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# Quantifying the propagation of distress and mental disorders in social networks

## Abstract

Heterogeneity of human beings leads to think and react differently to social phenomena. Awareness and homophily drive people to weigh interactions in social multiplex networks, influencing a potential contagion effect. To quantify the impact of heterogeneity on spreading dynamics, we propose a model of coevolution of social contagion and awareness, through the introduction of statistical estimators, in a weighted multiplex network. Multiplexity of networked individuals may trigger propagation enough to produce effects among vulnerable subjects experiencing distress, mental disorder, which represent some of the strongest predictors of suicidal behaviours. The exposure to suicide is emotionally harmful, since talking about it may give support or inadvertently promote it. To disclose the complex effect of the overlapping awareness on suicidal ideation spreading among disordered people, we also introduce a data-driven approach by integrating different types of data. Our modelling approach unveils the relationship between distress and mental disorders propagation and suicidal ideation spreading, shedding light on the role of awareness in a social network for suicide prevention. The proposed model is able to quantify the impact of overlapping awareness on suicidal ideation spreading and our findings demonstrate that it plays a dual role on contagion, either reinforcing or delaying the contagion outbreak.

## Model

In this work, we start from the model presented in25, generalizing and extending it. First, as well as in25, we consider a SIR-like model, ShIR, thought as a “composed” SIR, namely an extension of the classic “Susceptible-Infected-Recovered” (SIR) model8,13,61, where Sh represents the heterogeneous susceptibility of each node in the layers of multiplex structure (see eq. 1). The second spreading process, coexisting and coevolving with the first one, is an extension of the “Unaware-Aware-Faded” (UAF)25, denoted by UAF(Aπ), where Aπ is the “overlapping awareness”, which derives from a non-zero probability ε of having an additional awareness correlated to the primary contagion phenomenon (see eq. 2). This represents an alternative state to F, as shown in the Dynamic Microscopic Markov Chain Approach (MMCA) (see details in Methods). This means that a node which is in the awareness state A may decide to acquire an awareness on another issue related to the primary contagion process, thus adding an extra awareness, rather than having a transition to the fading state F, where instead a node have a tendency to fade its attention over time until it completely vanishes. Differently from25, for the first time we consider a dual heterogeneity of nodes’ susceptibility and awareness in the layers of the weighted multiplex network. This results in a variation of the infection rate $${\beta }_{i}^{\alpha }$$ and the rate of awareness $${\lambda }_{i}^{\alpha }$$ for the generic node i at layer α, with α{1, ..., M}. In this work, we decide to consider weighted multiplex networks as network structure, and the heterogeneous factors, included in the analytic definition of the infection rate and the rate of awareness, are obtained from properties of the weighted multiplex networks37 (see details in Model). Heterogeneity and overlapping awareness are introduced in this model in order to describe a realistic spreading scenario and disentangle the complex coevolution of two interdependent processes, the social contagion and the awareness spreading on the contagious phenomenon, without neglecting the crucial influence of other aspects related to the contagion. Let us consider a weighted multiplex network of M layers α = {1, ..., M} and N nodes i = {1, ..., N}, which is a set of M weighted networks G α = (V, E α ) (see Fig. 1). The set of nodes V is the same for each layer, whereas the set of links E changes according to the layer37,62. Each network G α is described by the adjacency matrix, denoted by aα with elements $${a}_{ij}^{\alpha }$$, where $${a}_{ij}^{\alpha }={w}_{ij}^{\alpha } > 0$$, if there is a link between i and j, with a weight w ij , otherwise $${a}_{ij}^{\alpha }=0$$. The heterogeneity of weights’ distribution in the multiplex network can be evaluated by means of the two following local properties37,62: the strength of nodes, $${s}_{i}^{\alpha }$$, that is the sum of the weights of the links incident upon node i in layer α, and the inverse participation ratio, $${Y}_{i}^{\alpha }$$, which indicates how the weights are distributed in the layer α37,62.

