Given a network represented by a weighted directed graph G, we consider the problem of finding a bounded cost set of nodes S such that the influence spreading from S in G, within a given time bound, is as large as possible. The dynamic that governs the spread of influence is the following: initially only elements in S are influenced; subsequently at each round, the set of influenced elements is augmented by all nodes in the network that have a sufficiently large number of already influenced neighbors. We prove that the problem is NP-hard, even in simple networks like complete graphs and trees. We also derive a series of positive results. We present exact pseudo-polynomial time algorithms for general trees, that become polynomial time in case the trees are unweighted. This last result improves on previously published results. We also design polynomial time algorithms for general weighted paths and cycles, and for unweighted complete graphs.
 This paper provides an overview of recent developments concerning the process of local majority voting in graphs, and its basic properties, from graph theoretic and algorithmic standpoints. We consider a well-known distributed colouring game played on a simple connected graph: initially, each vertex is coloured black or white; at each round, each vertex simultaneously recolours itself by the colour of the simple (strong) majority of its neighbours. A set of vertices M is said to be a dynamo, if starting the game with only the vertices of M coloured black, the computation eventually reaches an all-black configuration.The importance of this game follows from the fact that it models the spread of faults in point-to-point systems with majority-based voting; in particular, dynamos correspond to those sets of initial failures which will lead the entire system to fail. Investigations on dynamos have been extensive but restricted to establishing tight bounds on the size (i.e., how small a dynamic monopoly might be).In this paper we start to study dynamos systematically with respect to both the size and the time (i.e., how many rounds are needed to reach all-black configuration) in various models and topologies.We derive tight tradeoffs between the size and the time for a number of regular graphs, including rings, complete d-ary trees, tori, wrapped butterflies, cube connected cycles and hypercubes. In addition, we determine optimal size bounds of irreversible dynamos for butterflies and shuffle-exchange using simple majority and for DeBruijn using strong majority rules. Finally, we make some observations concerning irreversible versus reversible monotone models and slow complete computations from minimal dynamos. Over the past decade there has been a growing public fascination with the complex connectedness of modern society. This connectedness is found in many incarnations: in the rapid growth of the Internet, in the ease with which global communication takes place, and in the ability of news and information as well as epidemics and financial crises to spread with surprising speed and intensity. These are phenomena that involve networks, incentives, and the aggregate behavior of groups of people; they are based on the links that connect us and the ways in which our decisions can have subtle consequences for others. This introductory undergraduate textbook takes an interdisciplinary look at economics, sociology, computing and information science, and applied mathematics to understand networks and behavior. It describes the emerging field of study that is growing at the interface of these areas, addressing fundamental questions about how the social, economic, and technological worlds are connected.