Epidemic Spreading on Activity-Driven Networks with Attractiveness

TL;DR

This paper analyzes how the interplay of activity and attractiveness in time-varying networks influences epidemic spreading, deriving thresholds and confirming findings through simulations, revealing the impact of heterogeneity and correlations.

Contribution

It introduces a generalized activity-driven network model incorporating attractiveness, deriving epidemic thresholds analytically for various distributions and correlations.

Findings

Heterogeneous attractiveness affects contagion dynamics.

Positive correlations facilitate spreading, negative correlations hinder it.

Analytical results are validated by numerical simulations.

Abstract

We study SIS epidemic spreading processes unfolding on a recent generalisation of the activity-driven modelling framework. In this model of time-varying networks each node is described by two variables: activity and attractiveness. The first, describes the propensity to form connections. The second, defines the propensity to attract them. We derive analytically the epidemic threshold considering the timescale driving the evolution of contacts and the contagion as comparable. The solutions are general and hold for any joint distribution of activity and attractiveness. The theoretical picture is confirmed via large-scale numerical simulations performed considering heterogeneous distributions and different correlations between the two variables. We find that heterogeneous distributions of attractiveness alter the contagion process. In particular, in case of uncorrelated and positive correlations between the two variables, heterogeneous attractiveness facilitates the spreading. On the contrary, negative correlations between activity and attractiveness hamper the spreading. The results presented contribute to the understanding of the dynamical properties of time-varying networks and their effects on contagion phenomena unfolding on their fabric.

Equations

$$⟨b⟩=\int dadbH(a,b)b.$$

$$\frac{\lambda }{\mu }⟨k⟩>\frac{2⟨a⟩}{⟨a⟩+\sqrt{⟨a^2⟩}}\equiv T_{AD},$$

$$\begin{split}&I_{a,b}(t+1)=(1-\mu)I_{a,b}(t)+\\ &\frac{\lambda m}{N\langle b\rangle}S_{a,b}(t)\Bigl{[}a\sum_{a^{\prime},b^{\prime}}b^{\prime}I_{a^{\prime},b^{\prime}}(t)+b\sum_{a^{\prime},b^{\prime}}a^{\prime}I_{a^{\prime},b^{\prime}}(t)\Bigr{]}.\end{split}$$

$$\theta (t)\equiv \sum _{a,b}a{I}_{a,b}(t),$$

$$\varphi (t)\equiv \sum _{a,b}b{I}_{a,b}(t).$$

$$I_{a,b}(t+1)\simeq (1-\mu )I_{a,b}(t)+\frac{\lambda m}{N⟨b⟩}{N}_{a,b}[a\varphi (t)+b\theta (t)].$$

$$\frac{\partial I(t)}{\partial t}\simeq -\mu I(t)+\frac{\lambda m}{⟨b⟩}[⟨a⟩\varphi (t)+⟨b⟩\theta (t)],$$

$$\frac{\partial \theta (t)}{\partial t}\simeq -\mu \theta (t)+\frac{\lambda m}{⟨b⟩}[⟨a^2⟩\varphi (t)+⟨ab⟩\theta (t)],$$

$$\frac{\partial \varphi (t)}{\partial t}\simeq -\mu \varphi (t)+\frac{\lambda m}{⟨b⟩}[⟨ab⟩\varphi (t)+⟨b^2⟩\theta (t)],$$

$${\kappa }_0=-\mu ,{\kappa }_{\pm }=\frac{\lambda m}{⟨b⟩}\left(⟨ab⟩\pm \sqrt{⟨a^2⟩⟨b^2⟩}\right)-\mu .$$

$$\frac{\beta }{\mu }>\frac{2⟨a⟩⟨b⟩}{⟨ab⟩+\sqrt{⟨a^2⟩⟨b^2⟩}}\equiv T_{ADA},$$

$$T_{ADA}=\frac{2}{1+\sqrt{\frac{⟨a^2⟩⟨b^2⟩}{⟨a{⟩}^2⟨b{⟩}^2}}}.$$

$$H(a,b)=F(a)\delta (b-q(a)),$$

$$G(b)=F(q^{-1}(b))\left|\frac{d{q}^{-1}(b)}{db}\right|.$$

$$⟨a^n{b}^m⟩=⟨a^{n+{\gamma }_{c}m}⟩,$$

$$T_{ADA}=\frac{2⟨a⟩⟨a^{{\gamma }_c}⟩}{⟨a^{1+{\gamma }_c}⟩+\sqrt{⟨a^2⟩⟨a^{2{\gamma }_c}⟩}}.$$

