scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A tutorial on modeling and analysis of dynamic social networks. Part I

TL;DR: The aim of this tutorial is to highlight a novel chapter of control theory, dealing with applications to social systems, to the attention of the broad research community.
About: This article is published in Annual Reviews in Control.The article was published on 2017-01-01 and is currently open access. It has received 382 citations till now. The article focuses on the topics: Social network & Social dynamics.

Summary (5 min read)

1. Introduction

  • The 20th century witnessed a crucial paradigm shift in social and behavioral sciences, which can be described as “moving from the description of social bodies to dynamic problems of changing group life” [1].
  • 24], mathematical methods of SNA have focused on graph-theoretic properties of social networks, paying much less attention to dynamics over them.
  • These models and results are mainly concerned with opinion formation under social influence.
  • The authors also discuss the relations between these models and modern multiagent control, where some of them have been subsequently rediscovered.

2.1. Approaches to opinion dynamics modeling

  • The authors primarily deal with models of opinion dynamics.
  • Up to now, system-theoretic studies on opinion dynamics have primarily focused on models with realvalued (“continuous”) opinions, which can attain continuum of values and are treated as some quantities of interest, e.g. subjective probabilities [38, 39].
  • These models obey systems of ordinary differential or difference equations and can be examined by conventional control-theoretic techniques.
  • Models of social dynamics can be divided into two major classes: macroscopic and microscopic models.
  • Unlike statistical models, adequate for very large groups (mathematically, the number of agents goes to infinity), agent-based models can describe both small-size and large-scale communities.

2.2. Basic notions from graph theory

  • Social interactions among the agents are described by weighted (or valued) directed graphs.
  • The authors introduce only basic definitions regarding graphs and their properties; a more detailed exposition and examples of specific graphs can be found in textbooks on graph theory, networks or SNA, e.g. [4, 9, 10, 53–55].
  • In graph theory, the arc (i, j) usually corresponds to the positive entry aij >.
  • In multi-agent control [56, 57] and opinion formation modeling it is however convenient1 to identify the arc (i, j) with the entry aji >.
  • The graph in Fig. 3a has the single root node 4, constituting its own strong component, all other strong components are not closed.

If A is irreducible, then ρ(A) is a simple eigenvalue and v is strictly positive vi > 0∀i.

  • Obviously, Theorem 1 is also applicable to the transposed matrix A⊤, and thus A also has a left nonnegative eigenvector w⊤, such that w⊤A = ρ(A)w⊤.
  • Besides ρ(A), a nonnegative matrix can have other eigenvalues λ of maximal modulus |λ| = ρ(A).
  • Hence, cyclic irreducible matrices correspond to periodic strong graphs.
  • Many models of opinion dynamics employ stochastic and substochastic nonnegative matrices.

2.4. M-matrices and Laplacians of weighted graphs

  • 61] that are closely related to nonnegative matrices and have some important properties.the authors.
  • These vectors are positive if the graph G[−Z] is strongly connected.
  • Non-singular M -matrices are featured by the following important property [59, 61].
  • The zero eigenvalue is simple if and only if the graph G[A] has a directed spanning tree (quasi-strongly connected).
  • A more general relation between the kernel’s dimension dimkerL[A] = n− rankL[A] and the graph’s structure has been established3 in [58, 65].

3. The French-DeGroot Opinion Pooling

  • One of the first agent-based models4 of opinion formation was proposed by the social psychologist French in his influential paper [68], binding together SNA and systems theory.
  • Along with its generalization, suggested by DeGroot [38] and called “iterative opinion pooling”, this model describes a simple procedure, enabling several rational agents to reach consensus [69–71]; it may also be considered as an algorithm of non-Bayesian learning [72, 73].
  • The original 3As discussed in [55, Section 6.6], the first studies on the Laplacian’s rank date back to 1970s and were motivated by the dynamics of compartmental systems in mathematical biology.
  • 4As was mentioned in Section 2, a few statistical models of social systems had appeared earlier, see in particular [50, 51].
  • Goal of French, however, was not to study consensus and learning mechanisms but rather to find a mathematical model for social power [68, 74, 75].