In our model, we consider the coevolution of two spreading processes on a weighted multiplex network (see Fig. 2). The first is the process of social contagion spreading, ShIR, which is a SIR-like model8,9, where Sh indicates heterogeneous susceptible state25, which means that each node has a different infection rate β i (see eq. 5). As second spreading process, we consider the UAF(Aπ) model, SIR-like, that is the “Unaware - Aware – Faded/Overlapping Aware”, which is an extension of the UAF model25, where U indicates the condition of unawareness, A is the aware state where nodes begin to have an interest in the social contagion phenomenon, increasing their attention, while in the F state, nodes tend to decrease their attention over time up to the point that it completely vanishes. When a node reaches this state, it maintains the same awareness, but it has no interest in increasing its acquired awareness on the phenomenon. The more susceptible are nodes that reach the faded state, the more vulnerable they become due to their low resilience against the phenomenon. Alternatively, if they have a transition from A to Aπ, an alternative state to F, they have the opportunity to increase their awareness also about other issues correlated with the primary contagion phenomenon.

$$\begin{array}{l}{S}^{h}IR\Rightarrow {S}^{h}\mathop{\to }\limits^{{\beta }_{i}^{\alpha }}I\mathop{\to }\limits^{\mu }R;\end{array}$$
(1)
$$UAF({A}^{\pi })=\{\begin{array}{cc}U\mathop{\to }\limits^{{\lambda }_{i}^{\alpha }}A\mathop{\to }\limits^{\delta }F, & if\quad \varepsilon =0\\ U\mathop{\to }\limits^{{\lambda }_{i}^{\alpha }}A\mathop{\to }\limits^{\varepsilon }{A}^{\pi },\, & otherwise\end{array}$$
(2)

We introduce a new definition of weight in the multiplex network, as follows: $${w}_{ij}^{\alpha }={h}_{ij}^{\alpha }\cdot |a{w}_{i}-a{w}_{j}|+1$$. Weights are function of h ij , which is the homophily between nodes, that is the tendency to associate and interact more with similar people34,35, and the absolute difference of awareness, |aw i  − aw j |, between nodes i and j. Thus, when this difference of awareness is equal to zero, nodes will have a weight $${w}_{ij}^{\alpha }=1$$, only if there is a link between i and j. Homophily is defined as follows:

$$\begin{array}{l}{h}_{ij}^{\alpha }=\frac{1}{1+{\delta }_{ij}^{\alpha }}\end{array}$$
(3)

where $${\delta }_{ij}^{\alpha }$$ is the measure of the homophily difference between nodes i and j. To bind this type of weighted network structure with the coevolving spreading processes, showed in eq. 1 and 2, we define the rate of awareness, $${\lambda }_{i}^{\alpha }$$, and the infection rate, $${\beta }_{i}^{\alpha }$$, for each node i at each layer α of the multiplex, as follows:

$$\begin{array}{l}\begin{array}{l}{\lambda }_{i}^{\alpha }={\gamma }_{i}^{\alpha }\lambda \end{array}\end{array}$$
(4)
$$\begin{array}{l}\begin{array}{l}{\beta }_{i}^{\alpha }={{\psi }}_{i}^{\alpha }\beta +s\end{array}\end{array}$$
(5)

The rate of awareness and the infection rate are interdependent since $${\beta }_{i}^{\alpha }$$ depends on $${\lambda }_{i}^{\alpha }$$ (see eq. 7)25. Both rates are characterised by the heterogeneous factors, $${\gamma }_{i}^{\alpha }$$ and $${\psi }_{i}^{\alpha }$$, defined as follows:

$$\begin{array}{c}{\gamma }_{i}^{\alpha }=\frac{{s}_{i}^{\alpha }}{1+{s}_{i}^{\alpha }}\end{array}$$
(6)
$$\begin{array}{c}{\psi }_{i}^{\alpha }=\frac{1}{1+{\lambda }_{i}^{\alpha }}\cdot \frac{1}{{Y}_{i}^{\alpha }}\end{array}$$
(7)

