Let ǫ 1 , . . . , ǫ n be i.i.d. Rademacher random variables taking values ±1 with probability 1/2 each. Given an integer vector a = (a 1 , . . . , a n ), its concentration probability is the quantityThe Littlewood-Offord problem asks for bounds on ρ(a) under various hypotheses on a, whereas the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a characterization of all vectors a for which ρ(a) is large. In this paper, we study the associated counting problem: How many integer vectors a belonging to a specified set have large ρ(a)? The motivation for our study is that in typical applications, the inverse Littlewood-Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood-Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first 'exponential-type' (i.e., exp(−n c ) for some positive constant c) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best known bound is O(n −1/4 ) due to Cook; and (ii) dense rowregular {0, 1}-matrices, for which the previous best known bound is O C (n −C ) for any constant C > 0 due to Nguyen.

In this article we identify social communities among gang members in the Hollenbeck policing district in Los Angeles, based on sparse observations of a combination of social interactions and geographic locations of the individuals. This information, coming from LAPD Field Interview cards, is used to construct a similarity graph for the individuals. We use spectral clustering to identify clusters in the graph, corresponding to communities in Hollenbeck, and compare these with the LAPD's knowledge of the individuals' gang membership. We discuss different ways of encoding the geosocial information using a graph structure and the influence on the resulting clusterings. Finally we analyze the robustness of this technique with respect to noisy and incomplete data, thereby providing suggestions about the relative importance of quantity versus quality of collected data.

Let ε1,…,εn be independent and identically distributed Rademacher random variables taking values ±1 with probability 1/2 each. Given an integer vector a=(a1,…,an), its concentration probability is the quantity ρfalse(bold-italicafalse):=trueprefixsupx∈ZPrfalse(ε1a1+⋯+εnan=xfalse). The Littlewood–Offord problem asks for bounds on ρ(a) under various hypotheses on bold-italica, whereas the inverse Littlewood–Offord problem, posed by Tao and Vu, asks for a characterization of all vectors bold-italica for which ρ(a) is large. In this paper, we study the associated counting problem: How many integer vectors bold-italica belonging to a specified set have large ρ(a)? The motivation for our study is that in typical applications, the inverse Littlewood–Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood–Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first ‘exponential‐type’ (that is, exp(−cnc) for some positive constant c) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best‐known bound is O(n−1/4), due to Cook; and (ii) dense row‐regular {0,1}‐matrices, for which the previous best‐known bound is OCfalse(n−Cfalse) for any constant C>0, due to Nguyen.

In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $\Delta\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree $\Delta$. We show that a random graph $G_{n,p}$ with high probability contains a copy of $H$, provided that $p\gg (n^{-1}\log^{1/\Delta}n)^{2/(\Delta+1)}$. Our assumption on $p$ is optimal up to the $polylog$ factor. We note that this $polylog$ term matches the conjectured threshold for the spanning case.Comment: Incorporated referee comments. To appear in Bulletin of the London Mathematical Societ

Let Mn be a class of symmetric sparse random matrices, with independent entries Mij = δij ξij for i ≤ j. δij are i.i.d. Bernoulli random variables taking the value 1 with probability p ≥ n −1+δ for any constant δ > 0 and ξij are i.i.d. centered, subgaussian random variables. We show that with high probability this class of random matrices has simple spectrum (i.e. the eigenvalues appear with multiplicity one). We can slightly modify our proof to show that the adjacency matrix of a sparse Erdős-Rényi graph has simple spectrum for n −1+δ ≤ p ≤ 1 − n −1+δ . These results are optimal in the exponent. The result for graphs has connections to the notorious graph isomorphism problem. * K. Luh is supported by the National Science Foundation under Award No. 1702533. † V. Vu is supported by NSF grant DMS 1307797 and AFORS grant FA9550-12-1-0083.

For a family of graphs F \mathcal {F} , a graph G G is F \mathcal {F} -universal if G G contains every graph in F \mathcal {F} as a (not necessarily induced) subgraph. For the family of all graphs on n n vertices and of maximum degree at most two, H ( n , 2 ) \mathcal {H}(n,2) , we prove that there exists a constant C C such that for p ≥ C ( log ⁡ n n 2 ) 1 3 , p \geq C \left ( \frac {\log n}{n^2} \right )^{\frac {1}{3}}, the binomial random graph G ( n , p ) G(n,p) is typically H ( n , 2 ) \mathcal {H}(n,2) -universal. This bound is optimal up to the constant factor as illustrated in the seminal work of Johansson, Kahn, and Vu for triangle factors. Our result improves significantly on the previous best bound of p ≥ C ( log ⁡ n n ) 1 2 p \geq C \left (\frac {\log n}{n}\right )^{\frac {1}{2}} due to Kim and Lee. In fact, we prove the stronger result that for H ℓ ( n , 2 ) \mathcal {H}^{\ell }(n,2) , the family of all graphs on n n vertices, of maximum degree at most two and of girth at least ℓ \ell , G ( n , p ) G(n,p) is typically H ℓ ( n , 2 ) \mathcal H^{\ell }(n,2) -universal when p ≥ C ( log ⁡ n n ℓ − 1 ) 1 ℓ . p \geq C \left (\frac {\log n}{n^{\ell -1}}\right )^{\frac {1}{\ell }}. This result is also optimal up to the constant factor. Our results verify (in a weak form) a classical conjecture of Kahn and Kalai.

We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability from work of the second author and Vu. We also show that for sparse Erdős-Rényi graphs, weak and strong nodal domains are the same, answering a question of Dekel, Lee, and Linial.

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