We study online learning in an oblivious changing environment. The standard measure of regret bounds the difference between the cost of the online learner and the best decision in hindsight. Hence, regret minimizing algorithms tend to converge to the static best optimum, clearly a suboptimal behavior in changing environments. On the other hand, various metrics proposed to strengthen regret and allow for more dynamic algorithms produce inefficient algorithms.We propose a different performance metric which strengthens the standard metric of regret and measures performance with respect to a changing comparator. We then describe a series of datastreaming-based reductions which transform algorithms for minimizing (standard) regret into adaptive algorithms albeit incurring only poly-logarithmic computational overhead.Using this reduction, we obtain efficient low adaptive-regret algorithms for the problem of online convex optimization. This can be applied to various learning scenarios, i.e. online portfolio selection, for which we describe experimental results showing the advantage of adaptivity.
Community structure plays a significant role in the analysis of social networks and similar graphs, yet this structure is little understood and not well captured by most models. We formally define a community to be a subgraph that is internally highly connected and has no deeper substructure. We use tools of combinatorics to show that any such community must contain a dense Erdős-Rényi (ER) subgraph. Based on mathematical arguments, we hypothesize that any graph with a heavytailed degree distribution and community structure must contain a scale free collection of dense ER subgraphs. These theoretical observations corroborate well with empirical evidence. From this, we propose the Block Two-Level Erdős-Rényi (BTER) model, and demonstrate that it accurately captures the observable properties of many real-world social networks. INTRODUCTIONGraph analysis is becoming increasingly prevalent in the quest to understand diverse phenomena like social relationships, scientific collaboration, purchasing behavior, computer network traffic, and more. We refer to graphs coming from such scenarios collectively as interaction networks. A significant amount of investigation has been done to understand the graph-theoretic properties common to interaction networks. Of particular importance is the notion of community structure. Interaction networks typically decompose into internally wellconnected sets referred to as low conductance or high modularity cuts [1,2]. Moreover, many graphs have high clustering coefficients [3], which is indicative of underlying community structure. Communities occur in a variety of sizes, though the largest community is often much smaller than the graph itself [4,5]. Community analysis can reveal important patterns, decomposing large collections of interactions into more meaningful components. A Theory of CommunitiesOne metric of the quality of a community is the modularity metric [2]. There are other measures such as conductance [6], but they are equivalent to modularity in terms of our intentions. Consider a graph G (undirected) with n vertices and degreesdenote the number of edges. We say a subgraph S has high modularity if S contains many more internal edges than predicted by a null model, which says vertices i and j are connected with probability d i d j /2m. (Technically, the probability is min(1, d i d j /2m), but we keep the notation simple for clarity.) We refer to the null model as the CL model, based on its formalization by Chung and Lu [7,8]; see also Aiello et al. [9] and the edge-configuration model of Newman et al. [10].Given a high modularity subgraph S, we say it is a module if it does not contain any further substructures of interest; in other words, it is internally well-modeled by CL. Formally, assume S has r nodes with internal degreesd 1 ,d 2 , . . . ,d r and let the number of edges in S be denoted by s = 1 2 r i=1d i . Consider the CL model on S, where edge (i, j) occurs with probabilityd idj /2s. We call S a module if the induced subgraph on S (the subgraph internal to S) is mod...
Network data is ubiquitous and growing, yet we lack realistic generative network models that can be calibrated to match real-world data. The recently proposed Block Two-Level Erdss-Renyi (BTER) model can be tuned to capture two fundamental properties: degree distribution and clustering coefficients. The latter is particularly important for reproducing graphs with community structure, such as social networks. In this paper, we compare BTER to other scalable models and show that it gives a better fit to real data. We provide a scalable implementation that requires only O(d_max) storage where d_max is the maximum number of neighbors for a single node. The generator is trivially parallelizable, and we show results for a Hadoop MapReduce implementation for a modeling a real-world web graph with over 4.6 billion edges. We propose that the BTER model can be used as a graph generator for benchmarking purposes and provide idealized degree distributions and clustering coefficient profiles that can be tuned for user specifications
The communities of a social network are sets of vertices with more connections inside the set than outside. We theoretically demonstrate that two commonly observed properties of social networks, heavy-tailed degree distributions and large clustering coefficients, imply the existence of vertex neighborhoods (also known as egonets) that are themselves good communities. We evaluate these neighborhood communities on a range of graphs. What we find is that the neighborhood communities can exhibit conductance scores that are as good as the Fiedler cut. Also, the conductance of neighborhood communities shows similar behavior as the network community profile computed with a personalized PageRank community detection method. Neighborhood communities give us a simple and powerful heuristic for speeding up local partitioning methods. Since finding good seeds for the PageRank clustering method is difficult, most approaches involve an expensive sweep over a great many starting vertices. We show how to use neighborhood communities to quickly generate a small set of seeds.
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