arXiv subjects

Abbas Mehrabian

Publications and source records attributed to Abbas Mehrabian.

32 records · Page 2Linked to original sources

arXiv2015

On the push&pull protocol for rumour spreading

The asynchronous push&pull protocol, a randomized distributed algorithm for spreading a rumour in a graph $G$, works as follows. Independent Poisson clocks of rate 1 are associated with the vertices of $G$. Initially, one vertex of $G$ knows the rumour. Whenever the clock of a vertex $x$ rings, it calls a random neighbour $y$: if $x$ knows the rumour and $y$ does not, then $x$ tells $y$ the rumour (a push operation), and if $x$ does not know the rumour and $y$ knows it, $y$ tells $x$ the rumour (a pull operation). The average spread time of $G$ is the expected time it takes for all vertices to know the rumour, and the guaranteed spread time of $G$ is the smallest time $t$ such that with probability at least $1-1/n$, after time $t$ all vertices know the rumour. The synchronous variant of this protocol, in which each clock rings precisely at times $1,2,\dots$, has been studied extensively. We prove the following results for any $n$-vertex graph: In either version, the average spread time is at most linear even if only the pull operation is used, and the guaranteed spread time is within a logarithmic factor of the average spread time, so it is $O(n\log n)$. In the asynchronous version, both the average and guaranteed spread times are $Ω(\log n)$. We give examples of graphs illustrating that these bounds are best possible up to constant factors. We also prove theoretical relationships between the guaranteed spread times in the two versions. Firstly, in all graphs the guaranteed spread time in the asynchronous version is within an $O(\log n)$ factor of that in the synchronous version, and this is tight. Next, we find examples of graphs whose asynchronous spread times are logarithmic, but the synchronous versions are polynomially large. Finally, we show for any graph that the ratio of the synchronous spread time to the asynchronous spread time is $O(n^{2/3})$.

cs.DC↗

arXiv2014

Randomized Rumor Spreading in Poorly Connected Small-World Networks

Push-Pull is a well-studied round-robin rumor spreading protocol defined as follows: initially a node knows a rumor and wants to spread it to all nodes in a network quickly. In each round, every informed node sends the rumor to a random neighbor, and every uninformed node contacts a random neighbor and gets the rumor from her if she knows it. We analyze this protocol on random $k$-trees, a class of power law graphs, which are small-world and have large clustering coefficients, built as follows: initially we have a $k$-clique. In every step a new node is born, a random $k$-clique of the current graph is chosen, and the new node is joined to all nodes of the $k$-clique. When $k>1$ is fixed, we show that if initially a random node is aware of the rumor, then with probability $1-o(1)$ after $O\left((\log n)^{1+2/k}\cdot\log\log n\cdot f(n)\right)$ rounds the rumor propagates to $n-o(n)$ nodes, where $n$ is the number of nodes and $f(n)$ is any slowly growing function. Since these graphs have polynomially small conductance, vertex expansion $O(1/n)$ and constant treewidth, these results demonstrate that Push-Pull can be efficient even on poorly connected networks. On the negative side, we prove that with probability $1-o(1)$ the protocol needs at least $Ω\left(n^{(k-1)/(k^2+k-1)}/f^2(n)\right)$ rounds to inform all nodes. This exponential dichotomy between time required for informing almost all and all nodes is striking. Our main contribution is to present, for the first time, a natural class of random graphs in which such a phenomenon can be observed. Our technique for proving the upper bound carries over to a closely related class of graphs, random $k$-Apollonian networks, for which we prove an upper bound of $O\left((\log n)^{c_k}\cdot\log\log n\cdot f(n)\right)$ rounds for informing $n-o(n)$ nodes with probability $1-o(1)$ when $k>2$ is fixed. Here, $c_k=(k^2-3)/(k-1)^2<1 + 2/k$.

cs.SI↗

arXiv2014

Justifying the small-world phenomenon via random recursive trees

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three-fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank-based selection model, the Aiello-Chung-Lu models, the generalized linear preference model, directed scale-free graphs, the Cooper-Frieze model, and random unordered increasing $k$-trees. Our results shed light on why the small-world phenomenon is observed in so many real-world graphs.

cs.DM↗

arXiv2014

It's a Small World for Random Surfers

We prove logarithmic upper bounds for the diameters of the random-surfer Webgraph model and the PageRank-based selection Webgraph model, confirming the small world phenomenon holds for them. In the special case when the generated graph is a tree, we provide close lower and upper bounds for the diameters of both models.

cs.DM↗

arXiv2014

Longest paths in random Apollonian networks and largest $r$-ary subtrees of random $d$-ary recursive trees

Let $r$ and $d$ be positive integers with $r<d$. Consider a random $d$-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it $d$ newly created offspring. Let ${\mathcal T}_t$ be the tree produced after $t$ steps. We show that there exists a fixed $δ<1$ depending on $d$ and $r$ such that almost surely for all large $t$, every $r$-ary subtree of ${\mathcal T}_t$ has less than $t^δ$ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After $t$ steps, we obtain a random triangulated plane graph with $t+3$ vertices, which is called a random Apollonian network. We prove that there exists a fixed $δ<1$, such that eventually every path in this graph has length less than $t^δ$, which verifies a conjecture of Cooper and Frieze.

math.PR↗

arXiv2013

On the Longest Paths and the Diameter in Random Apollonian Networks

We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n-3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) every path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a path of length (2n-5)^{log 2/log 3}, and that the expected length of its longest path is Omega(n^0.88). Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to c log n, where c \approx 1.668 is the solution of an explicit equation.

