# 阈值模型

Probability Pinf of infection for a susceptible individual after K contacts with infected individuals. (a) Independent interaction (e.g., SIR-type) model. (b) Stochastic threshold model. (c) Deterministic threshold model. Adapted from Dodds and Watts, 2004.

For disease epidemics it is customary to assume that a susceptible individual has a constant probability to receive the infection from a peer upon every exposure, independently of whether other infected individuals are simultaneously in contact or other exposures have occurred in the past. While generally reasonable for the transmission of pathogens [although exceptions may occur [1] ], this hypothesis is clearly unrealistic in most situations where a social meme is spreading: a piece of information is more credible if arriving from different sources; the push to adopt a technological innovation is stronger if neighboring nodes in the social network have already adopted it. These considerations lead naturally to the introduction of “threshold models” for spreading phenomena, where the effect of multiple exposures changes from low to high as a function of their number. Figure 14 displays the probability of infection (adoption) Pinf after K attempts in the different scenarios. In the case of SIR each attempt has a fixed probability $p$ of success and $P_{inf}=1-(1-p)^K$.

## Watts阈值模型

Phase diagram of Watts’s threshold model. The dashed line encloses the region of the $(\Phi,\left<k\right>)$ plane in which the condition for the existence of global cascades is satisfied for a uniform random graph with uniform threshold $\Phi$. The solid circles outline the region in which global cascades occur for the same parameter settings in the full dynamical model for $N=10 000$ (averaged over 100 random single-node perturbations). Adapted from Watts, 2002.

Threshold models have a long tradition in the social and economical sciences [2] [3]. In the context of spreading phenomena on complex networks, a seminal role has been played by the model introduced by Watts (2002)[4]. Each individual can be in one of two states (S and I) and is endowed with a quenched, randomly chosen threshold value $\Phi_i$. In an elementary step an individual agent in state S observes the current state of its neighbors and adopts state I if at least a threshold fraction $\Phi_i$ of its neighbors is in state I; else it remains in state S.No transition from I back to S is possible. Initially all nodes except for a small fraction are in state S. Out of these initiators a cascade of transitions to the I state is generated. The nontrivial question concerns whether the cascade remains local, i.e., restricted to a finite number of individuals, or it involves a finite fraction of the whole population. Given an initial seed, the spreading can occur only if at least one of its neighbors has a threshold such that $\Phi_i\leq1/k_i$. A cascade is possible only if a cluster of these “vulnerable” vertices is connected to the initiator. For global cascades to be possible it is then conjectured that the subnetwork of vulnerable vertices must percolate throughout the network. The condition for global cascades can then be derived applying on locally treelike networks the machinery of generating functions for branching processes. In the simple case of a uniform threshold $\Phi$ and an Erdős-Rényi pattern of interactions the phase diagram as a function of the threshold $\Phi$ and of the average degree $\left<k\right>$ is reported in Fig. 15. For fixed $\Phi$, global cascades occur only for intermediate values of the mean connectivity $1<\left<k\right><1/\Phi$. The transition occurring for small $\left<k\right>$ is trivial and is not due to the spreading dynamics: the average cascade size is finite for $\left<k\right><1$ because the network itself is composed of small disconnected components: the transition is percolative with power-law distributed cascade size. For large $\left<k\right>>1/\Phi$ instead, the propagation is limited by the local stability of nodes.

## 参考文献 References

1. Joh R I, Wang H, Weiss H, et al. Dynamics of indirectly transmitted infectious diseases with immunological threshold[J]. Bulletin of mathematical biology, 2009, 71(4): 845-862.
2. Granovetter M. Threshold models of collective behavior[J]. American journal of sociology, 1978, 83(6): 1420-1443.
3. Morris S. Contagion[J]. The Review of Economic Studies, 2000, 67(1): 57-78.
4. Watts D J. A simple model of global cascades on random networks[J]. Proceedings of the National Academy of Sciences, 2002, 99(9): 5766-5771.