The spread of behavior in a society has two major features: the synergy of multiple spreaders and the dominance of hubs. While strong synergy is known to induce mixed-order transitions (MOTs) at percolation, the effects of hubs on the phenomena are yet to be clarified. By analytically solving the generalized epidemic process on random scale-free networks with the power-law degree distribution pk∼k−α, we clarify how the dominance of hubs in social networks affects the conditions for MOTs. Our results show that, for α<4, an abundance of hubs drive MOTs, even if a synergistic spreading event requires an arbitrarily large number of adjacent spreaders. In particular, for 2<α<3, we find that a global cascade is possible even when only synergistic spreading events are allowed. These transition properties are substantially different from those of cooperative contagions, which are another class of synergistic cascading processes exhibiting MOTs.
Introduction. There has been a growing body of literature on mixed-order transitions (MOTs), which qualify as both continuous and discontinuous phase transitions depending on the chosen order parameter. Such transitions appear in many different contexts, such as DNA unzipping [1–3], Ising spins with long-range interactions , and various percolation models with biased merger of clusters . A common aspect of these systems is the existence of long-range interactions which encourage global ordering over a finite fraction of the system at criticality .
Recently added to the list are various models of cascades with synergistic spreading rules involving cooperation between different contagions [6–9], weakened individuals [10–16], or multiple spreading thresholds . If each transmission occurs independently without synergy, the cascade exhibits a continuous percolation transition . In contrast, with sufficiently strong synergy, the transition can be a MOT: a continuous transition of the probability of a global cascade coincides with a discontinuous jump of the cascade size. Moreover, the lines of MOTs and purely continuous transitions join at a tricritical point (TCP) with its own critical properties . Again, the long loops of the substrate, through which different spreading pathways cross each other, facilitate global cascades at the MOTs [8,11].
A natural question arises on how the conditions for MOTs depend on the structure of the underlying substrate. In homogeneous structures, such as lattices [6,7,13–15], Poissonian random networks [6–8,10–13,17], and modular networks , a MOT requires sufficiently strong synergy between two spreaders and dimension greater than two [13,14]. However, cascades typically occur on heterogeneous structures: for instance, social networks feature a significant fraction of highly connected individuals called hubs, whose existence is typically modeled by scale-free networks (SFNs) with a power-law distribution (with ) of the number of neighbors (called degree ) . Since SFNs with a greater variance of contain more loops , can be a major determinant of the conditions for MOTs. For cooperative contagions on SFNs, a heterogeneous mean-field approach  showed that a discontinuous jump of the cascade size is possible for given sufficiently strong synergy, but not for ; however, whether the same statement holds for general kinds of synergy remains to be clarified.
In this study, we show that the synergistic spread of behavior exhibits substantially different transition phenomena for small values of . As empirically observed , social reinforcement induces a large boost in the spread of a behavior if the target individual has sufficiently many adjacent spreaders. As a simple model incorporating this feature, we study the generalized epidemic process (GEP) with the synergy threshold, in which the spreading probability changes when the number of spreading neighbors is greater than or equal to , extending the original version limited to . In the sense that the cluster is formed by a mixture of single-node and multinode mechanisms, our model can be considered a cascading-process analog of the heterogeneous -core percolation , which is a pruning process. We analytically show that, for , an abundance of hubs enable MOTs for arbitrarily large . In contrast to cooperative contagions, the cascade size exhibits a discontinuous jump even for in a manner similar to the abrupt appearance of a giant heterogeneous core with on the same SFNs . While the near-TCP scaling exponents for remain identical to those of cooperative contagions , a new set of exponents can be identified for .
Dynamics. In the GEP, a node can be susceptible (), weakened (), infected (), or removed (). All nodes are initially , except for one randomly chosen node (the “seed”) starting the spread. At each time step, a random node attempts to spread the behavior to all of its or neighbors, each of the former (latter) with probability (). Upon success, the target becomes . A failed attempt does not affect the target unless it is the attempt on the same node, in which case the node becomes . After then, the chosen node immediately deactivates and becomes , permanently removing itself from the dynamics. The process goes on until the network runs out of nodes. The GEP with on a five-node network is illustrated in Fig. 1(a).
Substrate. The GEP spreads on an ensemble of infinitely large random SFNs constrained by two conditions. First, the degree distribution obeys a power law for and , where the generalized zeta function , defined as the analytic continuation of for , normalizes the distribution. The assumed range of ensures that the mean degree is finite. Second, there is no correlation between the degrees of adjacent nodes. Given these two conditions, one may assume that a node and each of its neighbors have mutually independent statistics, which makes the problem analytically tractable.
