On Feb. 4^{th}, IJAC (International Journal of Automation and Computing) and CASIA (Institute of Automation, Chinese Academy of Sciences) co-hosted the first seminar “Beyond consensus and polarisation: complex social phenomena in social networks”, which attracted more than 4500 audiences around the world. Review the wonderful talk now!

**【Seminar Recording】**

__https://www.bilibili.com/video/BV1Fv411v7xj__

**【Speakers】**

**Brian D. O. Anderson**Research School of Engineering, Australian National University, Canberra, Australia

Bio: Brian Anderson was born in Sydney, Australia. He received the B.Sc. degree in pure mathematics in 1962, and B.E. in electrical engineering in 1964, from Sydney University, Sydney, Australia, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, USA, in 1966.

He is an Emeritus Professor at the Australian National University. His awards include the IEEE Control Systems Award of 1997, the 2001 IEEE James H Mulligan, Jr Education Medal, and the Bode Prize of the IEEE Control System Society in 1992, as well as several IEEE and other best paper prizes. He is a Fellow of the Australian Academy of Science, the Australian Academy of Technological Sciences and Engineering, the Royal Society, and a foreign member of the US National Academy of Engineering. He holds honorary doctorates from a number of universities, including Université Catholique de Louvain, Belgium, and ETH, Zürich. He is a past president of the International Federation of Automatic Control and the Australian Academy of Science. His current research interests are in distributed control, sensor networks and social networks.

**Mengbin Ye **School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Australia

Bio: Mengbin Ye was born in Guangzhou, China. He received the B.E. degree (with First Class Honours) in mechanical engineering from the University of Auckland, Auckland, New Zealand in 2013, and the Ph.D. degree in systems and control engineering at the Australian National University, Canberra, Australia in 2018. From 2018-2020, he was a postdoctoral researcher with the Faculty of Science and Engineering, University of Groningen, Netherlands. Since 2020, he has been an Optus Fellow at the Optus--Curtin Centre of Excellence in Artificial Intelligence, Curtin University, Perth, Australia.

He has been awarded the 2018 J.G. Crawford Prize (Interdisciplinary), ANU's premier award recognising graduate research excellence. He has also received the 2019 Springer PhD Thesis Prize, and was Highly Commended in the Best Student Paper Award at the 2016 Australian Control Conference. His current research interests include opinion dynamics and decision making in complex social networks, epidemic modelling, coordination of multi-agent systems, and localisation using bearing measurements.

**【Introduction】**

Mathematical models of opinion dynamics on social networks grew from the seminal works by French and Harary (1950s) and DeGroot (1970s). Since then, the development and analysis of such models has received extensive interest from a broad range of scientific communities, including sociology, computer science, physics, economics, and engineering. Many efforts have focused on developing models and identifying conditions that can explain various social phenomena, such as a group of individuals reaching a consensus, becoming polarised, or breaking into clusters.

In this talk, we review three recent developments in modelling complex social phenomena using opinion dynamics models. First, we introduce a model in which an individual may express an opinion different from his/her true opinion due to a pressure to conform to the expressed opinions of others. Besides theoretical results, we use the model to revisit and explain famed Asch conformity experiments from the 1950s. Second, we present the DeGroot-Friedkin model of the evolution of social power, and provide convergence results when the network topology is time-varying. Last, we examine a model in which multiple logically interdependent topics are simultaneously discussed. We show that the opinions of the individuals can converge to a state of persistent disagreement as a consequence of individuals having different understandings of the logical interdependency relations between topics.

**【Slides Sharing】**https://mp.weixin.qq.com/s/FQVK2U6f34BiwSJUj6oqIQ

**【Q&A Part】**

**【Question 1】**

Is it possible to combine opinion consensus with a math tool "belief and plausibility functions"? which is a "non-traditional" probability framework and can be used for statistics of opinions.

**【****Answer****】**

Our understanding is that some, but not all, of “belief and plausibility” theory presupposes that some form of probabilities are known for “related” questions. The only possibility of parallel with those frequently arising aspects of belief and plausibility could occur with our treatment of interdependent topics, as opposed to single topics.

