, the transition probability PAIk or PAAk is as a power-law function of degree k. The greater the degree of active agents at the current time step, the smaller the probability they become inactive at the next time step. That is to say, agents with high social activity (high degree) have more tendency to maintain the active status all the time. However, we cannot find the relation between the transition probability PIAk (or PIIk) and degree k since the degree of nodes in inactive state is 0. Hence, we study the transition property of inactive nodes with the firing rate b=Nit∩Nat+1/Nit, which is the probability of an inactive node becoming active at one time step. The average firing rate PIA is expressed as(2)PIA=1T−1∑t=1T−1Nit∩Nat+1Nit.Figure 2
Transition probabilities between active state and inactive state as functions of the current degreek of the node, for the three empirical datasets. Solid lines represent the fitting curves: (a) PAIk=0.78k−0.84 (Sex6yr), (b) PAIk=0.45k−0.56 (MIT_RM), and (c) PAIk=0.33k−0.91 (School). (d) The firing rate PIA with 95% confidence intervals in the three empirical datasets. The average firing rates PIA are 0.04 (Sex6yr), 0.16 (MIT_RM), and 0.27 (School).
(a)(b)(c)(d)Figure2(d) demonstrates that the variance of firing rates at different time steps is small.


### 3.2. Contact Establishment of Active Individuals

Previous studies [22–25] assume that all active individuals contact with s other individuals at each time step, where s=m is a constant. However, we find that the number of contacts (degree) at each time step follows a different distribution in empirical datasets (see Figure 3(a)). Figure 3 suggests that the assumption of the same number of contacts cannot reflect the real properties; thus, the distribution Fs of contact number s should be considered for the temporal network models with different datasets.Figure 3
(a) Degree distributionsPk at each time step in an empirical dataset and (b) degree distribution Pk at each time step in the activity-driven (AD) model [22] (to be described in Section 5.1) with s=5 and N=1000, and node activity xi is sampled from Fxi∝xi−ν with ν=2.1, 10−2≤xi≤1.
(a)(b)An active individual either establishes a contact by social keeping, connecting to a node which has already been connected to at previous time steps, or social exploring, connecting to a new one. The average fraction of social keeping contacts of each individual at each time step is(3)p=1NT∑i=1N∑t=1Tpit,in which the fraction pit of reconnected links of node i is(4)pit=∑j∈VitδijtVit,where δijt=0 if agent i never interacted with agent j before; otherwise, δijt=1. The Vit is the number of direct neighbors of agent i at time step t. The fraction pit quantifies the inclination of an agent to keep previous established contacts. Figure 4(a) shows that in the context of sex trade, individuals dominantly adopt the strategy of social exploring, while in the context of physical proximity (MIT_RM) and face-to-face communication (School), individuals tend to reconnect with the nodes who have already contacted by social keeping.Figure 4
(a) Average fraction of social keeping contacts of each individual at each time step, withp=19% (in Sex6yr), p=92% (in MIR_RM), and p=92% (in School). (b) Probability distributions PΔi,j of the recurrence intervals Δi,j of the contacts i,j for each individual i. Here, we plot the distribution for 10 individuals as an example. (c) The intercontact time distributions PΔ in empirical datasets, where Δ is the recurrence interval for any contact.
(a)(b)(c)Moreover, the recurrence interval of human contacts is analysed. The recurrence intervalΔi,j of individuals i and j refers to the number of time steps between any two consecutive contacts. We can deduce that, if an active node i has a degree ki at each time step and ki contacts are established randomly with other nodes, the distribution of recurrence interval is an exponential distribution as PΔi,j=ki/N1−ki/NΔi,j−1. Note that this is the basic assumption in previous works [14, 26–28]. However, we find that the probability distributions of recurrence intervals of contacts between node i and all others follow a power-law distribution in three datasets (see Figure 4(b)). The finding might give a support for the burst property of temporal networks that the intercontact time distribution has a power-law form (see Figure 4(c)), which has also been reported recently in [15, 41, 42]. The results imply that agents do not establish contacts randomly with others regardless of their pervious contacts. Besides, we have identified the statistical law in real-world systems that agents preferentially contact individuals who have recently been in interact with [29].To summarize, two typical characteristics coexist in the process of realistic network evolution, individual activity state transition, and contact establishment. In the next section, we propose a temporal network model based on the mechanisms.


