Modeling Information Diffusion in Online Social Networks with Partial Differential Equations

© Springer Nature Switzerland AG 2020

H. Wang et al.Modeling Information Diffusion in Online Social Networks with Partial Differential EquationsSurveys and Tutorials in the Applied Mathematical Sciences7https://doi.org/10.1007/978-3-030-38852-2_1

1. Introduction

Haiyan Wang¹ , Feng Wang¹ and Kuai Xu¹

(1)

School of Mathematical & Natural Sciences, Arizona State University, Phoenix, AZ, USA

Abstract

Online social networks (OSNs) such as Twitter and Facebook, emerging as the model organism of Big Data, have gained tremendous popularity for the platforms they provided for information exchange. Much of prior work on information diffusion over online social networks has been based on empirical and statistical approaches. The majority of dynamical models arising from information diffusion over online social networks are ordinary differential equations (ODEs). Recently, the authors proposed to use partial differential equations (PDEs) to model information diffusion in online social networks and introduced a new transdisciplinary architecture for modeling information diffusion. These studies demonstrate fascinating connections between advanced mathematics and online social networks.

Online social networks (OSNs) such as Twitter and Facebook, emerging as the model organism of Big Data, have gained tremendous popularity for the platforms they provided for information exchange. Much of prior work on information diffusion over online social networks has been based on empirical and statistical approaches. The majority of dynamical models arising from information diffusion over online social networks are ordinary differential equations (ODEs). Recently, the authors proposed to use partial differential equations (PDEs) to model information diffusion in online social networks and introduced a new transdisciplinary architecture for modeling information diffusion. These studies demonstrate fascinating connections between advanced mathematics and online social networks.

A significant body of research about online social networks has focused on analysis of such networks with empirical approaches that use data mining and statistical modeling schemes [9, 18, 19, 25, 29, 34, 35, 41, 43, 47, 50, 55, 60–62, 68, 69, 72, 73, 85, 88, 89, 105, 111, 120, 121, 138–141, 144]. Mathematical models have played a significant role in understanding and predicting information diffusion in online social networks over time. In particular, epidemiological models have influenced the research on information diffusion [8, 51, 80, 89, 90, 124, 145, 146]. However, the deterministic models proposed for online social networks in the literature are largely based on ordinary differential equations (ODEs) that deal with collective social processes over time.

In a paper [128], the authors proposed to use partial differential equations (PDEs) built on intuitive cyber-distance among online users to study both temporal and spatial patterns of information diffusion process in social media. One of the simple, yet fundamental questions that the models address is this: for a piece of given information m initiated from a particular user called source s, what is the density of influenced users at network distance x from the source at any time t. We validate the models with real datasets collected from two popular social media sites, Twitter and Digg. The experiment results show that the models can achieve over 90% accuracy and effectively predict the density of influenced users.

This paper [128] is the first attempt to propose a PDE-based model for characterizing and predicting the temporal and spatial patterns of information diffusion over online social networks, which is also indicated in a survey by Zhang et al. [146]. In Guille et al.’s survey [38] on information diffusion over online social networks, the PDE model in [128] is reported as one of the three non-graph predictive models: epidemiological models, linear influence model (LIM), and PDE approach. The LIM approach developed in [140] focuses on predicting the temporal dynamics of information diffusion through solving non-negative least squares problems. The PDE-based models are dynamic systems that take into account the influence of the underlying network structure as well as information contents for predicting information diffusion over both temporal and spatial dimensions. We shall propose a number of spatio-temporal epidemiological models to describe information diffusion in online social networks in this book.

The book lies at the interface of mathematics, social media analysis, network science, and data science. A key challenge of this interdisciplinary research is to integrate big data from online social networks into the framework of partial differential equations. We use clustering analysis from data mining to aggregate big data for the validation of the PDE models. There are many clustering methods such as k-means clustering. We focus on spectral clustering analysis in this book as our data are graph-structured. The book integrates the research efforts carried out collaboratively by mathematicians specialized in partial differential equations, computer scientists focused on network theory and data mining, and researchers in social media. The extension of partial differential equations into online social networks presents new opportunities and challenges for mathematicians as well as computer scientists and researchers in social media.

