Brian Karrer

Stochastic blockmodels and community structure in networks

Stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks. Most blockmodels, however, ignore variation in vertex degree, making them unsuitable for applications to real-world networks, which typically display broad degree distributions that can significantly affect the results. Here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks. We also propose a heuristic algorithm for community detection using this objective function or its non-degree-corrected counterpart and show that the degree-corrected version dramatically outperforms the uncorrected one in both real-world and synthetic networks.

Efficient and principled method for detecting communities in networks.

A fundamental problem in the analysis of network data is the detection of network communities, groups of densely interconnected nodes, which may be overlapping or disjoint. Here we describe a method for finding overlapping communities based on a principled statistical approach using generative network models. We show how the method can be implemented using a fast, closed-form expectation-maximization algorithm that allows us to analyze networks of millions of nodes in reasonable running times. We test the method both on real-world networks and on synthetic benchmarks and find that it gives results competitive with previous methods. We also show that the same approach can be used to extract nonoverlapping community divisions via a relaxation method, and demonstrate that the algorithm is competitively fast and accurate for the nonoverlapping problem.

Robustness of community structure in networks

The discovery of community structure is a common challenge in the analysis of network data. Many methods have been proposed for finding community structure, but few have been proposed for determining whether the structure found is statistically significant or whether, conversely, it could have arisen purely as a result of chance. In this paper we show that the significance of community structure can be effectively quantified by measuring its robustness to small perturbations in network structure. We propose a suitable method for perturbing networks and a measure of the resulting change in community structure and use them to assess the significance of community structure in a variety of networks, both real and computer generated.

Social influence and the diffusion of user-created content

Social influence determines to a large extent what we adopt and when we adopt it. This is just as true in the digital domain as it is in real life, and has become of increasing importance due to the deluge of user-created content on the Internet. In this paper, we present an empirical study of user-to-user content transfer occurring in the context of a time-evolving social network in Second Life, a massively multiplayer virtual world. We identify and model social influence based on the change in adoption rate following the actions of ones friends and find that the social network plays a significant role in the adoption of content. Adoption rates quicken as the number of friends adopting increases and this effect varies with the connectivity of a particular user. We further find that sharing among friends occurs more rapidly than sharing among strangers, but that content that diffuses primarily through social influence tends to have a more limited audience. Finally, we examine the role of individuals, finding that some play a more active role in distributing content than others, but that these influencers are distinct from the early adopters.

Competing epidemics on complex networks

Human diseases spread over networks of contacts between individuals and a substantial body of recent research has focused on the dynamics of the spreading process. Here we examine a model of two competing diseases spreading over the same network at the same time, where infection with either disease gives an individual subsequent immunity to both. Using a combination of analytic and numerical methods, we derive the phase diagram of the system and estimates of the expected final numbers of individuals infected with each disease. The system shows an unusual dynamical transition between dominance of one disease and dominance of the other as a function of their relative rates of growth. Close to this transition the final outcomes show strong dependence on stochastic fluctuations in the early stages of growth, dependence that decreases with increasing network size, but does so sufficiently slowly as still to be easily visible in systems with millions or billions of individuals. In most regions of the phase diagram we find that one disease eventually dominates while the other reaches only a vanishing fraction of the network, but the system also displays a significant coexistence regime in which both diseases reach epidemic proportions and infect an extensive fraction of the network.

Message passing approach for general epidemic models

In most models of the spread of disease over contact networks it is assumed that the probabilities per unit time of disease transmission and recovery from disease are constant, implying exponential distributions of the time intervals for transmission and recovery. Time intervals for real diseases, however, have distributions that in most cases are far from exponential, which leads to disagreements, both qualitative and quantitative, with the models. In this paper, we study a generalized version of the susceptible-infected-recovered model of epidemic disease that allows for arbitrary distributions of transmission and recovery times. Standard differential equation approaches cannot be used for this generalized model, but we show that the problem can be reformulated as a time-dependent message passing calculation on the appropriate contact network. The calculation is exact on trees (i.e., loopless networks) or locally treelike networks (such as random graphs) in the large system size limit. On non-tree-like networks we show that the calculation gives a rigorous bound on the size of disease outbreaks. We demonstrate the method with applications to two specific models and the results compare favorably with numerical simulations.

Coauthorship and citation patterns in the Physical Review.

A large number of published studies have examined the properties of either networks of citation among scientific papers or networks of coauthorship among scientists. Here, using an extensive data set covering more than a century of physics papers published in the Physical Review, we study a hybrid coauthorship/citation network that combines the two, which we analyze to gain insight into the correlations and interactions between authorship and citation. Among other things, we investigate the extent to which individuals tend to cite themselves or their collaborators more than others, the extent to which they cite themselves or their collaborators more quickly after publication, and the extent to which they tend to return the favor of a citation from another scientist.

Percolation on sparse networks.

We study percolation on networks, which is used as a model of the resilience of networked systems such as the Internet to attack or failure and as a simple model of the spread of disease over human contact networks. We reformulate percolation as a message passing process and demonstrate how the resulting equations can be used to calculate, among other things, the size of the percolating cluster and the average cluster size. The calculations are exact for sparse networks when the number of short loops in the network is small, but even on networks with many short loops we find them to be highly accurate when compared with direct numerical simulations. By considering the fixed points of the message passing process, we also show that the percolation threshold on a network with few loops is given by the inverse of the leading eigenvalue of the so-called nonbacktracking matrix.

Random graphs containing arbitrary distributions of subgraphs

Traditional random graph models of networks generate networks that are locally treelike, meaning that all local neighborhoods take the form of trees. In this respect such models are highly unrealistic, most real networks having strongly nontreelike neighborhoods that contain short loops, cliques, or other biconnected subgraphs. In this paper we propose and analyze a class of random graph models that incorporates general subgraphs, allowing for nontreelike neighborhoods while still remaining solvable for many fundamental network properties. Among other things we give solutions for the size of the giant component, the position of the phase transition at which the giant component appears, and percolation properties for both site and bond percolation on networks generated by the model.

We know who you followed last summer: inferring social link creation times in twitter

Understanding a networks temporal evolution appears to require multiple observations of the graph over time. These often expensive repeated crawls are only able to answer questions about what happened from observation to observation, and not what happened before or between network snapshots. Contrary to this picture, we propose a method for Twitters social network that takes a single static snapshot of network edges and user account creation times to accurately infer when these edges were formed. This method can be exact in theory, and we demonstrate empirically for a large subset of Twitter relationships that it is accurate to within a few hours in practice. We study users who have a very large number of edges or who are recommended by Twitter. We examine the graph formed by these nearly 1,800 Twitter celebrities and their 862 million edges in detail, showing that a single static snapshot can give novel insights about Twitters evolution. We conclude from this analysis that real-world events and changes to Twitters interface for recommending users strongly influence network growth.