Tutorial ArticleA tutorial on modeling and analysis of dynamic social networks. Part II☆
Introduction
Originating from the early studies on sociometry (Moreno, 1934, Moreno, 1951), Social Network Analysis (SNA) has quickly grown into an interdisciplinary science (Freeman, 2004, Scott, 2000, Scott, Carrington, 2011, Wasserman, Faust, 1994) that has found applications in political sciences (Knoke, 1993, Lazer, 2011), medicine (O’Malley & Marsden, 2008), economics (Easley, Kleinberg, 2010, Jackson, 2008), crime prevention and security (Bichler, Malm, 2015, Masys, 2014). The recent breakthroughs in algorithms and software for big data analysis have made SNA an efficient tool to study online social networks and media (Arnaboldi, Passarella, Conti, Dunbar, 2015, Kazienko, Chawla, 2015) with millions of users. The development of SNA has inspired many important concepts of modern network science (Newman, 2003, Newman, Barabasi, Watts, 2006, Strogatz, 2001, Van Mieghem, 2006) such as cliques and communities, centrality measures, resilience, graph’s density and clustering coefficient.
Employing many mathematical and algorithmic tools, SNA has however benefited little from the recent progress in systems and control (Annaswamy, Chai, Engell, Hara, Isaksson, Khargonekar, et al., 2017, Murray, 2003, Samad, Annaswamy, 2011). The realm of social sciences has remained almost untouched by control theory, despite the long-term studies on social group dynamics (Diani, McAdam, 2003, Lewin, 1947, Sorokin, 1947) and “sociocybernetics” (Bailey, 2006, Geyer, 1995, Geyer, van der Zouwen, 2001, Wiener, 1954). This gap between SNA and control can be explained, to a great extent, by the lack of dynamic models of social processes and mathematical armamentarium for their analysis. Focusing on topological properties of networks, SNA and network science have paid much less attention to dynamics over them, except for some special processes such as e.g. random walks, branching and queueing processes, percolation and contagion dynamics (Newman, Barabasi, Watts, 2006, Van Mieghem, 2006).
The recent years have witnessed an important tendency towards filling the gap between SNA and control theory, enabled by the rapid progress in multi-agent systems and dynamic networks. The emerging branch of control theory, studying social processes, is very young and even has no name yet. However, the interest of sociologists to this new area and understanding that “coordination and control of social systems is the foundational problem of sociology” Friedkin (2015) leaves no doubt that it should become a key instrument to examine social networks and dynamics over them. Without aiming to provide a exhaustive survey of “social control theory” at its dawn, this tutorial focuses on the most “mature” results, primarily dealing with mechanisms of opinion formation (Acemoglu, Dahleh, Lobel, Ozdaglar, 2011, Castellano, Fortunato, Loreto, 2009, Dong, Zhan, Kou, Ding, Liang, 2018, Friedkin, 2015, Hołyst, Kacperski, Schweitzer, 2001, Xia, Wang, Xuan, 2011).
In the first part of this tutorial (Proskurnikov & Tempo, 2017), the most classical models of opinion formation have been discussed that have anticipated and inspired the “boom” in multi-agent and networked control, witnessed by the past decades. This paper is the second part of the tutorial and deals with more recent dynamic models, taking into account effects of time-varying graphs, homophily, negative influence, asynchronous interactions and quantization. The theory of such models and multi-agent control have been developed concurrently, inspiring and reinforcing each other.
Whereas analysis of the classical models addressed in Proskurnikov and Tempo (2017) is mainly based on linear algebra and matrix analysis, the models discussed in this part of the tutorial require more sophisticated and diverse mathematical tools. The page limit makes it impossible to include the detailed proofs of all results discussed in this part of the tutorial; for many of them, we have to omit the proofs or provide only their brief sketches.
The paper is organized as follows. Section 2 introduces preliminary concepts and some notation used throughout the paper. Section 3 considers basic results, concerned with properties of the non-stationary French–DeGroot and Abelson models. In Section 4 we consider bounded confidence models, where the interaction graph is opinion-dependent. Section 5 is devoted to dynamic models based on asynchronous gossiping interactions. Section 6 introduces some models, exploiting the idea of negative influence. Section 7 concludes the tutorial.
Section snippets
Preliminaries and notation
In this section we introduce some notation; basic concepts regarding opinion formation modeling are also recollected for the reader’s convenience.
The models by French–DeGroot and Abelson with time-varying interaction graphs
Non-stationary counterparts of the models (1) and (2) have been thoroughly studied in regard to consensus and synchronization in multi-agent networks. In this tutorial, only some results are considered that directly related to social dynamics; detailed overview of consensus algorithms can be found e.g. in the recent monographs and surveys (Li, Duan, Chen, Huang, 2010, Olfati-Saber, Fax, Murray, 2007, Proskurnikov, Cao, 2016a, Proskurnikov, Fradkov, 2016, Ren, Beard, 2008, Ren, Cao, 2011, Wu,
Opinion dynamics with bounded confidence
The well-known adage “birds of a feather flock together” prominently manifests the principle of homophily (McPherson, Smith-Lovin, & Cook, 2001): similar individuals interact more often and intensively than dissimilar people. Distancing from the members of other social groups, e.g. rejection of cultural forms they like Mark (2003), is an important factor of social segregation and cleavage. Humans readily assimilate opinions of like-minded individuals, accepting dissimilar opinions with
Randomized gossip-based models
The models considered in the previous sections adopt an implicit assumption of synchronous interactions among the agents. The agents simultaneously display their opinions to each other and simultaneously update them. Evidently, even for small-group discussions this assumption is unrealistic; as noticed in Friedkin and Johnsen (1999), “interpersonal influences do not occur in the simultaneous way... and there are more or less complex sequences of interpersonal influences in the group”. One
Disagreement via negative influence
The models discussed in the previous section extend the basic French–DeGroot and Abelson models, inheriting however the key idea of iterative averaging. Even though agreement is not always possible (due to the effects of stubborness, homophily etc.), the agents cooperate in order to reach it, always changing their opinions towards each other. In many systems, arising in economics, natural sciences and robotics such positive (attractive) couplings among the agents coexist with negative
Conclusions and future works
The models describing social processes are numerous. It will not be an exaggeration to say that almost every week a novel model appears. When this tutorial was started, many of the papers referred in it had not been even written. Confining ourselves to a special class of dynamic models, we clearly realize that even this class remain partially uncovered by this tutorial. For instance, we do not consider models with quantized communication among the agents (that is, information an agent displays
Afterword by Anton V. Proskurnikov
This paper was conceived by Dr. Roberto Tempo and myself in 2016 as a survey, giving an overview of social dynamics models, scattered in mathematical, physical, sociological and engineering literature, from the systems and control viewpoint. Soon we realized that such a survey will be appreciated only by researchers, working on consensus and coordination of multi-agent networks, whereas our purpose was to open the exciting field of “social systems theory” to the broadest audience. It was
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The paper is supported by Russian Science Foundation (RSF) grant 14-29-00142, hosted by IPME RAS.