arXiv subjects

Nikolaos Fountoulakis

Publications and source records attributed to Nikolaos Fountoulakis.

35 records · Page 2Linked to original sources

arXiv2015

Bootstrap percolation in inhomogeneous random graphs

A bootstrap percolation process on a graph G is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round every uninfected node which has at least r infected neighbours becomes infected and remains so forever. The parameter r > 1 is fixed. We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank 1. Assuming that initially every vertex is infected independently with probability p > 0, we provide a law of large numbers for the number of vertices that will have been infected by the end of the process. We also focus on a special case of such random graphs which exhibit a power-law degree distribution with exponent in (2,3). The first two authors have shown the existence of a critical function a_c(n) such that a_c(n)=o(n) with the following property. Let n be the number of vertices of the underlying random graph and let a(n) be the number of the vertices that are initially infected. Assume that a set of a(n) vertices is chosen randomly and becomes externally infected. If a(n) << a_c(n), then the process does not evolve at all, with high probability as n grows, whereas if a(n)>> a_c(n), then with high probability the final set of infected vertices is linear. Using the techniques of the previous theorem, we give the precise asymptotic fraction of vertices which will be eventually infected when a(n) >> a_c (n) but a(n) = o(n). Note that this corresponds to the case where p approaches 0.

math.PR↗

arXiv2014

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least $r\geq 2$ infected neighbours becomes infected and remains so forever. Assume that initially $a(t)$ vertices are randomly infected, where $t$ is the total number of vertices of the graph. Suppose also that $r < m$, where $2m$ is the average degree. We determine a critical function $a_c(t)$ such that when $a(t) \gg a_c(t)$, complete infection occurs with high probability as $t \rightarrow \infty$, but when $a(t) \ll a_c (t)$, then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to $a(t)$.

math.PR↗

arXiv2014

On the Average-case Complexity of Parameterized Clique

The k-Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most well-known NP-complete and W[1]-complete problems. Moreover, its average-case complexity analysis has created a long thread of research already since the 1970s. Here, we continue this line of research by studying the dependence of the average-case complexity of the k-Clique problem on the parameter k. To this end, we define two natural parameterized analogs of efficient average-case algorithms. We then show that k-Clique admits both analogues for Erdős-Rényi random graphs of arbitrary density. We also show that k-Clique is unlikely to admit neither of these analogs for some specific computable input distribution.

cs.DS↗

arXiv2013

On the Insertion Time of Cuckoo Hashing

Cuckoo hashing is an efficient technique for creating large hash tables with high space utilization and guaranteed constant access times. There, each item can be placed in a location given by any one out of k different hash functions. In this paper we investigate further the random walk heuristic for inserting in an online fashion new items into the hash table. Provided that k > 2 and that the number of items in the table is below (but arbitrarily close) to the theoretically achievable load threshold, we show a polylogarithmic bound for the maximum insertion time that holds with high probability.

cs.DS↗

arXiv2013

The Multiple-orientability Thresholds for Random Hypergraphs

A $k$-uniform hypergraph $H = (V, E)$ is called $\ell$-orientable, if there is an assignment of each edge $e\in E$ to one of its vertices $v\in e$ such that no vertex is assigned more than $\ell$ edges. Let $H_{n,m,k}$ be a hypergraph, drawn uniformly at random from the set of all $k$-uniform hypergraphs with $n$ vertices and $m$ edges. In this paper we establish the threshold for the $\ell$-orientability of $H_{n,m,k}$ for all $k\ge 3$ and $\ell \ge 2$, i.e., we determine a critical quantity $c_{k, \ell}^*$ such that with probability $1-o(1)$ the graph $H_{n,cn,k}$ has an $\ell$-orientation if $c < c_{k, \ell}^*$, but fails doing so if $c > c_{k, \ell}^*$. Our result has various applications including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

cs.DM↗

arXiv2013

Bootstrap percolation in power-law random graphs

A bootstrap percolation process on a graph $G$ is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least $r$ infected neighbours becomes infected and remains so forever. The parameter $r\geq 2$ is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse bootstrap percolation process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are $a(n)$ randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent $β$, where $2 < β< 3$, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function $a_c(n)$ such that $a_c(n)=o(n)$ with the following property. Assuming that $n$ is the number of vertices of the underlying random graph, if $a(n) \ll a_c(n)$, then the process does not evolve at all, with high probability as $n$ grows, whereas if $a(n)\gg a_c(n)$, then there is a constant $\eps>0$ such that, with high probability, the final set of infected vertices has size at least $\eps n$. It turns out that when the maximum degree is $o(n^{1/(β-1)})$, then $a_c(n)$ depends also on $r$. But when the maximum degree is $Θ(n^{1/(β-1)})$, then $a_c (n)=n^{β-2 \over β-1}$.

math.PR↗

arXiv2012

On the evolution of random graphs on spaces of negative curvature

In this work, we study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on a hyperbolic space and any two of them are joined by an edge with probability that depends on their hyperbolic distance, independently of every other pair. In particular, when the positions of the points have been fixed, the distribution over the set of graphs on these points is the Boltzmann distribution, where the Hamiltonian is given by the sum of weighted indicator functions for each pair of points, with the weight being proportional to a real parameter β>0 (interpreted as the inverse temperature) as well as to the hyperbolic distance between the corresponding points. This class of random graphs was introduced by Krioukov et al. We provide a rigorous analysis of aspects of this model and its dependence on the parameter β, verifying some of their observations. We show that a phase transition occurs around β=1. More specifically, we show that when β> 1 the degree of a typical vertex is bounded in probability (in fact it follows a distribution which for large values exhibits a power-law tail whose exponent depends only on the curvature of the space), whereas for β<1 the degree is a random variable whose expected value grows polynomially in N. When β= 1, we establish logarithmic growth. For the case β> 1, we establish a connection with a class of inhomogeneous random graphs known as the Chung-Lu model. Assume that we use the Poincaré disc representation of a hyperbolic space. If we condition on the distance of each one of the points from the origin, then the probability that two given points are adjacent is expressed through the kernel of this inhomogeneous random graph.

