\section*{Distinguishing simple and complex contagion processes on networks: Supplementary Material}


\section{Data description and generation of interaction networks and hypergraphs}
\label{SMsec_data}

The data sets we consider describe interactions between individuals, either in terms of physical proximity or online, and are temporally resolved.
We thus preprocess these data in order to build the  three types of structures needed for the numerical simulations: the temporally aggregated pairwise network, the aggregated hypergraph with all orders of interactions, and the aggregated triangle network.

First we build the aggregated network: we consider each interaction between two individuals as a pairwise link of a static network with weight given by the number of times (consecutive or not) that the connection has appeared in the temporal data. The aggregated network is used to simulate the simple and threshold contagion models, and the contagion events of the simplicial contagion model involving pairwise links.

We then generate the hypergraph by considering as hyperedges all the groups of simultaneous all-to-all interactions in the temporal data. Each hyperedge is stored with its weight given by the number of temporal repetitions. This is used to simulate the NL-hyper contagion process, where each hyperedge is characterized by its own infection probability.

Finally, we consider each hyperedge of order higher than 2 (involving more than 3 nodes) and we decompose it into multiple groups of three nodes (in other words we project these interactions on hyperedges of size 3). The set of the obtained three-nodes interactions, together with the hyperedges of order 2, forms the aggregated triangle network. This is used to simulate the infections taking place on triangles in the simplicial contagion model (those occurring with probability $\beta^{\Delta}$). The union of the aggregated triangle network and the pairwise network forms indeed a simplicial complex.

The unweighted networks are obtained by setting all the weights equal to 1.

\begin{table}[hb]
\centering
\begin{tabular}{|c|c|c|c|c|}
	\hline
	data set & Nb. of nodes & Nb. of pairwise & Nb. of 2nd order & Max. hyperedge \\
    &  & connections & connections
    (projected) & size\\
	\hline
	Workplace & 217 & 4274 & 772 & 4\\
	Hospital  & 75 & 1139 & 703 & 5\\
	Conference & 403 & 9565 & 2634 & 9\\
	Primary school France & 242 & 8317 & 5138 & 5\\
	High school France & 327 & 5818 & 2370 & 5\\
	University campus (48h) & 495 & 3486 & 2981 & 6\\
	Elementary school Utah (1d) & 339 & 8763 & 6868 & 8\\
	Middle school Utah (1d) & 591 & 13550 & 4453 & 7\\
	Email & 143 & 1800 & 6578 & 18\\
	\hline
\end{tabular}
\caption{\textbf{Size details about the used data sets}. The third column reports the number of pairwise connections in the aggregated network, i.e. all the connections of the networks reduced to pairwise interactions. The fourth column reports the number of 2nd order connections in the triangle network, i.e. the simplicial complex obtained by projecting higher orders on simplices of 3 nodes. The fifth column reports the number of nodes in the largest hyperedge.}
\label{tab_datasets}
\end{table}

We use the following publicly available data sets of social interactions (see also Table \ref{tab_datasets}):
\begin{itemize}
	\item \textbf{Sociopatterns}: data sets describing pairwise face-to-face interactions among individuals, collected using RFID devices \cite{SP}. These data have a temporal resolution of 20 seconds and describe contacts between individuals collected during several successive days. We consider five of these data sets: a workplace~\cite{Genois2018}, a conference~\cite{Genois2018}, a hospital~\cite{vanhems2013estimating}, a high school~\cite{mastrandrea2015contact}, and a primary school~\cite{Gemmetto2014,stehle2011high}.
	%
	\item \textbf{University campus}~\cite{sapiezynski2019interaction}: data set describing pairwise co-location interactions among  students at the campus of the Technical University of Denmark for one month, recorded by the exchange of Bluetooth radio packets between smartphones every 5 minutes for 1 month. We consider a reduced version of the original data set, involving only the first 48 hours, in order to obtain an aggregated network with a moderate density.
	%
	\item \textbf{Contacts among Utah's School-age Population}~\cite{toth2015role}: data set describing pairwise interactions among students in an elementary and a middle school in Utah. The interactions were collected with a temporal resolution of 20 seconds for 2 days via Wireless Ranging Enabled Nodes. We consider a limited version of the original data set, involving only the first day.
	%
	\item  \textbf{Email Enron}~\cite{klimt2004enron}: data set describing group interactions among a company employees via email messages: each node corresponds to an email address and a hyperedge includes the sender and all receivers of an email.
\end{itemize}






