Potential-based analysis of social, communication, and distributed networks

In recent years, there has been a wide range of studies on the role of social and distributed networks in various disciplinary areas. In particular, availability of large amounts of data from online social networks and advances in control of distributed systems have drawn the attention of many researchers to exploit the connection between evolutionary behaviors in social, communication and distributed networks. In this thesis, we first revisit several well-known types of social and distributed networks and review some relevant results from the literature. Building on this, we present a set of new results related to four different types of problems, and identify several directions for future research. The study undertaken and the approaches adopted allow us to analyze the evolution of certain types of social and distributed networks and also to identify local and global patterns of their dynamics using some novel potential-theoretic techniques. Following the introduction and preliminaries, we focus on analyzing a specific type of distributed algorithm for quantized consensus known as an unbiased quantized algorithm where a set of agents interact locally in a network in order to reach a consensus. We provide tight expressions for the expected convergence time of such dynamics over general static and time-varying networks. Following this, we introduce new protocols using a special class of Markov chains known as Metropolis chains and obtain the fastest (as of today) randomized quantized consensus protocol. The bounds provided here considerably improve the state of the art over static and dynamic networks. We make a bridge between two classes of problems, namely distributed control problems and game problems. We analyze a class of distributed averaging dynamics known as Hegselmann-Krause opinion dynamics. Modeling such dynamics as a non-cooperative game problem, we elaborate on some of the evolutionary properties of such dynamics. In particular, we answer an open question related to the termination time of such dynamics by connecting the convergence time to the spectral gap of the adjacency matrices of underlying dynamics. This not only allows us to improve the best known upper bound, but also removes the dependency of termination time from the dimension of the ambient space. The approach adopted here can also be leveraged to connect the rate of increase of a so-called kinetic-s-energy associated with multi-agent systems to the spectral gap of their underlying dynamics. We describe a richer class of distributed systems where the agents involved in the network act in a more strategic manner. More specifically, we consider a class of resource allocation games over networks and study their evolution to some final outcomes such as Nash equilibria. We devise some simple distributed algorithms which drive the entire network to a Nash equilibrium in polynomial time for dense and hierarchical networks. In particular, we show that such games benefit from having low price of anarchy, and hence, can be used to model allocation systems which suffer from lack of coordination. This fact allows us to devise a distributed approximation algorithm within a constant gap of any pure-strategy Nash equilibrium over general networks. Subsequently we turn our attention to an important problem related to competition over social networks. We establish a hardness result for searching an equilibrium over a class of games known as competitive diffusion games, and provide some necessary conditions for existence of a pure-strategy Nash equilibrium in such games. In particular, we provide some concentration results related to the expected utility of the players over random graphs. Finally, we discuss some future directions by identifying several interesting problems and justify the importance of the underlying problems

Tight Bounds for Asymptotic and Approximate Consensus

We study the performance of asymptotic and approximate consensus algorithms under harsh environmental conditions. The asymptotic consensus problem requires a set of agents to repeatedly set their outputs such that the outputs converge to a common value within the convex hull of initial values. This problem, and the related approximate consensus problem, are fundamental building blocks in distributed systems where exact consensus among agents is not required or possible, e.g., man-made distributed control systems, and have applications in the analysis of natural distributed systems, such as flocking and opinion dynamics. We prove tight lower bounds on the contraction rates of asymptotic consensus algorithms in dynamic networks, from which we deduce bounds on the time complexity of approximate consensus algorithms. In particular, the obtained bounds show optimality of asymptotic and approximate consensus algorithms presented in [Charron-Bost et al., ICALP'16] for certain dynamic networks, including the weakest dynamic network model in which asymptotic and approximate consensus are solvable. As a corollary we also obtain asymptotically tight bounds for asymptotic consensus in the classical asynchronous model with crashes. Central to our lower bound proofs is an extended notion of valency, the set of reachable limits of an asymptotic consensus algorithm starting from a given configuration. We further relate topological properties of valencies to the solvability of exact consensus, shedding some light on the relation of these three fundamental problems in dynamic networks

Analyzing the opinion dynamics models discrete & continuous

In this thesis we analyze some of the opinion dynamics in both discrete and continuous cases. In the discrete case, we will find some criteria under which we can say more about the behavior of the dynamics such as convergence of the agents to the same opinion, or consensus. For this purpose, we first consider the agent-based bounded confidence model of the Hegselmann-Krause where multiple agents want to agree on a common scalar, or they can be divided in several subgroups, with each subgroup having its own agreement value. In this model, we restrict ourselves to the case when all the agents have the same bound of confidence, often referred to as homogeneous case. We are interested to study the number of iterations which is enough for the termination of the Hegselmann-Krause algorithm. In other words, we want to give an upper bound on the number of iterations which guarantees the termination of the algorithm independently of reaching a consensus or not. Assuming the consensus is achieved in the Hegselmann-Krause model, we first give an upper bound on the number of iterations and then we provide another upper bound without any assumption. In chapter 3 we use some analysis based on Lyapunov function theory to improve our upper bound substantially. In our analysis we use two differnt type of Lyapunov functions which each of them gives us a polynomial upper bound for the termination time. In chapter 4 we consider the Hegselmann-Krause model in higher dimensions. We will see that in higher dimensions we don’t have lots of nice properties which exist in the scalar case. Then, we will find some upper bounds for the termination time. Also, at the end we will consider an extension of the Hegselmann-Krause model to continuous case such that the time is discrete but the density of the agents is continuous over the real line. In chapter 5 we use the matrix representation for the discrete dynamics and we provide some conditions on a chain of stochastic matrices based on their decomposition by permutation matrices such that it can guarantee the convergence of the chain to a consensus matrix. Also, we provide some examples and one necessary condition for finite time convergence of an especial case of averaging gossip algorithms

Influence of clustering on the opinion formation dynamics in online social networks

With the advent of Online Social Networks (OSNs), opinion formation dynamics continuously evolves, mainly because of the widespread use of OSNs as a platform of social interactions and our growing exposure to others’ opinions instantly. When presented with neighbours’ opinions in OSNs, the natural clustering ability of human agents enables them to perceive the grouping of opinions formed in the neighbourhood. A group with similar opinions exhibits stronger influence on an agent than the individual group members. Distance-based opinion formation models only consider the influence of neighbours who are within a confidence bound threshold in the opinion space. However, a bigger group formed outside this distance threshold can exhibit stronger influence than a group within the bound, especially when that group contains influential or popular agents like leaders. To the knowledge of the authors, the proposed model is the first to consider the impact of clustering capability of agent and incorporates the influence of opinion clusters (groups) formed outside the confidence bound. Simulation results show that our model can capture several characteristics of real-world opinion dynamics. © Springer Nature Switzerland AG 2018