Abstract We study the two-player safe game of Competitive Diffusion, a game-theoretic model for the diffusion of technologies or influence through a social network. In game theory, safe strategies are mixed strategies with a minimum expected gain against unknown strategies of the opponents. Safe strategies for competitive diffusion lead to maximum spread of influence in the presence of uncertainty about the other players. We study the safe game on two specific classes of trees, spiders and complete trees, and give tight bounds on the minimum expected gain. We then use these results to give an algorithm that suggests a safe strategy for a player on any tree. We test this algorithm on randomly generated trees and show that it finds strategies that are close to optimal.
In cascade models, as a vertex becomes activated, it activates each of its neighbours with a given probability. The most well-known is the Independent Cascade Model (see @cite_17 and @cite_9 ). In this model, we also start with an initial set of activated vertices. Here, each edge @math is assigned a probability @math . If vertex @math becomes activated, its neighbour @math will become activated in the next round with probability @math . The spread of influence in competitive diffusion can be seen as a cascade model where the activation probability equals 1.
@cite_9 @cite_17
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In this paper we consider a fundamental problem in the area of viral marketing, called T ARGET S ET S ELECTION problem. In a a viral marketing setting, social networks are modeled by graphs with potential customers of a new product as vertices and friend relationships as edges, where each vertex @math is assigned a threshold value @math . The thresholds represent the different latent tendencies of customers (vertices) to buy the new product when their friend (neighbors) do. Consider a repetitive process on social network @math where each vertex @math is associated with two states, active and inactive, which indicate whether @math is persuaded into buying the new product. Suppose we are given a target set @math . Initially, all vertices in @math are inactive. At time step 0, we choose all vertices in @math to become active. Then, at every time step @math , all vertices that were active in time step @math remain active, and we activate any vertex @math if at least @math of its neighbors were active at time step @math . The activation process terminates when no more vertices can get activated. We are interested in the following optimization problem, called T ARGET S ET S ELECTION : Finding a target set @math of smallest possible size that activates all vertices of @math . There is an important and well-studied threshold called strict majority threshold, where for every vertex @math in @math we have @math and @math is the degree of @math in @math . In this paper, we consider the T ARGET S ET S ELECTION problem under strict majority thresholds and focus on three popular regular network structures: cycle permutation graphs, generalized Petersen graphs and torus cordalis. In this paper, we consider the problem of maximizing the spread of influence through a social network. Given a graph with a threshold value thr(v) attached to each vertex v, the spread of influence is modeled as follows: A vertex v becomes ''active'' (influenced) if at least thr(v) of its neighbors are active. In the corresponding optimization problem the objective is then to find a fixed number k of vertices to activate such that the number of activated vertices at the end of the propagation process is maximum. We show that this problem is strongly inapproximable in time f(k)@?n^O^(^1^), for some function f, even for very restrictive thresholds. In the case that the threshold of each vertex equals its degree, we prove that the problem is inapproximable in polynomial time and it becomes r(n)-approximable in time f(k)@?n^O^(^1^), for some function f, for any strictly increasing function r. Moreover, we show that the decision version parameterized by k is W[1]-hard but becomes fixed-parameter tractable on bounded degree graphs. We consider the popular and well-studied push model, which is used to spread information in a given network with n vertices. Initially, some vertex owns a rumour and passes it to one of its neighbours, which is chosen randomly. In each of the succeeding rounds, every vertex that knows the rumour informs a random neighbour. It has been shown on various network topologies that this algorithm succeeds in spreading the rumour within O(log n) rounds. However, many studies are quite coarse and involve huge constants that do not allow for a direct comparison between different network topologies. In this paper, we analyse the push model on several important families of graphs, and obtain tight runtime estimates. We first show that, for any almost-regular graph on n vertices with small spectral expansion, rumour spreading completes after log2n + log n+o(log n) rounds with high probability. This is the first result that exhibits a general graph class for which rumour spreading is essentially as fast as on complete graphs. Moreover, for the random graph G(n,p) with p=c log n n, where c > 1, we determine the runtime of rumour spreading to be log2n + γ (c)log n with high probability, where γ(c) = clog(c (c−1)). In particular, this shows that the assumption of almost regularity in our first result is necessary. Finally, for a hypercube on n=2d vertices, the runtime is with high probability at least (1+β) ⋅ (log2n + log n), where β > 0. This reveals that the push model on hypercubes is slower than on complete graphs, and thus shows that the assumption of small spectral expansion in our first result is also necessary. In addition, our results combined with the upper bound of O(log n) for the hypercube (see [11]) imply that the push model is faster on hypercubes than on a random graph G(n, clog n n), where c is sufficiently close to 1. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph @math from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and --in particular-- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing.