Complex contagion in social systems with distrust

Contagion is the process through which a disease, an opinion, a behavior or a technological innovation spreads in a population of interacting individuals [1]. While some spreading processes only require a single exposure for a contagion to occur, others are only triggered by multiple exposures. In the first case, the spreading process is referred to as simple contagion, while in the latter case it is known as complex contagion [2]. The spread of an infectious disease in a population is a typical example of a simple contagion [3], while the propagation of a social behavior, such as the adoption of a social norm, of an opinion, or even a novel item can be characterized either as a simple [4] or a complex contagion [5]. In particular in social contagion there are cases where the individuals/agents of a social system can either accept or refuse transmission, based on the type of their social interactions, instead of just passively receive it as in the case of a virus [6].

The spread of a social behavior in a population is utterly determined by the precise patterns of interactions among individuals. Usually, the structure of contacts in a social system is mathematically modeled as a graph, whose nodes represent the individuals, the social agents, while the edges encode for their pairwise interactions [7]. Without surprise thus, the contagion process is shaped by the structure of such graph, for instance by its modular organization, the existence of short paths, weak ties or influential nodes [8, 9, 10, 11]. Complex contagion on networks has been notably analyzed using threshold models, where a susceptible individual becomes infected if a sufficiently large fraction of its neighbors is infected [12, 13].

Although network science provides powerful tools to model dynamical processes in social systems, the network approach has strong limitations in dealing with group interactions. The latter are indeed a fundamental ingredient of social systems and result to be a main driver for social contagion [14]. As a consequence, last years have seen a growing interest in higher-order networks, namely mathematical structures generalizing complex networks and accounting for group interactions [15, 16, 17, 18, 19]. Recent works on social contagion on higher-order networks have demonstrated that peer pressure and group reinforcement can induce novel phenomena, such as the appearance of abrupt phase transitions and bistability [14, 20, 21, 22, 23, 24, 25]. One aspect that still has not been considered and explored exhaustively is the role that trust/distrust can have on the diffusion of a novel idea or item, or in the adoption of a social norm. Trust among individuals is a key driver of contagion in social systems: agents are more prone to adopt a novel behavior if it comes from trusted sources. Moreover, infected agents, i.e., agents having adopted a given item, when they interact with other infected agents that they do not trust, can decide to stop adopting the item. The rationale behind the latter process is that two agents that distrust each other would not be inclined to share the same item. Some researches have already started to investigate the impact of trust on social contagion and epidemic spreading on networks [26, 27, 28], but no work has yet considered contagion processes on signed higher-order networks.

In this paper we propose a model of complex contagion on signed higher-order networks, which allows to investigate the impact of trust and distrust among individuals on the spread of contagion in a social system, by properly taking into account interactions in groups of two or more individuals at the microscopic level. Our model is inspired by the model of complex contagion on simplicial complexes proposed in Ref. [14]. Indeed, as the underlying structure of the social system we consider a simplicial complex instead of a network. A simplicial complex is a made not only by nodes and by links connecting pairs of nodes, and representing pairwise interactions between individuals, but also by more complex objects, which represent interactions in groups of more than two individuals. We then extend the model of Ref. [14] by introducing positive and negative signs over the links of the simplicial complex. It is however a common experience that the trust/distrust relations existing between two individuals can change once the two individuals are interacting into a group of larger size. For this reason we propose and analyze two variants of our social contagion model. In the first version of the model we assume that the signs of the pairwise interactions remain the same also in higher-order interactions. In the second one, we relax such an assumption allowing agents to exhibit pairwise trust/distrust relations different from those experienced in groups involving more than two individuals. Namely, in order to determine the distribution of signed relations in a group interactions, we recur to social balance theory [29, 30, 31, 32, 33]. According to this theory a group of three individuals, a triad, is balanced when there is an odd number of relations among them. Thus we have two distinct balanced configurations that can be interpreted either as “the friend of my friend is my friend” in case of three trust relations, or “the enemy of my enemy is my friend” in presence of a single trust relation. Both of these configurations ensure the consistency of the members’ attitude with respect to each other. At contrast, a triad in which there is an even number of trust relations leads unavoidably to some kind of frustration among the individuals in the triad as it implies that the “friend of my friend is my enemy” or that “the enemy of my enemy is my enemy”, both cases being unwanted by its members. For this reason we expect these types of triads to appear less frequently.

With the first version of our model, we show how the level of distrust in the social network determines the nature of the transition, i.e., continuous or discontinuous, from a state where no individual adopts the social behavior, i.e., disease-free state, to a regime where a fraction of agents spread it, i.e., to an endemic state. Particularly, we show that the size of the endemic state can be a non-monotonic function of the average node degree, depending of the level of distrust in the population. With the second version of the model, we analyze the ambivalent role of social balance, showing that simplicial complexes where balanced triads are overrepresented compared to the random case can either promote or impede contagion. We corroborate our findings by comparing the mean-field predictions to stochastic simulations performed by using a Gillespie algorithm [34]. In the framework of social balance theory, namely the second proposed model, we show that a structurally balanced network - a network in which balanced triads are overrepresented compared to the random case - can either promote or impede contagion. In particular, we find that the abundance of triangles with three positive signs enhances the emergence of a bistable regime and thus lets the system easily switch to an endemic state. We also show that the other balanced configuration, i.e., two negative signs, induces instead the opposite behavior. Similar results are also presented when investigating the role of unbalanced triangles.

The paper is organized as follows. In Section II, we introduce our model of complex contagion on signed simplicial complexes. In Section III, we derive the mean-field equations of the model, while in Section IV, we determine the equilibrium solutions of the system and examine their stability, assuming a random distribution of edge signs, namely under the assumption of the first model. In section V, we introduce the second version of the model, and we investigate how structural balance affects contagion. We then conclude with some future perspectives.

In the context of pairwise interactions, trust and distrust relations between individuals are represented as links associated with signs, respectively positive or negative. Inspired by this, to model group interactions with trust or distrust we will use signed simplices. A $q$-simplex, $\sigma$, is a collection of $q+1$ distinct nodes. With such a definition, a $0$-simplex is a node, a $1$-simplex is a link, corresponding to a dyadic interaction, a $2$-simplex is a triangle, representing an interaction among three individuals, and so forth. To model trust and distrust, each node in a $q$-simplex, $q\geq 1$, is endowed with a positive or negative sign toward each of the remaining $q$ nodes of the simplex. For simplicity, we assume (dis)trust relations to be reciprocal. Let us observe that in real social systems the existence of a (dis)trust relation between two individuals in a pairwise interaction does not imply that the same (dis)trust relation exists when a third person is involved in the interaction.

In our model, individuals are assumed to be either in an infected state, i.e., they are spreading a given social behavior, or in a susceptible state, i.e., they are not. Note that we will hereafter keep using the vocabulary of epidemic processes. In particular, we will denote endemic state the condition of the system where a positive fraction of individuals are infected, i.e., they have adopted and spread the social behavior. Also, we will refer to the threshold above which an endemic state emerges as epidemic threshold. However, compared to epidemic spreading, in our model we assume that individuals can adopt but also refuse to adopt the social behavior under study, namely because they can change their mind by interacting with others. This is impossible in the case of disease spreading.

In our model, we assume that infection and recovery processes can take place at three distinct levels, namely at the level of single individuals, i.e., $0$-simplices, at the level of pairwise contacts, i.e., $1$-simplices, and also at the level of three-body interactions, $2$-simplices. To present our results in a simple and direct way, we limit our analysis here to simplices up to order $q=2$. This assumption is motivated by the fact that larger interacting groups are less abundant compared to smaller groups [35]. However, our modeling approach can be straightforwardly extended to larger group sizes [20]. At the level of $0$-simplices, we only have recovery processes, with infected individuals coming back to the susceptible state with a rate $\mu$, namely they stop spreading after an average time $1/\mu$. At the level of $1$-simplices, two processes are at play, an infection process and a recovery one. For the infection process, a susceptible individual gets infected with a rate $\beta_{1}>0$ when interacting with a trusted infected individual (see top panel of Fig. 1). For the recovery process, if an infected individual interacting with a distrusted infected individual becomes susceptible with a rate $\beta_{2}>0$ (see bottom panel of Fig. 1). Two-body recovery processes have not been extensively adopted, with a few exceptions [36], and in our case the idea behind it is that individuals that do not trust each other are less willing to adopt and spread the same social behavior.

In the case of three-body interactions, we consider five different processes, namely three types of infections and two types of recoveries. An individual can change state by interacting with two infected individuals, with a rate that depends on the trust/distrust relations existing among the three agents. This comes from the assumption that, when interacting in group, each agent has full knowledge of all the trust/distrust relations existing among the group members. Note that this is a reasonable assumption in the case of small groups. Let us first consider the cases in which a susceptible individual, $i$, participates in a three-body interaction with two infected ones, $j$ and $k$ (see Fig. 2).

Case 1: If all the agents trust each other, i.e., there are three “$+$” signs in the $2$-simplex, then the susceptible individual $i$ becomes infected with a rate $\gamma_{1}>0$.

Case 2: If $i$ trusts $j$ and $k$ but there is a distrust relation between $j$ and $k$, $i$ can still be infected, but with a positive rate $\gamma_{2}\leq\gamma_{1}$. This comes from the idea that the distrust relation existing among $j$ and $k$ can negatively influence the propensity of $i$ to adopt the social behavior.

Case 3: If a susceptible agent $i$ trusts agent $j$ but distrusts agent $k$, while there is a trust relation between $j$ and $k$, then $i$ becomes infected with a positive rate $\gamma_{3}\leq\gamma_{2}$. As in the previous case, the presence of a distrust relation determines a decrease in the adoption rate of agent $i$, even more than in the case of the second mechanism, as agent $i$ directly experience distrust.

We now consider the cases where three infected individuals, experiencing distrust relations, interact (see Fig. 3).

Case 4: If an agent $i$ distrusts both $j$ and $k$, perceiving a trust relation between them, it will become susceptible with a rate $\gamma_{4}\geq 0$.

Case 5: If the three agents distrust each other, one of them can still become susceptible, but at a smaller rate, i.e., $0\leq\gamma_{5}\leq\gamma_{4}$. In both cases, the rationale is that distrust can induce doubt, pushing the agents to stop adopting the social behavior.

So far, we have described the microscopic mechanisms, occurring at the level of $q$-simplices, underlying our model of social contagion with trust and distrust. However, the macroscopic spreading of social behaviors within a population is determined by the precise patterns of interaction among individuals. To model the complex structure of interactions in a real social systems, we rely on signed simplicial complexes. Given a set $\mathcal{V}=\{1,\dots,N\}$ of nodes, a simplicial complex $\mathcal{K}$ is a collection of signed simplices, with an inclusion requirement, meaning that for any simplex $\sigma\in\mathcal{K}$, all simplices $\tau\subset\sigma$ are also contained in $\mathcal{K}$. Let us observe that the inclusion property means, for example, that if three people interact altogether, then each of them will also interact pairwise with the two others. In the following, we will denote by $\eta\in[0,1]$ the fraction of $1$-simplices (the links) with a minus sign in the simplicial complex, i.e., the fraction of dyadic relations based on distrust. Let us observe that besides the epidemic parameters and the number of trust/distrust relations, our model of social contagion will also depend on the topological features of the simplicial complex representing the interactions among the individuals, e.g., the arrangement of trust/distrust relations, the average number of $1$ and $2$-simplices incident to a node, and so on. In the following we will focus on the impact of the distrust using mean-field equations and stochastic simulations.

Let us now cast the microscopic mechanisms discussed so far into an ODE model for the fraction $\rho(t)$ of infected agents at time $t$. Working in a mean-field approximation that neglects $2$-nodes correlations, we get:

where $\ell_{1}$ (resp. $\ell_{0}$) is the mean number of trust links, i.e. links with a “$+$” sign (resp. distrust links, i.e. with a “$-$” sign), which are incident in a node. We can write $\ell_{1}=(1-\eta)\langle k\rangle$, and thus $\ell_{0}=\eta\langle k\rangle$, where $\langle k\rangle$ is the average node degree, and $\eta$ the fraction of negative links existing in the social network. For $m\in\{0,1,2,3\}$, the mean number of triangles with $m$ trust relations, i.e., $m$ “$+$” signs, is denoted by $\tau_{m}$.

The different terms in the first line of Eq. (1) respectively model (from the left to right) the standard (single-node) recovery, the infection due to a pairwise interaction between a susceptible and an infected person trusting each other, and finally the recovery of an infected agent because of a dyadic interaction involving two distrusting infected agents (i.e., the processes described in Fig. 1). The second line, again from left to right, describes a contagion due to a triadic interaction involving two infected and one susceptible agent, all trusting each other (see top panel of Fig. 2). A contagion occurring in a triadic group with two infected and one susceptible people, the latter trusting the two others who do not trust each other; the factor $1/3$ represents the fact that, focusing on one node, only one third of all triangles with two “$+$” and one “$-$” has the right configuration of signs (see middle panel of Fig. 2). Finally, a contagion due to again a triadic interaction with two infected individuals and a susceptible one, where the susceptible person trusts one person but not the other one. The factor $2/3$ again is due to the fact that only a fraction $2/3$ of such triangles has the right signs setting (see bottom panel of Fig. 2). The third line of Eq. (1) denotes the recovery of an infected person because of a triadic interaction involving three infected people, the former one distrusting the latter two, from which the factor $1/3$ follows. Then a recovery process involving three mutually distrusting infected people (see Fig. 3).

Gathering together powers of $\rho$ we can rewrite Eq.  (1) as follows:

where

We are now interested in finding the stationary solutions of Eq. (1) and in studying their stability. This will allow us to investigate the asymptotic behavior of our system and to draw conclusion on the conditions leading to the emergence on an endemic state. Let us start by assuming that the three agents taking part in a three-body interaction maintain the same trust and distrust relations they have in pairwise interactions, meaning that the signs of the three-body interaction are directly “inherited” from the three underlying pairs. This will allow us to draw some preliminary conclusions on the role of trust on contagion. we will then relax such an hypothesis and present a more general setting in the following section. Under this working assumption, we can write

where $\langle k_{\Delta}\rangle$ is the average number of $2$-simplices incident to a node. We can thus determine the stationary solutions of the mean-field equations and their stability. Beside the disease free equilibrium $\rho_{0}^{*}=0$, that is always a solution, there can be other two solutions given by:

if $(\Lambda-\tilde{\lambda}_{1}-\lambda_{2})^{2}-4(\Lambda+\tilde{\Lambda})(1-\tilde{\lambda}_{1})>0$ and the parameters are such that $\rho_{\pm}^{*}\in(0,1)$, where $\Lambda:=\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$ and $\tilde{\Lambda}:=\Lambda_{4}+\Lambda_{5}$ with:

We notice that, in the special case where all agents trust each other, i.e. when $\eta=0$, we have $\tilde{\Lambda}=0,\lambda_{2}=0$, $\Lambda=\lambda_{\Delta}:=\frac{\gamma_{1}k_{\Delta}}{\mu}$ and $\tilde{\lambda}_{1}=\lambda_{1}:=\frac{\beta_{1}\langle k\rangle}{\mu}$ and we recover the fixed points obtained in [14]. In this case, it has been shown that the epidemic threshold is given by $\lambda_{1}=1$ and that the transition at the epidemic threshold is continuous when $\lambda_{\Delta}\leq 1$ and discontinuous otherwise; in that case the system shows bistability for an interval of values of $\lambda_{1}$, i.e., it admits as stable equilibria both the disease free state and the endemic state. In the following we show how the analysis can be extended to the general case where $0<\eta\leq 1$.

