Complexity

Volume 2018, Article ID 4732491, 11 pages

https://doi.org/10.1155/2018/4732491

## Complexity of a Microblogging Social Network in the Framework of Modern Nonlinear Science

School of Business Informatics, National Research University Higher School of Economics, Moscow 101000, Russia

Correspondence should be addressed to Andrey Dmitriev; ur.esh@veirtimd.a

Received 9 August 2018; Accepted 11 November 2018; Published 2 December 2018

Guest Editor: Piotr Brodka

Copyright © 2018 Andrey Dmitriev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Recent developments in nonlinear science have caused the formation of a new paradigm called the paradigm of complexity. The self-organized criticality theory constitutes the foundation of this paradigm. To estimate the complexity of a microblogging social network, we used one of the conceptual schemes of the paradigm, namely, the system of key signs of complexity of the external manifestations of the system irrespective of its internal structure. Our research revealed all the key signs of complexity of the time series of a number of microposts. We offer a new model of a microblogging social network as a nonlinear random dynamical system with additive noise in three-dimensional phase space. Implementations of this model in the adiabatic approximation possess all the key signs of complexity, making the model a reasonable evolutionary model for a microblogging social network. The use of adiabatic approximation allows us to model a microblogging social network as a nonlinear random dynamical system with multiplicative noise with the power-law in one-dimensional phase space.

#### 1. Introduction

Social networks have been studied longer than any other type of networks. It is remarkable that one of the signs of network complexity—a power law of nodes’ degree distribution [1]—was first empirically formulated by D. Price in 1965 for social networks. In 1999, A. L. Barabasi, a physicist from the University of Notre Dame (USA), and his graduate student R. Albert determined [2, 3] that, for many networks, instead of the expected Poisson probability distribution of nodes’ degree (i.e., the number of connections a node has to other nodes), the distribution they obtained approximately followed a power law as all critical states do. In many real networks, a small number of nodes have a large number of connections, whereas a large number of nodes have just a few connections. Such networks are called scale-free networks. This name was not invented specifically for this type of networks. It came from the theory of critical phenomena, where fluctuations in critical states also follow a power law. The theory of scale-free networks is considered to be one of the scenarios complex systems follow when they come into a critical state. As of late, such networks are more often called complex networks.

Some other relevant works in this area are those of refs. [4–8].

An extensive body of research on the modeling of the structure and functioning of social networks is available today. This research has two directions. The first direction relates to the analysis of the social networks data (see one of the latest reviews [9]), while the second concerns the development of models of the structure, dynamics, and evolution of social networks. The distinction between these two directions is somewhat arbitrary, since in most cases these directions overlap (see, e.g., [10, 11]).

Starting from the second half of the century, the ideas and methods of physics have tended to infiltrate natural sciences and traditional humanities. Methods of physical modeling are often used in such areas of science as demographics, sociology, and linguistics. As a result, sociophysical models of social networks, such as the Ising model [12–15], Bose-Einstein condensate model [3, 16], Quantum walk model [17], Ground state and community detection [18, 19], among others, were developed.

Despite having a variety of sociophysical models, the results and theories of nonlinear science, with some exclusions (see, e.g., [20, 21]), are not used to model the evolution of social networks. First of all, we are talking about the complexity and self-organized criticality theory describing the mechanism of complexity [22–24]. Mechanisms of self-organized criticality in social knowledge creation process are presented in the paper [25]. It is noteworthy that the key sign of complexity of a system regardless of its internal structure, i.e., one based solely on its external characteristics, was formulated in the framework of this theory. According to this theory, a system is considered to be complex if it is able to generate unexpected and/or extraordinary events (for instance, bursts of values in time series). This motivated our research. The purpose of the research is a nonlinear dynamical interpretation of the complexity of a microblogging network and the development of an appropriate network model that could explain its complexity using the third paradigm of nonlinear science called the complexity paradigm. Another motivation for the research was the results presented in [26–31] where the time series of a number of microposts are characterized by the majority of key signs of the system complexity (a detailed description of the key signs of the system complexity is presented in Section 2).

This paper is organized as follows. Section 2 deals with the key signs of the system complexity according to the complexity paradigm. Section 3 presents the results of the analysis of an empiric time series of a number of microposts, including the results of the calculation of the key signs of the complexity. Section 4 presents a model of a microblogging social network as a nonlinear deterministic dynamical system including its capabilities and restrictions. Section 5 presents a generalized model of a microblogging social network, modified by the consideration of stochastic sources and a decrease in the order parameter, as well as the results of an analysis in the adiabatic approximation. Section 6 contains the main results of the research and a discussion.

