Abstract: We consider conditioned Galton–Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 2014

Abstract: In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the k-dimensional Laplacian for 1 ≤ k ≤ d. We study an example of random walks on simplicial complexes in the context of a semi-supervised learning problem. Specifically, we consider a label propagation algorithm on oriented edges, which applies to a generalization of the partially labelled classification problem on graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 379-405, 2016.

Abstract: Let r be a fixed constant and let H be an r-uniform, D-regular hypergraph on N vertices. Assume further that D>Nϵ for some ϵ>0. Consider the random greedy algorithm for forming an independent set in H. An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly at random from the collection of vertices that could be added to the independent set (i.e. the collection of vertices v with the property that v is not in the current independent set I and I∪{v} contains no edge in H). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of H; that is, the process terminates at a maximal independent set. We prove that if H satisfies certain degree and codegree conditions then there are Ω(N·((logN)/D)1r−1) vertices in the independent set produced by the random greedy algorithm with high probability. This result generalizes a lower bound on the number of steps in the H-free process due to Bohman and Keevash and produces objects of interest in additive combinatorics. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 479–502, 2016

Abstract: The classical Corr adi-Hajnal theorem claims that every n-vertex graph G with (G) 2n=3 contains a triangle factor, when 3jn. In this paper we asymptotically determine the minimum degree condition necessary to guarantee a triangle factor in graphs with sublinear independence number. In particular, we show that if G is an n-vertex graph with (G) = o(n) and (G) (1=2 + o(1))n, then G has a triangle factor and this is asymptotically best possible. Furthermore, it is shown for every r that if every linear size vertex set of a graph G spans quadratic many edges, and (G) (1=2 +o(1))n, then G has a Kr-factor for n suciently large. We also propose many related open problems whose solutions could show a relationship with RamseyTur an theory. Additionally, we also consider a fractional variant of the Corr adi-Hajnal Theorem, settling a conjecture of Balogh-Kemkes-Lee-Young. Let t2 (0; 1) and w : E(Kn)! [0; 1]. We call a triangle in Kn heavy if the sum of the weights on its edges is more than 3t. We prove that if 3jn and w is such that for every vertex v the sum of w(e)

Abstract: We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for -cohomology of a random k-dimensional simplicial complex within a narrow transition window. In particular, we show that if Y is a random k-dimensional simplicial complex with each k-simplex appearing i.i.d. with probability with and fixed, then the dimension of cohomology is asymptotically Poisson distributed with mean . In the k = 2 case we also prove that in an accompanying growth process, with high probability, vanishes exactly at the moment when the last -simplex gets covered by a k-simplex, a higher-dimensional analogue of a “stopping time” theorem about connectivity of random graphs due to Bollobas and Thomason. Random Struct. Alg., 2015

Abstract: In this paper we determine the threshold for d-collapsibility in the probabilistic model Xdn,p of d-dimensional simplicial complexes. A lower bound for this threshold p=i¾?dn was established in Aronshtam and Linial, Random Struct. Algorithms 46 2015 26-35, and here we show that this is indeed the correct threshold. Namely, for every c>i¾?d, a complex drawn from Xdn,cn is asymptotically almost surely not d-collapsible. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 260-269, 2016

Abstract: In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of . Let the maximum density of a graph H be the maximum average degree of all the subgraphs of H. First, we show that for , a graph w.h.p. contains copies of all spanning graphs H with maximum degree at most Δ and maximum density at most d. For , this improves a result of Dellamonica, Kohayakawa, Rodl and Rucincki. Next, we show that if we additionally restrict the spanning graphs to have girth at least 7 then the random graph contains w.h.p. all such graphs for . In particular, if , the random graph therefore contains w.h.p. every spanning tree with maximum degree bounded by Δ. This improves a result of Johannsen, Krivelevich and Samotij. Finally, in the same spirit, we show that for any spanning graph H with constant maximum degree, and for suitable p, if we randomly color the edges of a graph with colors, then w.h.p. there exists a rainbow copy of H in G (that is, a copy of H with all edges colored with distinct colors). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Abstract: We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of for the two-sided-error randomized decision tree complexity of evaluating height h formulae with error . This improves the lower bound of given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most . The previous best known algorithm achieved complexity . The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel “interleaving” of two recursive algorithms. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Abstract: Let HPn,m,k be drawn uniformly from all m-edge, k-uniform, k-partite hypergraphs where each part of the partition is a disjoint copy of [n]. We let HPn,m,k(κ) be an edge colored version, where we color each edge randomly from one of κ colors. We show that if κ=n and m=Knlogn where K is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if n is even and m=Knlogn where K is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in Gn,m(n). Here Gn,m(n) denotes a random edge coloring of Gn,m with n colors. When n is odd, our proof requires m=ω(nlogn) for there to be a rainbow Hamilton cycle. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 503–523, 2016