By rescaling time by $\mu$ and linearizing (2) close to the equilibrium $\rho_{0}^{*}=0$, we can realize that the latter is stable if and only if $\tilde{\lambda}_{1}=\lambda_{1}(1-\eta)<1$. Hence, we find that the epidemic threshold for the disease free equilibrium is given by $\lambda_{1}=1/(1-\eta)$. Above this threshold, the infection will always spread and reach a non-zero fraction of the population, while below this threshold the infection will die out unless the fraction of initially infected nodes is sufficiently large, i.e., above a critical mass. In that case nonlinear terms might sustain the disease. Crucially, we observe that the epidemic threshold increases as $\eta$ becomes larger (see top panel of Fig. 4). This means that distrust can limit the spreading of the infection, as a large value of $\lambda_{1}$ is needed to initiate the contagion process. Let us now analyze the existence and the stability of the aforementioned fixed points $\rho_{\pm}^{*}$, as a function of $\tilde{\lambda}_{1}$.

• •

  By looking at the definition of $\rho^{*}_{\pm}$ one can realize that if $0<\tilde{\lambda}_{1}<\lambda_{\mathit{crit}}$ then $\rho^{*}_{\pm}$ assume complex values, where $\lambda_{\mathit{crit}}=-(\lambda_{2}+\Lambda+2\tilde{\Lambda})+2\sqrt{(\Lambda+\tilde{\Lambda})(1+\lambda_{2}+\tilde{\Lambda})}<1$ as one can observe by rewriting it as follows

  	$\displaystyle\lambda_{\mathit{crit}}$	$\displaystyle=$	$\displaystyle-(\lambda_{2}+1+\tilde{\Lambda})-(\Lambda+\tilde{\Lambda})+1$
  			$\displaystyle+2\sqrt{(\Lambda+\tilde{\Lambda})(1+\lambda_{2}+\tilde{\Lambda})}$
  		$\displaystyle=$	$\displaystyle-\left[\sqrt{\lambda_{2}+\Lambda+\tilde{\Lambda}}-\sqrt{\Lambda+\tilde{\Lambda}}\right]^{2}+1<1\,.$

• •

  If $\lambda_{\mathit{crit}}<\tilde{\lambda}_{1}<1$ and $1+\lambda_{2}>{\Lambda}$, then $\rho^{*}_{\pm}$ are real but negative, so they are not physically acceptable.

• •

  If $\lambda_{\mathit{crit}}<\tilde{\lambda}_{1}<1$ and $1+\lambda_{2}<{\Lambda}$, then $\rho^{*}_{\pm}$ are real and positive, so they are both physically acceptable, moreover $\rho^{*}_{-}$ is unstable while $\rho^{*}_{+}$ is stable.

• •

  If $\tilde{\lambda}_{1}>1$, $\rho^{*}_{\pm}$ are real but $\rho^{*}_{-}$ is negative, thus the only physically acceptable solution is $\rho^{*}_{+}$ which is stable.

By considering the behavior of the solutions $\rho^{*}_{\pm}$ for large $\tilde{\lambda}_{1}$, we can prove that $\rho^{*}_{+}\rightarrow 1$. In Table 1 we summarize the different asymptotic solutions of Eq. (2) and their stability as a function of the involved parameters.

The above analysis allows us to conclude that when $\Lambda>1+\lambda_{2}$, the prevalence of infected individuals has a discontinuous phase transition at $\tilde{\lambda}_{1}=\lambda_{\mathit{crit}}$ and it is bistable for $\lambda_{\mathit{crit}}<\tilde{\lambda}_{1}<1$. On the other hand when $\Lambda<1+\lambda_{2}$, the transition from the disease-free state to the endemic state is instead continuous.