In eq. 5 we indicate with s the spontaneous contagion, which evaluates the realistic condition to contract the contagion spontaneously regardless the interactions on the whole multiplex network4. We define the awareness matrix Λ, where each element is calculated based on eq. 4, as follows:

$${\rm{\Lambda }}=[\begin{array}{llll}{\lambda }_{1}^{1} & {\lambda }_{1}^{2} & \mathrm{..}. & {\lambda }_{1}^{M}\\ {\lambda }_{2}^{1} & {\lambda }_{2}^{2} & \mathrm{..}. & {\lambda }_{2}^{M}\\ \mathrm{..}. & \mathrm{..}. & \mathrm{..}. & \mathrm{..}.\\ {\lambda }_{N}^{1} & {\lambda }_{N}^{2} & \mathrm{..}. & {\lambda }_{N}^{M}\end{array}]\,\in {{\mathbb{R}}}^{N\times M}$$
(8)

and, the matrix B, whose elements are the infection rate for each node in each layer (see eq. 5).

$$B=[\begin{array}{llll}{\beta }_{1}^{1} & {\beta }_{1}^{2} & \mathrm{..}. & {\beta }_{1}^{M}\\ {\beta }_{2}^{1} & {\beta }_{2}^{2} & \mathrm{..}. & {\beta }_{2}^{M}\\ \mathrm{..}. & \mathrm{..}. & \mathrm{..}. & \mathrm{..}.\\ {\beta }_{N}^{1} & {\beta }_{N}^{2} & \mathrm{..}. & {\beta }_{N}^{M}\end{array}]\,\in {{\mathbb{R}}}^{N\times M}$$
(9)

In the second process spreading process, UAF(Aπ), we introduce an alternative state Aπ, where if π = 1 the awareness is only referred to the primary contagion phenomenon. In the presence of variously correlated issues with the main contagion process, we define the overlapping awareness as follows:

$$\begin{array}{l}\begin{array}{l}\overline{a{w}_{i}}=\mathop{\overbrace{{a{w}_{i}|}_{\pi \mathrm{=1}}}}\limits^{awareness}+\mathop{\overbrace{\sum _{\pi \mathrm{=2}}^{T}a{w}_{i}^{\pi }}}\limits^{awareness\,on\,correlated\,\,issues}\end{array}\end{array}$$
(10)

with $$a{w}_{i}^{\pi }={\varphi }_{1,\pi }\cdot a{w}_{i}$$, where ϕ1,π is the ϕ-correlation between the primary contagion phenomenon and the other issues on a space of issues T. Based on the previous definition of overlapping awareness, the $${w}_{ij}^{\alpha }$$ becomes:

$$\begin{array}{l}{w}_{ij}^{\alpha }={h}_{ij}^{\alpha }\cdot \mathop{\overbrace{|\overline{a{w}_{i}}-\overline{a{w}_{j}}|}}\limits^{difference\,\,of\,awareness}+1\end{array}$$
(11)

considering also the awareness on T. In order to capture the potential heterogeneity of the network structure in terms of weights, we introduce a measure of centrality of both nodes and layers, X i and zα as defined in63, to obtain the simultaneous ranking of nodes and layers.

These measures are coupled to get a simultaneous ranking of nodes X i and layers zα, an overall measure of centrality for nodes and layers. In our model, it is dependent on the weights of the multiplex network, therefore including awareness and homophily (see Supplementary Figure S1). We exploit this kind of measures because we apply a rewiring process64, in which we choose the fraction of the links to be rewired considering the less central nodes in the less central layer, based on the previously defined ranking (see Simulation Results).