Whether or not there are independent topics, belief and plausibility theory appears not to dwell on questions of consensus, nor on dynamic processes such as those we have described. Rather, the theory is more concerned with formulating major variations on the use of Bayes’ Theorem. To this extent, the scenarios in which the theory is used seem rather different to the scenarios being considered in opinion dynamics.

Further, examination of the underlying mathematical frameworks used in addressing opinion dynamics equations shows almost no similarity with those frameworks used in belief and plausibility theory, even though it is common in both to restrict values of the variables of interest to the interval [0,1].

Neither of the speakers would claim any depth of knowledge of belief and plausibility theory however, and it may be that beyond the rather superficial remarks above, a genuine connection to opinion dynamics could be formed.

**【Question 2】**

How about the application of real-world data?

**【****Answer****】**

We want to stress that there is much real-world data available in reports of laboratory and field experiments, and that Friedkin’s work on stubbornness and the dynamics of self-appraisal is backed by controlled laboratory experiments of small groups (4-10 people) [1, 2]. As indicated in the talk, the suggested model for private and expressed opinion evolution matches the observations from the laboratory experiments of Asch. Field experiments get closer to reality than does laboratory work, and those involving alcohol consumption at Princeton university involved hundreds of people, and recorded dynamic behavior (as the school year progressed) [3]. The theory we have advanced on private and expressed opinions is also well validated by this Princeton data.

However, a large open research domain remains in connecting the opinion dynamics models with real-world online social media data. Challenges include extracting a time series of real-valued opinion values from text-based social media posts, and inferring the underlying influence network (since social media ties are not one-to-one mappings to influence weights in the opinion models).

[1] N. E. Friedkin, P. Jia, and F. Bullo. A Theory of the Evolution of Social Power: Natural Trajectories of Interpersonal Influence Systems along Issue Sequences. *Sociological Science*, 3:444-472, 2016.

[2] N. E. Friedkin, and E. C. Johnsen. *Social Influence Network Theory. *Cambridge University Press, 2011

[3] D. A. Prentice, and D. T. Miller. Pluralistic ignorance and alcohol use on campus: some consequences of misperceiving the social norm. *Journal of Personality and Social Psychology*, 64(2), 243, 1993.

**【Question 3】**

How C be determined? (in part 3)

**【****Answer****】**

Our work mainly focuses on the following problem: Assuming that C_i are known for all the individuals, establish a method to determine whether a given topic will reach a consensus or fail to do so. Thus, determination of C_i is a complementary, but important, problem that we have not considered.

Identification can be quite a challenge, and we do not have any clear suggestion on how to determine C. One could perhaps obtain this using surveys, or laboratory experiments to estimate C.

**【Question 4】**

Sorry, I didn't keep up with all the content due to network delay. I want to know about how topics is determined. Is it preset in advance?

**【****Answer****】**

The question is not clear to us. We can say that our work is not dependent on assuming particular topics for group discussion. Rather, our work studies the dynamics of the system for a given sequence of topics (in Part II) and for a given set of topics (in Part III), and we study convergence and analyse the opinion distributions at steady state. In a sense, we assume some parameters (such as the relative interaction matrices in Part II and logic matrices in Part III) are pre-determined. However whether the topics associated with these matrices deal with music, sport, the environment or whatever is irrelevant. Determining these matrices is a different, open, and important question.

**【Question 5】**

How about the underlying structure in the role of opinion dynamics beyond influence matrix?

**【****Answer****】**

Again, the question is not clear to us. We offer some remarks which may address the intended query.

The underlying structure in the basic opinion dynamics model of DeGroot, *before* one says what the influence matrix is, is a linear structure. After one says what the influence matrix is, it is almost always the case that a connectivity property is required by the structure. The linearity assumption is heroic, and is more one made for convenience in our opinion than one made in the light of concrete substantiating evidence from experiments. On the other hand, the connectivity property, at least for a small group, is very reasonable.