$$⟨a^n{b}^m⟩=⟨a^{n-m}⟩,$$

$$T_{ADA}=\frac{2}{1+\sqrt{\frac{⟨a^2⟩⟨a^{-2}⟩}{⟨a{⟩}^2⟨a^{-1}{⟩}^2}}},$$

Full text

PACS numbers

89.75.-k, 64.60.aq, 87.23.Ge

Many social, natural and technological systems can be modelled as networks. The structure of such systems is often not fixed and exhibits complex temporal dynamics Holme (2015); ”Holme and Saramäki (2012); Masuda and Lambiotte (2016); Porter and Gleeson (2016). However, the large majority of studies revolve around representations that neglect the role of time Barrat et al. (2008); Boccaletti et al. (2006); Newman (2010). In particular, connections are typically approximated as either static or annealed Vespignani (2012a); Boguñá et al. (2013). Since networks are often used as an environment for the study of dynamical processes, the choice concerning which approximation to adopt is a matter of time scales: when the process is faster than the network evolution, the network structure can be assumed static; in the opposite conditions, networks can be effectively described by annealed representations. When, however, the time scale of the process studied is comparable to the one characterising the network evolution, static or annealed approximations are not viable and can lead to incorrect conclusions such as misrepresenting i) the spreading potential of a disease Morris (1993, 2007); Valdano et al. (2015); Prakash et al. (2010); Vespignani (2012b); Rocha et al. (2011); Bajardi et al. (2011); Vanhems et al. (2013); Stehlé et al. (2011); Perra et al. (2012a); Takaguchi et al. (2013); Holme and Liljeros (2014); Karsai et al. (2011); Toroczkai and Guclu (2007); Liu et al. (2013); Holme and Masuda (2015); Wang et al. (2016); Sun et al. (2015); Han et al. (2015); Rizzo et al. (2016), ii) the exploring capabilities of random walkers Starnini et al. (2012a, b); Ribeiro et al. (2013a); Perra et al. (2012b); Hoffmann et al. (2012); Masuda et al. (2016), iii) the features of social interactions Karsai et al. (2014a); Clauset and Eagle (2007); Isella et al. (2011); Pfitzner et al. (2013); Starnini and Pastor-Satorras (2013a); Takaguchi et al. (2012); Gonçalves and Perra (2015); Fournet and Barrat (2014); Barrat and Cattuto (2015); Sekara et al. (2016); Holme (2003); Jo et al. (2012); Ubaldi et al. (2016, 2017), or the processes of iv) information spreading Miritello et al. (2011); Kivela et al. (2011); Panisson et al. (2011); Weng et al. (2013); Gleeson et al. (2016), v) synchronisation Fujiwara et al. (2011), vi) percolation Parshani et al. (2010), vii) consensus Baronchelli and Díaz-Guilera (2012), viii) competition Artime et al. (2017), ix) social contagion Liu et al. (2017), and x) innovation Rizzo and Porfiri (2016).

Thanks to the unprecedented availability of large and longitudinal datasets, in recent years a great effort has been put into the development of temporal network representations and models. See References Holme (2015); ”Holme and Saramäki (2012); Masuda and Lambiotte (2016) for detailed reviews on the subject.

