Critical behavior of a two-step contagion model with multiple seeds

Abstract

A two-step contagion model with a single seed serves as a cornerstone for understanding the critical behaviors and underlying mechanism of discontinuous percolation transitions induced by cascade dynamics. When the contagion spreads from a single seed, a cluster of infected and recovered nodes grows without any cluster merging process. However, when the contagion starts from multiple seeds of O(N ) where N is the system size, a node weakened by a seed can be infected more easily when it is in contact with another node infected by a different pathogen seed. This contagion process can be viewed as a cluster merging process in a percolation model. Here, we show analytically and numerically that when the density of infectious seeds is relatively small but O(1), the epidemic transition is hybrid, exhibiting both continuous and discontinuous behavior, whereas when it is sufficiently large and reaches a critical point, the transition becomes continuous. We determine the full set of critical exponents describing the hybrid and the continuous transitions. Their critical behaviors differ from those in the single-seed case.

Competing contagion processes: Complex contagion triggered by simple contagion

“…46 Such hybrid phase transition also known as mixed phase transition has been observed widely in cooperative percolation in networks such as k-core percolation, 46,50,51 bootstrap percolation, 48 percolation of interdependent networks, [52][53][54] and cooperative epidemic processes. [22][23][24][25] A hybrid transition is also predicted in a model of spin chains with long-range interactions, 55 DNA denaturation, 56 and jamming, 57,58 and recently observed experimentally in a colloidal crystal. 59 An example of how to identify a phase transition is shown in terms of the graphical solution of f (R) with z = 10 and n = 4 in the limit ρ → 0 [ Fig.…”

“…The other class of contagion processes is complex contagion representing spreading phenomena in which multiple exposures to a spreading entity are needed for changing agents' state. 8,9 Models of complex contagion processes encompass a wide range of contagious models such as threshold model, 4,6,20 generalized epidemic model, [21][22][23][24][25] diffusion percolation, 26 threshold learning, 27,28 and bootstrap percolation. 29 The spread of fads, ideas, and new technologies in our society is better described by complex contagion rather than by simple contagion due to a collective effect in social contagion.…”

Memory-induced complex contagion in epidemic spreading

“…In recent years, an important amount of research has focused on overcoming these modeling limitations. Regarding memory, a wide array of modifications has been analyzed, such as two-step infection models [19,20], nonexponential distributions [21][22][23], and time-varying transmission probabilities [24,25]. Conversely, a plethora of complex contagion schemes has been proposed to mediate the assumption of independent transmissions.…”

Role of hubs in the synergistic spread of behavior

“…Recently added to the list are various models of cascades with synergistic spreading rules involving cooperation between different contagions [6][7][8][9], weakened individuals [10][11][12][13][14][15][16], or multiple spreading thresholds [17]. If each transmission occurs independently without synergy, the cascade exhibits a continuous percolation transition [18].…”

“…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][14][15], Poissonian random networks [6][7][8][10][11][12][13]17], and modular networks [16], 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 p k ∼ k −α (with α > 2) of the number of neighbors k (called degree) [20].…”