Diversity of graphs with highly variable connectivity

A popular approach for describing the structure of many complex networks focuses on graph theoretic properties that characterize their large-scale connectivity. While it is generally recognized that such descriptions based on aggregate statistics do not uniquely characterize a particular graph and also that many such statistical features are interdependent, the relationship between competing descriptions is not entirely understood. This paper lends perspective on this problem by showing how the degree sequence and other constraints (e.g., connectedness, no self-loops or parallel edges) on a particular graph play a primary role in dictating many features, including its correlation structure. Building on recent work, we show how a simple structural metric characterizes key differences between graphs having the same degree sequence. More broadly, we show how the (often implicit) choice of a background set against which to measure graph features has serious implications for the interpretation and comparability of graph theoretic descriptions

Optimal Control for LQG Systems on Graphs---Part I: Structural Results

In this two-part paper, we identify a broad class of decentralized output-feedback LQG systems for which the optimal control strategies have a simple intuitive estimation structure and can be computed efficiently. Roughly, we consider the class of systems for which the coupling of dynamics among subsystems and the inter-controller communication is characterized by the same directed graph. Furthermore, this graph is assumed to be a multitree, that is, its transitive reduction can have at most one directed path connecting each pair of nodes. In this first part, we derive sufficient statistics that may be used to aggregate each controller's growing available information. Each controller must estimate the states of the subsystems that it affects (its descendants) as well as the subsystems that it observes (its ancestors). The optimal control action for a controller is a linear function of the estimate it computes as well as the estimates computed by all of its ancestors. Moreover, these state estimates may be updated recursively, much like a Kalman filter

SPoT: Representing the Social, Spatial, and Temporal Dimensions of Human Mobility with a Unifying Framework

Modeling human mobility is crucial in the analysis and simulation of opportunistic networks, where contacts are exploited as opportunities for peer-topeer message forwarding. The current approach with human mobility modeling has been based on continuously modifying models, trying to embed in them the mobility properties (e.g., visiting patterns to locations or specific distributions of inter-contact times) as they came up from trace analysis. As a consequence, with these models it is difficult, if not impossible, to modify the features of mobility or to control the exact shape of mobility metrics (e.g., modifying the distribution of inter-contact times). For these reasons, in this paper we propose a mobility framework rather than a mobility model, with the explicit goal of providing a exible and controllable tool for modeling mathematically and generating simulatively different possible features of human mobility. Our framework, named SPoT, is able to incorporate the three dimensions - spatial, social, and temporal - of human mobility. The way SPoT does it is by mapping the different social communities of the network into different locations, whose members visit with a configurable temporal pattern. In order to characterize the temporal patterns of user visits to locations and the relative positioning of locations based on their shared users, we analyze the traces of real user movements extracted from three location-based online social networks (Gowalla, Foursquare, and Altergeo). We observe that a Bernoulli process effectively approximates user visits to locations in the majority of cases and that locations that share many common users visiting them frequently tend to be located close to each other. In addition, we use these traces to test the exibility of the framework, and we show that SPoT is able to accurately reproduce the mobility behavior observed in traces. Finally, relying on the Bernoulli assumption for arrival processes, we provide a throughout mathematical analysis of the controllability of the framework, deriving the conditions under which heavy-tailed and exponentially-tailed aggregate inter-contact times (often observed in real traces) emerge

ChoiceRank: Identifying Preferences from Node Traffic in Networks

Understanding how users navigate in a network is of high interest in many applications. We consider a setting where only aggregate node-level traffic is observed and tackle the task of learning edge transition probabilities. We cast it as a preference learning problem, and we study a model where choices follow Luce's axiom. In this case, the $O(n)$ marginal counts of node visits are a sufficient statistic for the $O(n^2)$ transition probabilities. We show how to make the inference problem well-posed regardless of the network's structure, and we present ChoiceRank, an iterative algorithm that scales to networks that contains billions of nodes and edges. We apply the model to two clickstream datasets and show that it successfully recovers the transition probabilities using only the network structure and marginal (node-level) traffic data. Finally, we also consider an application to mobility networks and apply the model to one year of rides on New York City's bicycle-sharing system.Comment: Accepted at ICML 201

Pregelix: Big(ger) Graph Analytics on A Dataflow Engine

There is a growing need for distributed graph processing systems that are capable of gracefully scaling to very large graph datasets. Unfortunately, this challenge has not been easily met due to the intense memory pressure imposed by process-centric, message passing designs that many graph processing systems follow. Pregelix is a new open source distributed graph processing system that is based on an iterative dataflow design that is better tuned to handle both in-memory and out-of-core workloads. As such, Pregelix offers improved performance characteristics and scaling properties over current open source systems (e.g., we have seen up to 15x speedup compared to Apache Giraph and up to 35x speedup compared to distributed GraphLab), and makes more effective use of available machine resources to support Big(ger) Graph Analytics

A simple yet effective baseline for non-attributed graph classification

Graphs are complex objects that do not lend themselves easily to typical learning tasks. Recently, a range of approaches based on graph kernels or graph neural networks have been developed for graph classification and for representation learning on graphs in general. As the developed methodologies become more sophisticated, it is important to understand which components of the increasingly complex methods are necessary or most effective. As a first step, we develop a simple yet meaningful graph representation, and explore its effectiveness in graph classification. We test our baseline representation for the graph classification task on a range of graph datasets. Interestingly, this simple representation achieves similar performance as the state-of-the-art graph kernels and graph neural networks for non-attributed graph classification. Its performance on classifying attributed graphs is slightly weaker as it does not incorporate attributes. However, given its simplicity and efficiency, we believe that it still serves as an effective baseline for attributed graph classification. Our graph representation is efficient (linear-time) to compute. We also provide a simple connection with the graph neural networks. Note that these observations are only for the task of graph classification while existing methods are often designed for a broader scope including node embedding and link prediction. The results are also likely biased due to the limited amount of benchmark datasets available. Nevertheless, the good performance of our simple baseline calls for the development of new, more comprehensive benchmark datasets so as to better evaluate and analyze different graph learning methods. Furthermore, given the computational efficiency of our graph summary, we believe that it is a good candidate as a baseline method for future graph classification (or even other graph learning) studies.Comment: 13 pages. Shorter version appears at 2019 ICLR Workshop: Representation Learning on Graphs and Manifolds. arXiv admin note: text overlap with arXiv:1810.00826 by other author