Many TT copies in HH-free graphs

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) $ex(n,K_3,C_5) \leq (1+o(1)) \frac{\sqrt 3}{2} n^{3/2},$ (ii) For any fixed $m$, $s \geq 2m-2$ and $t \geq (s-1)!+1$, $ex(n,K_m,K_{s,t})=\Theta(n^{m-\binom{m}{2}/s})$ and (iii) For any two trees $H$ and $T$, $ex(n,T,H) =\Theta (n^m)$ where $m=m(T,H)$ is an integer depending on $H$ and $T$ (its precise definition is given in Section 1). The first result improves (slightly) an estimate of Bollob\'as and Gy\H{o}ri. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques

Thresholds and expectation-thresholds of monotone properties with small minterms

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $\mu_p$ on the power set of $N$, where for $\mathcal{A} \subseteq 2^N$ $\mu_p(\mathcal{A}) := Pr[N_p \in \mathcal{A}]$. In this paper we study upward-closed families $\mathcal{A}$ for which all minimal sets in $\mathcal{A}$ have size at most $k$, for some positive integer $k$. We prove that for such a family $\mu_p(\mathcal{A}) / p^k$ is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables to identify in $\mathcal{A}$ either a substantial subfamily $\mathcal{A}_0$ for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than $k$. Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using duality of linear programming. This is related to the threshold conjecture of Kahn and Kalai

Unjamming Lightning: A Systematic Approach

Users of decentralized financial networks suffer from inventive security exploits. Identity-based fraud prevention methods are inapplicable in these networks, as they contradict their privacy-minded design philosophy. Novel mitigation strategies are therefore needed. Their rollout, however, may damage other desirable network properties. In this work, we introduce an evaluation framework for mitigation strategies in decentralized financial networks. This framework allows researchers and developers to examine and compare proposed protocol modifications along multiple axes, such as privacy, security, and user experience. As an example, we focus on the jamming attack in the Lightning Network. Lightning is a peer-to-peer payment channel network on top of Bitcoin. Jamming is a cheap denial-of-service attack that allows an adversary to temporarily disable Lightning channels by flooding them with failing payments. We propose a practical solution to jamming that combines unconditional fees and peer reputation. Guided by the framework, we show that, while discouraging jamming, our solution keeps the protocol incentive compatible. It also preserves security, privacy, and user experience, and is straightforward to implement. We support our claims analytically and with simulations. Moreover, our anti-jamming solution may help alleviate other Lightning issues, such as malicious channel balance probing

Greedy maximal independent sets via local limits

The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science, and even chemistry. The algorithm builds a maximal independent set by inspecting the graph's vertices one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge. In this paper, we present a natural and general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a notion of local convergence. We use this framework both to give short and simple proofs for results on previously studied families of graphs, such as paths and binomial random graphs, and to study new ones, such as random trees and random planar graphs. We conclude our work by analysing the random greedy algorithm more closely when the base graph is a tree. We show that in expectation, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order.Comment: 26 pages. This is an extended and revised version of a conference version presented at the 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA2020

Interactive Proofs for Social Graphs

We consider interactive proofs for social graphs, where the verifier has only oracle access to the graph and can query for the $i^{th}$ neighbor of a vertex $v$, given $i$ and $v$. In this model, we construct a doubly-efficient public-coin two-message interactive protocol for estimating the size of the graph to within a multiplicative factor $\epsilon>0$. The verifier performs $\tilde{O}(1/\epsilon^2 \cdot \tau_{mix} \cdot \Delta)$ queries to the graph, where $\tau_{mix}$ is the mixing time of the graph and $\Delta$ is the average degree of the graph. The prover runs in quasi-linear time in the number of nodes in the graph. Furthermore, we develop a framework for computing the quantiles of essentially any (reasonable) function $f$ of vertices/edges of the graph. Using this framework, we can estimate many health measures of social graphs such as the clustering coefficients and the average degree, where the verifier performs only a small number of queries to the graph. Using the Fiat-Shamir paradigm, we are able to transform the above protocols to a non-interactive argument in the random oracle model. The result is that social media companies (e.g., Facebook, Twitter, etc.) can publish, once and for all, a short proof for the size or health of their social network. This proof can be publicly verified by any single user using a small number of queries to the graph