arXiv subjects

Pawel Pralat

Publications and source records attributed to Pawel Pralat.

At least 73 records · Page 4Linked to original sources

arXiv2015

Almost all 5-regular graphs have a 3-flow

Tutte conjectured in 1972 that every 4-edge connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge-connected graph has an edge orientation in which every out-degree is either 1 or 4. We show that the assertion of the conjecture holds asymptotically almost surely for random 5-regular graphs. It follows that the conjecture holds for almost all 4-edge connected 5-regular graphs.

math.CO↗

arXiv2014

Meyniel's conjecture holds for random graphs

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 $C \sqrt{|V(G)|}$. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. 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 $d = d(n) \ge 3$.

math.CO↗

arXiv2014

Firefighting on square, hexagonal, and triangular grids

In this paper, we consider the \emph{firefighter problem} on a graph $G=(V,E)$ that is either finite or infinite. Suppose that a fire breaks out at a given vertex $v \in V$. In each subsequent time unit, a firefighter protects one vertex which is not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible (if $G$ is finite) or to stop the fire from spreading (for an infinite case). The surviving rate $ρ(G)$ of a finite graph $G$ is defined as the expected percentage of vertices that can be saved when a fire breaks out at a vertex of $G$ that is selected uniformly random. For a finite square grid $P_n \square P_n$, we show that $5/8 + o(1) \le ρ(P_n \square P_n) \le 67243/105300 + o(1)$ (leaving the gap smaller than 0.014) and conjecture that the surviving rate is asymptotic to 5/8. We define the surviving rate for infinite graphs and prove it to be 1/4 for the infinite square grid, even in the case of finitely many initial fires. For the infinite hexagonal grid we provide a winning strategy if two additional vertices can be protected at any point of the process, and we conjecture that the firefighter has no strategy to stop the fire without additional help. We also show how the speed of the spreading fire can be reduced by a constant factor.

math.CO↗

arXiv2014

Revolutionaries and spies on random graphs

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of $r$ revolutionaries tries to hold an unguarded meeting consisting of $m$ revolutionaries. A team of $s$ spies wants to prevent this forever. For given $r$ and $m$, the minimum number of spies required to win on a graph $G$ is the spy number $σ(G,r,m)$. We present asymptotic results for the game played on random graphs $G(n,p)$ for a large range of $p = p(n), r=r(n)$, and $m=m(n)$. The behaviour of the spy number is analyzed completely for dense graphs (that is, graphs with average degree at least $n^{1/2+\eps}$ for some $\eps > 0$). For sparser graphs, some bounds are provided.

math.CO↗

arXiv2014

Almost all k-cop-win graphs contain a dominating set of cardinality k

We consider $k$-cop-win graphs in the binomial random graph $G(n,1/2).$ It is known that almost all cop-win graphs contain a universal vertex. We generalize this result and prove that for every $k \in N$, almost all $k$-cop-win graphs contain a dominating set of cardinality $k$. From this it follows that the asymptotic number of labelled $k$-cop-win graphs of order $n$ is equal to $(1+o(1)) (1-2^{-k})^{-k} {n \choose k} 2^{n^2/2 - (1/2-\log_2(1-2^{-k})) n}$.

math.CO↗

arXiv2014

Chasing robbers on random geometric graphs---an alternative approach

We study the vertex pursuit game of \emph{Cops and Robbers}, in which cops try to capture a robber 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 $G$. We focus on $G_{d}(n,r)$, a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^d$, and two vertices are adjacent if the Euclidean distance between them is at most $r$. The main result is that if $r^{3d-1}>c_d \frac{\log n}{n}$ then the cop number is $1$ with probability that tends to $1$ as $n$ tends to infinity. The case $d=2$ was proved earlier and independently in \cite{bdfm}, using a different approach. Our method provides a tight $O(1/r^2)$ upper bound for the number of rounds needed to catch the robber.

math.CO↗

arXiv2014

On the hyperbolicity of random graphs

Let $G=(V,E)$ be a connected graph with the usual (graph) distance metric $d:V \times V \to N \cup \{0 \}$. Introduced by Gromov, $G$ is $δ$-hyperbolic if for every four vertices $u,v,x,y \in V$, the two largest values of the three sums $d(u,v)+d(x,y), d(u,x)+d(v,y), d(u,y)+d(v,x)$ differ by at most $2δ$. In this paper, we determinate the value of this hyperbolicity for most binomial random graphs.