One proposal for an analytical model of temporal network comes from the activity-driven model Perra et al. (2012a), which relates the temporal structure of the connections to one fundamental quantity, the activity. This feature represents the propensity of a node to establish connections per unit time. In the model, each node $i$ is equipped with an activity $a_{i}$ extracted from a distribution $F(a)$. At any time step $t$, nodes are active and thus willing to establish connections with probability proportional to their activity. One praise of this simple mechanism is that it relates the contact dynamics to the structure of the time-integrated network: the resulting degree distribution $P(k)$ depends on the form of $F(a)$, and in particular a power-law distributed activity produces a power-law degree distribution Perra et al. (2012a). This fact is particularly significant in relation with social networks, which are known to exhibit distributions of this kind both for the degree Barabási (2016); Barrat et al. (2008) and for the activity Perra et al. (2012a); Ribeiro et al. (2013b); Ubaldi et al. (2016); Karsai et al. (2014b); Tomasello et al. (2014).

In its original form, the activity-driven model is extremely simple, thus relatively lightweight for performing calculations. Nonetheless it gives rise to a non-trivial temporal structure having an impact on the unfolding of dynamical processes Perra et al. (2012a, c); Ribeiro et al. (2013b); Liu et al. (2014). Precisely because of its simplicity, and in particular of its reliance on only one node property (the activity), the original activity-driven is not able to reproduce other widespread properties of social networks, namely finite clustering, assortative mixing, a bursty contact sequence and memory effects Granovetter (1973); Newman (2003); Barabási (2005). For this reasons, and also thanks to its flexibility, modifications to the original model have been introduced and investigated Karsai et al. (2014b); Laurent et al. (2015); Sun et al. (2015); Moinet et al. (2015); Ubaldi et al. (2016, 2017).

Here, we consider a recent extension of the model Alessandretti et al. (2017) in which beyond the activity distribution, the network is characterised by an attractiveness distribution Alessandretti et al. (2017).This feature accounts for the fact that some nodes may be a preferential target of interactions (i.e. be more popular), in the same way activity accounts for the fact that some nodes are more inclined to be their initiators. The attractiveness $b_{i}$ of a node $i$ is a measure of its propensity to attract contacts. Therefore it is to some extent the reciprocal of the activity, and a natural complement to it within the model. Heterogeneous attractiveness distributions have been observed in different networks such as online communities Ghoshal and Holme (2006); Valverde and Solé (2007); Alessandretti et al. (2017), face-to-face interactions Starnini et al. (2013), and animal hierarchies Sapolsky (2005).

In Ref. Alessandretti et al. (2017), besides the introduction of the model, the authors studied the effects of the attractiveness and of its interplay with the activity on the fundamental dynamical processes: the random walk. Here, we keep investigating how attractiveness and activity affect spreading processes on temporal networks focusing on contagion phenomena. In particular, we consider SIS epidemic models in the case of a generic joint activity-attractiveness distribution deriving an analytic expression for the epidemic threshold. We give a detailed treatment of three scenarios. First, we examine the case of uncorrelated activity and attractiveness. Second, inspired by observations on real data Alessandretti et al. (2017), we study the case of positive correlation between the two variables. Finally, we complete our analysis considering the case of negative correlation. In all cases we use numerical simulations to validate our results.

The shape of the activity distribution, and its second moment in particular, has been shown to influence the unfolding of different kinds of processes unfolding on activity-driven networks Perra et al. (2012a, c); Starnini and Pastor-Satorras (2014); Liu et al. (2014); Sun et al. (2015); Starnini and Pastor-Satorras (2013b). In case of spreading phenomena, the more heterogeneous the activity distribution (i.e. the larger its variance), the easier it is for a disease to reach a finite portion of the network Perra et al. (2012a); Starnini and Pastor-Satorras (2014); Liu et al. (2014); Sun et al. (2015). Here, we found how the presence of a heterogeneous attractiveness has an analogous impact. The presence of positive correlations between activity and attractiveness further facilitates the contagion, while the presence of negative correlations, conversely, hinders it.

The paper is structured as follows: in Section I we present the model and we discuss the attractiveness and its correlation with the activity; in Section II we study the epidemic threshold for an SIS process in the case of absence of activity-attractiveness correlation (II.1), and in the case of deterministic correlation (II.2), treating all cases both analytically and with simulations; Section III contains the conclusions and an address of possible future works.