Abstract: In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most C|VG|. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph Gn,p, which improves upon existing results showing that asymptotically almost surely the cop number of Gn,p is Onlogn provided that pni¾?2+elogn for some e>0. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random d-regular graphs, where we show that the conjecture holds asymptotically almost surely when d=dni¾?3. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 396-421, 2016

Abstract: In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Combin Probab Comput 9 (2000), 125–148] can be adapted from random graphs to random r-uniform hypergraphs and provide sufficient conditions when a random r-uniform hypergraph H(r)(n,p) contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube-hypergraphs, lattices, spheres, and Hamilton cycles in hypergraphs. Moreover, we study universality, i.e. when does an r-uniform hypergraph contain any hypergraph on n vertices and with maximum vertex degree bounded by Δ? For H(r)(n,p) it is shown that this holds for p≫(lnn/n)1/Δ a.a.s. by combining approaches taken by Dellamonica, Kohayakawa, Rodl and Rucinski [Random Struct Algorithms 46 (2015), 274–299] and of Ferber, Nenadov and Peter [Random Struct Algorithms 48 (2016), 546–564] and of Kim and Lee [SIAM J Discrete Math 28 (2014), 1467–1478]. Furthermore it is shown that the random graph G(n, p) for appropriate p and explicit constructions of universal graphs due to Alon, Capalbo, Kohayakawa, Rodl, Rucinski and Szemeredi [Lecture Notes in Comput. Sci., Vol. 2129, Springer, Berlin, 2001, pp. 170–180] and Alon and Capalbo [Random Struct Algorithms 31 (2007), 123–133; Proceedings of the 9th Annual ACM-SIAM Symposium Society of Industrial and Applied Mathematics, 2008, pp. 373–378] yield constructions of universal hypergraphs that are sparser than the random hypergraph H(r)(n,p) with p≫(lnn/n)1/Δ © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Abstract: We present estimates on the small singular values of a class of matrices with independent Gaussian entries and inhomogeneous variance profile, satisfying a broad-connectedness condition. Using these estimates and concentration of measure for the spectrum of Gaussian matrices with independent entries, we prove that for a large class of graphs satisfying an appropriate expansion property, the Barvinok–Godsil-Gutman estimator for the permanent achieves sub-exponential errors with high probability. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 183–212, 2016

Abstract: A wide class of problems in combinatorics, computer science and physics can be described along the following lines. There are a large number of variables ranging over a finite domain that interact through constraints that each bind a few variables and either encourage or discourage certain value combinations. Examples include the k-SAT problem or the Ising model. Such models naturally induce a Gibbs measure on the set of assignments, which is characterised by its partition function. The present paper deals with the partition function of problems where the interactions between variables and constraints are induced by a sparse random (hyper)graph. According to physics predictions, a generic recipe called the “replica symmetric cavity method” yields the correct value of the partition function if the underlying model enjoys certain properties [Krzkala et al., PNAS (2007) 10318–10323]. Guided by this conjecture, we prove general sufficient conditions for the success of the cavity method. The proofs are based on a “regularity lemma” for probability measures on sets of the form Ωn for a finite Ω and a large n that may be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Abstract: We study randomized gossip-based processes in dynamic networks that are motivated by information discovery in large-scale distributed networks such as peer-to-peer and social networks. A well-studied problem in peer-to-peer networks is resource discovery, where the goal for nodes (hosts with IP addresses) is to discover the IP addresses of all other hosts. Also, some of the recent work on self-stabilization algorithms for P2P/overlay networks proceed via discovery of the complete network. In social networks, nodes (people) discover new nodes through exchanging contacts with their neighbors (friends). In both cases the discovery of new nodes changes the underlying network --- new edges are added to the network --- and the process continues in the changed network. Rigorously analyzing such dynamic (stochastic) processes in a continuously changing topology remains a challenging problem with obvious applications.This paper studies and analyzes two natural gossip-based discovery processes. In the push discovery or triangulation process, each node repeatedly chooses two random neighbors and connects them (i.e., "pushes" their mutual information to each other). In the pull discovery process or the {\em two-hop walk}, each node repeatedly requests or "pulls" a random contact from a random neighbor and connects itself to this two-hop neighbor. Both processes are lightweight in the sense that the amortized work done per node is constant per round, local, and naturally robust due to the inherent randomized nature of gossip.Our main result is an almost-tight analysis of the time taken for these two randomized processes to converge. We show that in any undirected n-node graph both processes take O(n log2 n) rounds to connect every node to all other nodes with high probability, whereas Ω(n log n) is a lower bound. We also study the two-hop walk in directed graphs, and show that it takes O(n2 log n) time with high probability, and that the worst-case bound is tight for arbitrary directed graphs, whereas Ω(n2) is a lower bound for strongly connected directed graphs. A key technical challenge that we overcome in our work is the analysis of a randomized process that itself results in a constantly changing network leading to complicated dependencies in every round. We discuss implications of our results and their analysis to discovery problems in P2P networks as well as to evolution in social networks.