The model presented in the previous section is based on the assumption that agents maintain the same pairwise trust/distrust relations when they are involved in interactions in group of more than two. This assumption implies that the distribution of signed triangles depends on the fraction of negative links in the network, $\eta$, and not on a peculiar combination of “$+$” and “$-$”. However social balance theory affirms that in a social network, the fraction of balanced triangles, i.e., triadic relations with an odd number of positive links, is larger than the one we would obtain if signs were randomly distributed [29, 30, 37]. In this section we extend the previous model in order to investigate the role of balanced and unbalanced full triangles on social contagion. This can be achieved by introducing new parameters, $p_{i}\geq 0$, where $i=0,\dots,3$, refers to the number of positive links in the triangles, to express the excess or the deficiency of peculiar signed $2$-simplices with respect to the case where these inherit the signs of the links. This allows us to write

The model presented in the previous section, i.e., where all the signs were randomly attributed to the edges, can be recovered with the choice $p_{i}=1$ for all $i$. If there were only balanced triangles, then $p_{0}=p_{2}=0$, while $p_{1}=p_{3}=0$ denotes the complete absence of balanced triangles. Let us note that the parameters $p_{i}$ are not independent from each other as we must have $\sum_{i}\tau_{i}=\langle k_{\Delta}\rangle$, hence

Let us observe that the solutions of Eq. (1) are still given by $\rho^{*}_{0}=0$ and $\rho^{*}_{\pm}$ obtained by (3), of course with new parameters $\Lambda_{i}$, $i=1,\dots,5$ depending on the new definitions of $\tau_{k}$, $k=0,\dots,3$ given by (V). The same analysis above performed concerning the existence and stability of the stationary densities can be straightforwardly adapted to the present case.

In the following, we examine how the distribution of distrust relations influences the contagion outcome. As a first analysis, we set the bias parameters relative to balanced triangles to be equal, i.e., $p_{1}=p_{3}$, and we investigate the impact on the contagion of the parameters relative to unbalanced triangles, i.e., $p_{0}$ and $p_{2}$. In Fig. 6 we show the prevalence curves for two values of $p_{0}$, i.e., $p_{0}=1$ (red curve) and $p_{0}=5$ (blue curve), while fixing $p_{2}=1$, the values of the other parameters are reported in the caption. Notice that the value of $p_{1}=p_{3}$ is automatically adapted as to satisfy Eq. (8). Note also that $\eta$ is fixed, meaning that the average number of negative links is constant. In the same Figure we compare the mean-field solution (curves) with the results of stochastic simulations performed using the Gillespie algorithm (points); the agreement is again satisfactory, the discrepancy being due to finite size effects that enhance correlations, neglected in the construction of the mean-field model (see Appendix A for more details).

We can observe two opposite behaviors. On the one hand, for small values of $\lambda_{1}$, the prevalence $\rho^{*}$ increases when reducing $p_{0}$, i.e., decreasing the number of unbalanced triangles (the red curve is above the blue one). On the other hand, for large values of $\lambda_{1}$, we find on the contrary that $\rho^{*}$ increases as we increase $p_{0}$ (the red curve is below the blue one). This implies the existence of a value $\lambda_{\mathrm{cross}}$ at which both curves have the same prevalence. The prevalence at the particular value $\lambda_{\text{cross}}$ can be computed explicitly (see Appendix C) and is given, by assuming $\eta=1/2$, $p_{1}=p_{3}$ and $p_{2}=1$ by:

provided the parameters $\gamma_{4}>4\gamma_{5}$ to ensure the prevalence to lie into $(0,1)$. For the parameters used in Fig. 6, we obtain $\rho^{*}(\lambda_{\text{cross}})=7/17\sim 0.41$.

