Evolution in finite populations is often modelled using the classical Moran process. Over the last ten years this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population, is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original...

We present an overview of the applications of graph theory in power system modeling, dynamics, coherency, and control. We first study synchronization of generator dynamics using both nonlinear and small-signal representations of classical structure-preserving models of power systems in light of their network structure and the weights associated with the nodes and edges of the network graph. We ...

This book objective is to develop an algebraization of graph grammars. Equivalently, we study graph dynamics. From the point of view of a computer scientist, graph grammars are a natural generalization of Chomsky grammars for which a purely algebraic approach does not exist up to now. A Chomsky (or string) grammar is, roughly speaking, a precise description of a formal language (which in essenc...

Causal graph dynamics are transformations over graphs that capture two important symmetries of physics, namely causality and homogeneity. They can be equivalently defined as continuous and translation invariant transformations or functions induced by a local rule applied simultaneously on every vertex of the graph. Intrinsic universality is the ability of an instance of a model to simulate ever...

Graphs are often used in artificial intelligence as means for symbolic knowledge representation. A graph is nothing more than a collection of symbols connected to each other in some fashion. For example, in computer vision a graph with five nodes and some edges can represent a table – where nodes correspond to particular shape descriptors for legs and a top, and edges to particular spatial rela...

We consider graph states of an arbitrary number of particles undergoing generic decoherence. We present methods to obtain lower and upper bounds for the system's entanglement in terms of that of considerably smaller subsystems. For an important class of noisy channels, namely, the Pauli maps, these bounds coincide and thus provide the exact analytical expression for the entanglement evolution. ...

Studies of the Internet have typically focused either on the routing system, i.e. the paths chosen to reach a given destination, or on the evolution of traffic on a physical link. In this paper, we combine routing and traffic, and study for the first time the evolution of the traffic on the Internet topology. We rely on the traffic and routing data of a large transit provider, spanning almost a...

29 2 Suresh Dara, S.M. Hegde, V. Deva, S.B. Rao and T. Zaslavsky In 1966, Cummins introduced the “tree graph”: the tree graph T(G) 30 of a graph G (possibly infinite) has all its spanning trees as vertices, and 31 distinct such trees correspond to adjacent vertices if they differ in just one 32 edge, i.e., two spanning trees T1 and T2 are adjacent if T2 = T1 − e+ f for 33 some edges e ∈ T1 and ...

Graph dynamics arise naturally in many contexts. For instance in peer-to-peer networks, a participating peer may replace an existing connection with one neighbour by a new connection with a neighbour’s neighbour. Several such local rewiring rules have been proposed to ensure that peer-to-peer networks achieve good connectivity properties (e.g. high expansion) in equilibrium. However it has rema...

Skeletal animation is a concept that has been used in the areas of motion pictures and computer games to create realistic motion for the animation of articulated characters. Recent work has applied skeletal animation techniques from inverse kinematics to the field of graph interaction. The previous work introduced an interesting idea suggesting a skeletal graph interaction system would be intui...