Clique counts reveal important properties about the structure of massive graphs, especially social networks. The simple setting of just 3-cliques (triangles) has received much attention from the research community. For larger cliques (even, say 6-cliques) the problem quickly becomes intractable because of combinatorial explosion.Most methods used for triangle counting do not scale for large cliques, and existing algorithms require massive parallelism to be feasible.We present a new randomized algorithm that provably approximates the number of k-cliques, for any constant k. The key insight is the use of (strengthenings of) the classic Turán's theorem: this claims that if the edge density of a graph is sufficiently high, the k-clique density must be non-trivial. We define a combinatorial structure called a Turán shadow, the construction of which leads to fast algorithms for clique counting.We design a practical heuristic, called Turán-shadow, based on this theoretical algorithm, and test it on a large class of test graphs. In all cases, Turán-shadow has less than 2% error, and runs in a fraction of the time used by well-tuned exact algorithms. We do detailed comparisons with a range of other sampling algorithms, and find that Turán-shadow is generally much faster and more accurate. For example, Turán-shadow estimates all clique numbers up to size 10 in social network with over a hundred million edges. This is done in less than three hours on a single commodity machine.

Clique counting is a fundamental task in network analysis, and even the simplest setting of 3-cliques (triangles) has been the center of much recent research. Getting the count of k-cliques for larger k is algorithmically challenging, due to the exponential blowup in the search space of large cliques. But a number of recent applications (especially for community detection or clustering) use larger clique counts. Moreover, one often desires local counts, the number of k-cliques per vertex/edge.Our main result is Pivoter, an algorithm that exactly counts the number of k-cliques, for all values of k. It is surprisingly effective in practice, and is able to get clique counts of graphs that were beyond the reach of previous work. For example, Pivoter gets all clique counts in a social network with a 100M edges within two hours on a commodity machine. Previous parallel algorithms do not terminate in days. Pivoter can also feasibly get local per-vertex and per-edge k-clique counts (for all k) for many public data sets with tens of millions of edges. To the best of our knowledge, this is the first algorithm that achieves such results.The main insight is the construction of a Succinct Clique Tree (SCT) that stores a compressed unique representation of all cliques in an input graph. It is built using a technique called pivoting, a classic approach by Bron-Kerbosch to reduce the recursion tree of backtracking algorithms for maximal cliques. Remarkably, the SCT can be built without actually enumerating all cliques, and provides a succinct data structure from which exact clique statistics (k-clique counts, local counts) can be read off efficiently.

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Growing fuel costs, environmental awareness, government directives, an aggressive push to deploy Electric Vehicles (EVs) (a single EV consumes the equivalent of 3 to 10 homes) have led to a severe strain on a grid already on the brink. Maintaining the stability of the grid requires automatic agent based control of these loads and rapid coordination between them. In the literature, a number of iterative pricing, signaling and tâtonnement (or bargaining) approaches have been proposed to allow smart homes, storage devices and the autonomous agents that control them to be responsive to the state of the grid in a distributed manner. These existing approaches are not scalable due to slow convergence and moreover the approaches are not incentive compatible. In this paper, we present a tâtonnement framework for resource allocation among intelligent agents in the smart grid, that non-trivially generalizes past work in this area. Our approach based on the work in server load balancing involves communicating carefully chosen, centrally verifiable constraints on the set of actions available to agents and cost functions, leading to distributed, incentive compatible protocols that converge in a constant number of iterations, independent of the number of users. These protocols can work on the top of prior approaches and result in a substantial speed-up, while ensuring that it is in the best interests of the agents to be truthful. We demonstrate this theoretically and through extensive simulations for three important scenarios that have been discussed in the literature. We extend the techniques to account for capacity limits in each time slot, the EV charging problem and the distributed storage