3.2. History of the French-DeGroot model

  • A special case of the model (2) has been introduced by French in his seminal paper [68].
  • This paper first introduces a graph G, whose nodes correspond to the agents; it is assumed that each node has a self-loop.
  • French formulated without proofs several conditions for reaching a consensus, i.e. the convergence xi(k) −−−→ k→∞ x∗ of all opinions to a common “unani- mous opinion” [68] that were later corrected and rigorously proved by Harary [53, 76].
  • To obtain a simpler algorithm of reaching consensus, a heuristical algorithm was suggested in [80], replacing the convex optimization by a very simple procedure of weighted averaging, or opinion pooling [81].
  • Unlike the French model [68], the matrix W can be an arbitrary stochastic matrix and the opinions are vector-valued.

3.3. Algebraic convergence criteria

  • The authors discuss convergence properties of the French-DeGroot model (2); the properties for the multidimensional model (4) are the same.
  • The first question, regarding the model (4), is whether the opinions converge or oscillate.
  • Fully regular matrices are also referred to as SIA (stochastic indecomposable aperiodic) matrices [56, 57, 84, 85].
  • Since an imprimitive irreducible matrix W has eigenvalues {e2πki/h}h−1k=0, where h > 1, for almost all 7 initial conditions the solution of (2) oscillates.

3.4. Graph-theoretic conditions for convergence

  • For large-scale social networks, the criterion from Lemma 9 cannot be easily tested.
  • The model (2) is convergent if and only if all closed strong components in G[W ] are aperiodic.
  • 7“Almost all” means “all except for a set of zero measure”.
  • As shown in the next subsection, Theorem 12 can be derived from the standard results on the Markov chains convergence [82], using the duality between Markov chains and the French-DeGroot opinion dynamics.
  • It should be noted that the existence of a directed spanning tree is in general not sufficient for consensus in the case where W has zero diagonal entries.

3.5. The dual Markov chain and social power

  • The closed strong components in G[W ] correspond to essential classes of states, whereas the remaining nodes correspond to inessential (or non-recurrent) states [94].
  • The standard ergodicity condition is that the essential class is unique and aperiodic, which is in fact equivalent to the second part of Theorem 12.
  • The greater this weight is, the more influential is the ith individual’s opinion.
  • A more detailed discussion of social power and social influence mechanism is provided in [68, 75].
  • This centrality measure is similar to the eigenvector centrality [95], which is defined as the left eigenvector of the conventional binary adjacency matrix of a graph instead of the “normalized” stochastic adjacency matrix.

3.6. Stubborn agents in the French-DeGroot model

  • There are situations when the opinions do not reach consensus but split into several clusters.
  • Theorem 12 implies that if G[W ] has the only source, being also a root (Fig. 3a), then the opinions reach a consensus (the source node is the only closed strong component of the graph).
  • Theorem 12 implies, however, that typically the opinions in such a group converge.
  • Let the group have s ≥ 1 stubborn agents, influencing all other individuals (i.e. the set of source nodes is connected by walks to all other nodes of G[W ]).
  • Example 2. Consider the French-DeGroot model, corresponding to the weighted graph in Fig. 5 x1(k) x2(k) x3(k).

4.2. Convergence and consensus conditions

  • Note that Corollary 6, applied to the M-matrix L[A] and λ0 = 0, implies that all Jordan blocks, corresponding to the eigenvalue λ0 = 0, are trivial and for any other eigenvalue λ of the Laplacian L[A] one has Reλ >.
  • He concluded that “compactness” is necessary and sufficient for consensus; the proof, however, was given only for diagonalizable Laplacian matrices.
  • The full proof of Theorem 16 was given only in [66]; the case of strong graph was earlier considered in [100].
  • Lemma 17 implies that the vectors x̃(k) = x(τk) satisfy a special French-DeGroot model with W = Wτ and allows to derive Theorem 16 from Corollary 13; this lemma can also be used for analysis of time-varying extensions of Abelson’s model [56].
  • More advanced results on nonlinear consensus algorithms [105, 112, 113] allow to examine the nonlinear Abelson model (11) under different assumptions on the coupling function g(·).