\section{Models of contagion}
\label{secSM_models}

We describe in details the four different contagion models used in the main text. For the sake of generality we define the models on weighted networks, making use of the weighted adjacency matrix $W$, which reduces to the adjacency matrix for non-weighted networks.
In order to make clear the distinction between models we give for each of them the node infection probability.

In simple contagion processes, infections take place independently on each link connecting infected to susceptible nodes at rate $\beta$ multiplied by the weight of the link (normalized such that $\max_{(i,j)}W_{ij} = 1$). The probability that a susceptible node $i$ is infected at a specific time $t$ is hence given by:
\begin{equation}
	p_i(t) = 1 - \prod_{j\in I(t)}(1-\beta W_{ij}),
	\label{eq_simple_W}
\end{equation}
where $I(t)$ is the set of infected nodes at time $t$, so the second term is the product of the independent probabilities that node $i$ is not infected by each of its infected neighbors.

\smallskip


The threshold model is a deterministic model where a susceptible node is infected when the fraction of its neighbors that are infected exceeds a  threshold $\theta$. More generally, if the network is weighted, a susceptible node is infected when the weight of its connections with infected nodes divided by the total weight of its connections exceeds the  threshold. The general probability can be described by a Heaviside function:
\begin{equation}
	p(t) = H\biggl(\frac{1}{s_i}\sum_{j\in I(t)}W_{ij}-\theta \biggr) = 
	\begin{cases}
		1 \text{ if } \frac{1}{s_i}\sum_{j\in I(t)}W_{ij} \geq \theta\\
		0 \text{ if } \frac{1}{s_i}\sum_{j\in I(t)}W_{ij} < \theta
	\end{cases}\,. 
	\label{eq_thresh_W}
\end{equation}
where $s_i=\sum_jW_{ij}$ is the total strength of node $i$.

\smallskip


The simplicial contagion model can be considered as a generalization of the simple model on simplicial complexes, where nodes are connected via simplices instead of only pairwise links. We only consider simplices up to the second order, i.e. three nodes interactions, neglecting structures of higher orders (which will only appear decomposed into edges of first and second order). Each susceptible node can be infected via its first-order connections, at rate $\beta$ (like in simple contagion), or via its second-order connections, if present, at rate $\beta^{\Delta}$. Second order contagions can only happen if the susceptible node is part of one (or more) closed triangles where  the other two nodes are both infected. Notice that in this case, as the model is defined on a simplicial complex, the two nodes additionally contribute independently to the first-order contagion. All infection probabilities are multiplied by the weight of the involved simplex, via the matrix $W$ at first order and tensor $W^{\Delta}$ at second order, where the element $W^{\Delta}_{ikl}$ is the weight of the closed triangle $(i,k,l)$, normalized by the same factor as the link weights. The probability per unit time that a susceptible node is infected is hence given by the combination of its first and second order interactions with infected nodes: 
\begin{equation}
	p_i(t) = 1 - 
 \prod_{j\in I(t)}(1-\beta W_{ij})
 \prod_{k,l\in I(t)}(1-\beta^{\Delta} W^{\Delta}_{ikl}) \ .
	\label{eq_simplicial_W}
\end{equation}

