In this paper, we tackle a challenging problem inherent in a series of applications: tracking the influential nodes in dynamic networks. Specifically, we model a dynamic network as a stream of edge weight updates. This general model embraces many practical scenarios as special cases, such as edge and node insertions, deletions as well as evolving weighted graphs. Under the popularly adopted linear threshold model and independent cascade model, we consider two essential versions of the problem: finding the nodes whose influences passing a user specified threshold and finding the top- @math most influential nodes. Our key idea is to use the polling-based methods and maintain a sample of random RR sets so that we can approximate the influence of nodes with provable quality guarantees. We develop an efficient algorithm that incrementally updates the sample random RR sets against network changes. We also design methods to determine the proper sample sizes for the two versions of the problem so that we can provide strong quality guarantees and, at the same time, be efficient in both space and time. In addition to the thorough theoretical results, our experimental results on five real network data sets clearly demonstrate the effectiveness and efficiency of our algorithms.
Aggarwal @cite_15 explored how to find a set of nodes that has the highest influence within a time window @math . They modeled influence propagation as a non-linear system which is very different from triggering models like the Linear Threshold model or the Independent Cascade model. The algorithm in @cite_15 is heuristic and the results produced do not come with any provable quality guarantee.
@cite_15
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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. Given a social network G and a constant @math , the influence maximization problem asks for k nodes in G that (directly and indirectly) influence the largest number of nodes under a pre-defined diffusion model. This problem finds important applications in viral marketing, and has been extensively studied in the literature. Existing algorithms for influence maximization, however, either trade approximation guarantees for practical efficiency, or vice versa. In particular, among the algorithms that achieve constant factor approximations under the prominent independent cascade (IC) model or linear threshold (LT) model, none can handle a million-node graph without incurring prohibitive overheads. This paper presents TIM, an algorithm that aims to bridge the theory and practice in influence maximization. On the theory side, we show that TIM runs in O((k+ l) (n+m) log n e2) expected time and returns a (1-1 e-e)-approximate solution with at least 1 - n-l probability. The time complexity of TIM is near-optimal under the IC model, as it is only a log n factor larger than the Ω(m + n) lower-bound established in previous work (for fixed k, l, and e). Moreover, TIM supports the triggering model, which is a general diffusion model that includes both IC and LT as special cases. On the practice side, TIM incorporates novel heuristics that significantly improve its empirical efficiency without compromising its asymptotic performance. We experimentally evaluate TIM with the largest datasets ever tested in the literature, and show that it outperforms the state-of-the-art solutions (with approximation guarantees) by up to four orders of magnitude in terms of running time. In particular, when k = 50, e = 0.2, and l = 1, TIM requires less than one hour on a commodity machine to process a network with 41.6 million nodes and 1.4 billion edges. This demonstrates that influence maximization algorithms can be made practical while still offering strong theoretical guarantees. Given a network represented by a graph @math G=V,E, we consider a dynamical process of influence diffusion in G that evolves as follows: Initially only the nodes of a given @math S⊆V are influenced; subsequently, at each round, the set of influenced nodes is augmented by all the nodes in the network that have a sufficiently large number of already influenced neighbors. The question is to determine a small subset of nodes Sa target set that can influence the whole network. This is a widely studied problem that abstracts many phenomena in the social, economic, biological, and physical sciences. It is known [6] that the above optimization problem is hard to approximate within a factor of @math 2log1-∈|V|, for any @math ∈>0. In this paper, we present a fast and surprisingly simple algorithm that exhibits the following features: 1 when applied to trees, cycles, or complete graphs, it always produces an optimal solution i.e., a minimum size target set; 2 when applied to arbitrary networks, it always produces a solution of cardinality matching the upper bound given in [1], and proved therein by means of the probabilistic method; 3 when applied to real-life networks, it always produces solutions that substantially outperform the ones obtained by previously published algorithms for which no proof of optimality or performance guarantee is known in any class of graphs. Influence maximization is the problem of finding a small set of most influential nodes in a social network so that their aggregated influence in the network is maximized. In this paper, we study influence maximization in the linear threshold model, one of the important models formalizing the behavior of influence propagation in social networks. We first show that computing exact influence in general networks in the linear threshold model is #P-hard, which closes an open problem left in the seminal work on influence maximization by Kempe, Kleinberg, and Tardos, 2003. As a contrast, we show that computing influence in directed a cyclic graphs (DAGs) can be done in time linear to the size of the graphs. Based on the fast computation in DAGs, we propose the first scalable influence maximization algorithm tailored for the linear threshold model. We conduct extensive simulations to show that our algorithm is scalable to networks with millions of nodes and edges, is orders of magnitude faster than the greedy approximation algorithm proposed by and its optimized versions, and performs consistently among the best algorithms while other heuristic algorithms not design specifically for the linear threshold model have unstable performances on different real-world networks.