Characterizing pairwise contact patterns in human contact networks

Abstract

We use a counting process representation of the pairwise contact process to analyze pairwise contact patterns. Studying two real-world traces, we find that the pairwise contact patterns have three characteristics. First, human contact patterns are influenced by daily and weekly cycles of activity. Second short time intervals with intensive contact event (bursts) are separated by long periods with few contact events. Third, the pairwise contact process exhibits long range dependence. We introduce a Markov modulated Poisson process (MMPP) as a flexible model for pairwise contact process exhibiting both regular structure and irregular bursts of activity. Using standard statistical techniques, we demonstrate that the proposed model is consistent with the empirical data. Our work has significant implication for mobility modeling and performance analysis in human contact networks.

Introduction

As radio-equipped mobile devices of tomorrow become both more pervasive and more powerful, it might be possible to form a dynamic network based only on pairwise contact of users in daily life [1]. In this network, mobile devices are sparsely distributed and form a network that is often partitioned due to human mobility, geographical distribution of human beings and signal decay [2]. As opposed to conventional wireless network that relies on infrastructure, mobile devices can use contact opportunities to allow users to communicate with each other. Because the performance of networking applications highly depends on human mobility characteristics [3], [4], [5], especially the characteristic of contact between users, we claim that it is critical to study first the human contact patterns.

Many theoretical and empirical studies have examined the diverse aspects of human contact patterns including contact, inter-contact times, and pause durations. Especially, most of these studies have typically focused on the distribution of inter-contact times by computing the relative frequency of occurrence of inter-contact times as a function of inter-contact times. This measure highlights the behavior of the times between adjacent events, but reveals none of the information contained in the relationships among these times, such as correlation between adjacent time intervals. This paper addresses this issue.

In this paper, we use a counting process representation of the pairwise contact process to analyze pairwise contact patterns. By using a counting process representation, we can reliably reveal the second-order statistics (such as burstiness and correlations) of the pairwise contact patterns. The goal of this paper is to propose a model characterizing pairwise contact patterns, not a mobility model.

We study two experimental datasets MIT [6] and UCSD [7]. The former recorded contact directly between mobile devices (e.g., imotes), while the latter included client based logs of the visibility of access points (APs). We define the contact between two mobile devices as a contact event. In our context, the term “inter-contact-event time” refers to the time between the start of contact between two devices and the start of the next contact between the same two devices. Note that this definition is different from inter-contact time used in [8], [9], which accounts for the time duration between a contact event and the subsequent one.

Human contact processes are often related to human activity and behavior. Human activity follows periodicities—daily, weekly, monthly and yearly—often in combination with the burstiness [10]. Indeed, from the analysis of our dataset, we observe that the human contact process have two bursty characteristics: human contact patterns are influenced by daily and weekly cycles of activity; short time intervals with intensive contact events (bursts) are separated by long periods of minimal to no contact events. Extensive works have been published on process related to human activity, which often generates bursty traffic. For example, computational jobs in grid systems [11], blog networks, and post networks [10] all exhibit similar bursty characteristics.

Furthermore, we find that pairwise inter-contact-event times follow a heavy-tailed distribution. Two main classes of mechanisms are responsible for its emergence [12]. The first class belongs to human behavior governed by human decisions. Barabási [13] proposed a model based on priority queues, which captures the bursty nature of human behavior. The second class of behavior is primarily driven by external factors such as circadian and weekly cycles, which has been observed in e-mail communication [12] and the stock exchange [14]. In this paper, we demonstrate that the bursty nature of human behavior is a consequence of circadian and weekly cycles of activity, which introduces a set of distinct characteristic time scales, thereby giving rise to a heavy-tailed distribution of the pairwise inter-contact-event times.

By means of the Allan factor statistics [15], we show that at certain time scales pairwise contact time follows non-Poisson statistics and is not temporally independent, but correlated over the long-term. In this respect the traditional models, which mainly focus on the distribution of inter-contact times, need to be changed in favor of new models which take into account the occurrence of burstiness and long-range correlations in human contact process.

Here, we model pairwise contact process as a discrete MMPP, which incorporates the hypothesized periodic and bursty features of human contact process. We subsequently utilize the method provided by Malmgren et al. [16] to estimate the model parameters from empirical data. We then use the Kolmogorov Smirnov statistical hypothesis test to demonstrate that the predictions of our model are consistent with the empirical cumulative distribution of inter-contact-event times. Finally, we validate that our model can capture the second-order statistical features.

The remaining of the paper is organized as follows: in Section 2 we discuss related work. Section 3 introduces the definition and methodology used in the analysis. Section 4 describes the real-world datasets and presents a comprehensive analysis on it. In Section 5 we describe in detail the MMPP. Then we estimate the parameters for this model from the empirical data and validate it. Finally, we present our concluding remarks and identify directions for future work in Section 5.