\smallskip

The non-linear hypergraph contagion model is defined on hypergraphs rather than on simplicial complexes, and  multi-body interactions are modelled through a non-linear infection term. In this case, we consider the SI epidemic model on a hypergraph with hyperedges of arbitrary size (i.e. links, triangles and also higher-order structures up to the maximum hyperedge size $M$): if in a hyperedge $h=(i_1,i_2,...i_m)$ of size $m$ there are $n_h$ infected individuals, each of the remaining $(m-n_h)$ susceptible nodes is independently infected with probability $\lambda n_h^{\nu} W^m_h$. Note that $\nu$ sets the non-linearity, $\lambda \in [0,1]$ and $m \in [2,M]$. All the infection terms are multiplied by the weight of the involved hyperedge, via the tensor $W^m$ at the $(m-1)$th order, whose element $W^m_h=W^m_{i_1,i_2,...,i_m}$ is the weight of the hyperedge $h=(i_1,i_2,...i_m)$. By assuming that each hyperedge produces an independent contagion process, the probability per unit time that a susceptible node $i$ is infected is hence given by:
\begin{equation}
	p_i(t)=1-\prod_{m=2}^M\prod_{h \in \mathcal{E}^m_i} (1-\lambda n_h^{\nu} W^m_h)
\end{equation}
where $\mathcal{E}^m_i$ is the set of all hyperedges of size $m$ containing $i$ and $n_h$ is the number of infected individuals in $h$.



\section{Classification tasks for processes on all the data sets considered}

\subsection{SI models on unweighted networks}

The analyses reported in the main text using the workplace network can be repeated with the networks described in section~\ref{SMsec_data}. The results for SI model on unweighted networks are shown in Figs.~\ref{figSM_distr_unweighted1} and \ref{figSM_distr_unweighted2}, which report the distributions of the correlations $C_1$ and $C_2$, the resulting ROC curves and the confusion matrices, 
similarly to Fig.~\ref{fig2} and Fig.~\ref{fig3}(a,b) in the main text. It is evident that the results obtained in the main text can be generalized to other networks, showing the versatility of the proposed classifier (with a classification accuracy always above 0.7).
We however notice that the classification performances are worse for the networks of the two schools in France and the elementary school in Utah, with in particular 
a lower ability to distinguish between simple and simplicial model when using $C_2$ (even if the AUC is still above 0.7). This could be due to the existence of a strong community structure in these networks, but future work is needed to investigate how different network topologies affect the classification based on the measure of correlations.

\begin{figure*}[thb]
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_conf.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_hosp.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_highSc.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_PrSc.pdf}
\caption{\textbf{Unweighted SI model.} 
Analogous of Fig.~\ref{fig2} and Fig.~\ref{fig3}(a) of the main text for four other data sets: conference, hospital, high school, and primary school. For each data set, the top panels report the distributions of $C_1$ and $C_2$. The three panels below give: in orange, the ROC curve when using $C_1$ to classify 
threshold model processes against the other three; in light green, the ROC curve when using $C_2$ to classify the simplicial model against simple and threshold; in green, the ROC curve when using $C_2$ to classify the NL-hyper model against simple and threshold. The bottom panels report the confusion matrices with 4 classes (simple, complex threshold, complex simplicial and complex NL-hyper) and with 3 classes (simple, complex threshold and complex higher-order).
For the stochastic models (simple, simplicial and NL-hyper) 1000 realizations are implemented for each parameter setting, always starting with one random infected node. For the threshold model, due to its deterministic nature, we use only one realization for each different initial condition, i.e. one for each network node, for each parameter value. Confusion matrices are obtained using the same number of instances for each model.
Parameter values: $\beta \in \{0.005, 0.008, 0.014, 0.023, 0.039\}$ for both simple and simplicial models, $\beta^{\Delta} = 0.8$, $\lambda \in \{0.0001, 0.001, 0.006, 0.011, 0.015\}$, $\nu=4$,
$\theta \in \{0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10\}$. Additional data sets are shown in Fig.~\ref{figSM_distr_unweighted2}.
\label{figSM_distr_unweighted1}}
\end{figure*}



\begin{figure*}[thb]
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_DTU.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_Utahelem.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_Utahmid.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_email.pdf}
\caption{\textbf{Unweighted SI model.} 
Same as Fig.~\ref{figSM_distr_unweighted1}
for the data sets: University campus, elementary school in Utah, middle school in Utah, and email.
\label{figSM_distr_unweighted2}}
\end{figure*}



\subsection{SI models on weighted networks}
\label{secSM_weight}

The classifier can be applied to spreading processes occurring on weighted networks: the processes spread according to the links' and hyperedges' weights, 
as described in section~\ref{secSM_models}.
The classifier takes as input the correlations between the order of infection of the nodes and their strengths instead of their degrees.

