We study the critical behavior of a general contagion model where nodes are either active (e.g., with opinion A, or functioning) or inactive (e.g., with opinion B, or damaged). The transitions between these two states are determined by (i) spontaneous transitions independent of the neighborhood, (ii) transitions induced by neighboring nodes, and (iii) spontaneous reverse transitions. The resulting dynamics is extremely rich including limit cycles and random phase switching. We derive a unifying mean-field theory. Specifically, we analytically show that the critical behavior of systems whose dynamics is governed by processes (i)–(iii) can only exhibit three distinct regimes: (a) uncorrelated spontaneous transition dynamics, (b) contact process dynamics, and (c) cusp catastrophes. This ends a long-standing debate on the universality classes of complex contagion dynamics in mean field and substantially deepens its mathematical understanding.
In 1972 Schlögl proposed two models describing autocatalytic chemical reactions  that are commonly known today as Schlögl’s first and Schlögl’s second model (henceforth referred to as Schlögl I and Schlögl II). Schlögl I, also known as contact process , comprises the important case of simple contagion, i.e., the susceptible-infected-susceptible (SIS) model where healthy individuals can be infected due to the exposure to a single infectious source, eventually leading to the spread of an epidemic disease [2–5]. In contrast, Schlögl II, also known as quadratic contact process , requires contact with two sources. Later studies on Schlögl II sparked a debate on its critical behavior and Grassberger noticed in 1982 that a relation to the Ising universality class “would be a most remarkable extension of the universality hypothesis, from models with detailed balance to models without it” to conclude that Schlögl II “is not an example of universality between models with and without detailed balance” .
Closely related to this debate, but more recently, a generalized model of Schlögl II has been proposed where an arbitrary number of sources is necessary to induce a transition . The study of the model’s mean-field critical behavior led the authors to conjecture that such general failure-recovery dynamics belong to the Ising universality class . This model is of particular interest since it not only includes simple contagions but also complex contagion phenomena such as the diffusion of innovations [10,11], political mobilization , and viral marketing  that require social reinforcement, i.e., the connection to multiple sources [14,15]. The model displays an intricate and very rich dynamics including hysteresis effects, limit cycles, and cusp catastrophes [9,16–20]. Thus, a unifying mean-field theory of the critical behavior is essential for a broad range of dynamical systems.
However, the relation to contact process dynamics and cusp catastrophes has only been shown for specific values of the model’s parameters . But, given the model’s parameter regime, can we generally predict the dynamics type? And does the model’s mean-field critical behavior belong to the Ising universality class or not? Here we answer these questions and analytically demonstrate that the mean-field critical behavior of the model is restricted to only three possible regimes: (a) uncorrelated spontaneous transition dynamics, (b) contact process dynamics, and (c) cusp catastrophes. Cusp catastrophes can display abrupt transitions and hysteresis effects—phenomena that can harm the proper functioning of real-world networked systems since small variations in the system’s control parameters may cause catastrophic transitions from a seemingly well-functioning state to global malfunction or severe outages [21–30].
Model.— The general contagion dynamics is defined in a network whose constituents (i.e., nodes) are regarded as either active (e.g., not damaged) or inactive (e.g., failed). Three fundamental processes define the transitions between these two states [9,20]: (i) nodes undergo a spontaneous transition from an active () to an inactive state () in a time interval with probability ; (ii) if fewer than or equal to nearest neighbors of a node are active, the node becomes inactive () due to an induced transition, i.e., , with probability , and (iii) a spontaneous reverse transition with probability if or probability if . The inactive states and only differ in their reverse transitions and are equivalent if . Process (ii) describes that a node with degree can become inactive if its number of inactive neighbors is larger or equal to . Similar to threshold models describing complex contagion phenomena, the threshold defines the number of contacts to inactive nodes that is necessary to induce a transition as defined by process (ii) [14,31–33]. A low value of corresponds to the situation where many inactive neighbors are required to sustain spreading. In contrast, for a large value of only a few inactive neighbors can sustain the spreading process. Processes (i)–(iii) are illustrated in Fig. 1.
