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Graph models are widely used to analyse diffusion processes embedded in social contacts and to develop applications. A range of graph models are available to replicate the underlying social structures and dynamics realistically. However, most of the current graph models can only consider concurrent interactions among individuals in the co-located interaction networks. However, they do not account for indirect interactions that can transmit spreading items to individuals who visit the same locations at different times but within a certain time limit. The diffusion phenomena occurring through direct and indirect interactions is called same place different time (SPDT) diffusion. This paper introduces a model to synthesize co-located interaction graphs capturing both direct interactions, where individuals meet at a location, and indirect interactions, where individuals visit the same location at different times within a set timeframe. We analyze 60 million location updates made by 2 million users from a social networking application to characterize the graph properties, including the space-time correlations and its time evolving characteristics, such as bursty or ongoing behaviors. The generated synthetic graph reproduces diffusion dynamics of a realistic contact graph, and reduces the prediction error by up to 82% when compare to other contact graph models demonstrating its potential for forecasting epidemic spread.
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Random walk-based sampling methods are gaining popularity and importance in characterizing large networks. While powerful, they suffer from the slow mixing problem when the graph is loosely connected, which results in poor estimation accuracy. Random walk with jumps (RWwJ) can address the slow mixing problem but it is inapplicable if the graph does not support uniform vertex sampling (UNI). In this work, we develop methods that can efficiently sample a graph without the necessity of UNI but still enjoy the similar benefits as RWwJ. We observe that many graphs under study, called target graphs, do not exist in isolation. In many situations, a target graph is related to an auxiliary graph and a bipartite graph, and they together form a better connected {em two-layered network structure}. This new viewpoint brings extra benefits to graph sampling: if directly sampling a target graph is difficult, we can sample it indirectly with the assistance of the other two graphs. We propose a series of new graph sampling techniques by exploiting such a two-layered network structure to estimate target graph characteristics. Experiments conducted on both synthetic and real-world networks demonstrate the effectiveness and usefulness of these new techniques.
Based on an expert systems approach, the issue of community detection can be conceptualized as a clustering model for networks. Building upon this further, community structure can be measured through a clustering coefficient, which is generated from the number of existing triangles around the nodes over the number of triangles that can be hypothetically constructed. This paper provides a new definition of the clustering coefficient for weighted networks under a generalized definition of triangles. Specifically, a novel concept of triangles is introduced, based on the assumption that, should the aggregate weight of two arcs be strong enough, a link between the uncommon nodes can be induced. Beyond the intuitive meaning of such generalized triangles in the social context, we also explore the usefulness of them for gaining insights into the topological structure of the underlying network. Empirical experiments on the standard networks of 500 commercial US airports and on the nervous system of the Caenorhabditis elegans support the theoretical framework and allow a comparison between our proposal and the standard definition of clustering coefficient.
Graph models have long been used in lieu of real data which can be expensive and hard to come by. A common class of models constructs a matrix of probabilities, and samples an adjacency matrix by flipping a weighted coin for each entry. Examples include the ErdH{o}s-R{e}nyi model, Chung-Lu model, and the Kronecker model. Here we present the HyperKron Graph model: an extension of the Kronecker Model, but with a distribution over hyperedges. We prove that we can efficiently generate graphs from this model in order proportional to the number of edges times a small log-factor, and find that in practice the runtime is linear with respect to the number of edges. We illustrate a number of useful features of the HyperKron model including non-trivial clustering and highly skewed degree distributions. Finally, we fit the HyperKron model to real-world networks, and demonstrate the models flexibility with a complex application of the HyperKron model to networks with coherent feed-forward loops.
Random graph models are important constructs for data analytic applications as well as pure mathematical developments, as they provide capabilities for network synthesis and principled analysis. Several models have been developed with the aim of faithfully preserving important graph metrics and substructures. With the goal of capturing degree distribution, clustering coefficient, and communities in a single random graph model, we propose a new model to address shortcomings in a progression of network modeling capabilities. The Block Two-Level Erd{H{o}}s-R{e}nyi (BTER) model of Seshadhri et al., designed to allow prescription of expected degree and clustering coefficient distributions, neglects community modeling, while the Generalized BTER (GBTER) model of Bridges et al., designed to add community modeling capabilities to BTER, struggles to faithfully represent all three characteristics simultaneously. In this work, we fit BTER and two GBTER configurations to several real-world networks and compare the results with that of our new model, the Extended GBTER (EGBTER) model. Our results support that EBGTER adds a community-modeling flexibility to BTER, while retaining a satisfactory level of accuracy in terms of degree and clustering coefficient. Our insights and empirical testing of previous models as well as the new model are novel contributions to the literature.
Over the last two decades, alongside the increased availability of large network datasets, we have witnessed the rapid rise of network science. For many systems, however, the data we have access to is not a direct description of the underlying network. More and more, we see the drive to study networks that have been inferred or reconstructed from non-network data---in particular, using time series data from the nodes in a system to infer likely connections between them. Selecting the most appropriate technique for this task is a challenging problem in network science. Different reconstruction techniques usually have different assumptions, and their performance varies from system to system in the real world. One way around this problem could be to use several different reconstruction techniques and compare the resulting networks. However, network comparison is also not an easy problem, as it is not obvious how best to quantify the differences between two networks, in part because of the diversity of tools for doing so. The netrd Python package seeks to address these two parallel problems in network science by providing, to our knowledge, the most extensive collection of both network reconstruction techniques and network comparison techniques (often referred to as graph distances) in a single library (https://github.com/netsiphd/netrd). In this article, we detail the two main functionalities of the netrd package. Along the way, we describe some of its other useful features. This package builds on commonly used Python packages and is already a widely used resource for network scientists and other multidisciplinary researchers. With ongoing open-source development, we see this as a tool that will continue to be used by all sorts of researchers to come.
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