The results are reported in Figs.~\ref{figSM_distr_weighted0}, \ref{figSM_distr_weighted1} and \ref{figSM_distr_weighted2}.
Overall, the classifier performs well in these cases too (accuracy always above 0.7). For some data sets, we observe however a reduced performance with respect to the unweighted case, especially for the ability to distinguish between higher-order complex contagion and simple contagion. We have checked that the weights of hyperedges of size larger than 2 are typically much smaller than the weights of the links, so that the spreading processes along links dominate even for higher order processes: as a result, higher-order complex and simple contagion become more similar processes.


\begin{figure*}[thb]
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_workplace.pdf}
\caption{\textbf{Weighted SI model.} Analogous to Figs.~\ref{fig2} and \ref{fig3} for the workplace data set, using the weighted aggregated networks for the spreading processes. The parameters are chosen as in Fig.~\ref{figSM_distr_unweighted1}, except for $\theta$, which becomes a threshold for nodes strength instead of nodes degree and is consequently adapted, taking the values $\{0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, 0.20, 0.22\}$. 
\label{figSM_distr_weighted0}}
\end{figure*}



\begin{figure*}[thb]
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_conf.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_hosp.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_highSc.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_PrSc.pdf}
\caption{\textbf{Weighted SI model.} Analogous of Fig.~\ref{figSM_distr_weighted0} for other networks: conference, hospital, high school in France, and primary school in France. Additional data sets are shown in Fig.~\ref{figSM_distr_weighted2}. 
\label{figSM_distr_weighted1}}
\end{figure*}

\begin{figure*}[thb]
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_DTU.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_Utahelem.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_Utahmid.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_w_email.pdf}
\caption{\textbf{Weighted SI model.} Same as Fig.~\ref{figSM_distr_weighted1} for the data sets: University campus, elementary school in Utah, middle school in Utah, and email.
\label{figSM_distr_weighted2}}
\end{figure*}

\subsection{SIR models on unweighted networks}
\label{secSM_SIR}

The above analyses can be repeated using an SIR instead of an SI model, i.e. including the possibility for nodes to recover (with a fixed probability per unit time $\mu$) after they have been infected. A recovered node cannot infect other nodes or be infected again. The process ends when none of the nodes is infected (they are all susceptible or recovered). 

The results of the classifier in this case are reported for unweighted networks in Figs.~\ref{figSM_distr_SIR0}, \ref{figSM_distr_SIR1} and \ref{figSM_distr_SIR2}. 
The classifier performances are slightly reduced with respect to the SI case. This can be understood by the fact that the nodes remain infected only a limited amount of time (independent of their topological properties and of their neighbors' properties). When a node recovers, it does not anymore contribute to the risk of infection of its neighbours. When reinforcement effects are needed (complex contagion), this can thus alter substantially the relationship between the structure of a node's neighborhood and its risk of becoming infected, especially if the recovery is fast. 

Here, as a preliminary analysis, we have considered 
simulations with $\mu=0.1$, and 
the classifier performances remain good, with accuracy values always above 0.7. 
A future deeper analysis will investigate in more details the impact of the recovery parameter on the classifier performance.



\begin{figure*}[thb]
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_workplace.pdf}
\caption{\textbf{Unweighted SIR model.} Analogous of Figs.~\ref{fig2} and \ref{fig3} for the SIR model with $\mu=0.1$ with the workplace network. Additional data sets are shown in Figs.~\ref{figSM_distr_SIR1} and \ref{figSM_distr_SIR2}.
\label{figSM_distr_SIR0}}
\end{figure*}



\begin{figure*}[thb]
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_conf.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_hosp.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_highSc.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_PrSc.pdf}
\caption{\textbf{Unweighted SIR model.} Analogous of Fig.~\ref{figSM_distr_unweighted1} for the SIR model with $\mu=0.1$. Results for additional data sets are shown in Fig.~\ref{figSM_distr_SIR2}.
\label{figSM_distr_SIR1}}
\end{figure*}



\begin{figure*}[thb]
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_DTU.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_Utahelem.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_Utahmid.pdf}
\includegraphics[width=0.41\textwidth]{figs/SM/distr+conf_SIR_email.pdf}
\caption{\textbf{Unweighted SIR model.} 
Same as Fig.~\ref{figSM_distr_SIR1} for
additional data sets: University campus, elementary school in Utah, middle school in Utah, and email.
\label{figSM_distr_SIR2}}
\end{figure*}