## Results

### Simulation Results

Simulations have been carried out considering a multiplex network with M = 3 layers, where each layer is modeled as a scale-free network65 with N = 1000 nodes. In Fig. 3, each curve corresponds to a different value of the ϕ-correlation of the primary contagion phenomenon with the other issue, in both cases of anti-correlation and positive correlation. The plots show how the density of infected nodes depends on to what extent the specific issue is correlated with the social contagion of the primary phenomenon. In (a), where nodes maintain a high attention to the contagion (see details in Model), we can observe how the density of infected nodes for an anti-correlated issue is lower than the case of a positively correlated issue. This extremely interesting result is due to the fact that exceeding in information on issues positively correlated to the contagion phenomenon may produce a negative influence on it, in fact encouraging the contagion rather than curbing it. In (b) nodes’ attention to contagion fades quickly over time, so this vanishes the effect of correlation and the density of infected nodes in the two cases of anti-correlation and positive correlation results approximately the same. Finally, in (c), the two dynamics are close and the high probability of getting into the faded state causes a scarce interest in the main contagion. It produces an overall decrease in the density of infected and in some points the anti-correlated curve is better than the positive correlated one because the dynamics after contagion is faster. In Fig. 4, we show how the double heterogeneity, in terms of both infection rates and rates of awareness, allows delaying the contagion outbreak compared to the homogeneous case, where nodes have a uniform susceptibility and rate of awareness. Comparing the phase diagrams before and after applying the rewiring, we can observe that the contagion threshold is more delayed in the post-rewiring cases, as we expected. Overall, the gap among the contagion thresholds between homogeneous and heterogeneous cases is wider in the anti-correlated case. In other words, the figure highlights the effect due to the presence of overlapping awareness, depending on the type of correlation with the primary contagion phenomenon. Although the impact is overall positive delaying the threshold, it is more evident in the anti-correlated case. In Fig. 5, we show the results of the data-driven approach with regards to a population of nodes (see details in Methods), according to the data on suicidal ideation spreading, taking into account the two temporal windows before (pre-event) and after a specific suicide event (post-event). In (a), where the overlapping awareness is referred to a positive correlated keyword, the more vulnerable nodes (small-sized nodes) show a high infection rate, and this is more evident in the post-event case, as highlighted by blue circle, as the rate of awareness increases. The red circle emphasises the area with more vulnerable people in the pre-event case, while the yellow circles show the effect of the overlapping awareness’ increasing. In (b), in the case of anti-correlated keyword, the overall infection rate is lower than the previous case, and as the rate of awareness increases, the distribution of the more vulnerable nodes remains confined in a region of low infection rates. This means that, differently from the previous case, the rate of awareness does not boost the contagion, but bounds the more vulnerable people within a range of low infection rates, thanks to the spread of positive contents, such as prevention, related to suicide. By using the red circle and the blue circle we highlight the high density area of vulnerable people in the pre-event case and post-event case, respectively. Yellow circles underline the effect of the overlapping awareness. We shed light on how the overlapping awareness apparently could act on the less vulnerable people (high-sized nodes), but influences the overall network, through social contagion dynamics. This demonstrates the dual role of overlapping awareness in the case of a social contagion phenomenon, such as suicidal ideation (see Discussion).

## Methods

### Dynamic Microscopic Markov Chain

To explore the dynamics of the coevolution of social contagion and awareness spreading on the weighted multiplex network, we take into account the Dynamic Microscopic Markov Chain Approach (MMCA). Initially, we assign to each node a state probability to be in one of the initial states. At the beginning, each node in the weighted multiplex network can occupy only one of the following states: susceptible and unaware (SU), infected and aware (AI), and susceptible and aware (SA). Some states are not reachable or do not exist, such as IU (Infected Unaware), IF (Infected Faded), SAπ (Susceptible - Overlapping Aware) and FAπ (Faded - Overlapping aware) (see Fig. 6). At time step t each node i can occupy one of the initial three states, with probabilities $${p}_{i}^{SU}(t)$$, $${p}_{i}^{SA}(t)$$ and $${p}_{i}^{IA}(t)$$ respectively. Moreover, we define: q i (t), probability of node i not being infected at time step t and r i (t), probability of unaware node i staying unaware at time step t, as follows:

$$\begin{array}{l}{q}_{i}(t)=\mathrm{(1}-{\bar{\beta }}_{i})\prod _{j}\mathrm{[1}-{a}_{ji}{p}_{j}^{I}(t){\bar{\beta }}_{i}]\end{array}$$
(12)
$$\begin{array}{l}{r}_{i}(t)=\mathrm{(1}-{\bar{\lambda }}_{i})\prod _{j}\mathrm{[1}-{a}_{ji}{p}_{j}^{A}(t)\overline{{\lambda }_{j}]}\end{array}$$
(13)