#### 2. Nonlinear Dynamical Interpretation of Complexity

The development of any branch of science leads to the formulation of paradigms, namely, initial conceptual schemes, models of problem statements, and solutions of the problems. At this time, three paradigms have been developed in nonlinear science. The first paradigm is that of self-organization. The second is the paradigm of deterministic chaos. The most recent development of nonlinear science is closely linked to the third paradigm, which could be defined as a paradigm of complexity that has the theory of self-organized criticality as its foundation. The paradigm of complexity lies at the junction of the first two paradigms. If the first two paradigms deal with order and chaos, respectively, the third is usually described as “life on the edge of chaos” [32].

Since it is impossible to rigorously define complexity, our research is limited to consideration of the key signs of system complexity defined in the publications by Per Bak and co-authors [22–24], and their application to the interpretation of the complexity of microblogging social networks. As stated in the introduction, first of all, we consider the complexity of external system manifestations regardless of internal structure. For the purposes of this research, we define “external system manifestations” as signals (the time series of a number of microposts) of a microblogging social network generated as a result of nontrivial interactions within a very large pool of users.

One of the key signs of complexity is its inclination to the occurrence of catastrophic events—either unexpected (i.e., nonpredictable) or extraordinary (i.e., prominent among similar events), or both. Importantly, in either case we can conclude that the system that has generated such an event is complex. From simple systems, we could expect predictability and similarities in their behavior. As for the signals of a microblogging system, such events qualitatively correspond to considerable bursts seen on a plot of value increments of the time series of a number of microposts. One of the quantitative criteria of the existence of catastrophic events is the existence of power low of the probability density function (PDF) for the values of the time series. It is worth mentioning that, in the majority of cases, the occurrence of such events on the network signal level corresponds to the qualitative restructuring of the system, i.e., a transition from a polycentric state to a monocentric state, and vice versa (such transitions are thoroughly described in [33]).

Another key sign of complexity is scale invariance, meaning that events or objects lack their own characteristic dimensions, durations, energies, etc. At the level of external manifestations of a microblogging network, scale invariance means that the time series of a number of microposts are fractal or multifractal time series (such time series are described in detail in [34]).

In a general case, a power low for PDF is a statistical expression of scale invariance of the time series:where usually . PDF (1) belongs to the class of fat-tailed PDFs. For statistical description of catastrophic events, PDF (1) is a rule with almost no exceptions. PDF (1) differs from compact distributions (for example, Gaussian distribution) because the events corresponding to the tail of the distribution are not rare enough to be neglected. PDF (1) reflects a strong interdependence of the events. For example, such distribution may be caused by an avalanche-like increase of the number of microposts in the network as a result of a “chain reaction” caused by reposting.

Another manifestation of the scale invariance of the time series is the existence of the power spectral density (PSD) specific for flicker noise:where . The existence of PSD (2) means that a considerable part of the energy is linked to very slow processes. For a microblogging network, the existence of PSD (2) means that it is impossible to predict the behavior of the time series of a number of microposts without considering global information exchange processes.

The aforementioned features of PDF and PSD are not the only criteria of scale invariance. Besides PDF and PSD, we used a fractal dimension and a Hurst exponent along with other quantitative measures and criteria. It is important to stress that the scale invariance and an inclination to catastrophes are typical only for systems that are far from equilibrium. Therefore, a nonequilibrium state of the system and, therefore, a nonlinearity are the necessary conditions for the complexity of the system.

Lastly, the third key sign that characterizes complex systems is their integrity. The integral properties of a system usually are statistically described by power-law space and time correlations. These correlations are known as distant space and time correlations. The existence of distant time correlations or long memory in time series is characterized by the autocorrelation function (ACF) in the following form [34]:where . The existence of the relationship (3) implies the absence of characteristic times at which the information about the previous events could be lost. A catastrophic behavior and integrity are connected in the following way: for the catastrophic behavior, part of the system should be able to function in coordination. For a microblogging network, an avalanche-like increase of the number of microposts is possible when a user and his followers, followers of these followers, etc., are working in coordination. Integrity is possible in complex systems only due to the processes of self-organization. Here we talk about coarse scale properties of the system, since minor changes in system parameters do not affect its integrity.

Therefore, a microblogging network is a complex system when all the key signs of complexity listed above are satisfied. This statement forms the foundation of our research and is key to the construction of a model of microblogging network evolution.

#### 3. Analysis of Empirical Data from Twitter

Empirical data used for our research is a sample of more than 3 million microposts (tweets, retweets, and links) about the first US presidential debates of 2016. The sample includes microposts posted by more than 1 million users from 13:45 on September 26, 2016, to 11:00 on September 27, 2016, with 1-second increments.

Figure 1 shows the total number of microposts vs. time (Twitter time series, ). It is easy to see that has extraordinary events and unexpected events (bursts).