I THE MODEL

In the original activity-driven model (AD), a network of $N$ nodes is characterised by an activity distribution $F(a)$ from which the activity, $a_{i}$, of each node $i$ is extracted. The model uses a discrete time framework, with time steps of duration $\Delta t$. At the beginning of each time step, a node $i$ may activate; the activation happens with probability $a_{i}\Delta t$; if a node activates, it will form a fixed number $m$ of connections towards randomly selected nodes (multiple connections, as well as self-connections, are forbidden, and in general $m\ll N$); the connections remain active for the duration of the time step, at the end of which they are all reset, and the process starts again.

The above depicted is the original version of the model, as proposed in Perra et al. (2012a). There, when a node activates, it will choose the targets of its connections among all other members of the network with equal probability. In the version of the model we consider here, which we will call activity-driven with attractiveness (ADA), the network is also characterised by an attractiveness distribution Alessandretti et al. (2017). In general, the two values of activity $a_{i}$ and attractiveness $b_{i}$ for the same node $i$ are not necessarily uncorrelated, and are sampled from a joint probability distribution $H(a,b)$. Interestingly, recent observations on online social networks have shown both variables to behave according to a power-law with similar exponents and an approximately linear correlation Alessandretti et al. (2017).

The ADA works like the AD, except that when a node $i$ activates it will choose another node $j$ as a target of one of its $m$ connections with probability proportional to the second node’s attractiveness, $b_{j}$. As the probability of choosing any node must be equal to one, the correct normalisation for the probability is $b_{j}/\langle b\rangle N$, where $\langle b\rangle$ is the mean value of the attractiveness:

[TABLE]

The model thus behaves similarly to a linear preferential attachment, as the overall number of contacts received by a node during any time-window is linearly proportional to its attractiveness; the total number of contacts (received and initiated), on the other hand, depends on both activity and attractiveness.

By time-integrating the connections obtained at different time steps we can study the emergent topological properties of the network. In particular, the time-integrated network over $T$ time steps is defined as the union of the instantaneous networks obtained at $T$ different time steps, i.e. two nodes figure as connected in the integrated network if they are connected in at least one of the $T$ instantaneous networks. A weight, equal to the number of instantaneous networks in which the edge appears, can also be associated to each edge. For the AD, the degree distribution of the integrated network is connected to the activity distribution through the relation $P(k)\sim F(k/Tm)$, as long as it holds $k\ll T\ll N$ Perra et al. (2012a). The study of the time-integrated properties of ADA networks will be instead matter of future work.

FIG. 1 illustrates the degree distribution $P(k)$ and the edge weight distribution $P(w)$ obtained for two ADA and an AD networks of size $N=10^{5}$ after a time-integration of $T=10^{3}$ time steps. We used an activity distribution $F(a)\propto a^{-2.4}$. In the AD model all nodes have equal attractiveness; in the uncorrelated ADA we used an attractiveness distribution independent on, but identical to, the activity distribution: $G(b)\propto a^{-2.4}$; for the correlated ADA we set the attractiveness of every node to be proportional to its activity: $b_{i}\propto a_{i},\,i=1,\dots,N$. The exponents are chosen to be representative of typical values encountered in social networks Newman (2003); Boccaletti et al. (2006). The plot of the degree distribution shows that the presence of heterogeneous attractiveness in ADA networks does not induce dramatic changes. However, the inspection of the weight distribution highlights the difference between the two models. The presence of heterogeneous attractiveness induces heterogeneity in the partner selection that reverberate in the weight distribution. As we will see later, such heterogeneity favours the contagion process.

The ADA model differs from recent generalisations of AD networks introduced in Karsai et al. (2014b) and further expanded in Laurent et al. (2015); Ubaldi et al. (2016, 2017). In fact, in these extensions local reinforcement mechanisms have been used as a way to model the emergence and evolution of strong/weak ties Granovetter (1973). However, local mechanisms alone cannot explain the dynamics of ties especially in the current social media landscape where people can easily be in contact with celebrities or access information provided by popular accounts. The attractiveness describes scenarios of global popularity, as opposed to cases of local reinforcement where the perceived attractiveness of a node may change between its peers, so that the contact probability is encoded in pairs rather than in the single nodes; also, we model the attractiveness as constant in time, not being strengthened nor weakened by the occurrence of contacts - or its lack.