where a ij are the elements of the adjacency matrix of each layer of the weighted multiplex network. $${\bar{\beta }}_{i}$$ and $${\bar{\lambda }}_{i}$$ are the “elected infection rate” and the “elected rate of awareness” of the node i, respectively. Once calculated the centrality measures of nodes and layers X i and zα, from this heterogeneous ranking we extract the “elected” layer, that is the most central layer and in both matrices B and Λ, we select the corresponding column. We consider the most central layer because it is the most influential in the evaluation of the transition dynamics. The following MMCA equations represent the probability of each node of being in one of the states at time step t + 1, as showed in Fig. 6:

$$\begin{array}{rcl}{p}_{i}^{SA}(t+1) & = & {q}_{i}(t){p}_{i}^{SA}(t)+(1-{r}_{i}(t))(1-\delta ){p}_{i}^{SU}(t);\\ {p}_{i}^{IA}(t+1) & = & (1-{q}_{i}(t))(1-\varepsilon ){p}_{i}^{SA}(t)+(1-\mu ){p}_{i}^{IA}(t);\\ {p}_{i}^{I{A}^{\pi }}(t+1) & = & \varepsilon (1-{q}_{i}(t))(1-\mu ){p}_{i}^{SA}(t);\\ {p}_{i}^{R{A}^{\pi }}(t+1) & = & \mu \varepsilon (1-{q}_{i}(t)){p}_{i}^{SA}(t)+\mu \varepsilon (1-\delta ){p}_{i}^{IA}(t);\\ {p}_{i}^{SU}(t+1) & = & {r}_{i}(t){p}_{i}^{SU}(t);\\ {p}_{i}^{SF}(t+1) & = & \delta (1-{r}_{i}(t)){p}_{i}^{SU}(t);\\ {p}_{i}^{RA}(t+1) & = & \mu (1-\delta )(1-\varepsilon ){p}_{i}^{IA}(t);\\ {p}_{i}^{RF}(t+1) & = & \mu \delta {p}_{i}^{IA}(t);\end{array}$$
(14)

To obtain the contagion threshold, we explore the steady state solution of the system constituted by the previous equations. When time t → +∞, there exists a contagion threshold β C for the two coevolving processes, so that the contagion can outbreak only if ββ C . Following the same conditions of25, the contagion threshold is given by the order parameter ρ i and it is defined as follows:

$$\begin{array}{l}{\rho }^{I}=\frac{1}{N}\sum _{i\mathrm{=1}}^{N}{p}_{i}^{I}=\frac{1}{N}\sum _{i\mathrm{=1}}^{N}{p}_{i}^{IA}\end{array}$$
(15)

Thus, starting from equation $${p}_{i}^{IA}(t+\mathrm{1)}$$ (see eq. (14)), at steady state we have:

$$\begin{array}{l}{p}_{i}^{IA}=(1-{q}_{i})(1-\varepsilon ){p}_{i}^{SA}\end{array}$$
(16)

Since around the contagion threshold β C , the infected probability is close to zero ($${p}_{i}^{IA}={\eta }_{i}\ll 1$$), the probabilities of being infected can be approximated as follows:

$$\begin{array}{l}{q}_{i}=(1-\overline{{\beta }_{i}})[1-\overline{{\beta }_{j}}\sum _{j}{a}_{ij}{\eta }_{j}]=(1-\overline{{\beta }_{i}})(1-{\omega }_{i})\end{array}$$
(17)

where:

$$\begin{array}{l}{\omega }_{i}=\overline{{\beta }_{j}}\sum _{j}{a}_{ij}{\eta }_{j}\end{array}$$
(18)

Furthermore, close to the contagion onset we have that the fading rate is approximately close to zero ($$\delta \simeq 0$$). Considering this approximation into eq. 16 and omitting higher order items, equation 16 is reduced to the following form:

$$\begin{array}{l}\begin{array}{l}\mu {\eta }_{i}\simeq (1-\varepsilon ){p}_{i}^{SA}\overline{{\beta }_{i}}\overline{{\beta }_{j}}\sum _{j}{a}_{ij}{\eta }_{j}\end{array}\end{array}$$
(19)