4.3. The community cleavage problem

  • Admitting that in general the outcome of consensus is “too strong to be realistic” [52], Abelson formulated a fundamental problem, called the community cleavage problem [19] or Abelson’s diversity puzzle [99].
  • The informal formulation, stated in [34], was: “Since universal ultimate agreement is an ubiquitous outcome of a very broad class of mathematical models, the authors are naturally led to inquire what on earth one must assume in order to generate the bimodal outcome of community cleavage studies.”.
  • This requires to find mathematical models of opinion formation that are able to capture the complex behavior of real social groups, yet simple enough to be rigorously examined.
  • 6, one of the reasons for opinion clustering is the presence of stubborn agents, whose opinions are invariant.
  • In the next sections the authors consider more general models with “partially” stubborn, or prejudiced, agents.

5.3. The Taylor model and containment control

  • A multidimensional extension of the Taylor model (13) arises in the containment control problems [57, 116–118].
  • The other agents can be either “P-dependent” (indirectly influenced by one or several leaders) or “P-independent”.
  • In the recent literature on containment control [57, 116, 117, 119, 120] Theorem 20 has been extended in various directions: the leaders may be dynamic (and thus their convex hull is time-varying S = S(t)), the interaction graph may also be time-varying and the agents may have non-trivial dynamics.
  • Furthermore, the polyhedron S can be replaced by an arbitrary closed convex set; the relevant problem is sometimes referred to as the target aggregation [121] and is closely related to distributed optimization [122].

6.1. Convergence and stability conditions

  • In since otherwise (20) reduces to the French-DeGroot model (2).
  • Below the authors give the sketch of the proof of Theorem 21, retracing the proof of Theorem 18.
  • The Friedkin-Johnsen model (20) is asymptotically stable if and only if all agents are Pdependent.
  • In the second case (Fig. 6b) agents 1, 2, 4 move their opinions towards the stubborn agent 3’s opinion, however, the visible cleavage of their opinions is observed.

6.2. Friedkin’s influence centrality and PageRank

  • A natural question arises whether the concept of social power, introduced for the French-DeGroot model, can be extended to the model (20).
  • The authors discuss such an extension, introduced by Friedkin [19, 129] and based on the equality (23).
  • Recall that the definition of French’s social power assumed that the agents converge to the same consensus opinion x1(∞) = . . . = xn(∞); the social power of agent i is defined as the weight of its initial opinion xi(0) in this final opinion of the group.
  • To avoid this problem, the Markov process of random surfing is modified, allowing the teleportation [131] from each node to a randomly chosen webpage.
  • Another extension of the PageRank centrality, based on the general model (20), has been proposed in [138].

6.3. Alternative interpretations and extensions

  • The French-DeGroot model with stubborn agents, examined in Subsect.
  • In the original system (2) the prejudice is a part of the state vector, destroying the asymptotic stability.
  • An important open problem is to find the relation between the behavior of opinions in the models from Sections 3-6 and the structure of communities or modules in the network’s graph [150].

Did you find this useful? Give us your feedback

Citations
More filters
01 Jan 2012

3,692 citations

Journal ArticleDOI
TL;DR: In this paper, the authors highlight a novel chapter of control theory, dealing with dynamic models of social networks and processes over them, to the attention of the broad research community, and focus on more recent models of complex networks that have been developed concurrently with MAS theory.
Abstract: Recent years have witnessed a significant trend towards filling the gap between Social Network Analysis (SNA) and control theory. This trend was enabled by the introduction of new mathematical models describing dynamics of social groups, the development of algorithms and software for data analysis and the tremendous progress in understanding complex networks and multi-agent systems (MAS) dynamics. The aim of this tutorial is to highlight a novel chapter of control theory, dealing with dynamic models of social networks and processes over them, to the attention of the broad research community. In its first part [1], we have considered the most classical models of social dynamics, which have anticipated and to a great extent inspired the recent extensive studies on MAS and complex networks. This paper is the second part of the tutorial, and it is focused on more recent models of social processes that have been developed concurrently with MAS theory. Future perspectives of control in social and techno-social systems are also discussed.