Abstract: We show that, for , the largest set in a p-random sub-family of the power set of containing no k-chain has size with high probability. This confirms a conjecture of Osthus. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2016

Abstract: We study the Maker-Breaker H-game played on the edge set of the random graph . In this game two players, Maker and Breaker, alternately claim unclaimed edges of , until all edges are claimed. Maker wins if he claims all edges of a copy of a fixed graph H; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an H-game is given by the threshold of the corresponding Ramsey property of with respect to the graph H. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Abstract: Os coleopteros (Coleoptera) se distribuem em diferentes areas e profundidades do solo e sao importantes executores de servicos ambientais. Seu estudo representa um excelente foco para elucidar os efeitos da perturbacao antropica sobre a biodiversidade e funcoes dos ecossistemas. O objetivo desse estudo foi avaliar a diversidade de familias de coleopteros em sistemas de uso do solo (SUS), bem como a relacao destes com os atributos edaficos. Os sistemas estudados envolvem: floresta nativa (FN), reflorestamento de eucalipto (RE), pastagem (PA), integracao lavoura-pecuaria (ILP) e lavoura com plantio direto (PD). As amostras foram coletadas em grade de amostragem de 3 × 3 pontos, distanciados entre si em 30 m, nos periodos de inverno e verao, em tres municipios do Planalto Sul Catarinense, considerados replicas verdadeiras. Os invertebrados edaficos foram coletados pelos metodos Tropical Soil Biology and Fertility (TSBF) e Pitfall traps (armadilhas de queda) . Nos mesmos pontos coletaram-se amostras para determinacao dos atributos fisicos e quimicos do solo. Utilizou-se analise estatistica multivariada para a composicao da comunidade, sendo, as variaveis ambientais consideradas como explicativas. Foram estudados 1.437 individuos, sendo a Familia Staphylinidae a mais representativa. Os sistemas RE e PD apresentaram maior diversidade de acordo com o indice de Shannon (H). As analises de componentes principais demonstraram distincao na distribuicao dos invertebrados entre os diferentes SUS. As propriedades do solo contribuiram para explicar essa variacao, dando destaque aos atributos materia orgânica e porosidade que favoreceram a maior abundância de Coleoptera em FN e a ocorrencia de Staphylinidae, potencial bioindicador das condicoes do ambiente.

Abstract: Let f be an edge ordering of Kn: a bijection . For an edge , we call f(e) the label of e. An increasing path in Kn is a simple path (visiting each vertex at most once) such that the label on each edge is greater than the label on the previous edge. We let S(f) be the number of edges in the longest increasing path. Chvatal and Komlos raised the question of estimating m(n): the minimum value of S(f) over all orderings f of Kn. The best known bounds on m(n) are , due respectively to Graham and Kleitman, and to Calderbank, Chung, and Sturtevant. Although the problem is natural, it has seen essentially no progress for three decades. In this paper, we consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, S(f) often takes its maximum possible value of n – 1 (visiting all of the vertices with an increasing Hamiltonian path). We prove that this occurs with probability at least about 1/ e. We also prove that with probability 1- o(1), there is an increasing path of length at least 0.85 n, suggesting that this Hamiltonian (or near-Hamiltonian) phenomenon may hold asymptotically almost surely. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Abstract: The transformation to a more market-oriented steering approach in European higher education challenges universities and other higher education institutions to consider developing branding or image management activities. The existing literature focuses either on the content or the style, but we argue that an integra­ted perspective is needed to fully grasp the processes underlying branding. In a comparative case study of ten UK higher education institutions with varying re­putations – five highly reputed versus five low(er) reputed institutions – we de­monstrate how and why branding is deployed in welcome addresses of institutional leaders. Our findings indicate that isomorphic tendencies are visible, although brand differentiation could also be identified between highly and lowly reputed institutions. Our findings provide support for the competitive group perspective on branding activities.

Abstract: We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of G(n,p) in order to typically find a subgraph possessing a given target property. We show that if p≥lnn+lnlnn+ω(1)n, then one can find a Hamilton cycle with high probability after exposing (1+o(1))n edges. Our result is tight in both p and the number of exposed edges. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Abstract: For k | n let Combn,k denote the tree consisting of an (n/k)-vertex path with disjoint k-vertex paths beginning at each of its vertices. An old conjecture says that for any k = k(n) the threshold for the random graph G(n, p) to contain Combn,k is at p logn n . Here we verify this for k ≤ C logn with any fixed C > 0. In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for k ≥ κ0 logn (with κ0 ≈ 4.82).