## 3.1. Individual Activity State Transition

We here assume that each node at time stept in the network is in one of two states: active state, connecting with other nodes, denoted by a and inactive state i. The sets of nodes in active state and in inactive state at time step t are denoted by Nat and Nit, respectively. We here study whether the activity state of an individual j at time step t+1 depends on the state at previous step t and how the degree k of individual j at time step t influences the activity state transition. We denote the number of nodes with degree k in active state at time step t by Nk,at=Nk,at, where N represents the number of nodes in a node set N. The transition probability of nodes with degree k transferring from active state a to inactive state i at one time step is(1)PAIk=∑t=1T−1Nk,at∩Nit+1∑t=1T−1Nk,at.The transition probability of nodes with degreek staying at active state a in one time step is PAAk=1−PAIk. Note that the degree of an individual might change over time since the individual might take part in or leave from the interactions of the system across time.The transition probabilities characterize the relation between the activity state of nodei with degree k at the current time step and that at next time step. Although it has been observed in empirical networks (e.g., Facebook) that the strong link between activity and degree [18], the detail about how they are related and how to apply it to network modeling is not fully discussed previously. Here, we study how the transition probability, i.e., PAIk or PAAk, varies with the degree k. Figures 2(a)–2(c) show the transition probabilities of remaining in the active status or switching to the inactive status calculated on three empirical networks. Remarkably, the transition probability PAIk or PAAk is as a power-law function of degree k. The greater the degree of active agents at the current time step, the smaller the probability they become inactive at the next time step. That is to say, agents with high social activity (high degree) have more tendency to maintain the active status all the time. However, we cannot find the relation between the transition probability PIAk (or PIIk) and degree k since the degree of nodes in inactive state is 0. Hence, we study the transition property of inactive nodes with the firing rate b=Nit∩Nat+1/Nit, which is the probability of an inactive node becoming active at one time step. The average firing rate PIA is expressed as(2)PIA=1T−1∑t=1T−1Nit∩Nat+1Nit.Figure 2
Transition probabilities between active state and inactive state as functions of the current degreek of the node, for the three empirical datasets. Solid lines represent the fitting curves: (a) PAIk=0.78k−0.84 (Sex6yr), (b) PAIk=0.45k−0.56 (MIT_RM), and (c) PAIk=0.33k−0.91 (School). (d) The firing rate PIA with 95% confidence intervals in the three empirical datasets. The average firing rates PIA are 0.04 (Sex6yr), 0.16 (MIT_RM), and 0.27 (School).
(a)(b)(c)(d)Figure2(d) demonstrates that the variance of firing rates at different time steps is small.


## 3.2. Contact Establishment of Active Individuals

Previous studies [22–25] assume that all active individuals contact with s other individuals at each time step, where s=m is a constant. However, we find that the number of contacts (degree) at each time step follows a different distribution in empirical datasets (see Figure 3(a)). Figure 3 suggests that the assumption of the same number of contacts cannot reflect the real properties; thus, the distribution Fs of contact number s should be considered for the temporal network models with different datasets.Figure 3
(a) Degree distributionsPk at each time step in an empirical dataset and (b) degree distribution Pk at each time step in the activity-driven (AD) model [22] (to be described in Section 5.1) with s=5 and N=1000, and node activity xi is sampled from Fxi∝xi−ν with ν=2.1, 10−2≤xi≤1.
(a)(b)An active individual either establishes a contact by social keeping, connecting to a node which has already been connected to at previous time steps, or social exploring, connecting to a new one. The average fraction of social keeping contacts of each individual at each time step is(3)p=1NT∑i=1N∑t=1Tpit,in which the fraction pit of reconnected links of node i is(4)pit=∑j∈VitδijtVit,where δijt=0 if agent i never interacted with agent j before; otherwise, δijt=1. The Vit is the number of direct neighbors of agent i at time step t. The fraction pit quantifies the inclination of an agent to keep previous established contacts. Figure 4(a) shows that in the context of sex trade, individuals dominantly adopt the strategy of social exploring, while in the context of physical proximity (MIT_RM) and face-to-face communication (School), individuals tend to reconnect with the nodes who have already contacted by social keeping.Figure 4
(a) Average fraction of social keeping contacts of each individual at each time step, withp=19% (in Sex6yr), p=92% (in MIR_RM), and p=92% (in School). (b) Probability distributions PΔi,j of the recurrence intervals Δi,j of the contacts i,j for each individual i. Here, we plot the distribution for 10 individuals as an example. (c) The intercontact time distributions PΔ in empirical datasets, where Δ is the recurrence interval for any contact.
(a)(b)(c)Moreover, the recurrence interval of human contacts is analysed. The recurrence intervalΔi,j of individuals i and j refers to the number of time steps between any two consecutive contacts. We can deduce that, if an active node i has a degree ki at each time step and ki contacts are established randomly with other nodes, the distribution of recurrence interval is an exponential distribution as PΔi,j=ki/N1−ki/NΔi,j−1. Note that this is the basic assumption in previous works [14, 26–28]. However, we find that the probability distributions of recurrence intervals of contacts between node i and all others follow a power-law distribution in three datasets (see Figure 4(b)). The finding might give a support for the burst property of temporal networks that the intercontact time distribution has a power-law form (see Figure 4(c)), which has also been reported recently in [15, 41, 42]. The results imply that agents do not establish contacts randomly with others regardless of their pervious contacts. Besides, we have identified the statistical law in real-world systems that agents preferentially contact individuals who have recently been in interact with [29].To summarize, two typical characteristics coexist in the process of realistic network evolution, individual activity state transition, and contact establishment. In the next section, we propose a temporal network model based on the mechanisms.