Let denote the total fraction of inactive nodes. Thus, with and being the fractions of nodes that are inactive due to spontaneous and induced transitions, respectively. The total fraction of inactive nodes in the stationary state is referred to as . In accordance with Ref.  we derive the mean-field rate equations by assuming a system with homogeneous degrees in the thermodynamic limit that exhibits perfect mixing. Here, perfect mixing either refers to a network of randomly connected nodes with a sufficiently large mean degree or dynamical rewiring [3,34]. For the fraction of nodes that spontaneously became inactive we find
Induced transitions [process (ii)] can only occur for nodes whose number of active neighbors is smaller than or equal to . Under the assumption of a perfectly mixed population, the probability that a node of degree is located in such a neighborhood is [9,20]. The time evolution of the fraction of nodes that are inactive due to induced transitions is therefore given by
The coupled equations (1) and (2) admit oscillatory behavior for  as a dynamical feature that does not belong to the critical behavior . The equations describing the critical behavior, i.e., and , can be decoupled by multiplying one of them with an appropriate constant excluding limit cycles —tantamount to setting . This yields
Class (a): Uncorrelated spontaneous transitions.—We start with the case where the number of active nodes necessary to sustain spreading has to be smaller or equal to the node’s degree according to the definition of process (ii). This describes the regime where spreading occurs independently of the neighborhood’s state such as in exogenously driven adoption dynamics [36,37],
Class (b): Contact dynamics.—By definition implies that or fewer neighbors of a node have to be active to induce a transition. This case describes a contact process where one inactive neighbor is sufficient to sustain spreading . As demonstrated in Supplemental Material , we find for that there exists a critical separating an absorbing and an active phase, i.e., as and as . In the limit of Eq. (3) takes the form
Class (c): Cusp catastrophes.—For some values of , we find a metastable region as illustrated in Fig. 3 (right). Inside this hysteresis region two stable fixed points coexist. Phase switching is observed when fluctuations in systems of finite size push the dynamics close to the unstable fixed point, cf., Fig. 2 (right). Between the switching events the dynamics remains in one of the two phases for some time. The waiting times thus depend on the fluctuation strength and the distance from one phase to the unstable state in the phase portrait, cf., Fig. 2 (right). For where either two or more inactive neighbors are necessary to induce a transition, we show that the corresponding metastable regions always exist due to the relation to cusp catastrophes . For a detailed analytical treatment we refer the reader to Supplemental Material . The cusp point where the two bifurcation lines intersect [cf., Fig. 3 (right)] is given by together with the corresponding control parameters,
Final remarks.— We find that the critical behavior of the general contagion model as formulated in Eq. (3) does not belong to the Ising universality class but to exactly three regimes. The first regime, , corresponds to purely spontaneous failure and recovery dynamics. For the model recovers the critical behavior of the contact process. A cusp catastrophe is found for all with the typical critical behavior at the cusp point [Eqs. (9) and (10)]. This sheds analytical light onto a broad range of spreading processes that are determined by the network’s connectivity and the threshold parameter , cf., examples in Table I.
|Exogenous factors influencing adoption of innovations a Social response to exogenous factors b||Schlögl I [1,7] Contact process [2,3,38]a SIS model [4,5]a Reggeon field theory a Directed percolation a Bass model [37,41]a||Schlögl II [1,7]a Quadratic contact process a General contact process a Behavioral adoption b Threshold models of complex contagions [11,13–15,31–33]ab or coordination games b|
We have demonstrated that the phase diagram corresponds to a cusp catastrophe, when two or more inactive nodes are needed to trigger induced node transitions. This scenario typically implies dramatic and uncontrollable global transitions in the network for many systems involving complex contagion dynamics. One could naively expect that it could be beneficial for failure control to design systems such that a component only fails if many of its neighbors already failed, i.e., delaying the failure dynamics. Our results suggest, however, that this delaying procedure might facilitate uncontrollable transitions, hence achieving exactly the opposite as initially intended. This result agrees well with previous findings on delaying procedures that have been applied to a SIS model [44,45]. For low spatial dimensions or highly structured networks, the assumptions of perfect mixing or independent node-to-node interactions are not guaranteed. Still mean-field approximations qualitatively describe a given dynamics [3,4,46]; see examples given in Supplemental Material .
Future work should establish the behavior of transients as a function of threshold parameter and the topology of the network. It has been demonstrated that opinions as well as coinfections may spread faster in clustered networks compared to random ones [42,47]. This links our result to the multiple exposure condition in complex contagion phenomena.
In the study of collective behaviors, such as the adoption of innovations, the distinction between exogenous and endogenous factors is of great interest but often solely based on a contact processlike adoption model [36,37]. Our results suggest studying these processes within our more general framework that incorporates contact processlike adoption as one special case and can account for spreading that relies on multiple contacts.
We acknowledge financial support from the ETH Risk Center (Grant No. RC SP 08-15) and European Research Council (ERC) Advanced Grant No. FP7-319968 FlowCCS. We thank Shlomo Havlin for fruitful discussions. We also thank the Instituto Nacional de Ciência e Tecnologia de Sistemas Complexos (INCT-SC) for financial support.