## 1 Introduction  
The continuous use of the Internet for all kinds of human activities has resulted in an unprecedented proliferation of data across multiple platforms, which can be used to attack everybody's privacy [1]. As a result, experts and researchers have recommended employing privacy-preserving technologies to protect society from digital threats [2]. Moreover, it has been argued that privacy protection should not be optional but mandatory, even when there is a trade-off between security and liberty [3]. More worrisome is the fact that people's information is currently stored in many databases, and if these can be correlated, the produced metadata may be exploited to measure individuals' behaviors, preferences, attitudes, health, stress levels, demographics, sleep habits, exercising, etc. [4; 5; 6]. Personal data are any information referring to some identified or identifiable person. Collecting personal data about multiple people has tremendous value for data brokers, because they are usually avid to gather as much information as possible in order to sell it. It is already known that free services over the Internet have a hidden cost, since our metadata are exposed [7; 8]. Personal data are considered an article of trade by technological platforms, which exchange it in return for a free service. It has been stated that young people surrender their personal information more easily to obtain a service, get more followers on social media, establish an attractive profile, and engage in other apparently naive activities [9]. Email networks, constructed by the exchange of messages between people, can be used as a proxy for the professional relationship between individuals in an organization. Therefore, other researchers have classified it as a social network [10; 11; 12] that can provide information about personal acquaintances, how groups of people work to complete a task, or how information flows across an organization. In other words, the exchange of emails may reveal private information about individuals and their workplace. It has been proved that traffic analysis can be used to infer personal data, even when privacy protection mechanisms are used in an anonymous communication channel [13]. Also, some of the authors in this paper have carried out previous research on the mechanisms and consequences of inferring information from the communication data produced by anonymous social networks [14]. This research focuses on developing an algorithm to infer information even when the communication channels are protected with a mix network [15]. A mix network is a mechanism used to hide end-to-end communication from third parties. This mechanism stores messages from multiple senders, shuffles them, and sends them to their destinations, with the objective of breaking the relationship between senders and receivers. The algorithm proposed here is called Threading Statistical Disclosure Attack using the Expectation-Maximization Algorithm (TSDA-EM), and it is an improvement over two previous statistical disclosure attack algorithms [16; 17]: The one in [16], which will be referred to as the Original Algorithm (OA) in this paper, infers communication within an anonymous social network but does not include threading and does not infer the frequency of messages. This research compares the results from the OA against the proposed TSDA-EM algorithm. On the other hand, the algorithm introduced in [17], called the Base Algorithm (BA) in here, describes how to calculate feasible tables and, using likelihood values, determines who is communicating with whom in an anonymous social network. This research has implemented threads to speed up the algorithms. In computer programming, a thread is a short set of instructions planned to be scheduled and executed independently of the parent process. For instance, a program can have a thread waiting for a concrete incident to happen or running a different task, allowing the main program to perform other tasks.