Anthony Bonato

The game of cops and robbers on graphs

This book is the first and only one of its kind on the topic of Cops and Robbers games, and more generally, on the field of vertex pursuit games on graphs. The book is written in a lively and highly readable fashion, which should appeal to both senior undergraduates and experts in the field (and everyone in between). One of the main goals of the book is to bring together the key results in the field; as such, it presents structural, probabilistic, and algorithmic results on Cops and Robbers games. Several recent and new results are discussed, along with a comprehensive set of references. The book is suitable for self-study or as a textbook, owing in part to the over 200 exercises. The reader will gain insight into all the main directions of research in the field and will be exposed to a number of open problems.

Dominating Biological Networks

Proteins are essential macromolecules of life that carry out most cellular processes. Since proteins aggregate to perform function, and since protein-protein interaction (PPI) networks model these aggregations, one would expect to uncover new biology from PPI network topology. Hence, using PPI networks to predict protein function and role of protein pathways in disease has received attention. A debate remains open about whether network properties of “biologically central (BC)” genes (i.e., their protein products), such as those involved in aging, cancer, infectious diseases, or signaling and drug-targeted pathways, exhibit some topological centrality compared to the rest of the proteins in the human PPI network. To help resolve this debate, we design new network-based approaches and apply them to get new insight into biological function and disease. We hypothesize that BC genes have a topologically central (TC) role in the human PPI network. We propose two different concepts of topological centrality. We design a new centrality measure to capture complex wirings of proteins in the network that identifies as TC those proteins that reside in dense extended network neighborhoods. Also, we use the notion of domination and find dominating sets (DSs) in the PPI network, i.e., sets of proteins such that every protein is either in the DS or is a neighbor of the DS. Clearly, a DS has a TC role, as it enables efficient communication between different network parts. We find statistically significant enrichment in BC genes of TC nodes and outperform the existing methods indicating that genes involved in key biological processes occupy topologically complex and dense regions of the network and correspond to its “spine” that connects all other network parts and can thus pass cellular signals efficiently throughout the network. To our knowledge, this is the first study that explores domination in the context of PPI networks.

A survey of models of the web graph

The web graph has been the focus of much recent attention, with several stochastic models proposed to account for its various properties. A survey of these models is presented, focussing on the models which have been defined and analyzed rigorously.

A spatial web graph model with local influence regions

The web graph may be considered as embedded in a topic space, with a metric that expresses the extent to which web pages are related to each other. Using this assumption, we present a new model for the web and other complex networks, based on a spatial embedding of the nodes, called the Spatial Preferred Attachment (SPA) model. In the SPA model, nodes have influence regions of varying size, and new nodes may only link to a node if they fall within its influence region. We prove that our model gives a power law in-degree distribution, with exponent in (2, ∞) depending on the parameters, and with concentration for a wide range of in-degree values. We also show that the model allows for edges that span a large distance in the underlying space, modelling a feature often observed in real-world complex networks.

The capture time of a graph

We consider the game of Cops and Robbers played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to a new parameter, the capture time of a graph. While the capture time of a cop-win graph on n vertices is bounded above by n-3, half the number of vertices is sufficient for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have capture time within a small additive constant of the number of vertices. We consider the ratio of the capture time to the number of vertices, and extend this notion of capture time density to infinite graphs. For the infinite random graph, the capture time density can be any real number in [0,1]. We also consider the capture time when more than one cop is required to win. While the capture time can be calculated by a polynomial algorithm if the number k of cops is fixed, it is NP-complete to decide whether k cops can capture the robber in no more than t moves for every fixed t.

Pursuit-Evasion in Models of Complex Networks

Pursuit-evasion games, such as the game of Cops and Robbers, are a simplified model for network security. In this game, cops try to capture a robber loose on the vertices of the network. The minimum number of cops required to win on a graph G is its number. We present asymptotic results for the game of Cops and Robbers played in various stochastic network models, such as in G(n, p) with nonconstant p and in random power-law graphs. We find bounds for the cop number of G(n, p) for a large range p as a function of n. We prove that the cop number of random power-law graphs with n vertices is asymptotically almost surely Θ(n). The cop number of the core of random power-law graphs is investigated, and it is proved to be of smaller order than the order of the core.

A geometric model for on-line social networks

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 dimension parameter of an OSN may eventually be used as a tool to group users with similar attributes using only the link structure of the network.

Geometric Protean Graphs

Abstract We study the link structure of online social networks (OSNs) and introduce a new model for such networks that may help in inferring 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.

Infinite Limits of Copying Models of the Web Graph

Several stochastic models were proposed recently to model the dynamic evolution of the web graph. We study the infinite limits of the stochastic processes proposed to model the web graph when time goes to infinity. We prove that deterministic variations of the so-called copying model can lead to several nonisomorphic limits. Some models converge to the infinite random graph R, while the convergence of other models is sensitive to initial conditions or minor changes in the rules of the model. We explain how limits of the copying model of the web graph share several properties with R that seem to reflect known properties of the web graph.

On an adjacency property of almost all graphs

Abstract A graph is called n-existentially closed or n-e.c. if it satisfies the following adjacency property: for every n-element subset S of the vertices, and for every subset T of S, there is a vertex not in S which is joined to all of T and to none of S ⧹ T . The unique countable random graph is known to be n-e.c. for all n. Equivalently, for any fixed n, almost all finite graphs are n-e.c. However, few examples of n-e.c. graphs are known other than large Paley graphs and examples of 2-e.c. graphs given in (Cacetta, et al., Ars Combin . 19 (1985) 287–294). An n-e.c. graph is critical if deleting any vertex leaves a graph which is not n-e.c. We classify the 1-e.c. critical graphs. We construct 2-e.c. critical graphs of each order ⩾9, and describe a 2-e.c.-preserving operation: replication of an edge. We also examine which of the well-known binary operations on graphs preserve n-e.c. for n =1,2,3.