\clearpage
\newpage



\section{Classification across data sets}
\label{sec_train1_test2}

We report here the results of classification experiments in which we train and test the classifier 
on instances of the spreading processes taking place on distinct networks.

In a first experiment, we train the classifier using instances of spreading processes run on one data set, and we test on instances simulated on a second data set. The classification accuracy depends on the  networks chosen for training and testing, as reported in the table in Fig.~\ref{figSM_net_of_net}(a), where the results above 0.7 are highlighted (diagonal elements represent the cases where training and testing are implemented using the same data set). 
The same results are also depicted on the right as a network of data sets: arrows start from the data set used for training and end at the data set used for testing. Only arrows associated to an accuracy above 0.6 are drawn, in purple for accuracy between 0.6 and 0.7, in turquoise above 0.7.

The second experiment instead involves training on eight different data sets and testing on the ninth one. The table in Fig.~\ref{figSM_net_of_net}(b) reports the accuracy obtained for testing on each data set. The highest accuracy is obtained when the testing data set is the workplace (we report the corresponding confusion matrix on the right of the figure).

In all these experiments we consider the same model parameter values as in Fig.~\ref{fig2}:

$\beta \in \{0.005, 0.008, 0.014, 0.023, 0.039\}$ for both simple and simplicial models, 

$\beta^{\Delta} = 0.8$, 

$\lambda \in \{0.0001, 0.001, 0.006, 0.011, 0.015\}$, 

$\nu=4$,

$\theta \in \{0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10\}$.

\begin{figure*}[thb]
\includegraphics[width=1\textwidth]{figs/SM/Train_net1_test_net2.pdf}
\caption{(a): Train on one data set, test on another one. The table reports the accuracy for each pair of data sets, while the network only reports the pairs with accuracy above 0.6, where arrows go from the training to the testing data set. (b): Train on eight data sets, test on the ninth. The table reports the accuracies and the confusion matrix is shown for the best case: testing on workplace with training implemented on the other data sets.}
\label{figSM_net_of_net}
\end{figure*}

\clearpage
\newpage


\section{Robustness with respect to incomplete observations}

The results presented in the main text have been obtained by observing the entire infection chains, i.e. all the infected nodes with their infection order. Here we explore a more realistic case where the access to infection data is limited and only a partial observation of infection chains is possible. In the two following sections we show how the proposed classification changes with respect to a reduction in the number of observed infections: first observing just a random percentage of nodes, secondly observing just the first infected.


\subsection{Observing only a fraction of the nodes}

Let us assume that only a random subset of nodes can be observed. We rely on the same numerical simulations that are presented in the main text, with the difference that the correlation between infection order and topological features is computed only for an a priori chosen subset of nodes, ignoring all the others. We repeated the experiment for 10 realizations of the random subsets, and for each
of the following subset sizes: $20\%$, $40\%$, $60\%$, $80\%$ and $100\%$ of the network nodes. For each experiment we computed the Random Forest accuracy of the classification in three classes $\{$Simple/Threshold/Higher-order$\}$, the area under the ROC curve when using $C_1$ to single out the threshold model vs. the others and the AUC when using $C_2$ to single out the higher-order models vs.
the simple and threshold ones. The results are reported in Fig.~\ref{fig_perc} for the nine data sets considered: the classification remains very efficient 
in all cases.