II EPIDEMIC THRESHOLD

As we discussed above, the presence of a heterogenous attractiveness affects the temporal structure of contacts. We want to quantify this phenomenon by studying its impact on a dynamical process; namely, we choose to evaluate the epidemic threshold for an SIS process. The fact that the analytical value of such threshold has already been calculated and tested for the original activity-driven in Perra et al. (2012a) allows us to straightforwardly draw a comparison between the AD and the ADA.

The SIS is an example of a compartmental epidemic model Kermack and McKendrick (1927); Keeling and Rohani (2008); Barrat et al. (2008); in this framework, every node belongs to a certain class with respect to the disease status: susceptible (S), or infected (I). When a susceptible node contacts (or is contacted by) an infected one, it may become itself infected, with probability $\lambda$. Meanwhile, infected nodes can undergo spontaneous recovery with rate $\mu$ and become susceptible again.

In general contagion processes are characterised by a threshold which determines whether the disease is able to spread in the system affecting a macroscopic fraction of nodes Kermack and McKendrick (1927); Keeling and Rohani (2008); Barrat et al. (2008); Pastor-Satorras et al. (2015); Wang et al. (2016). In the limit of static networks the epidemic threshold of a SIS processes is determined by the spectral properties of the adjacency matrix Wang et al. (2003). In the limit of annealed networks and uncorrelated topologies the threshold is defined by the moments of the degree distribution Pastor-Satorras and Vespignani (2001). Interestingly, a closed expression for the threshold of a SIS process unfolding on a general time-varying networks has been obtained Prakash et al. (2010); Valdano et al. (2015). This can be expressed in terms of the spectral properties of the system-matrix which is defined as $\mathbf{S}=\prod_{t}[(1-\mu)\mathbf{A_{t}}+\lambda]$ (where $A_{t}$ is the adjacency matrix at time $t$). Despite its generality, this expression hides the physics of the process behind the computation of eigenvalues which is typically done numerically.

The AD framework allows an explicit mathematical derivation Perra et al. (2012a). In particular, the threshold, for SIS models, depends on the moments of the activity distribution:

[TABLE]

at the first order in $N^{-1}$ and activity Perra et al. (2012a). We have introduced $T_{AD}$, that denotes the value of the epidemic threshold for the activity-driven model and depends on the properties the network, which in turn are determined by the moments of the activity distribution.

It is important to stress how in the expression of the threshold the time integrated properties of the network (as the degree distribution) do not appear. The dynamical properties are defined only by the activity distribution.

Here, we extend the literature providing an explicit analytical expression for the epidemic threshold for an SIS process in the ADA model for any form of the probability distribution $H(a,b)$. To do so, we assume nodes to be characterised by their activity and attractiveness values alone, and accordingly group them in classes; nodes within each class are considered statistically equivalent (mean-field assumption). We also assume that the two variables are discretely distributed, but the derivation would apply as well to the case of continuous variables. We denote with $N_{a,b}$ the number of nodes of activity $a$ and attractiveness $b$, with the condition $\sum_{a,b}N_{a,b}=N$. The number of susceptible and infected nodes of activity $a$ and attractiveness $b$ at time $t$ is indicated as $S_{a,b}(t)$ and $I_{a,b}$ respectively. A master equation for the temporal evolution of the number of infected nodes in each class can be written, again, in the limit of large size $N\gg 1$, where the probability of having repeated contacts between the same two nodes can be neglected. Without lack of generality in the following we will set $\Delta t=1$. The master equation reads:

[TABLE]

The first term on the right side accounts for the infected inherited from the previous time step, minus the cases of spontaneous recovery. The two terms in bracket represent, the first, the probability for a susceptible node in the class $(a,b)$ to activate and contact an infected node in any other class, and the second term represents the probability for an infected node in any other class to activate and contact a susceptible node in the class $(a,b)$; the difference with the AD model is that now the probability for a node in the class $(a,b)$ to be contacted depends on $b$, and the probability for it to contact a node in the class $(a^{\prime},b^{\prime})$ depends on $b^{\prime}$. We can define two auxiliary functions to simplify what follows:

[TABLE]

In considering the initial phase of the spreading, when $I_{a,b}(t)\ll N_{a,b}$, we can take $S_{a,b}(t)\simeq N_{a,b}$; the master equation becomes:

[TABLE]

From the last one, we can obtain three more equations: one by summing aver all classes,and two more by first multiplying by $a$ and $b$ respectively, and then summing.