Abstract: A fauna edafica e sensivel a perturbacoes ambientais e sua resposta pode indicar o estado de conservacao do solo em locais com diferentes uso e manejo. O objetivo deste estudo foi avaliar a diversidade de grupos da fauna invertebrada e sua relacao com atributos edaficos em tres sistemas de uso do solo (SUS): Floresta Nativa (FN), Reflorestamento de Pinus (RP) e Campo Nativo Melhorado (CNM), no municipio de Lages, SC. A amostragem consistiu em tres pontos distanciados entre si por 30 metros, estabelecidos ao longo de transectos, em cada SUS. Avaliaram-se os atributos fisicos e quimicos do solo e da abundância e diversidade da fauna, coletada pelos metodos Pitfall traps e Tropical Soil Biology and Fertility (TSBF). Os dados foram submetidos a analise estatistica multivariada. Identificaram-se 1210 organismos do solo, pertencentes a 17 grupos taxonomicos, sendo, os mais representativos Collembola e Coleoptera, independente do SUS. A FN apresentou maior riqueza e diversidade da fauna edafica em comparacao aos outros sistemas. Os maiores teores de materia orgânica e pH demonstraram relacao com Oligochaeta, Enchytraeidae e Collembola. A umidade do solo contribuiu para explicar a abundância dos grupos em CNM e FN. Contudo, a fauna mostrou respostas diferentes na sua distribuicao para cada SUS, logo, as variaveis ambientais podem limitar o estabelecimento dos invertebrados mais frequentes no solo. Os sistemas FN e CNM apresentaram melhores condicoes dos atributos edaficos e por isso maior biodiversidade, quando comparados ao RP.

Abstract: © 2016 Wiley Periodicals, Inc. The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(n4) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is O(nlogn) and O(n2). We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.

Abstract: We prove that the empirical spectral distribution of a dL, dR-biregular, bipartite random graph, under certain conditions, converges to a symmetrization of the Marcenko-Pastur distribution of random matrix theory. This convergence is not only global on fixed-length intervals but also local on intervals of increasingly smaller length. Our method parallels the one used previously by Dumitriu and Pal 2012. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 313-340, 2016

Abstract: We apply the graph container method to prove a number of counting results for the Boolean latticeP(n). In particular, we: (i) We give a partial answer to a question of Sapozhenko estimating the number of t error correcting codes inP(n), and we also give an upper bound on the number of transportation codes; (ii) Provide an alternative proof of Kleitman’s theorem on the number of antichains inP(n) and give a two-coloured analogue; (iii) Give an asymptotic formula for the number of (p;q)-tilted Sperner families inP(n); (iv) Prove a random version of Katona’s t-intersection theorem. In each case, to apply the container method, we rst prove corresponding supersaturation results. A number of open questions are also given.

Abstract: The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In this paper, we study essentially the most fundamental judicious partitioning problem for directed graphs, which naturally extends the classical Max Cut problem to this setting: we seek bipartitions in which many edges cross in each direction. It is easy to see that a minimum outdegree condition is required in order for the problem to be nontrivial, and we prove that every directed graph with m edges and minimum outdegree at least two admits a bipartition in which at least edges cross in each direction. We also prove that if the minimum outdegree is at least three, then the constant can be increased to . If the minimum outdegree tends to infinity with n, then the constant increases to . All of these constants are best-possible, and provide asymptotic answers to a question of Alex Scott. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Abstract: The foregoing notion, introduced by Goldreich and Ron STOC 2009, was originally defined with respect to c = 1, which corresponds to one-sided error proximity-oblivious testing. Here we study the two-sided error version of proximity-oblivious testers; that is, the general case of arbitrary c i¾? 0,1]. We show that, in many natural cases, two-sided error proximity-oblivious testers are more powerful than one-sided error proximity-oblivious testers; that is, many natural properties that have no one-sided error proximity-oblivious testers do have a two-sided error proximity-oblivious tester. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 341-383, 2016

Abstract: We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in (Angel and Ray, Ann Probab, in press). We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 213–234, 2016

Abstract: Given a sequence of independent random variables with a common continuous distribution, we consider the online decision problem where one seeks to minimize the expected value of the time that is needed to complete the selection of a monotone increasing subsequence of a prespecified length n. This problem is dual to some online decision problems that have been considered earlier, and this dual problem has some notable advantages. In particular, the recursions and equations of optimality lead with relative ease to asymptotic formulas for mean and variance of the minimal selection time. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016