Notations. The final fraction of nodes, denoted by , quantifies the cascade size. The probability of a global cascade with is denoted by . The percolation transition from the phase with zero and to the phase with positive and occurs at , and exhibits a continuous (discontinuous) transition at the point if (). The scaling behaviors near the TCP are characterized by three exponents , , and , so that , , and with and .
Transition of . For the SFNs defined above, multiple spreading pathways rarely cross at the same node unless the cascade has already reached a finite fraction of the network. For this reason, is completely irrelevant to the transition from to : only controls the transition by a bond-percolation mechanism. Thus one can simply apply the theory of bond percolation on the random SFNs  to obtain the transition point
Analytic calculation of . In contrast to , depends on as the crossing of spreading pathways is nonnegligible whenever . Here we present a calculation of the dependence based on a standard tree ansatz for random SFNs . For this aim, we consider the probability that a node at an end of a randomly chosen link is after the spread has stopped. For simplicity, we assume , which does not affect the main results. Then satisfies a self-consistency equation , where
Conditions for MOTs. A MOT occurs at when it coincides with a discontinuous jump of . Since Eq. (3) implies , the transitions of and should be of the same type. The latter are encoded in the small- expansion of Eq. (2), which for noninteger is given by (see  for the detailed derivation)
(i) If , for the solution is , which depends on only through . This is because the transition type is determined by the sign of in Eq. (4), which is a two-neighbor effect. On the other hand, for there is no solution because ; in other words, always holds, so the transition of is always continuous. Here comes into play only for three-or-more neighboring spreaders, so it cannot affect the sign of .
(ii) If , Eq. (6) is explicitly dependent on , reflecting the dominance of the hub-induced term. Here the solution exists for any , because the convergence of many spreading pathways at the hubs facilitates a MOT even if is arbitrarily large. We note that obtained from Eq. (6), depending on , can still be larger than 1 and thus impossible to achieve, as shown for in Fig. 1(c).
(iii) If , for any , is the only solution. This captures being positive (zero) for (); in other words, there are so many spreading pathways crossing at the hubs that, regardless of , synergistic spreading events by unaided by can induce a global cascade. This regime is where the cascades of the GEP differ most significantly from those of cooperative contagions . In the latter, a node should first be infected by one contagion with probability to experience a secondary infection with probability , so whenever . In the GEP, even if , a spreading event by can still occur because it only requires sufficiently many exposures to neighboring spreaders. This parallels the robust existence of a giant heterogeneous core with on the same SFNs even in the limit where the fraction of removed nodes approaches unity .
Based on these results, one can interpret the transition behaviors of the GEP with on the SFNs with illustrated in Fig. 1(b). For , both continuous and discontinuous transitions of are possible at with the boundary at , whereas for (see the inset for a magnified view) undergoes a continuous transition belonging to the bond percolation universality class () at even in the extreme case . Notably, there is a secondary discontinuous transition (marked by dotted vertical lines) at , whose possibility is not excluded by our argument. This phenomenon seems to be related to the double phase transitions reported in  and will be discussed in detail elsewhere .
Tricritical behaviors for . For small and positive , a Taylor expansion of Eq. (4) about yields
To numerically verify the scaling exponents derived above, we present the scaling form for , which converges to the average fraction of nodes, , readily obtained using random SFNs of nodes (see  for more details) in the limit. The scaling form is given by
Scaling behaviors for . As discussed above and illustrated in Figs. 3(a) and 3(b) (the latter providing a numerical verification of the tree ansatz, whose rigorous justification remains an open mathematical problem due to a diverging number of short loops ), holds in this regime. Due to the absence of the phase of localized cascades, it would be misleading to call the point a TCP; however, one can still identify universal scaling behaviors and the crossover between them from the leading-order terms of Eq. (4), identifying new scaling exponents previously unreported. We obtain
Summary. We examined the effects of the degree exponent on the percolation transitions of the GEP on uncorrelated random SFNs. All analytical results, based on the tree ansatz (2), are in good agreement with the numerics beyond the regime of strong finite-size effects. It is found that the hub-driven MOTs occur only for . In particular, for , we identified new transition behaviors stemming from the convergence of loops at the hubs. These imply that the spread of behavior and cooperative contagions  belong to different universality classes on typical social networks. Our results reveal fundamental principles underlying the formation of compact cultural subgroups fostered by the fat-tailed degree distribution of social networks. Interesting topics for future studies include the conditions for double phase transitions, the nature of finite-size effects, and connections to MOTs and TCPs reported in other percolation models [27,28].
Acknowledgments. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) (KR) [Grants No. NRF-2017R1D1A3A03000578 (M.H.) and No. NRF-2017R1A2B3006930 (H.J.)]. Y.B. is supported in part by the European Research Council under the Horizon 2020 Programme, ERC Grant Agreement No. 740269. We also thank Peter Grassberger for helpful comments on mixed-order transitions in DNA unzipping as well as the references.