## 4. Modelling Dynamic Contact Networks

### 4.1. Memory-Driven (MD) Model

In this section, we present a temporal network model, named the memory-driven (MD) model, of human contact networks. The empirical analysis of individual activity state transition and contact establishment are both the basic mechanisms for the MD model. We consider a set of agentsN in a human contact network G. For each time step, every agent has two possible activity states: active and inactive. We here assume that each individual has the maximum memory length L time steps; in other words, the contact establishment at current time step is only influenced by the connections at pervious L time steps, which is stored in the memory train Gℳ=Gt−L+1,Gt−L+2,…,Gt−1,Gt. We generate L random networks [43] to initialize the temporal network G=G1,G2,…,GL and the memory train Gℳ=G1,G2,…,GL. The generation of the temporal network G=G1,G2,…,GT is illustrated in Figure 5 and described as follows:(1)
Individual activity state transition: an active individuali with degree k at time step t becomes inactive at time step t+1 with probability PAIk=Ak−α (A∈0,1), while an inactive agent becomes active with a constant firing rate PIA.(2)
Contact establishment of active individuals: we assign each active individual withs contacts, where s is extracted from a given probability distribution Fs. The contacts are established by the following steps:(a)
Social keeping: With a keeping ratep∈0,1, agent i connects to a previous contacted agent j with probability pij=fΔi,j∝Δi,j−γ,1≤Δi,j≤L, which is a function of the time interval Δi,j since their last interaction.(b)
Social exploring: with an exploring rateq=1−p, the agent randomly interacts with a new agent who has never been contacted by agent i before (or the previous contacts have exceeded the memory length L).(3)
Generate the networkGt+1 and update the memory train as Gℳ=Gt−L+2,Gt−L+3,…,Gt,Gt+1.(4)
Repeat (1)–(3) until the end of time spanT of the temporal network.Figure 5
(a) Transition probabilities of individual activity states in the MD network model. (b) Contact establishment strategies of each agent in each time step: social keeping (blue) and social exploring (white).
(a)(b)


### 4.2. Analysis of the Model

We perform numerical simulations in terms of two model settings in Table2. In Figure 6, the network properties, i.e., the degree distribution ρk and the intercontact time distributions PΔ, of the memory-driven model are compared with those of the corresponding datasets. The results show that they exhibit pretty similar behaviors; that is, the models reproduce both the degree distributions of the aggregated empirical networks and the intercontact time distributions quite well. Furthermore, we compare the MD model with a null model, whose pij follows a uniform distribution instead of a power-law distribution. Figure 6(f) indicates a significant difference between the null model and the MD model, implying that the mechanisms of the MD model are responsible for the temporal properties in human contact networks.Figure 6
Distributions of the characteristic measures of the real-world networks and memory-driven networks. (a, b) The degree distributionsρk of the aggregated empirical networks and (d, e) the degree distributions ρk of the aggregated memory-driven networks. (c, f) The intercontact time distributions PΔ in the dataset “School,” the memory-driven networks with pij following a power-law distribution (in black), and the null model with a uniform distribution (in cyan). The synthetic networks are performed with parameters in Table 2.
(a)(b)(c)(d)(e)(f)Table 2
The parameters of the memory-driven model.
Memory-driven (MD) modelParametersSex6yrMIT_RMSchoolN16,70696236T64230104A0.780.450.33α0.840.560.91PIA0.040.160.27FsPl−2.6Norm9.0,5.1Norm7.0,3.4γ1.871.621.62L366012X∼Plβ represents the power-law ρx=β−1x−β, where x∈1,+∞, and X∼Normμ,σ2 represents the Gaussian distribution with the mean μ and variance σ2.Based on the above descriptions, we perform numerical simulations with two different parameter settings, that is, we assume thatFs follows a power-law distribution and a Gaussian distribution, respectively, to characterize the heterogeneity or homogeneity of human contact networks. The results in Figures 7(a) and 7(b) show that the degree distributions ρk of integrated networks generated by our model might be various for different Fs. If Fs follows a power-law distribution, ρk will have a power-law tail with the same exponent γ regardless of the keeping rates p (see Figure 7(a)). If Fs follows a Gaussian distribution, the functional form of the degree distribution ρk does not change with p; however, the average degree of the integrated network decreases with the increase in p (see Figure 7(b)). The intercontact time distributions PΔ of the two networks are shown in Figures 7(c) and 7(d). When the keeping rate p=0, the establishment of human contacts is random and memoryless, which leads to an exponential distributions PΔ of the intercontact time (insets of Figures 7(c) and 7(d)). With the increase in p, the intercontact time distribution asymptotically follows a power-law, indicating that the memory embedded in the process of contact establishment can induce the bursty interactive pattern of human activities.Figure 7
(a, b) The degree distributionsρk of integrated networks and (c, d) intercontact time distributions PΔ versus the keeping rate p for one simulation. We fix N=1000, A=1, α=1, PIA=0.1, L=40, and T=200. In (a) and (c), the number of contacts s is sampled from a power-law distribution Fs=s/smin−β with β=2.8, smin=1, and γ1=1.6. In (b) and (d), the distribution Fs satisfies a Gaussian distribution Normμ,σ2, with the mean μ=5 and variance σ2=1 and γ2=1.8. The intercontact time follows an exponent distribution when p=0 (insets of (c) and (d)).
(a)(b)(c)(d)