math.CO↗

arXiv2012

Largest sparse subgraphs of random graphs

For the Erdős-Rényi random graph G(n,p), we give a precise asymptotic formula for the size of a largest vertex subset in G(n,p) that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t = t(n) is not too large. In the case of fixed t and p, we find that this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.

math.CO↗

arXiv2010

The t-stability number of a random graph

Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets).

math.CO↗

arXiv2010

Rumor Spreading on Random Regular Graphs and Expanders

Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well-studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d, which is at least 3, i.e., the underlying graph is drawn uniformly at random from the set of all d-regular graphs with n vertices. We show that with probability 1 - o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In particular, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo-random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1+o(1))C ln n with probability 1 - o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.

math.CO↗

arXiv2009

Sharp Load Thresholds for Cuckoo Hashing

The paradigm of many choices has influenced significantly the design of efficient data structures and, most notably, hash tables. Cuckoo hashing is a technique that extends this concept. There,we are given a table with $n$ locations, and we assume that each location can hold one item. Each item to be inserted chooses randomly k>1 locations and has to be placed in any one of them. How much load can cuckoo hashing handle before collisions prevent the successful assignment of the available items to the chosen locations? Practical evaluations of this method have shown that one can allocate a number of elements that is a large proportion of the size of the table, being very close to 1 even for small values of k such as 4 or 5. In this paper we show that there is a critical value for this proportion: with high probability, when the amount of available items is below this value, then these can be allocated successfully, but when it exceeds this value, the allocation becomes impossible. We give explicitly for each k>1 this critical value. This answers an open question posed by Mitzenmacher (ESA '09) and underpins theoretically the experimental results. Our proofs are based on the translation of the question into a hypergraph setting, and the study of the related typical properties of random k-uniform hypergraphs.

cs.DS↗

arXiv2009

3-Connected Cores In Random Planar Graphs

The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints. In this paper we focus on the structure of random biconnected planar graphs regarding the sizes of their 3-connected building blocks, which we call cores. In fact, we prove a general theorem regarding random biconnected graphs. If B_n is a graph drawn uniformly at random from a class B of labeled biconnected graphs, then we show that with probability 1-o(1) B_n belongs to exactly one of the following categories: (i) Either there is a unique giant core in B_n, that is, there is a 0 < c < 1 such that the largest core contains ~ cn vertices, and every other core contains at most n^a vertices, where 0 < a < 1; (ii) or all cores of B_n contain O(log n) vertices. Moreover, we find the critical condition that determines the category to which B_n belongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 and n. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particular c = 0.765... and a = 2/3.

math.CO↗

arXiv2009

The Speed of Broadcasting in Random Networks: Density Does Not Matter

Broadcasting algorithms are of fundamental importance for distributed systems engineering. In this paper we revisit the classical and well-studied push protocol for message broadcasting. Assuming that initially only one node has some piece of information, at each stage every one of the informed nodes chooses randomly and independently one of its neighbors and passes the message to it. The performance of the push protocol on a fully connected network, where each node is joined by a link to every other node, is very well understood. In particular, Frieze and Grimmett proved that with probability 1-o(1) the push protocol completes the broadcasting of the message within (1 +/- ε) (log_2 n + ln n) stages, where n is the number of nodes of the network. However, there are no tight bounds for the broadcast time on networks that are significantly sparser than the complete graph. In this work we consider random networks on n nodes, where every edge is present with probability p, independently of every other edge. We show that if p > f(n)ln n/ n, where f(n) is any function that tends to infinity as n grows, then the push protocol broadcasts the message within (1 +/- ε) (log_2 n + ln n) stages with probability 1-o(1). In other words, in almost every network of density d such that d > f(n)ln n, the push protocol broadcasts a message as fast as in a fully connected network. This is quite surprising in the sense that the time needed remains essentially unaffected by the fact that most of the links are missing.

math.CO↗

arXiv2008

Embeddings and Ramsey numbers of sparse k-uniform hypergraphs

Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In previous work, we proved the same result for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random' hypergraphs.

math.CO↗

arXiv2007

Percolation on sparse random graphs with given degree sequence

We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards we focus on site percolation where the vertices are retained with probability p. We establish critical values for p above which a giant component emerges in both cases. Moreover, we show that in fact these coincide. As a special case, our results apply to power law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such graphs.

math.CO↗

arXiv2007

The Evolution of the Mixing Rate

In this paper we present a study of the mixing time of a random walk on the largest component of a supercritical random graph, also known as the giant component. We identify local obstructions that slow down the random walk, when the average degree d is at most (ln n lnln n)^{1/2}, proving that the mixing time in this case is O((ln n/d)^2) asymptotically almost surely. As the average degree grows these become negligible and it is the diameter of the largest component that takes over, yielding mixing time O(ln n/ln d). We proved these results during the 2003-04 academic year. Similar results but for constant d were later proved independently by I. Benjamini, G. Kozma and N. Wormald.

math.CO↗

arXiv2006

3-Uniform hypergraphs of bounded degree have linear Ramsey numbers

Chvátal, Rödl, Szemerédi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an embedding lemma for 3-uniform hypergraphs of bounded maximum degree into suitable 3-uniform `pseudo-random' hypergraphs.

math.CO↗