To further investigate the crossing behavior, we construct the phase diagram of $\rho^{*}$ as a function of $p_{0}$ and $\lambda_{1}$. This is shown in the inset Fig. 6. Note that we binned the values of $\rho^{*}$ so to better visualise the crossing. For large values of $\lambda_{1}$ (right of the dashed green line), the prevalence increases as we increase $p_{0}$, while the opposite holds for small values of $\lambda_{1}$ (left of the dashed green line). Note that the black region corresponds to the parameter set for which the disease-free state is the only stable equilibrium.

These results highlight how the effect of balanced and unbalanced triangles on the prevalence highly depends on the link infectivity of the process. As already stated, real social systems are more likely composed by balanced configurations, i.e., they are associated to low values of $p_{0}$ and $p_{2}$. So in our analysis, we expect a contagion spreading on them to show a behavior more similar to that of the red curve. Hence, we would expect real social systems to have larger (resp. smaller) prevalence compared to random sign configurations when the infectivity $\lambda_{1}$ is small (resp. large). This counter-intuitive behavior is due to the high nonlinearity of the contagion process, as infection and recovery mechanisms are associated to both balanced and unbalanced triangles (see Fig. 2 and 3).

Finally, we find that by tuning the bias parameter $p_{0}$ we can change the nature of the transition from the disease-free state to the endemic one. Indeed, we can observe that the system becomes bistable for low enough values of $p_{0}$ (the red curve shows bistability, while the blue curve does not).

We now allow $p_{2}$ to vary, thus studying the behavior of the system as a function of both $p_{0}$ and $p_{2}$. Still, we assume that $p_{1}=p_{3}$, with their value determined by Eq. (8). In Fig. 7, we show the phase diagram of $\rho^{*}$ as function of $p_{0}$ and $p_{2}$ for $\lambda_{1}=20$ (left) and $\lambda_{1}=2$ (right). In both panels, the white region denoted by $A$ is such that the values of $p_{0}$ and $p_{2}$ would imply $p_{1}=p_{3}<0$ which is not physically acceptable. In particular, the delimiting line of region $A$ is given by $p_{0}\eta^{3}+3p_{2}\eta(1-\eta)^{2}=1$, and it corresponds to the case where no balanced triangles are present, i.e., $p_{1}=p_{3}=0$. Coherently with the previous analysis, when $\lambda_{1}$ is large enough (left panel), the infection increases with $p_{0}$ and $p_{2}$. Note that the vertical dashed line represents $p_{2}=1$, corresponding to the case previously studied. For $\lambda_{1}=2$, the behavior of the system with respect to $p_{0}$ and $p_{2}$ is more complex. Indeed, for small values of $p_{2}$, the prevalence decreases as $p_{0}$ is increases, while the opposite is true for large values of $p_{2}$.

For this latter case, we distinguish two additional zones. In zone $B$, there is no bistability, while the latter is present in zone $C$. The separation line between the two is found for $\Lambda=1+\lambda_{2}$, and takes, for the above values of the parameters, the expression, $p_{0}=-\frac{5}{7}p_{2}+\frac{984}{175}$.

To conclude the analysis, we further investigate the impact of the particular configurations of balanced and unbalanced triangles present in the network. In particular, we first fix the values of $p_{1}$ and $p_{3}$, namely the bias parameters for the balanced triangles, and vary $p_{0}$ and $p_{2}$, so the bias parameters for the unbalanced configurations, then we do the opposite, fixing $p_{0}$ and $p_{2}$, and varying $p_{1}$ and $p_{3}$.