## 4.1. Memory-Driven (MD) Model

In this section, we present a temporal network model, named the memory-driven (MD) model, of human contact networks. The empirical analysis of individual activity state transition and contact establishment are both the basic mechanisms for the MD model. We consider a set of agentsN in a human contact network G. For each time step, every agent has two possible activity states: active and inactive. We here assume that each individual has the maximum memory length L time steps; in other words, the contact establishment at current time step is only influenced by the connections at pervious L time steps, which is stored in the memory train Gℳ=Gt−L+1,Gt−L+2,…,Gt−1,Gt. We generate L random networks [43] to initialize the temporal network G=G1,G2,…,GL and the memory train Gℳ=G1,G2,…,GL. The generation of the temporal network G=G1,G2,…,GT is illustrated in Figure 5 and described as follows:(1)
Individual activity state transition: an active individuali with degree k at time step t becomes inactive at time step t+1 with probability PAIk=Ak−α (A∈0,1), while an inactive agent becomes active with a constant firing rate PIA.(2)
Contact establishment of active individuals: we assign each active individual withs contacts, where s is extracted from a given probability distribution Fs. The contacts are established by the following steps:(a)
Social keeping: With a keeping ratep∈0,1, agent i connects to a previous contacted agent j with probability pij=fΔi,j∝Δi,j−γ,1≤Δi,j≤L, which is a function of the time interval Δi,j since their last interaction.(b)
Social exploring: with an exploring rateq=1−p, the agent randomly interacts with a new agent who has never been contacted by agent i before (or the previous contacts have exceeded the memory length L).(3)
Generate the networkGt+1 and update the memory train as Gℳ=Gt−L+2,Gt−L+3,…,Gt,Gt+1.(4)
Repeat (1)–(3) until the end of time spanT of the temporal network.Figure 5
(a) Transition probabilities of individual activity states in the MD network model. (b) Contact establishment strategies of each agent in each time step: social keeping (blue) and social exploring (white).
(a)(b)


## 4.2. Analysis of the Model

We perform numerical simulations in terms of two model settings in Table2. In Figure 6, the network properties, i.e., the degree distribution ρk and the intercontact time distributions PΔ, of the memory-driven model are compared with those of the corresponding datasets. The results show that they exhibit pretty similar behaviors; that is, the models reproduce both the degree distributions of the aggregated empirical networks and the intercontact time distributions quite well. Furthermore, we compare the MD model with a null model, whose pij follows a uniform distribution instead of a power-law distribution. Figure 6(f) indicates a significant difference between the null model and the MD model, implying that the mechanisms of the MD model are responsible for the temporal properties in human contact networks.Figure 6
Distributions of the characteristic measures of the real-world networks and memory-driven networks. (a, b) The degree distributionsρk of the aggregated empirical networks and (d, e) the degree distributions ρk of the aggregated memory-driven networks. (c, f) The intercontact time distributions PΔ in the dataset “School,” the memory-driven networks with pij following a power-law distribution (in black), and the null model with a uniform distribution (in cyan). The synthetic networks are performed with parameters in Table 2.
(a)(b)(c)(d)(e)(f)Table 2
The parameters of the memory-driven model.
Memory-driven (MD) modelParametersSex6yrMIT_RMSchoolN16,70696236T64230104A0.780.450.33α0.840.560.91PIA0.040.160.27FsPl−2.6Norm9.0,5.1Norm7.0,3.4γ1.871.621.62L366012X∼Plβ represents the power-law ρx=β−1x−β, where x∈1,+∞, and X∼Normμ,σ2 represents the Gaussian distribution with the mean μ and variance σ2.Based on the above descriptions, we perform numerical simulations with two different parameter settings, that is, we assume thatFs follows a power-law distribution and a Gaussian distribution, respectively, to characterize the heterogeneity or homogeneity of human contact networks. The results in Figures 7(a) and 7(b) show that the degree distributions ρk of integrated networks generated by our model might be various for different Fs. If Fs follows a power-law distribution, ρk will have a power-law tail with the same exponent γ regardless of the keeping rates p (see Figure 7(a)). If Fs follows a Gaussian distribution, the functional form of the degree distribution ρk does not change with p; however, the average degree of the integrated network decreases with the increase in p (see Figure 7(b)). The intercontact time distributions PΔ of the two networks are shown in Figures 7(c) and 7(d). When the keeping rate p=0, the establishment of human contacts is random and memoryless, which leads to an exponential distributions PΔ of the intercontact time (insets of Figures 7(c) and 7(d)). With the increase in p, the intercontact time distribution asymptotically follows a power-law, indicating that the memory embedded in the process of contact establishment can induce the bursty interactive pattern of human activities.Figure 7
(a, b) The degree distributionsρk of integrated networks and (c, d) intercontact time distributions PΔ versus the keeping rate p for one simulation. We fix N=1000, A=1, α=1, PIA=0.1, L=40, and T=200. In (a) and (c), the number of contacts s is sampled from a power-law distribution Fs=s/smin−β with β=2.8, smin=1, and γ1=1.6. In (b) and (d), the distribution Fs satisfies a Gaussian distribution Normμ,σ2, with the mean μ=5 and variance σ2=1 and γ2=1.8. The intercontact time follows an exponent distribution when p=0 (insets of (c) and (d)).
(a)(b)(c)(d)


## 5. Epidemic Processes in Temporal Human Contact Networks and Models

### 5.1. Dynamical Processes on Real-World Networks and Network Models

We study the susceptible-infected (SI) spreading dynamics [44, 45] on temporal human contact networks. The SI epidemic model, despite of its simplicity, has been pervasively operated as a useful tool to probe the topological structures and temporal characteristics of temporal networks [17, 19, 46]. In this model, each agent can be in two possible states: susceptible (S) or infected (I). A susceptible (S) agent is infected by an infected (I) one with probability λ if there is a contact between them. Once an agent is infected (I), it will not recover. We start each simulation with all agents in the susceptible state and randomly select a single agent as the infected “seed.” In order to uncover the effects of the two mechanisms, i.e., individual activity state transition and contact establishment, on the network topologies and further impacts on spreading dynamic processes on temporal networks, we proceed the SI spreading processes on real-world networks and five corresponding network models. The five network models include our memory-driven (MD) model, which has been introduced in Section 4.1, two null models (i.e., AR model and CR model), and two synthetic network models.We here first introduce two null models where each of the mechanisms is separately destroyed. Besides, as a contrast, we reproduce the temporal network of empirical datasets with two classes of the activity-driven model; that is, activity-driven networks without memory (AD model) and with memory (ADM model) [22, 26]. The former case has been widely used as a paradigmatic network example for the study of spreading dynamics taking place on the same time-scale of network evolution [23, 24, 47, 48]. The models are described in detail as follows:(i)
Agent randomized (AR) null modelThe AR null model only keeps the contact establishment mechanism that the recurrence intervals of the contacts for each individual are preserved. We keep all the human interaction time in the empirical datasets unchanged and replace each contact pair with two randomly selected individuals. Note that once a contact pairu1,v1 is replaced by u2,v2, all the contacts between individuals u1 and v1 will be replaced by contact pair u2 and v2.(ii).
Contact randomized (CR) null modelThe CR null model only keeps the individual activity state mechanism that the degree and activity state of individuals are unchanged in the networkGt at time step t. At each time step t, we randomly select two contacts associated with the four individuals and then rewire the two contacts. If the two contacts both have the same individual, we discard the contact pair and randomly select two new contacts. The rewiring steps are repeated more than 2LCt times to ensure the rewiring of most contacts in Gt, where LCt is the number of contacts in Gt. After the rewiring procedure, the interaction time and recurrence interval of human contacts are completely different from those of original interaction datasets.(iii)
Activity-driven (AD) modelThe AD model [22] is one of the most studied temporal network models. The generation of an AD model follows the rules. At each time step t, the Gt starts with N isolated nodes, and each node i is assigned an activity probability aitw to become active; then, the active nodes generate m links that are randomly connected to m other nodes, and the inactive nodes can only receive connections from active nodes; at next time step t+1, all connections in Gt are omitted, and the steps are repeated. Here, we apply the AD model to generate temporal networks to match the real-world networks. The activity probability ai=ηxi, where xi∈ε,1, ε is a lower cut-off that avoids possible divergences, and the xi is drawn from a given probability function Fxi. The probability function Fxi is statistically obtained from empirical data, and the average number of active agents per unit time is ηxN, where η=1 in this work. The AD model does not capture the above two mechanisms in empirical data.(iv)
Activity driven with memory (ADM) modelA major shortcoming of the AD model lies in the total absence of correlations between the connections in consecutive time steps. The ADM model [26] extends the AD model by introducing a memory mechanism into to link creation. In the ADM model, each node remembers all its previously connected nodes. At each time step t, the Gt starts as in the AD model with N disconnected nodes, and each node i becomes active with probability aitw. Each active node i interacts with one of the previously contacted n nodes with probability pADM=n/n+1 and randomly connects a new node with probability 1−pADM.In this work, the virus spreads on real human contact networks under a temporally periodic boundary condition (i.e., repeating the whole contact sequence) and network models until all the reachable agents are infected [17]. The five models have been introduced in detail in Sections 4.1 and 5.1. The relevant parameters of the MD model and AD model are measured in the corresponding datasets (see details on Tables 2 and 3). The parameters of the ADM model are the same as those of the AD model except the memory effect. We repeat the SI spreading process for all possible seeds and record the full prevalence time tf, i.e., the time to reach 100% infection in a connected network or the largest connected component (LCC) of a disconnected network.Table 3
The parameters of the activity-driven model.
Activity-driven (AD) modelParametersSex6yrMIT_RMSchoolN16,70696236T64230104FxPl−2.6U0,1U0,1ε10−210−210−2m153X∼Ua,b represents the uniform distribution from a to b.We calculate the ratiotfXλ/tfrealλ as a function of the infection rate λ, where tfreal and tfX represent the average spreading time of a full prevalence in the real-world network real and the temporal network model X, respectively. If the ratio tfXλ/tfrealλ=1, the full prevalence time of a spreading in the real-world network is the same as that in the corresponding network model X. In other words, X is a reasonable temporal model for human contact networks in the study of spreading processes. As shown in Figure 8, the results in three empirical datasets are similar. The ratio tfAR/tfreal is always larger than 1 regardless of the infection rate λ, and the ratio tfAR/tfreal increases with the increase in the infection rate λ. The results illustrate that the individual state transition mechanism might promote the spreading processes since the spread of virus in the AR model, which only keeps the contact establishment mechanism and omits the individual activity state transition mechanism, is faster than that in real human contact networks. Moreover, the promotion of spreading is even larger when the infection rate λ is larger. However, there is a very small difference between the CR model and AD model, where the ratios tfCR/tfreal and tfAD/tfreal are both much smaller than 1. Both models are built without the contact establishment mechanism, which might cause a slowing-down effect on the spreading dynamics. Besides, the ratios tfAR/tfreal and tfADM/tfreal are both larger than 1, provided that they both cannot capture the mechanism of individual activity state transition. We find that the ratio tfMD/tfreal always fluctuates around 1 and remains stable against the infection rate λ. Figure 8 confirms that the MD model is superior to all other models in characterizing human contact networks. This phenomenon highlights the crucial role of the two mechanisms considered in our model. They both serve as an indispensable component to characterize the evolution of the real-world networks. Binding them together enables us to accurately capture the contagion processes unfolding on empirical human contact networks.Figure 8
The ratiotfX/tfreal of the average full prevalence time as a function of the infection rate λ in three empirical datasets, where X=AR,CR,AD,ADM,MD represents the agents randomized null models (▽), contacts randomized null models (⋄), activity-driven models (△), activity-driven models with memory (◯), and memory-driven models (□), respectively. The simulations are performed on 103 realizations. The error bars here represent one binomial standard deviation from these averages with 95% confidence levels. All the models present the same critical spreading rate as the empirical counterparts.


### 5.2. Dynamical Processes on Memory-Driven Networks

We here study how the network evolution affects dynamic processes, such as the epidemic spreading on networks. The exploring rateq in the contact establishment mechanism of the MD model can tune the evolution of network topology. We perform the SI spreading processes with infection rates λ=0.2,0.4, and 1.0 on temporal networks generated by the MD model with N=1000 and the exploring rates q=0,0.1,…,1, respectively, and record the full prevalence time tf. Figure 9 indicates that tf is extremely large when q=0, which implies that all contacts in current time step are recurrences of previously established contacts; that is, no individual is willing to explore a new social relationship in the evolving network. Moreover, we observe that the average full prevalence time tf declines rapidly with the exploring rate q, when q is small. When the exploring rate q further increases and reaches a specific value (around 0.1), the spreading time decreases slightly and finally meets a saturation. The fact illustrates that even when individuals establish new social ties with a small exploring rate q, the spreading process will significantly speed up; however, when q is larger than a specific value, the influence of q on the spreading process is rather limited.Figure 9
The average full prevalence time as a function of the exploring rateq versus the infection rate λ. The temporal networks are generated by the MD model with N=1000, A=1, α=1, γ=1.6, PIA=0.1, L=40, T=3000, and s is sampled from Fs=s/smin−β with β=2.8 and smin=1. The simulation result is the average of 200 realizations.


