^{1}, Ming Tang

^{1,a)}, Kai Gong

^{1}and Ying Liu

^{1}

### Abstract

Weak ties play a significant role in the structures and the dynamics of community networks. Based on the contact process, we study numerically how weak ties influence the predictability of epidemic dynamics. We first investigate the effects of the degree of bridge nodes on the variabilities of both the arrival time and the prevalence of disease, and find out that the bridge node with a small degree can enhance the predictability of epidemic spreading. Once weak ties are settled, the variability of the prevalence will display a complete opposite trend to that of the arrival time, as the distance from the initial seed to the bridge node or the degree of the initial seed increases. More specifically, the further distance and the larger degree of the initial seed can induce the better predictability of the arrival time and the worse predictability of the prevalence. Moreover, we discuss the effects of the number of weak ties on the epidemic variability. As the community strength becomes very strong, which is caused by the decrease of the number of weak ties, the epidemic variability will change dramatically. Compared with the case of the hub seed and the random seed, the bridge seed can result in the worst predictability of the arrival time and the best predictability of the prevalence.

In community networks, the links that connect pairs of nodes belonging to different communities are defined as weak ties. The weak ties hypothesis, which is first proposed by Granovetter, is a central concept in the social network analysis. Weak ties not only play a role in effecting social cohesion but also are helpful for stabilizing complex systems under most conditions. Most recent research results showed that weak ties have significant impacts on spreading dynamics. But until now, no study on the effects of weak ties on the predictability of epidemic dynamics has been given to us. In this study, we investigate how the degree of bridge nodes and the number of weak ties influence the predictability of the epidemic dynamics on a local community. We show numerically that both the degree of bridge nodes and the network modularity are crucial in the predictability of the epidemic spreading on the local community. More importantly, we find out that the variability of the arrival time always displays a complete opposite trend to that of the prevalence, which implies that it is impossible to predict the epidemic patterns in the early stage of outbreaks accurately. This work provides us further understanding and new perspective in the effect of weak ties on epidemic spreading.

This work is supported by the NNSF of China (Grants No. 11105025), China Postdoctoral Science Foundation (Grant No. 20110491705), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20110185120021), the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2011J056), and China Postdoctoral Science Special Foundation (Grant No. 2012T50711).

I. INTRODUCTION

II. MODEL INTRODUCTION

A. The community network with degree heterogeneity

B. Dynamic process

C. Statistical parameter

III. THE EFFECT OF WEAK TIES

A. The effect of weak tie with different degrees

B. The effect of different initial seeds when d = 1

IV. THE EFFECT OF THE NUMBER OF WEAK TIES

V. CONCLUSIONS

### Key Topics

- Diseases and conditions
- 11.0
- Networks
- 10.0
- Poisson's equation
- 9.0
- Complex systems
- 3.0
- Numerical modeling
- 3.0

## Figures

The mean arrival time and its variability as a function of the degree of the bridge node where the “squares,” “circles,” “triangleups,” “triangledowns,” and “diamonds” denote the cases of the seeds with *d* = 0, 1, 2, 3, and 4, respectively. (a) versus , (b) versus . The parameters are chosen as . We perform the experiments on different networks, each of which are tested in independent realizations.

The mean arrival time and its variability as a function of the degree of the bridge node where the “squares,” “circles,” “triangleups,” “triangledowns,” and “diamonds” denote the cases of the seeds with *d* = 0, 1, 2, 3, and 4, respectively. (a) versus , (b) versus . The parameters are chosen as . We perform the experiments on different networks, each of which are tested in independent realizations.

Two spreading pathways through which the bridge node may be infected. The first one is a direct transmission from the seed to the bridge node, and the second one is an indirect transmission from the seed to node *i, j*, and then to the bridge node.

Two spreading pathways through which the bridge node may be infected. The first one is a direct transmission from the seed to the bridge node, and the second one is an indirect transmission from the seed to node *i, j*, and then to the bridge node.

At *T* = 20, the mean prevalence and its variability as a function of the degree of the bridge node where the “squares,” “circles,” “triangleups,” “triangledowns,” and “diamonds” denote the cases of the seeds with *d* = 0, 1, 2, 3, and 4, respectively. (a) versus , (b) versus . The parameters are chosen as . We perform the experiments on different networks, each of which are tested in independent realizations.

At *T* = 20, the mean prevalence and its variability as a function of the degree of the bridge node where the “squares,” “circles,” “triangleups,” “triangledowns,” and “diamonds” denote the cases of the seeds with *d* = 0, 1, 2, 3, and 4, respectively. (a) versus , (b) versus . The parameters are chosen as . We perform the experiments on different networks, each of which are tested in independent realizations.

When the distance between the initial seed and the bridge node *d* = 1, the mean arrival time and its variability as a function of the degree of the initial seed, where versus *k* for (a) and (b), versus *k* for (c) and (d). The results are averaged over independent realizations on networks.

When the distance between the initial seed and the bridge node *d* = 1, the mean arrival time and its variability as a function of the degree of the initial seed, where versus *k* for (a) and (b), versus *k* for (c) and (d). The results are averaged over independent realizations on networks.

When *d* = 1, the mean prevalence and its variability at *T* = 20 as a function of the degree of the initial seed, where versus *k*for (a) and (b), versus *k* for (c) and (d). The results are averaged over independent realizations in networks.

When *d* = 1, the mean prevalence and its variability at *T* = 20 as a function of the degree of the initial seed, where versus *k*for (a) and (b), versus *k* for (c) and (d). The results are averaged over independent realizations in networks.

The mean arrival time and its variability as a function of the modularity where the “squares,” “circles,” and “triangles” denote the cases of the bridge seed, the random seed, and the hub seed, respectively. (a) versus , (b) versus . The parameters are chosen as . We perform the experiments on different networks, each of which are tested in independent realizations.

The mean arrival time and its variability as a function of the modularity where the “squares,” “circles,” and “triangles” denote the cases of the bridge seed, the random seed, and the hub seed, respectively. (a) versus , (b) versus . The parameters are chosen as . We perform the experiments on different networks, each of which are tested in independent realizations.

At *T* = 2, the mean prevalence and its variability as a function of the modularity where the “squares,” “circles,” and “triangles” denote the cases of the bridge seed, the random seed, and the hub seed, respectively. (a) versus , (b) versus . We perform the experiments on different networks, each of which are tested in independent realizations.

At *T* = 2, the mean prevalence and its variability as a function of the modularity where the “squares,” “circles,” and “triangles” denote the cases of the bridge seed, the random seed, and the hub seed, respectively. (a) versus , (b) versus . We perform the experiments on different networks, each of which are tested in independent realizations.

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