A New Approximation Method for the Shapley Value Applied to the WTC 9/11 Terrorist Attack

The Shapley value (Shapley (1953)) is one of the most prominent one-point solution concepts in cooperative game theory that divides revenues (or cost, power) that can be obtained by cooperation of players in the game. The Shapley value is mathematically characterized by properties that have appealing real-world interpretations and hence its use in practical settings is easily justified. The down part is that its computational complexity increases exponentially with the number of players in the game. Therefore, in practical problems that consist of more that 25 players the calculation of the Shapley value is usually too time expensive. Among others the Shapley value is applied in the analysis of terrorist networks (cf. Lindelauf et al. (2013)) which generally extend beyond the size of 25 players. In this paper we therefore present a new method to approximate the Shapley value by refining the random sampling method introduced by Castro et al. (2009). We show that our method outperforms the random sampling method, reducing the average error in the Shapley value approximation by almost 30%. Moreover, our new method enables us to analyze the extended WTC 9/11 network of Krebs (2002) that consists of 69 members. This in contrast to the restricted WTC 9/11 network considered in Lindelauf et al. (2013), that only considered the operational cells consisting of the 19 hijackers that conducted the attack

A Survey of Social Network Forensics

Social networks in any form, specifically online social networks (OSNs), are becoming a part of our everyday life in this new millennium especially with the advanced and simple communication technologies through easily accessible devices such as smartphones and tablets. The data generated through the use of these technologies need to be analyzed for forensic purposes when criminal and terrorist activities are involved. In order to deal with the forensic implications of social networks, current research on both digital forensics and social networks need to be incorporated and understood. This will help digital forensics investigators to predict, detect and even prevent any criminal activities in different forms. It will also help researchers to develop new models / techniques in the future. This paper provides literature review of the social network forensics methods, models, and techniques in order to provide an overview to the researchers for their future works as well as the law enforcement investigators for their investigations when crimes are committed in the cyber space. It also provides awareness and defense methods for OSN users in order to protect them against to social attacks

Mapping networks of influence: tracking Twitter conversations through time and space

The increasing use of social media around global news events, such as the London Olympics in 2012, raises questions for international broadcasters about how to engage with users via social media in order to best achieve their individual missions. Twitter is a highly diverse social network whose conversations are multi-directional involving individual users, political and cultural actors, athletes and a range of media professionals. In so doing, users form networks of influence via their interactions affecting the ways that information is shared about specific global events. This article attempts to understand how networks of influence are formed among Twitter users, and the relative influence of global news media organisations and information providers in the Twittersphere during such global news events. We build an analysis around a set of tweets collected during the 2012 London Olympics. To understand how different users influence the conversations across Twitter, we compare three types of accounts: those belonging to a number of well-known athletes, those belonging to some well-known commentators employed by the BBC, and a number of corporate accounts belonging to the BBC World Service and the official London Twitter account. We look at the data from two perspectives. First, to understand the structure of the social groupings formed among Twitter users, we use a network analysis to model social groupings in the Twittersphere across time and space. Second, to assess the influence of individual tweets, we investigate the ageing factor of tweets, which measures how long users continue to interact with a particular tweet after it is originally posted. We consider what the profile of particular tweets from corporate and athletes’ accounts can tell us about how networks of influence are forged and maintained. We use these analyses to answer the questions: How do different types of accounts help shape the social networks? and, What determines the level and type of influence of a particular account

Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

Efficiently computing the Shapley value of connectivity games in low-treewidth graphs

The Shapley value is the solution concept in cooperative game theory that is most used in both theoretical and practical settings. Unfortunately, in general, computing the Shapley value is computationally intractable. This paper focuses on computing the Shapley value of (weighted) connectivity games. For these connectivity games, which are defined on an underlying (weighted) graph, computing the Shapley value is #P-hard, and thus (likely) intractable even for graphs with a moderate number of vertices. We present an algorithm that can efficiently compute the Shapley value if the underlying graph has bounded treewidth. Next, we apply our algorithm to several real-world (covert) networks. We show that our algorithm can quickly compute exact Shapley values for these networks, whereas in prior work these values could only be approximated using a heuristic method. Finally, it is demonstrated that our algorithm can also efficiently compute the Shapley value time for several larger (artificial) benchmark graphs from the PACE 2018 challenge