Switching to a continuous time regime, we obtain a system of three linear differential equations:

[TABLE]

of eigenvalues:

[TABLE]

The outbreak prevails when at least one eigenvalue is positive. The $\kappa_{\pm}$ recover the eigenvalues of the AD if we use a constant attractiveness; $\kappa_{0}$ is not a candidate for being the largest eigenvalue, unless $\kappa_{0}=\kappa_{+}$; so the epidemic threshold is determined by the positivity of $\kappa_{+}$, leading to:

[TABLE]

where we have used $\langle k\rangle=2m\langle a\rangle$ Perra et al. (2012a), and introduced $\beta\equiv\lambda\langle k\rangle$ as the per capita spreading rate. As for the AD (above), we use $T_{ADA}$ to denote the value of the epidemic threshold encoded in the features of nodes activity and attractiveness.

Eq. 11 is valid for any form of $H(a,b)$; in particular, the expression recovers the value from Eq. 2 if a constant attractiveness is used, so that $\langle b^{2}\rangle=\langle b\rangle^{2}$ and $\langle ab\rangle=\langle a\rangle\langle b\rangle$. In the reminder of this section we focus on two scenarios: first, when activity and attractiveness are uncorrelated; second, when they are instead deterministically correlated - either positively on negatively.

To provide a precise estimation of the threshold value when simulating an SIS process, we use the lifetime-based method introduced in Boguñá et al. (2013) and also used in Sun et al. (2015) for the same purpose. The lifetime $L$ is defined as the amount of time elapsed before the disease either dies out or spreads to a finite fraction $C$ of the network. The rationale behind this definition is the following: when the system is below threshold, the disease is not able to spread and dies out in a short time; above threshold, the lifetime is also short, as the disease can quickly reach a finite fraction of nodes. Only for values of $\beta/\mu$ close to the threshold we expect to observe a longer lifetime, as the contrasting effects of contagion and spontaneous recovery are almost equally strong and they struggle to prevail on each other; we can thus take the value of $\beta/\mu$ corresponding to the maximum in $L$ as an estimation of $T_{ADA}$. Indeed, the life time, obtained by averaging over many realisations, is equivalent to the susceptibility in standard percolation theory Boguñá et al. (2013), thus it provides a precise method to detect the threshold numerically.

The peak in the lifetime is more pronounced for larger values of $N$ as, for power-law activity and attractiveness distributions, the heterogeneity effects lowering the epidemic threshold are constrained by the finite size of the system Pastor-Satorras and Vespignani (2002). For such reason, we run our simulations on networks of increasing size and expect to observe the peak increase and occur at lower values of $\beta/\mu$, converging to the theoretical value as $N\to\infty$. In particular, here we chose to use $N$ equal to $10^{4}$, $10^{5}$ and $10^{6}$; we also set $C=0.25$ as the target value for the network coverage (fraction of infected nodes).

II.1 Uncorrelated distributions

In the absence of correlations, $H(a,b)$ can be written as a product $H(a,b)=F(a)G(b)$, where $F(a)$ is the activity distribution and $G(b)$ the attractiveness distribution. In Eq. 11, the mean value of the product can be factorised to obtain:

[TABLE]

We can see that, once fixed the average values, the threshold can be made arbitrarily small by increasing either or both the second moments, i.e. introducing heterogeneity. As the threshold depends on the moments of the two distributions in the same way, the case with constant attractiveness and generic $F(a)$ (the AD) can be mapped to the one with constant activity and attractiveness distribution $F(b)$.

As $\langle b^{2}\rangle\geq\langle b\rangle^{2}$ always holds, the threshold can only be lower than or equal to the one found in the AD; this means that the introduction of any amount of heterogeneity in the attractiveness helps the epidemic spreading.