The question may be asking about the effects of the underlying network structure beyond being tied to the influence matrix. First, we remark that the influence matrix provides influence weights on the social interactions (and defines the social network). How the influence weights are incorporated depends on the model itself: the DeGroot model makes a bold assumption about linearity, while newer models assume a nonlinear effect e.g. [1].

In terms of the underlying structure, the role often depends on the particular model. For instance, the final opinions in the DeGroot model are determined by individuals in the closed strongly connected components, while centrality plays a key role in determining who is more influential in the final opinion values [2].

[1] W. Mei, F. Bullo, G. Chen, J. Hendrickx, and F. Dorfler. Rethinking the Micro-Foundation of Opinion Dynamics: Rich Consequences of an Inconspicuous Change, __arXiv:1909.06474__,

2020.

[2] A. V. Proskurnikov and R. Tempo, “A tutorial on modeling and analysis of dynamic social networks. Part I,” *Annu. Rev. Control*, vol. 43, pp. 65–79, 2017.

**【Question 6】**

How do the stubbornness and self-appraisal affect the DeGroot-Friedkin model? Do they only affect the self-confidence parameter?

**【****Answer****】**

Stubbornness can be incorporated into the DeGroot-Friedkin model, by replacing the DeGroot opinion dynamics with the Friedkin-Johnsen dynamics. Experimental studies can be found in [1], and convergence analysis in [2]. Note that [2] does not provide a complete convergence analysis, so there are still open questions.

[1] N. E. Friedkin, P. Jia, and F. Bullo. A Theory of the Evolution of Social Power: Natural Trajectories of Interpersonal Influence Systems along Issue Sequences. *Sociological Science*, 3:444-472, 2016.

[2] Y. Tian, P. Jia, A. Mirtabatabaei, L. Wang, N. E. Friedkin, and F. Bullo. Social Power Evolution in Influence Networks with Stubborn Individuals. *IEEE Transactions on Automatic Control (to appear)*, 2021.

**【Question 7】**

When the dynamic network is considered, what is the intrinsic difference it will present compared to the static network?

**【****Answer****】**

We presume the term ‘dynamic network’ refers to a network modelling a group where the influence matrix is varying over time, whether by self-appraisal or change of topic or change of discussion participants or due to some other dynamical process. To begin, one must differentiate between dynamic topology due to an exogeneous factor (e.g. change of topic) or due to some dynamic process itself (e.g. self-appraisal). In both cases, the intrinsic difference is that the analysis can become significantly more challenging, and a wider range of dynamical behaviour may be possible: nonlinearity is often introduced into the system (as seen by the DeGroot-Friedkin model), and time-varying systems can often require more advanced theoretical tools to study.

Both convergence to a steady state (including a cyclic trajectory), and non-convergence can occur. Simulations may be used to identify dynamical behaviours, and then theoretical analysis can follow.

**【Question 8】**

Is it possible to consider some disturbing effects that might affect the developed models? the speed of changing these factors will have effects as well in this case.

**【****Answer****】**

Of course, it is reasonable to consider variations to the models, associated perhaps with parameter change, parameter inaccuracy, noise, and the like. One could envisage some kind of theory of robustness, that perhaps related the amount of variation in the models to the amount of variation in the quantities they were delivering (or for that matter the converse), or one could consider possibly larger variations and the question of whether such variations would destroy gross properties, such as a consensus outcome.

As a simple example, [1] looks at a DeGroot-Friedkin model with some stochastic properties for the relative interaction matrix, C. Noise is often encountered in real-life social situations, and this can lead to counter-intuitive outcomes (such as promoting consensus when a noiseless system would typically fail to reach a consensus) [2].

[1] G. Chen, X. Duan, N. E. Friedkin, and F. Bullo. Social Power Dynamics over Switching and Stochastic Influence Networks. *IEEE Transactions on Automatic Control*, 64(2):582-597, 2019.

[2] M. Mäs, A. Flache, and J.A. Kitts. Cultural integration and differentiation in groups and organizations. *In Perspectives on culture and agent-based **simulations* (pp. 71–90) Springer, 2014.

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