Macroscopic network circulation for planar graphs

The analysis of networks, aimed at suitably defined functionality, often focuses on partitions into subnetworks that capture desired features. Chief among the relevant concepts is a 2-partition, that underlies the classical Cheeger inequality, and highlights a constriction (bottleneck) that limits accessibility between the respective parts of the network. In a similar spirit, the purpose of the present work is to introduce a new concept of maximal global circulation and to explore 3-partitions that expose this type of macroscopic feature of networks. Herein, graph circulation is motivated by transportation networks and probabilistic flows (Markov chains) on graphs. Our goal is to quantify the large-scale imbalance of network flows and delineate key parts that mediate such global features. While we introduce and propose these notions in a general setting, in this paper, we only work out the case of planar graphs. We explain that a scalar potential can be identified to encapsulate the concept of circulation, quite similarly as in the case of the curl of planar vector fields. Beyond planar graphs, in the general case, the problem to determine global circulation remains at present a combinatorial problem

Analysis of Social and Flow Networks

ABSTRACT OF THE DISSERTATIONAnalysis of Social and Flow Networks By Zahra Askarzadeh Doctor of Philosophy in Mechanical and Aerospace Engineering University of California, Irvine, 2021 Professor Tryphon T. Georgiou, Chair In this thesis, we study problems in analysis of social and flow networks. Specifically, we study models of social interactions between individuals who discuss and form opinions about a sequence of issues. We also study quantifying macroscopic circulation in a given planar graph. A social network is a medium for the exchange of information, ideas, and influence among its members. In recent years, availability of large amounts of data from online social networks have drawn the attention of many researchers to study opinion formation and the evolutionary behaviors in social networks. In this thesis, we revisit several types of opinion dynamics models and review relevant results from the literature. We then present a set of new results related to both modeling and analysis of social networks. Starting with analyzing DeGroot-Friedkin model, we establish existence and uniqueness of its fixed point using local inverse function theorem and Hadamard’s global inverse function theorem. Motivated by DeGroot-Friedkin model, we then propose a group of nonlinear Markov chain models of social interaction for the purpose of assessing opinion evolu- tion in social networks. We seek and develop conditions that determine when such system display oscillations, manifest chaos, or lead to a stable equilibrium that represents consensus. We also provide extensions of proposed models to count for different subgroups of interacting individuals. Flow networks are typically used to model problems involving the transportation of mass between nodes, through routes that have limited capacity. Examples of such problems that motivate our research are modeling traffic on a network of roads and blood current in heart. Based on these examples, we introduce a new concept of maximal global circulation and explore 3-partitions that expose this type of macroscopic feature of networks. Herein, graph circulation is motivated by probabilistic flows (Markov chains) on graphs. Our goal is to quantify the large-scale imbalance of network flows and delineate key parts that mediate such global features. While we introduce and propose these notions in a general setting, here, we only work out the case of planar graphs. We explain that a scalar potential can be identified to encapsulate the concept of circulation, quite similarly as in the case of the curl of planar vector fields. Beyond planar graphs, in the general case, the problem to determine global circulation remains at present a combinatorial problem