As an example of uncorrelated distributions, let us consider two power-laws: $F(a)=Ca^{-\gamma_{a}}$ and $G(b)=Db^{-\gamma_{b}}$; in both cases $a$ and $b$ are bounded inside the interval $[\epsilon,1]$, to avoid divergences.

FIG. 2 illustrates the behaviour of the epidemic threshold obtained from Eq. 12; we report the values of $T_{ADA}^{-1}$ - so that the plot shows a maximum when the spreading potential is maximum (the threshold shows a minimum) - as a function of the two exponents $\gamma_{a}$ and $\gamma_{b}$. The threshold exhibits the same dependence on each of the two exponents, as the analytic expression also shows, with local maxima for $\gamma_{a}=2$ (and $\gamma_{b}=2$) and a global maximum in $\gamma_{a}=\gamma_{b}=2$. The function is symmetric around such value.

In FIG. 3 we show an analytical comparison between the ADA and the AD, by plotting $T_{AD}/T_{ADA}$ - the ratio between the epidemic threshold in the original model and the epidemic threshold with attractiveness computed following the analytical solution - as a function of the two power-law exponents. As expected from the equations we find that, for values of $\gamma_{b}$ either very large or close to zero, the ratio tends to one, as $\langle b^{2}\rangle\rightarrow\langle b\rangle^{2}$ and consequently $T_{ADA}\rightarrow T_{AD}$. Otherwise, the attractiveness always lowers the threshold thus facilitating the spreading of the epidemic phenomenon.

We tested the validity of our findings by simulating an SIS process with two different choices of the exponents: one with $\gamma_{a}=1.8$ and $\gamma_{b}=2.1$, the other with same $\gamma_{a}$ and $\gamma_{b}=2.8$; in both cases we set $m=2$ and took the median value over a number realisations of the process varying between $500$ and $5000$ (depending on the size). The results are shown in FIG. 4, where we plotted the lifetime of the process for different values of $\beta/\mu$. In particular, we let $\lambda$ vary while we keep fixed $\mu=0.01$ and $\langle k\rangle$ also does not change, being determined by the relation $\langle k\rangle=2m\langle a\rangle$. The lifetime exhibits a peak converging towards the theoretical prediction (dotted line) for increasing values of $N$. Also, the comparison with the AD (dashed line) shows that the epidemic threshold is appreciably lower in the ADA setting, as the heterogeneity due to the attractiveness distribution facilitates the spreading.

II.2 Deterministic correlation

As a second example, we study the case of a deterministic activity-attractiveness correlation, where the value of one variable uniquely determines the value of the other one for any given node. The joint distribution can be expressed in the form:

[TABLE]

where $\delta(x)$ is the Dirac delta and $q(a)$ is the function that determines the attractiveness of a node given its activity: $b_{i}=q(a_{i}),\,\forall i$. Using the relation $G(b)=F(a)|da/db|$ we can obtain an expression for $G(b)$ (provided $q(a)$ has an inverse):

[TABLE]

A case of interest is $q(a)\propto a^{\gamma_{c}},\,\gamma_{c}>0$, so that if one the variables is power-law distributed, the other is too, with different exponent: if for example $F(a)\propto a^{-\gamma_{a}}$, then the attractiveness will be distributed as $G(b)\propto b^{-1+\frac{1-\gamma_{a}}{\gamma_{c}}}$. This also includes the case of identical correlation, for $\gamma_{c}=1$. A generic moment of the joint distribution can be expressed as:

[TABLE]

and Eq. 11 becomes:

[TABLE]

FIG. 5 shows the behaviour of the threshold as a function of the exponents $\gamma_{a}$, governing the activity distribution, and $\gamma_{c}$, which determines the activity-attractiveness relation as depicted above. We report the values of the logarithm of $T_{ADA}^{-1}$: as in previous plots, we choose to show the reciprocal of the epidemic threshold. In this case we also choose to take the logarithm, as the value of $T^{-1}_{ADA}$ changes considerably in the range studied. For $\gamma_{c}=0$, which is equivalent to the AD as $G(b)$ is constant, the threshold shows a maximum for $\gamma_{a}=2$ as expected; as $\gamma_{c}$ increases, the maximum increases very quickly as the most active nodes become more and more popular, greatly facilitating the spreading of the disease.