math.CO↗

arXiv2014

An alternative proof of the linearity of the size-Ramsey number of paths

The size Ramsey number $\hat{r}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In 1983, Beck provided a beautiful argument that shows that $\hat{r}(P_n)$ is linear, solving a problem of Erdős. In this short note, we provide an alternative but elementary proof of this fact that actually gives a better bound, namely, $\hat{r}(P_n) < 137n$ for $n$ sufficiently large.

math.CO↗

arXiv2014

The Role of Visibility in Pursuit / Evasion Games

The cops-and-robber (CR) game has been used in mobile robotics as a discretized model (played on a graph G) of pursuit/evasion problems. The "classic" CR version is a perfect information game: the cops' (pursuer's) location is always known to the robber (evader) and vice versa. Many variants of the classic game can be defined: the robber can be invisible and also the robber can be either adversarial (tries to avoid capture) or drunk (performs a random walk). Furthermore, the cops and robber can reside in either nodes or edges of G. Several of these variants are relevant as models or robotic pursuit / evasion. In this paper, we first define carefully several of the variants mentioned above and related quantities such as the cop number and the capture time. Then we introduce and study the cost of visibility (COV), a quantitative measure of the increase in difficulty (from the cops' point of view) when the robber is invisible. In addition to our theoretical results, we present algorithms which can be used to compute capture times and COV of graphs which are analytically intractable. Finally, we present the results of applying these algorithms to the numerical computation of COV.

cs.DM↗

arXiv2013

The acquaintance time of (percolated) random geometric graphs

In this paper, we study the acquaintance time $\AC(G)$ defined for a connected graph $G$. We focus on $\G(n,r,p)$, a random subgraph of a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^2$, and two vertices are adjacent with probability $p$ if the Euclidean distance between them is at most $r$. We present asymptotic results for the acquaintance time of $\G(n,r,p)$ for a wide range of $p=p(n)$ and $r=r(n)$. In particular, we show that with high probability $\AC(G) = Θ(r^{-2})$ for $G \in \G(n,r,1)$, the "ordinary" random geometric graph, provided that $πn r^2 - \ln n \to \infty$ (that is, above the connectivity threshold). For the percolated random geometric graph $G \in \G(n,r,p)$, we show that with high probability $\AC(G) = Θ(r^{-2} p^{-1} \ln n)$, provided that $p n r^2 \geq n^{1/2+\eps}$ and $p < 1-\eps$ for some $\eps>0$.

math.CO↗

arXiv2013

Models of on-line social networks

We present a deterministic model for on-line social networks (OSNs) based on transitivity and local knowledge in social interactions. In the Iterated Local Transitivity (ILT) model, at each time-step and for every existing node $x$, a new node appears which joins to the closed neighbour set of $x.$ The ILT model provably satisfies a number of both local and global properties that were observed in OSNs and other real-world complex networks, such as a densification power law, decreasing average distance, and higher clustering than in random graphs with the same average degree. Experimental studies of social networks demonstrate poor expansion properties as a consequence of the existence of communities with low number of inter-community edges. Bounds on the spectral gap for both the adjacency and normalized Laplacian matrices are proved for graphs arising from the ILT model, indicating such bad expansion properties. The cop and domination number are shown to remain the same as the graph from the initial time-step $G_0$, and the automorphism group of $G_0$ is a subgroup of the automorphism group of graphs generated at all later time-steps. A randomized version of the ILT model is presented, which exhibits a tuneable densification power law exponent, and maintains several properties of the deterministic model.

cs.SI↗

arXiv2013

Independence densities of hypergraphs

We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as $F$-free densities of graphs for a given graph $F.$ In the case of $k$-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real number in $[0,1].$ Finally, we extend the notion of independence density via independence polynomials.

math.CO↗

arXiv2013

The robber strikes back

We consider the new game of Cops and Attacking Robbers, which is identical to the usual Cops and Robbers game except that if the robber moves to a vertex containing a single cop, then that cop is removed from the game. We study the minimum number of cops needed to capture a robber on a graph $G$, written $cc(G)$. We give bounds on $cc(G)$ in terms of the cop number of $G$ in the classes of bipartite graphs and diameter two, $K_{1,m}$-free graphs.

