Spreading predictability in complex networks

Na Zhao; Jian Wang; Yong Yu; Jun-Yan Zhao; Duan-Bing Chen

Spreading dynamics is an important issue in spreading and controlling [1] [2] [3] of 2 rumor [4] [5] [6] [7] and disease [8] [9] [10] [11] , marketing [12] , recommending [13] [14] [15] , source 3 detecting [16, 17] , and many other interesting topics [18] [19] [20] [21] [22] . How to predict the 4 infection probability [23] , infected scale [24, 25] , and even the infected nodes precisely 5 has been gotten much attention in recent years. 6 diverse collection of outbreaks and identified a fundamental entropy barrier for disease 16 time series forecasting through adopting permutation entropy as a model independent 17 measure of predictability. Funk et al [29] presented a stochastic semi-mechanistic 18 model of infectious disease dynamics that was used in real time during the 2013-2016 19 West African Ebola epidemic to fit the simulated trajectories in the Ebola Forecasting 20 Challenge, and to produce forecasts that were compared to following data points. 21 Venkatramanan et al [30] proposed a data-driven agent-based model framework for 22 forecasting the 2014-2015 Ebola epidemic in Liberia, and subsequently used during 23 the Ebola forecasting challenge. The data-driven approach can be refined and adapted 24 for future epidemics, and share the lessons learned over the course of the challenge. 25 Zhang et al [31] proposed a measurement to state the efforts of users on Twitter to get 26 their information spreading. They found that small fraction of users with special 27 performance on participation can gain great influence, while most other users play a 28 role as middleware during the information propagation. 29 Up to now, most researches are focused on macro level of spreading prediction, but 30 few on micro level. However, the detailed infected individuals should be known so as 31 to contain the spread of serious infectious diseases such as SARS [32, 33] and 32 H7N7 [34, 35] . Besides aspect of macro level of spreading, we should pay attention to 33 some more details besides the general infection coverage so as to achieve fine 34 prediction. Chen et al. did some interesting works on this area [23] . They presented 35 an iterative algorithm to estimate the infection probability of the spreading process 36 and then apply it to mean-field approach to predict the spreading coverage. 37 Combing mean-field or pair approximation models with infection probability 38 estimating strategy [23] , the number of infected nodes from a given snapshot of the 39 propagation on network can be predicted, but can not determine which nodes will be 40 infected. In this paper, we present a probability based prediction model to estimate 41 the infection probability of a node, further, to determine the nodes being infected in 42 the future. 44 For a given snapshot, a susceptible node can be infected by a probability in the future. 45 Denoting by P u (t) the score of node u at time t, we have,