We validated the case of identical correlation with computer simulations, by studying an SIS process. In FIG. 6 we plotted the lifetime for different values of $\beta/\mu$. For increasing values of $N$, the lifetime exhibits a peak that converges towards the predicted threshold (solid line) - which is significantly lower than the one obtained in the AD (dashed line), and also lower than the threshold for uncorrelated distributions (dotted line) - thus confirming our analytical predictions. We let $\lambda$ vary while keeping fixed $\mu=0.01$, $\gamma_{a}=2.8$, $m=2$, taking the median over a number of realisations ranging between $500$ and $3000$, depending on the size of the network.

As a second instance of the deterministically correlated case, we consider $q(a)\propto a^{-1}$. This form of the function accounts for a case of negative activity-attractiveness correlation, where the most active nodes are the least attractive and vice versa. The same formulae hold as for the case of positive $\gamma_{c}$ above, leading to the expressions:

[TABLE]

for a generic moment, and:

[TABLE]

for the threshold. In this scenario, opposite to the case of positive correlation considered above, we expect the correlations to work against the epidemic, as the most active potential spreaders are also the least attractive and hence the least likely to be infected in the first place.

Our expectation is corroborated by the experiments described in FIG. 7, where we study the lifetime of a SIS process on an ADA network characterised by an activity-attractiveness correlation of the form $b\propto a^{-1}$ for all nodes, the activity being distributed as a power-law with exponent $\gamma_{a}=2.8$. We used $m=2$, $\epsilon=10^{-3}$ and a number of realisations ranging from $2000$ to $3000$ for different sizes. The outcome of the simulations matches well the theoretical threshold (solid line), and the comparison with the case of identical correlation (i.e. same activity distribution with $\gamma_{a}=2.8$, and attractiveness $b\propto a$; dotted line) shows a stark difference between the two cases, highlighting the contrasting effects of the two phenomena. The threshold for an analogous AD network (with activity distribution of power $\gamma_{a}=2.8$ and constant attractiveness; dashed line) is close to the negatively correlated case.

III CONCLUSIONS

We studied a recent generalisation (labelled ADA for simplicity) of the activity-driven model where a second node’s property, called attractiveness, is added. This variable accounts for the fact that not all nodes are as likely to be the target of interactions initiated by others. The original activity-driven model (labelled AD for simplicity) would only account for heterogeneity in nodes’ behaviour by distinguishing between more and less active ones, while implicitly assuming constant attractiveness. Observations in different types of real networks show this is not always the case.

We studied the unfolding of epidemic processes on ADA networks. In particular, we derived analytically an expression for the epidemic threshold of SIS models. The analytical and numerical comparison between spreading dynamics unfolding on ADA and AD networks shows how the introduction of a new grade of heterogeneity due to a non-constant attractiveness can significantly alter the spreading of a disease. To precisely quantify the interplay between the activity and attractiveness we considered three cases. In the first case we used two power-law uncorrelated distributions. The results in this setting show that the introduction of heterogeneity in the attractiveness of nodes facilitates the spreading. In the second instead, we considered a scenario capturing observations in real networks Alessandretti et al. (2017) in which the two quantities follow heterogeneous and correlated distributions. In this case we found that correlations between the two variables facilitate the spreading process even further. Finally, we completed our analysis by considering a case of negative correlations; opposite to the previous case, we found that this type of correlations hinders the spreading phenomenon.

Many of the limits of the AD model are still present in the ADA, i.e. absence of high-order correlations, or the absence of burstiness. These properties will be the subject of future extensions of the model. An investigation of the topology of the time-integrated network could also provide some interesting insight, particularly by determining whether the introduction of the attractiveness, and of an appropriate activity-attractiveness correlation, can lead to the emergence of the desired properties that characterise most real social systems (assortativity and clustering).

Overall, our results contribute to the recent discussion around the effects of temporal connectivity patterns on dynamical processes unfolding on their fabrics.