math.CO↗

arXiv2012

Geometric Graph Properties of the Spatial Preferred Attachment model

The spatial preferred attachment (SPA) model is a model for networked information spaces such as domains of the World Wide Web, citation graphs, and on-line social networks. It uses a metric space to model the hidden attributes of the vertices. Thus, vertices are elements of a metric space, and link formation depends on the metric distance between vertices. We show, through theoretical analysis and simulation, that for graphs formed according to the SPA model it is possible to infer the metric distance between vertices from the link structure of the graph. Precisely, the estimate is based on the number of common neighbours of a pair of vertices, a measure known as {\sl co-citation}. To be able to calculate this estimate, we derive a precise relation between the number of common neighbours and metric distance. We also analyze the distribution of {\sl edge lengths}, where the length of an edge is the metric distance between its end points. We show that this distribution has three different regimes, and that the tail of this distribution follows a power law.

cs.SI↗

arXiv2012

Cops and Invisible Robbers: the Cost of Drunkenness

We examine a version of the Cops and Robber (CR) game in which the robber is invisible, i.e., the cops do not know his location until they capture him. Apparently this game (CiR) has received little attention in the CR literature. We examine two variants: in the first the robber is adversarial (he actively tries to avoid capture); in the second he is drunk (he performs a random walk). Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD), which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being the expected capture times in the adversarial and drunk CiR variants, respectively. We show that these capture times are well defined, using game theory for the adversarial case and partially observable Markov decision processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD for several special graph families such as $d$-regular trees, give some bounds for grids, and provide general upper and lower bounds for general classes of graphs. We also give an infinite family of graphs showing that iCOD can be arbitrarily close to any value in [2,infinty). Finally, we briefly examine one more CiR variant, in which the robber is invisible and "infinitely fast"; we argue that this variant is significantly different from the Graph Search game, despite several similarities between the two games.

cs.DM↗

arXiv2011

Geometric protean graphs

We study the link structure of on-line social networks (OSNs), and introduce a new model for such networks which may help infer their hidden underlying reality. In the geo-protean (GEO-P) model for OSNs nodes are identified with points in Euclidean space, and edges are stochastically generated by a mixture of the relative distance of nodes and a ranking function. With high probability, the GEO-P model generates graphs satisfying many observed properties of OSNs, such as power law degree distributions, the small world property, densification power law, and bad spectral expansion. We introduce the dimension of an OSN based on our model, and examine this new parameter using actual OSN data. We discuss how the geo-protean model may eventually be used as a tool to group users with similar attributes using only the link structure of the network.

physics.soc-ph↗

arXiv2011

Some remarks on cops and drunk robbers

The cops and robbers game has been extensively studied under the assumption of optimal play by both the cops and the robbers. In this paper we study the problem in which cops are chasing a drunk robber (that is, a robber who performs a random walk) on a graph. Our main goal is to characterize the "cost of drunkenness." Specifically, we study the ratio of expected capture times for the optimal version and the drunk robber one. We also examine the algorithmic side of the problem; that is, how to compute near-optimal search schedules for the cops. Finally, we present a preliminary investigation of the invisible robber game and point out differences between this game and graph search.

cs.DM↗

arXiv2009

Rank-based attachment leads to power law graphs

We investigate the degree distribution resulting from graph generation models based on rank-based attachment. In rank-based attachment, all vertices are ranked according to a ranking scheme. The link probability of a given vertex is proportional to its rank raised to the power -a, for some a in (0,1). Through a rigorous analysis, we show that rank-based attachment models lead to graphs with a power law degree distribution with exponent 1+1/a whenever vertices are ranked according to their degree, their age, or a randomly chosen fitness value. We also investigate the case where the ranking is based on the initial rank of each vertex; the rank of existing vertices only changes to accommodate the new vertex. Here, we obtain a sharp threshold for power law behaviour. Only if initial ranks are biased towards lower ranks, or chosen uniformly at random, we obtain a power law degree distribution with exponent 1+1/a. This indicates that the power law degree distribution often observed in nature can be explained by a rank-based attachment scheme, based on a ranking scheme that can be derived from a number of different factors; the exponent of the power law can be seen as a measure of the strength of the attachment.

math.CO↗