# Influence, Threshold, and Noise

My dear friend Itai Benjamini told me that he won’t be able to make it to my Tuesday talk on influence, threshold, and noise, and asked if I already have  the slides. So it occurred to me that perhaps I can practice the lecture on you, my readers, not just with the slides (here they are) but also roughly what I plan to say, some additional info, and some pedagogical hesitations. Of course, remarks can be very helpful.

I can also briefly report that there are plenty of exciting things happening around that I would love to report about – hopefully later in my travel-free summer. One more thing: while chatting with Yuval Rabani and Daniel Spielman I realized that there are various exciting things happening in algorithms (and not reported so much in blogs). Much progress has been made on basic questions: TSP, Bin Packing, flows & bipartite matching, market equilibria, and k-servers, to mention a few, and also new directions and methods. I am happy to announce that Yuval kindly agreed to write here an algorithmic column from time to time, and Daniel is considering contributing a guest post as well.

## The second AMS-IMU meeting

Since the early 70s, I have been a devoted participants in our annual meetings of the Israeli Mathematical Union (IMU), and this year we will have the second joint meeting with the American Mathematical Society (AMS). Here is the program. There are many exciting lectures. Let me mention that Eran Nevo, this year Erdős’ prize winner, will give a lecture about the g-conjecture. Congratulations, Eran! Among the 22 exciting special sessions there are a few related to combinatorics, and even one organized by me on Wednsday and Thursday.

 Combinatorics Contact person: Gil Kalai, gil.kalai@gmail.com TAU, Dan David building, Room 103 Wed, 10:50-11:30 Van H. Vu (Yale University) Real roots of random polynomials (abstract) Wed, 11:40-12:20 Oriol Serra (Universitat Politecnica de Catalunya, Barcelona) Arithmetic Removal Lemmas (abstract) Wed, 12:30-13:10 Tali Kaufman (Bar-Ilan University) Bounded degree high dimensional expanders (abstract) Wed, 16:00-16:40 Rom Pinchasi (Technion) On the union of arithmetic progressions (abstract) Wed, 16:50-17:30 Isabella Novik (University of Washington, Seattle) Face numbers of balanced spheres, manifolds, and pseudomanifolds (abstract) Wed, 17:40-18:20 Edward Scheinerman (Johns Hopkins University, Baltimore) On Vertex, Edge, and Vertex-Edge Random Graphs (abstract) Thu, 9:20-10:00 Yael Tauman Kalai (MSR, New England) The Evolution of Proofs in Computer Science (abstract) Thu, 10:10-10:50 Irit Dinur (Weitzman Institute) Lifting locally consistent solutions to global solutions (abstract) Thu, 11:00-11:40 Benny Sudakov (ETH, Zurich) The minimum number of nonnegative edges in hypergraphs (abstract)

And now for my own lecture.

# Nati’s Influence

When do we say that one event causes another? Causality is a topic of great interest in statistics, physics, philosophy, law, economics, and many other places. Now, if causality is not complicated enough, we can ask what is the influence one event has on another one.  Michael Ben-Or and Nati Linial wrote a paper in 1985 where they studied the notion of influence in the context of collective coin flipping. The title of the post refers also to Nati’s influence on my work since he got me and Jeff Kahn interested in a conjecture from this paper.

## Influence

The word “influence” (dating back, according to Merriam-Webster dictionary, to the 14th century) is close to the word “fluid”.  The original definition of influence is: “an ethereal fluid held to flow from the stars and to affect the actions of humans.” The modern meaning (according to Wictionary) is: “The power to affect, control or manipulate something or someone.”

## Ben-Or and Linial’s definition of influence

Collective coin flipping refers to a situation where n processors or agents wish to agree on a common random bit. Ben-Or and Linial considered very general protocols to reach a single random bit, and also studied the simple case where the collective random bit is described by a Boolean function $f(x_1,x_2,\dots,x_n)$ of n bits, one contributed by every agent. If all agents act appropriately the collective bit will be ‘1’ with probability 1/2. The purpose of collective coin flipping is to create a random bit R which is immune as much as possible against attempts of one or more agents to bias it towards ‘1’ or ‘0’. Continue reading