Globs in the Primordial Soup The Emergence of Connected Crowds in Mobile Wireless Networks Simon Heimlicher Computer Engineering and Networks Laboratory ETH Zurich, Switzerland heimlicher@tik.ee.ethz.ch Kavé Salamatian LISTIC-PolyTech Université de Savoie, France kave.salamatian@univ-savoie.fr ABSTRACT In many practical scenarios, nodes gathering at points of interest yield sizable connected components (clusters), which sometimes comprise the majority of nodes. While recent analysis of mobile networks focused on the process governing node encounters (“con- tacts”), this model is not particularly suitable for gathering behav- ior. In this paper, we propose a model of stochastic coalescence (merge) and fragmentation (split) of clusters. We implement this process as a Markov chain and derive analytically the exact station- ary distribution of cluster size. Further, we prove that, as the num- ber of nodes grows, the clustering behavior converges to a mean field, which is obtained as a closed-form expression. This expres- sion translates the empirical merge and split rate of a scenario, a microscopic property, to an important macroscopic property—the cluster size distribution—with surprising accuracy. We validate all results with synthetic as well as real-world mobility traces from conference visitors and taxicabs with several thousand nodes. Categories and Subject Descriptors C.2.1 [Network Architecture and Design]: Wireless Communi- cations; C.4 [Performance of Systems]: Modeling Techniques General Terms Performance, Theory 1. INTRODUCTION Up to now, the analysis of mobile networks was predominantly based on modeling individual nodes or node encounters. In partic- ular, research on delay-tolerant networks (DTN) made progress in characterizing the stochastic process underlying single-hop paths (“contacts”) and leveraging emerging “space-time paths” for com- munication in a disconnected network. Those studies laid an im- portant foundation toward understanding the temporal characteris- tics of contacts and designing contact-based forwarding schemes for delay-tolerant applications tailor-made for those networks. Yet, we argue that in addition to space-time paths, multi-hop paths may often exist in mobile wireless networks. Such multi- Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. MobiHoc’10, September 20–24, 2010, Chicago, Illinois, USA. Copyright 2010 ACM 978-1-4503-0183-1/10/09 ...$10.00. hop paths may enable some of the most popular applications, such as web browsing and email, in mobile wireless networks, even though the protocols used by those applications (in particular TCP [19]) have not been designed to be delay-tolerant. We argue that understanding the circumstances that lead to multi-hop paths is paramount to design routing and forwarding algorithms that lever- age those communication opportunities. In our discussion, we dis- tinguish between partial paths, which bridge only part of the way from the source to the destination, and full paths connecting source and destination; and we use the term partially-connected network to refer to a network that is disconnected but provides partial paths. Following those definitions, one may ask if disconnected net- works with partial paths exist. Continuum percolation theory [12] is often used to study connectivity; its main result is that if the node density is below the percolation threshold, almost all nodes are iso- lated; above the threshold, the network “percolates” and forms one giant connected component. Yet, [32] shows that in clustered net- works, such a phase transition does not occur. In delay-tolerant net- working, characterization of the contact process [9, 15] is mostly concerned with forwarding based on individual contacts, though there are proposals to leverage multi-hop paths [28] as well. In light of this, to analyze rigorously the existence and charac- teristics of partial paths, a new methodology seems in order. Node independence is at the core of most analytical models for mobile networks, but multi-hop paths are at odds with this assumption. Fortunately, if the behavior of individual nodes is abstracted from, the dependency between nodes on a multi-hop path can be hidden. Under this premise, one could consider multi-hop paths as the ba- sic entities, but those may still be connected to each other. There- fore we propose to lump together all connected multi-hop paths to connected components (clusters) and describe a mobile scenario through merge and split events between such clusters. In this paper, we introduce a model for arbitrary mobile wire- less networks based on the concept of stochastic coagulation and fragmentation [2]. In analogy to globs of particles coalescing and fragmenting in a system of particles in a solvent, we describe a sys- tem of N mobile nodes as a set of clusters that merge and split. The state of the system is represented by a vector whose elements i =1, 2,... are the number of clusters of size i. As N is constant, two primitive events may occur in this system. First, two clusters of sizes k and l may merge into a new cluster of size k + l, as de- scribed by the merge process. Second, a cluster of size k + l may split into two clusters of sizes k and l, according to the split pro- cess. It follows that the merge and the split process determine the stationary distribution of cluster size and thus, whether the network is connected, disconnected, or partially connected by partial paths. We implement this model as a Markov process over the finite state space of all partitions of N . Under certain conditions, this pro-