In the top panel of Fig. 8 we report the fraction of infected nodes as a function of $p_{0}$ and $p_{2}$ for fixed $p_{1}=p_{3}=1$; let us remember again that because of the constraint (8) $p_{0}$ and $p_{2}$ are not independent each other. We can observe that for a fixed $\lambda_{1}$, the larger $p_{2}$, i.e., the more triangles with two positive links, the larger the final infection state. Let us emphasize that $p_{0}$, i.e., the fraction of triangles with no positive links, impacts the system outcome in the opposite direction: the larger the value of the parameter, the smaller the fraction of infected. This comes from the fact that $p_{2}$ is associated to the three-body infection process having rates $\gamma_{2}$ and $\gamma_{3}$ (see Fig. 2), while $p_{0}$ is related instead to recoveries occurring at rate $\gamma_{5}$ (see Fig. 3). Let us also observe that by increasing $p_{2}$, and thus by decreasing $p_{0}$, $\lambda_{\mathrm{crit}}$ decreases and thus the system is more prone to exhibit a bistable regime. Moreover, for a given value of $\lambda_{1}$ in the bistability region, we find that the critical amount of infected people needed to converge to the endemic state decreases as $p_{2}$ increases. Stated differently, by increasing the number of unbalanced triangles with two positive links, the system can “easily” switch to an endemic state.

In the bottom panel of Fig. 8 we show the results corresponding to a variation of $p_{1}$, and thus $p_{3}$, for fixed $p_{0}=p_{2}=1$. Again the two parameters have opposite impact: by increasing $p_{3}$ (resp. $p_{1}$), the infection $\rho^{*}$ increases (resp. decreases). Similarly to the previous case, $p_{3}$ is associated to three-body infections occurring at rate $\gamma_{1}$, while $p_{1}$ is linked to recoveries occurring at rate $\gamma_{4}$. Coherently, when we increase the value of $p_{3}$ a bistable regime can emerge.

Overall these results suggest that the bias parameters $p_{i}$ strongly impact the outcome of the contagion process, both in terms of its final prevalence and of its dynamics, i.e., continuous and discontinuous transitions, in a nontrivial way. Also, our model highlights that not only the relative proportion of balanced and unbalanced triangles determines the contagion, but also the particular configurations within these two classes can have a crucial influence on the system.

In this paper we have introduced a model of complex contagion on signed higher-order networks, which has allowed us to investigate, both numerically and analytically, the combined effect of group interactions and distrust relations on the dynamics of social contagion. In our model, a social system with group interactions is represented as a signed simplicial complex, in which the edges are attributed either a “$+$” or a “$-$” sign to respectively denote trust or distrust between agents. In our model we have two compartments, i.e. nodes can either be in the infected state (if they have adopted an opinion or a social norm) or the susceptible state. The existence of signed relations allows for additional infection and recovery channels, which are different from those in standard model for the spreading of a disease. Namely an infected agent can become susceptible because of a signed three-body interaction. Two variants of the model were considered: in the first one, edge signs were attributed randomly and maintained the same during a group interaction, while in the second one the distribution of signs in the triadic groups was biased so as to account for social balance theory. With the first variant of the model, we have highlighted how trust and distrust relations affect the transition of a system from a disease-free state to an endemic one. Specifically, we have shown that increasing distrust can change the nature of the transition from discontinuous to continuous, by making the bistability region associated to the first-order transition vanish. Moreover, we have characterized the nontrivial interplay existing between the fraction of distrust relations and the connectivity of the social network in shaping the collective dynamics of the system. With the second variant of the model, we have analyzed how the precise patterns of trust/distrust relations impact complex contagion in higher-order networks. In particular, we have highlighted how the contagion is determined not only by the relative proportion of balanced and unbalanced triangles, but also by which configuration within these two classes is more frequent. Various extensions of the present work are possible. First of all, it would be interesting to investigate how trust and distrust influences the contagion process on real-world higher-order networks. Also, it would be useful to generalize the model by considering larger group structures, e.g., groups of size $4$, and the new infection and recovery channels related to those larger groups. A third research direction would be to allow the pattern of contacts to change over time or the signed relations and nodes states to co-evolve. Finally, it would be interesting to consider an heterogeneous mean-field approach to better investigate the impact of the structural properties of higher-order networks on contagion.