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© Springer Nature Switzerland AG 2020

H. Wang et al.Modeling Information Diffusion in Online Social Networks with Partial Differential EquationsSurveys and Tutorials in the Applied Mathematical Sciences7https://doi.org/10.1007/978-3-030-38852-2_2

2. Ordinary Differential Equation Models on Social Networks

Haiyan Wang¹ , Feng Wang¹ and Kuai Xu¹

(1)

School of Mathematical & Natural Sciences, Arizona State University, Phoenix, AZ, USA

Abstract

In this chapter we consider a number of ordinary differential equation models for diffusion of innovation and epidemiological models. We discuss the classical theory on diffusion of innovation, emphasizing online social networks and analyzing several ordinary differential equation models for innovation diffusion. We also present a number of basic compartment epidemiological models and their applications in online social networks, and finally we discuss SIR models and their extensions when the total population is not constant.

2.1 Introduction

Information diffusion over online social networks has become a fast growing research domain encompassing techniques from a plethora of sciences, among them mathematics, computer science, communications and marketing, etc. In addition, the method for studying the spread of infectious diseases among a population has been applied to understanding information spreading patterns. Information diffusion has become a subject with disparate views of what is an information diffusion process. We focus here on a particular case in which information diffuses in online social networks, and we define information diffusion as the process by which a piece of information (knowledge) spreads and reaches individuals through interactions in a network.

The theory of information diffusion can be traced back to the research on the diffusion of innovation over a population, advanced by a pioneering mass communication scholar, E. M. Rogers. Information diffusion in online social networks becomes a premier example to revisit innovation diffusion. In this chapter, we will focus on ODE models for diffusion of innovation and epidemics. Both diffusion of innovations and epidemic models provide a global view of how an innovation (e.g., news, a product) or a type of disease spreads through a population even when interactions among individuals are unavailable. We will extend innovation diffusion and epidemic models to spatial models where both local and global information between clusters in a network are available.

2.2 Diffusion of Innovations

Diffusion of innovations is a theory that seeks to explain how new ideas and technologies spread through cultures. E. M. Rogers was a professor of communication studies who popularized the theory with his book Diffusion of Innovations [104]. The origins of the diffusion of innovations theory span various disciplines including anthropology, early sociology, rural sociology, education, industrial sociology, and medical sociology. Rogers [104] theorized that information diffusion is the social process through which an innovation is communicated through certain channels over time among the participants in a social system. He identified four key elements that influence diffusion of a new idea: the innovation itself, communication channels, time, and a social system. The theory of diffusion of innovation has been applied to numerous contexts, including marketing, communications, health promotion, organizational studies, and complexity studies.

The rise of social media has provided a new platform to study diffusion of innovation. Rogers [104] defined an innovation as an idea, practice, or object if it is perceived as novel by an individual or other unit of adoption. A piece of news being reposted many times in online social networks is a typical example of innovations diffusing across online social networks. The theory of diffusion of innovations can be applied to online social networks to answer why and how information spreads fast and reaches a broad audience. It also can be used to reveal some key characteristics of information diffusion such as its motivations. With appropriate models, it can be used to further predict the rate at which ideas spread. In this section, we review some key characteristics of innovation diffusion that are important for the PDE modeling of information diffusion in online social networks. Finally, we present mathematical models that can be used to describe the process of innovation diffusion.

2.2.1 Characteristics of Innovation Diffusion

Rogers [104] explored many characteristics of innovations, individual adopters, and organizations. The nature of networks and the roles opinion leaders play determine the likelihood of innovation adoption. We here identify several characteristics that are related to mathematical modeling of information diffusion in online social networks. Our goal is to use mathematical models to identify key factors that influence the spread of information, and then explain how and why users adopt certain information.

2.2.1.1 Logistic Adoption Curve

In [104] Rogers argued that the diffusion process, depending on a combination of the four key elements, has a point at which an innovation reaches critical mass. He then studied the various stages and introduced an adopter category as a classification of individuals within a social system on the basis of innovativeness. Based on the order in which they adopt the innovations, it has been found that the five different types of adopters are: (1) Innovators (top 2.5%), (2) Early Adopters (13.5%), (3) Early Majority (34%), (4) Late Majority (34%), and (5) Laggards (16%). Numerous studies have shown that different types of adopters behave in significantly different ways in various