176 citations

Journal ArticleDOI
TL;DR: It is identified that for topics whose logical interdependencies take on a cascade structure, disagreement in opinions can occur if individuals have competing and/or heterogeneous views on how the topics are related, i.e., the logical interdependence structure varies between individuals.
Abstract: A fundamental aspect of society is the exchange and discussion of opinions between individuals, occurring in situations as varied as company boardrooms, elementary school classrooms and online social media. After a very brief introduction to the established results of the most fundamental opinion dynamics models, which seek to mathematically capture observed social phenomena, a brief discussion follows on several recent themes pursued by the authors building on the fundamental ideas. In the first theme, we study the way an individual′s self-confidence can develop through contributing to discussions on a sequence of topics, reaching a consensus in each case, where the consensus value to some degree reflects the contribution of that individual to the conclusion. During this process, the individuals in the network and the way they interact can change. The second theme introduces a novel discrete-time model of opinion dynamics to study how discrepancies between an individual′s expressed and private opinions can arise due to stubbornness and a pressure to conform to a social norm. It is also shown that a few extremists can create “pluralistic ignorance”, where people believe there is majority support for a position but in fact the position is privately rejected by the majority. Last, we consider a group of individuals discussing a collection of logically related topics. In particular, we identify that for topics whose logical interdependencies take on a cascade structure, disagreement in opinions can occur if individuals have competing and/or heterogeneous views on how the topics are related, i.e., the logical interdependence structure varies between individuals.

91 citations

Journal ArticleDOI
TL;DR: This paper summarizes these ground-breaking ideas and their fascinating extensions and introduces newly surfaced concepts in opinion dynamics over the last few years.
Abstract: Opinion dynamics have attracted the interest of researchers from different fields. Local interactions among individuals create interesting dynamics for the system as a whole. Such dynamics are important from a variety of perspectives. Group decision making, successful marketing, and constructing networks (in which consensus can be reached or prevented) are a few examples of existing or potential applications. The invention of the Internet has made the opinion fusion faster, unilateral, and on a whole different scale. Spread of fake news, propaganda, and election interferences have made it clear there is an essential need to know more about these dynamics. The emergence of new ideas in the field has accelerated over the last few years. In the first quarter of 2020, at least 50 research papers have emerged, either peer-reviewed and published or on preprint outlets such as arXiv. In this paper, we summarize these ground-breaking ideas and their fascinating extensions and introduce newly surfaced concepts.

83 citations

Journal ArticleDOI
TL;DR: In this article, a novel opinion dynamics model is proposed to study how a discrepancy can arise in general social networks of interpersonal influence, where each individual in the network has both a private and an expressed opinion: an individual's private opinion evolves under social influence from the expressed opinions of the individual's neighbours, while the individual determines his or her expressed opinion under a pressure to conform to the average expressed opinion of his or his neighbours, termed the local public opinion.

70 citations

References
More filters
Book
01 Jan 1985
TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.

23,986 citations

Book
01 Jan 1957
TL;DR: Cognitive dissonance theory links actions and attitudes as discussed by the authors, which holds that dissonance is experienced whenever one cognition that a person holds follows from the opposite of at least one other cognition that the person holds.
Abstract: Cognitive dissonance theory links actions and attitudes It holds that dissonance is experienced whenever one cognition that a person holds follows from the opposite of at least one other cognition that the person holds The magnitude of dissonance is directly proportional to the number of discrepant cognitions and inversely proportional to the number of consonant cognitions that a person has The relative weight of any discrepant or consonant element is a function of its Importance

22,553 citations

Journal ArticleDOI
TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

17,647 citations


"A tutorial on modeling and analysis..." refers background in this paper

  • ...Usually centrality measures are introduced as functions of the graph topology [104] while their relations to dynamical processes over graphs are not well studied....