The contagion threshold is obtained starting from the following condition:

$$\begin{array}{l}\sum _{j}|(1-\varepsilon )\overline{{\beta }_{i}}{p}_{i}^{SA}{a}_{ij}-\frac{\mu }{\overline{{\beta }_{j}}}{t}_{ji}|{\eta }_{j}=0\end{array}$$
(20)

where t ji are the elements of the Identity matrix. By defining the matrix H whose elements are given by: $${h}_{ij}=[(1-\varepsilon )\overline{{\beta }_{i}}{p}_{i}^{SA}]{a}_{ij}$$, the contagion threshold β c is the one that satisfies that Λmax(H), the largest eigenvalue of the matrix H is given by $${{\rm{\Lambda }}}_{\max }(H)=\mu /\overline{{\beta }_{j}}$$, and finally we get: β c = μmax(H).

### Data-driven analysis

In our model, we consider a data-driven approach for evaluating the overlapping awareness, which is the result of the different types of awareness on suicidal ideation spreading as a social contagion phenomenon66,67. First, we consider data derived from a machine classification dataset for suicide-related communications, where classes represent the types of suicidal communication with relative percentage proportion in dataset41,68. We decide to construct our population of N = 400 nodes based on these classes41,68 which represents the best representation of how people generally communicate on the topic of suicide. We associate an awareness score to each node which depends on three measures. The first measure is related to a distinct probability to post a text according to the associated class, that is an initial measure of awareness ranging from a low level to a high level. The second measure is associated with the Google search popularity of terms related to the classes of two geographical countries (see Supplementary Table S1, Figures S2, S3). Homophily corresponds to the geographical proximity of nodes, so that two individuals of the same country will have a high homophily. The third measure relates to the searches on Google Trends on issues either positively or anti-correlated with the primary contagion. Google Trends allows evaluating the time evolution of awareness and setting up a measure related to the interest in specific aspects of suicide contagion. In particular, we keep track of the total Google Trends search-volume of some of the most significant suicide keywords, such as ‘suicide’ and ‘suicide prevention’, in two temporal windows related to the period around a specific suicide event. We aim at shedding light on how these searches pre-event suicide and post-event suicide contribute to the contagion dynamics. The temporal window is that one around the Robin Williams’ suicide, occurred on August 11, 2014, so the two temporal windows before and after the event suicide are respectively from June 10, 2014 to August 10, 2014, and from August 12, 2014 to October 10, 2014. The target is to analyse the temporal evolution of the overlapping awareness, consisting of an aggregated measure of these sources. Furthermore, in order to extend our understanding on the importance of the Google Trends on the awareness about the suicide contagion, we choose three keywords, comparing the Google search popularity in different countries across the world of these terms in the subsequent year of the suicide event with the suicide rates of the same countries (see Supplementary Figure S3).

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## Acknowledgements

This work was partially supported by the Research Grant: Italian Ministry of University and Research - MIUR “Programma Operativo Nazionale Ricerca e Competitività 2007–2013” within the project “PON-03PE-00132-1” - Servify. We thank Dr. Mario Raspagliesi and the team of “Terra Amica” for their advice on the importance of prevention strategies in the medical field. Furthermore, we thank Dr. Luca Passamonti, Clinical Research Associate at Department of Clinical Neuroscience (University of Cambridge) and his team, the members of the consortium “Propag-Ageing”, whose one of the co-authors (PL) belongs to, and the Prof. Zoe Kourtzi, Professor of Experimental Psychology and her team, for the helpful discussions on the results during the revision process.

## Author information

Authors

### Contributions

M.S., A.D.S., A.L.C., P.L. conceived the model and performed simulations. M.S., A.D.S., A.L.C., P.L. wrote the paper and reviewed the manuscript.

### Corresponding author

Correspondence to Marialisa Scatà.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

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Reprints and Permissions

Scatà, M., Di Stefano, A., La Corte, A. et al. Quantifying the propagation of distress and mental disorders in social networks. Sci Rep 8, 5005 (2018). https://doi.org/10.1038/s41598-018-23260-2

• Accepted:

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