math.CO↗

arXiv2012

On the Maximum Density of Graphs with Good Edge-Labellings

A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on n vertices that admits a good edge-labelling has at most n log_2(n)/2 edges, and that this bound is tight for infinitely many values of n. Thus we significantly improve on the previously best known bounds. The main tool of the proof is a combinatorial lemma which might be of independent interest. For every n we also construct an n-vertex graph that admits a good edge-labelling and has n log_2(n)/2 - O(n) edges.

math.CO↗

arXiv2012

On a Bounded Budget Network Creation Game

We consider a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In our model, each link has unit price and each agent tries to minimize its cost, which is either its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two versions of the game are studied: in the MAX version, the cost incurred to a vertex is the maximum distance between the vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between the vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. We take the social cost of the created network to be its diameter, and next we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. When the sum of players' budgets is n-1, the equilibrium graphs are always trees, and we prove that their maximum diameter is Theta(n) and Theta(log n) in MAX and SUM versions, respectively. When each vertex has unit budget (i.e. can establish link to just one vertex), the diameter of any equilibrium graph in either version is Theta(1). We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Omega(sqrt(log n)). This interesting (and perhaps counter-intuitive) result shows that increasing the budgets may increase the diameter of equilibrium graphs and hence deteriorate the network structure. Then we prove that every equilibrium graph in the SUM version has diameter 2^O(sqrt(log n)). Finally, we show that if the budget of each player is at least k, then every equilibrium graph in the SUM version is k-connected or has diameter smaller than 4.

cs.GT↗

arXiv2012

On the Stretch Factor of Randomly Embedded Random Graphs

We consider a random graph G(n,p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u,v} of vertices have a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all u,v in V. We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for p not too close to 0 or 1, these bounds are best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if n(1-p) tends to 0, answering a question of O'Rourke.

cs.CG↗

arXiv2011

On the density of nearly regular graphs with a good edge-labelling

A good edge-labelling of a simple graph is a labelling of its edges with real numbers such that, for any ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. Say a graph is good if it admits a good edge-labelling, and is bad otherwise. Our main result is that any good n-vertex graph whose maximum degree is within a constant factor of its average degree (in particular, any good regular graph) has at most n^{1+o(1)} edges. As a corollary, we show that there are bad graphs with arbitrarily large girth, answering a question of Bode, Farzad and Theis. We also prove that for any Delta, there is a g such that any graph with maximum degree at most Delta and girth at least g is good.

math.CO↗

arXiv2011

Cops and Robber Game with a Fast Robber on Expander Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c_{infty}(G) denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c_{infty}(G)=1, and give an O(|V(G)|^2) algorithm for their detection. We prove a lower bound for c_{infty} of expander graphs, and use it to prove three things. The first is that if np > 4.2 log n then the random graph G = G(n,p) asymptotically almost surely has e1/p < c_{infty}(G) < e2 log (np)/p, for suitable constants e1 and e2. The second is that a fixed-degree random regular graph G with n vertices asymptotically almost surely has c_{infty}(G) = Theta(n). The third is that if G is a Cartesian product of m paths, then n / 4km^2 < c_{infty}(G) < n / k, where n=|V(G)| and k is the number of vertices of the longest path.

math.CO↗

arXiv2011

Lower Bounds for the Cop Number When the Robber is Fast

We consider a variant of the Cops and Robbers game where the robber can move t edges at a time, and show that in this variant, the cop number of a d-regular graph with girth larger than 2t+2 is Omega(d^t). By the known upper bounds on the order of cages, this implies that the cop number of a connected n-vertex graph can be as large as Omega(n^{2/3}) if t>1, and Omega(n^{4/5}) if t>3. This improves the Omega(n^{(t-3)/(t-2)}) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, 2011) when 1<t<7. We also conjecture a general upper bound O(n^{t/t+1}) for the cop number in this variant, generalizing Meyniel's conjecture.

math.CO↗

arXiv2011

Cops and Robber Game with a Fast Robber on Interval, Chordal, and Planar Graphs

We consider a variant of the Cops and Robber game, introduced by Fomin, Golovach, Kratochvil, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. We study this game on interval graphs, chordal graphs, planar graphs, and hypercube graphs. Let c_{\infty}(G) denote the number of cops needed to capture the robber in graph G in this variant. We show that if G is an interval graph, then c_{\infty}(G) = O(sqrt(|V(G)|)), and we give a polynomial-time 3-approximation algorithm for finding c_{\infty}(G) in interval graphs. We prove that for every n there exists an n-vertex chordal graph G with c_{\infty}(G) = Omega(n / \log n). Let tw(G) and Delta(G) denote the treewidth and the maximum degree of G, respectively. We prove that for every G, tw(G) + 1 \leq (Delta(G) + 1) c_{\infty}(G). Using this lower bound for c_{\infty}(G), we show two things. The first is that if G is a planar graph (or more generally, if G does not have a fixed apex graph as a minor), then c_{\infty}(G) = Theta(tw(G)). This immediately leads to an O(1)-approximation algorithm for computing c_{\infty} for planar graphs. The second is that if G is the m-hypercube graph, then there exist constants eta1, eta2>0 such that (eta1) 2^m / (m sqrt(m)) \leq c_{\infty}(G) \leq (eta2) 2^m / m.

math.CO↗

arXiv2010

The capture time of grids

We consider the game of Cops and Robber played on the Cartesian product of two trees. Assuming the players play perfectly, it is shown that if there are two cops in the game, then the length of the game (known as the 2-capture time of the graph) is equal to half the diameter of the graph. In particular, the 2-capture time of the m x n grid is proved to be floor ((m+n-2)/2).

math.CO↗