## 5.1. Dynamical Processes on Real-World Networks and Network Models

We study the susceptible-infected (SI) spreading dynamics [44, 45] on temporal human contact networks. The SI epidemic model, despite of its simplicity, has been pervasively operated as a useful tool to probe the topological structures and temporal characteristics of temporal networks [17, 19, 46]. In this model, each agent can be in two possible states: susceptible (S) or infected (I). A susceptible (S) agent is infected by an infected (I) one with probability λ if there is a contact between them. Once an agent is infected (I), it will not recover. We start each simulation with all agents in the susceptible state and randomly select a single agent as the infected “seed.” In order to uncover the effects of the two mechanisms, i.e., individual activity state transition and contact establishment, on the network topologies and further impacts on spreading dynamic processes on temporal networks, we proceed the SI spreading processes on real-world networks and five corresponding network models. The five network models include our memory-driven (MD) model, which has been introduced in Section 4.1, two null models (i.e., AR model and CR model), and two synthetic network models.We here first introduce two null models where each of the mechanisms is separately destroyed. Besides, as a contrast, we reproduce the temporal network of empirical datasets with two classes of the activity-driven model; that is, activity-driven networks without memory (AD model) and with memory (ADM model) [22, 26]. The former case has been widely used as a paradigmatic network example for the study of spreading dynamics taking place on the same time-scale of network evolution [23, 24, 47, 48]. The models are described in detail as follows:(i)
Agent randomized (AR) null modelThe AR null model only keeps the contact establishment mechanism that the recurrence intervals of the contacts for each individual are preserved. We keep all the human interaction time in the empirical datasets unchanged and replace each contact pair with two randomly selected individuals. Note that once a contact pairu1,v1 is replaced by u2,v2, all the contacts between individuals u1 and v1 will be replaced by contact pair u2 and v2.(ii).
Contact randomized (CR) null modelThe CR null model only keeps the individual activity state mechanism that the degree and activity state of individuals are unchanged in the networkGt at time step t. At each time step t, we randomly select two contacts associated with the four individuals and then rewire the two contacts. If the two contacts both have the same individual, we discard the contact pair and randomly select two new contacts. The rewiring steps are repeated more than 2LCt times to ensure the rewiring of most contacts in Gt, where LCt is the number of contacts in Gt. After the rewiring procedure, the interaction time and recurrence interval of human contacts are completely different from those of original interaction datasets.(iii)
Activity-driven (AD) modelThe AD model [22] is one of the most studied temporal network models. The generation of an AD model follows the rules. At each time step t, the Gt starts with N isolated nodes, and each node i is assigned an activity probability aitw to become active; then, the active nodes generate m links that are randomly connected to m other nodes, and the inactive nodes can only receive connections from active nodes; at next time step t+1, all connections in Gt are omitted, and the steps are repeated. Here, we apply the AD model to generate temporal networks to match the real-world networks. The activity probability ai=ηxi, where xi∈ε,1, ε is a lower cut-off that avoids possible divergences, and the xi is drawn from a given probability function Fxi. The probability function Fxi is statistically obtained from empirical data, and the average number of active agents per unit time is ηxN, where η=1 in this work. The AD model does not capture the above two mechanisms in empirical data.(iv)
Activity driven with memory (ADM) modelA major shortcoming of the AD model lies in the total absence of correlations between the connections in consecutive time steps. The ADM model [26] extends the AD model by introducing a memory mechanism into to link creation. In the ADM model, each node remembers all its previously connected nodes. At each time step t, the Gt starts as in the AD model with N disconnected nodes, and each node i becomes active with probability aitw. Each active node i interacts with one of the previously contacted n nodes with probability pADM=n/n+1 and randomly connects a new node with probability 1−pADM.In this work, the virus spreads on real human contact networks under a temporally periodic boundary condition (i.e., repeating the whole contact sequence) and network models until all the reachable agents are infected [17]. The five models have been introduced in detail in Sections 4.1 and 5.1. The relevant parameters of the MD model and AD model are measured in the corresponding datasets (see details on Tables 2 and 3). The parameters of the ADM model are the same as those of the AD model except the memory effect. We repeat the SI spreading process for all possible seeds and record the full prevalence time tf, i.e., the time to reach 100% infection in a connected network or the largest connected component (LCC) of a disconnected network.Table 3
The parameters of the activity-driven model.
Activity-driven (AD) modelParametersSex6yrMIT_RMSchoolN16,70696236T64230104FxPl−2.6U0,1U0,1ε10−210−210−2m153X∼Ua,b represents the uniform distribution from a to b.We calculate the ratiotfXλ/tfrealλ as a function of the infection rate λ, where tfreal and tfX represent the average spreading time of a full prevalence in the real-world network real and the temporal network model X, respectively. If the ratio tfXλ/tfrealλ=1, the full prevalence time of a spreading in the real-world network is the same as that in the corresponding network model X. In other words, X is a reasonable temporal model for human contact networks in the study of spreading processes. As shown in Figure 8, the results in three empirical datasets are similar. The ratio tfAR/tfreal is always larger than 1 regardless of the infection rate λ, and the ratio tfAR/tfreal increases with the increase in the infection rate λ. The results illustrate that the individual state transition mechanism might promote the spreading processes since the spread of virus in the AR model, which only keeps the contact establishment mechanism and omits the individual activity state transition mechanism, is faster than that in real human contact networks. Moreover, the promotion of spreading is even larger when the infection rate λ is larger. However, there is a very small difference between the CR model and AD model, where the ratios tfCR/tfreal and tfAD/tfreal are both much smaller than 1. Both models are built without the contact establishment mechanism, which might cause a slowing-down effect on the spreading dynamics. Besides, the ratios tfAR/tfreal and tfADM/tfreal are both larger than 1, provided that they both cannot capture the mechanism of individual activity state transition. We find that the ratio tfMD/tfreal always fluctuates around 1 and remains stable against the infection rate λ. Figure 8 confirms that the MD model is superior to all other models in characterizing human contact networks. This phenomenon highlights the crucial role of the two mechanisms considered in our model. They both serve as an indispensable component to characterize the evolution of the real-world networks. Binding them together enables us to accurately capture the contagion processes unfolding on empirical human contact networks.Figure 8
The ratiotfX/tfreal of the average full prevalence time as a function of the infection rate λ in three empirical datasets, where X=AR,CR,AD,ADM,MD represents the agents randomized null models (▽), contacts randomized null models (⋄), activity-driven models (△), activity-driven models with memory (◯), and memory-driven models (□), respectively. The simulations are performed on 103 realizations. The error bars here represent one binomial standard deviation from these averages with 95% confidence levels. All the models present the same critical spreading rate as the empirical counterparts.