\begin{figure*}[thb]
\includegraphics[width=0.9\textwidth]{figs/SM/cross_valid_random_h_unweighted.pdf}
\caption{Random Forest accuracy and AUC when using $C_1$ and $C_2$, as a function of the number of observed nodes in the different data sets. The $x$-axes report the number of observed nodes, corresponding to $20\%$, $40\%$, $60\%$, $80\%$ and $100\%$ of the total. The dots and bars show average and standard deviation for results collected on 10 realizations of the random subsets for each size.
\label{fig_perc}}
\end{figure*}



\clearpage
\newpage
\subsection{Observing only the first infected nodes}

As a second experiment, we try to classify contagion models by observing only the first infected nodes. The number of nodes to observe is
chosen as in the previous experiment, corresponding to  $20\%$, $40\%$, $60\%$, $80\%$ and $100\%$ of the total size.
In this case, Figure \ref{fig_first_h} shows that the performance depends more strongly on the fraction of observed nodes,
and is smaller with respect to the case where the same number of nodes have been chosen at random.
Combined with the results of Fig.~\ref{fig_perc}, this suggests that a reduction of the number of observed nodes does not have a drastic
effect if they are distributed along the entire infection chain, but the method is less robust if only the first phase of the infection pattern is observed. 

\begin{figure*}[thb]
\includegraphics[width=1\textwidth]{figs/SM/cross_valid_first_h_unweighted.pdf}
\caption{Random Forest accuracy and 
AUC when using $C_1$ and $C_2$, obtained by observing only the first infected nodes in each infection chain. The number of observed nodes is reported on the $x$-axes, corresponding to $20\%$, $40\%$, $60\%$, $80\%$ and $100\%$ of the total number of nodes.
}
\label{fig_first_h}
\end{figure*}


\clearpage
\newpage

\section{Influence of model parameters}
\label{secSM_param}

The classification experiments shown in the main text take into account the four different models of contagion with several choices of each model's parameters. In some cases the classifier performances depend on these parameters, while in others the parameters choice is less relevant, as discussed in the following. 

\subsection{Parameters of the higher-order models}

The parameters' choice is particularly relevant when 
the variation of these parameters can make two models' contagion mechanisms more similar. 
For instance the simplicial model is an extension of the simple one and reduces to a simple propagation for $\beta^{\Delta} \to 0$. 
The dynamics -- and hence the order at which nodes are infected -- in the simplicial model appears thus more similar to that of 
simple model for smaller values of $\beta^{\Delta}$ and larger values of $\beta$: in such cases, contagion events take place more via pairwise 
links than via triangles, weakening the importance of the higher-order mechanism.
This naturally affects the classification performances, as shown in Fig.~\ref{fig_paramHO}(a), 
which shows the area under the ROC curve when using $C_2$ to distinguish instances of the simplicial model from instances of the simple propagation, for 
varying values of $\beta$ and $\beta^{\Delta}$
(the instances of the simple model correspond to values of $\beta$ ranging between 0.005 and 0.1). 
The AUC decreases when $\beta$ increases and when $\beta^{\Delta}$ decreases.

Analogously, in Fig.~\ref{fig_paramHO}(b) we show the AUC obtained when using $C_2$ to classify instances of the NL-hyper model and instances of the simple model, at varying $\nu$ and $\lambda$. 
In this case, the AUC decreases slightly when $\lambda$ increases and $\nu$ decreases, but remains very high for all explored parameter values.

\begin{figure*}[thb]
\subfigure[]{\includegraphics[width=0.4\textwidth]{figs/SM/var_beta_c5_SI_simplicial_simple_all_beta_unweighted_workplace_lar.pdf}}
\subfigure[]{\includegraphics[width=0.4\textwidth]{figs/SM/var_lambda_c5_SI_NLHO_simple_all_beta_unweighted_workplace_lar.pdf}}
\caption{AUC of the ROC curve when using $C_2$ 
to distinguish simplicial vs. simple contagion models
(a) and NL-hyper vs. simple contagion (b), for 
 varying parameters. Each colored pixel is the result of 1000 experiments. The spreading processes are simulated on the workplace data set.}
\label{fig_paramHO}
\end{figure*}



In Fig.~\ref{fig3}(d) of the main text we showed the confusion matrix obtained when classifying the four contagion models using all the nine considered networks. We depict here the confusion matrix obtained when considering the two higher-order models as a single class, 
and we show the related accuracy obtained at varying the parameter $\beta$ in simple and simplicial contagion (Fig.~\ref{fig_accuracy}(a)).
Analogously we show the confusion matrix and the accuracy obtained at varying $\beta$ relative to the experiment reported in Fig.~\ref{fig3}(e), where the classifier is trained on the High School France data set and tested on the conference data set (Fig.~\ref{fig_accuracy}(b)).


\begin{figure*}[h]
\includegraphics[width=0.6\textwidth]{figs/SM/accuracy_fig_3.pdf}
\caption{Confusion matrix and classification accuracy vs. $\beta$ in simple and simplicial model in the case of classification with all data sets (a) 
and in the case of the classification on the conference data set after training on the High School France network (b). 
For (a) five different splits of the correlation measures in training and testing sets have been considered, the plot reports mean and standard deviation.}
\label{fig_accuracy}
\end{figure*}

\clearpage
\newpage


In Figs.~\ref{fig_params_simple}, \ref{fig_params_simplical}, and \ref{fig_params_NLhyper} the distributions of $C_1$ and $C_2$ are shown
at varying parameters $\beta$ and $\lambda$. While the correlation distributions obtained with the simple contagion model 
are not particularly sensitive to the value of its unique parameter $\beta$, 
the effect of $\beta$ is instead important in the simplicial model, where it plays a role in the competition between contagion 
via links and via triangles. Smaller values of $\beta$ (i.e. more contagions on triangles) lead to smaller values of the 
correlation $C_2$ (larger in absolute value), and thus to larger differences with respect to the simple contagion model.
A similar effect is observed in the NL-hyper model at decreasing $\lambda$. 

\begin{figure*}
{\includegraphics[width=1\textwidth]{figs/SM/distr_param_simple_unweighted.pdf}}
\caption{\textbf{Simple contagion model.} Distributions of correlations $C_1$ and $C_2$ at varying parameter $\beta$ in unweighted SI simulations on the nine considered networks.} 
\label{fig_params_simple}
\end{figure*}



\begin{figure*}
{\includegraphics[width=1\textwidth]{figs/SM/distr_param_simplicial_unweighted.pdf}}
\caption{\textbf{Simplicial contagion model.} Distributions of correlations $C_1$ and $C_2$ at varying parameter $\beta$ (while $\beta^{\Delta}$ is fixed at 0.8) in unweighted SI simulations on the nine considered networks.}
\label{fig_params_simplical}
\end{figure*}


\begin{figure*}
{\includegraphics[width=1\textwidth]{figs/SM/distr_param_NLhyper_unweighted.pdf}}
\caption{\textbf{NL-hyper model.} Distributions of correlations $C_1$ and $C_2$ at varying parameter $\lambda$ (while $\nu$ is fixed at 4) in unweighted SI simulations on the nine considered networks.}
\label{fig_params_NLhyper}
\end{figure*}


\clearpage
\newpage

\subsection{Threshold parameter in threshold model}

The correlations measured in the threshold model are almost always around zero, since the nodes susceptibility 
usually does not depend on their local neighborhood. This is hence almost independent on the threshold parameter $\theta$, 
as shown in Fig.~\ref{fig_params_thresh}. We however notice that for certain networks smaller values of $\theta$ correspond to 
smaller values of correlations $C_1$ and $C_2$. This may be due to the fact that the smaller is $\theta$ the less important is the 
threshold effect, and the more similar the threshold model becomes to the simple contagion one.
The effect is however small so overall $\theta$ does not affect significantly the classifier performance.



\begin{figure*}[th]
{\includegraphics[width=1\textwidth]{figs/SM/distr_param_thresh_unweighted.pdf}}
\caption{\textbf{Threshold model.} Distributions of correlations $C_1$ and $C_2$ at varying parameter $\theta$ in unweighted SI simulations on the nine considered networks.}
\label{fig_params_thresh}
\end{figure*}