    [...]

Book
25 Nov 1994
TL;DR: This paper presents mathematical representation of social networks in the social and behavioral sciences through the lens of Dyadic and Triadic Interaction Models, which describes the relationships between actor and group measures and the structure of networks.
Abstract: Part I. Introduction: Networks, Relations, and Structure: 1. Relations and networks in the social and behavioral sciences 2. Social network data: collection and application Part II. Mathematical Representations of Social Networks: 3. Notation 4. Graphs and matrixes Part III. Structural and Locational Properties: 5. Centrality, prestige, and related actor and group measures 6. Structural balance, clusterability, and transitivity 7. Cohesive subgroups 8. Affiliations, co-memberships, and overlapping subgroups Part IV. Roles and Positions: 9. Structural equivalence 10. Blockmodels 11. Relational algebras 12. Network positions and roles Part V. Dyadic and Triadic Methods: 13. Dyads 14. Triads Part VI. Statistical Dyadic Interaction Models: 15. Statistical analysis of single relational networks 16. Stochastic blockmodels and goodness-of-fit indices Part VII. Epilogue: 17. Future directions.

17,104 citations

Book
01 Jan 1969

16,023 citations

Frequently Asked Questions (13)
Q1. What is the aperiodic condition for the graph?

For reaching a consensus the graph G[W ] should have the only closed strong component (i.e. be quasi-strongly connected), which is aperiodic. 

With the aim to provide a basic introduction to social dynamics modeling and analysis, this tutorial is confined to agent-based models with real-valued scalar and vector opinions, whereas other models are either skipped or mentioned briefly. 

The spectral radius ρ(A) ≥ 0 of a nonnegative matrix A is an eigenvalue of A, for which a real nonnegative eigenvector existsAv = ρ(A)v for some v = (v1, . . . , vn) ⊤ 6= 0, vi ≥ 0. 

The class of Friedkin’s centrality measures fα contains the well-known PageRank [130–136], proposed originally for ranking webpages in Web search engines and used in scientometrics for journal ranking [133]. 

The linear Abelson model (12) reaches consensus if and only if G[A] is quasistrongly connected (i.e. has a directed spanning tree). 

To obtain a simpler algorithm of reaching consensus, a heuristical algorithm was suggested in [80], replacing the convex optimization by a very simple procedure of weighted averaging, or opinion pooling [81]. 

The update rule (20) implies that each agent i updates its opinion in a way to minimize Ji(x), assuming that xj(k), j 6= k, are constant. 

Given a positive influence weight wij > 0, agent j is able to influence the opinion of agent i at each step of the opinion iteration; the greater weight is assigned to agent j, the stronger is its influence on agent i. 

Note that Corollary 6, applied to the M-matrix L[A] and λ0 = 0, implies that all Jordan blocks, corresponding to the eigenvalue λ0 = 0, are trivial and for any other eigenvalue λ of the Laplacian L[A] one has Reλ > 

In general, the sufficiencypart requires to prove that the zero eigenvalue of L[A] is algebraically simple (statement 1 in Lemma 8). 

For this reason, the authors do not adopt these coupling conditions in this tutorial, allowing the prejudices and initial opinions to be independent; the same holds for the matrices Λ and W .Similar to the Taylor model, a generic FriedkinJohnsen model is asymptotically stable, i.e. the substochastic matrix ΛW is Schur stable ρ(ΛW ) < 1. 

Theorem 1 is also applicable to the transposed matrix A⊤, and thus A also has a left nonnegative eigenvector w⊤, such that w⊤A = ρ(A)w⊤. 

As discussed in Section 6, if the French social power vector p∞ is well-defined, then the probability distribution p(k) converges to p∞.