## 5.2. Dynamical Processes on Memory-Driven Networks

We here study how the network evolution affects dynamic processes, such as the epidemic spreading on networks. The exploring rateq in the contact establishment mechanism of the MD model can tune the evolution of network topology. We perform the SI spreading processes with infection rates λ=0.2,0.4, and 1.0 on temporal networks generated by the MD model with N=1000 and the exploring rates q=0,0.1,…,1, respectively, and record the full prevalence time tf. Figure 9 indicates that tf is extremely large when q=0, which implies that all contacts in current time step are recurrences of previously established contacts; that is, no individual is willing to explore a new social relationship in the evolving network. Moreover, we observe that the average full prevalence time tf declines rapidly with the exploring rate q, when q is small. When the exploring rate q further increases and reaches a specific value (around 0.1), the spreading time decreases slightly and finally meets a saturation. The fact illustrates that even when individuals establish new social ties with a small exploring rate q, the spreading process will significantly speed up; however, when q is larger than a specific value, the influence of q on the spreading process is rather limited.Figure 9
The average full prevalence time as a function of the exploring rateq versus the infection rate λ. The temporal networks are generated by the MD model with N=1000, A=1, α=1, γ=1.6, PIA=0.1, L=40, T=3000, and s is sampled from Fs=s/smin−β with β=2.8 and smin=1. The simulation result is the average of 200 realizations.


## 6. Conclusion

We have presented the analysis of several empirical datasets of human contacts. We observe two crucial mechanisms, i.e., the individual activity state transition and contact establishment, which create the evolving structures of human contact networks. The individual activity state transition governs the activation of agents. The active agents establish contacts with others according to the strategies of contact establishment; that is, agents tend to interact with recently contacted one. We find that the transition probabilityPAIk or PAAk between the active state and the inactive state is a power-law function of the degree k. Moreover, the empirical datasets show that not all active nodes have the number of contacts at each time step, and the recurrence intervals of individual contact establishment follow a power-law distribution. Considering the empirical observations, we propose a novel temporal network model (the memory-driven model) based on the two mechanisms. Furthermore, we study the effects of the two mechanisms on dynamical processes. We perform the SI spreading on the real-world human contact networks and four temporal network models. The full prevalence times of SI spreading on the networks and models are compared. Our model shows a good agreement with three empirical temporal networks, which implies that the two mechanisms enable us to capture the evolution of human interactions in temporal networks. Interestingly, the results demonstrate that the individual activity state transition accelerates the diffusion processes; contrarily, the specific contact establishment strategy slows down the spreading. Besides, we find that the exploration of new social ties effectively promotes the spreading processes, and even a small exploring rate is sufficient to induce an explosive spreading on networks. The study thus paves the way to a better understanding of the mechanisms driving the evolution of human contacts and their effects on dynamic processes in real-world social systems.



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*Source: 6643658-2021-02-02.xml*