Think Globally, Act Locally: On the Optimal Seeding for Nonsubmodular Influence Maximization

We study the r-complex contagion influence maximization problem. In the influence maximization problem, one chooses a fixed number of initial seeds in a social network to maximize the spread of their influence. In the r-complex contagion model, each uninfected vertex in the network becomes infected if it has at least r infected neighbors. In this paper, we focus on a random graph model named the stochastic hierarchical blockmodel, which is a special case of the well-studied stochastic blockmodel. When the graph is not exceptionally sparse, in particular, when each edge appears with probability omega (n^{-(1+1/r)}), under certain mild assumptions, we prove that the optimal seeding strategy is to put all the seeds in a single community. This matches the intuition that in a nonsubmodular cascade model placing seeds near each other creates synergy. However, it sharply contrasts with the intuition for submodular cascade models (e.g., the independent cascade model and the linear threshold model) in which nearby seeds tend to erode each others\u27 effects. Finally, we show that this observation yields a polynomial time dynamic programming algorithm which outputs optimal seeds if each edge appears with a probability either in omega (n^{-(1+1/r)}) or in o (n^{-2})

Beyond Worst-Case (In)approximability of Nonsubmodular Influence Maximization

We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds, formally referred to as the influence maximization problem. It admits a $(1-1/e)$-factor approximation algorithm if the influence function is submodular. Otherwise, in the worst case, the problem is NP-hard to approximate to within a factor of $N^{1-\varepsilon}$. This paper studies whether this worst-case hardness result can be circumvented by making assumptions about either the underlying network topology or the cascade model. All of our assumptions are motivated by many real life social network cascades. First, we present strong inapproximability results for a very restricted class of networks called the (stochastic) hierarchical blockmodel, a special case of the well-studied (stochastic) blockmodel in which relationships between blocks admit a tree structure. We also provide a dynamic-program based polynomial time algorithm which optimally computes a directed variant of the influence maximization problem on hierarchical blockmodel networks. Our algorithm indicates that the inapproximability result is due to the bidirectionality of influence between agent-blocks. Second, we present strong inapproximability results for a class of influence functions that are "almost" submodular, called 2-quasi-submodular. Our inapproximability results hold even for any 2-quasi-submodular $f$ fixed in advance. This result also indicates that the "threshold" between submodularity and nonsubmodularity is sharp, regarding the approximability of influence maximization.Comment: 53 pages, 20 figures; Conference short version - WINE 2017: The 13th Conference on Web and Internet Economics; Journal full version - ACM: Transactions on Computation Theory, 201

Parameterized Inapproximability of Target Set Selection and Generalizations

In this paper, we consider the Target Set Selection problem: given a graph and a threshold value $thr(v)$ for any vertex $v$ of the graph, find a minimum size vertex-subset to "activate" s.t. all the vertices of the graph are activated at the end of the propagation process. A vertex $v$ is activated during the propagation process if at least $thr(v)$ of its neighbors are activated. This problem models several practical issues like faults in distributed networks or word-to-mouth recommendations in social networks. We show that for any functions $f$ and $\rho$ this problem cannot be approximated within a factor of $\rho(k)$ in $f(k) \cdot n^{O(1)}$ time, unless FPT = W[P], even for restricted thresholds (namely constant and majority thresholds). We also study the cardinality constraint maximization and minimization versions of the problem for which we prove similar hardness results

The Complexity of Finding Effectors

The NP-hard EFFECTORS problem on directed graphs is motivated by applications in network mining, particularly concerning the analysis of probabilistic information-propagation processes in social networks. In the corresponding model the arcs carry probabilities and there is a probabilistic diffusion process activating nodes by neighboring activated nodes with probabilities as specified by the arcs. The point is to explain a given network activation state as well as possible by using a minimum number of "effector nodes"; these are selected before the activation process starts. We correct, complement, and extend previous work from the data mining community by a more thorough computational complexity analysis of EFFECTORS, identifying both tractable and intractable cases. To this end, we also exploit a parameterization measuring the "degree of randomness" (the number of "really" probabilistic arcs) which might prove useful for analyzing other probabilistic network diffusion problems as well.Comment: 28 page

Flow-based Influence Graph Visual Summarization

Visually mining a large influence graph is appealing yet challenging. People are amazed by pictures of newscasting graph on Twitter, engaged by hidden citation networks in academics, nevertheless often troubled by the unpleasant readability of the underlying visualization. Existing summarization methods enhance the graph visualization with blocked views, but have adverse effect on the latent influence structure. How can we visually summarize a large graph to maximize influence flows? In particular, how can we illustrate the impact of an individual node through the summarization? Can we maintain the appealing graph metaphor while preserving both the overall influence pattern and fine readability? To answer these questions, we first formally define the influence graph summarization problem. Second, we propose an end-to-end framework to solve the new problem. Our method can not only highlight the flow-based influence patterns in the visual summarization, but also inherently support rich graph attributes. Last, we present a theoretic analysis and report our experiment results. Both evidences demonstrate that our framework can effectively approximate the proposed influence graph summarization objective while outperforming previous methods in a typical scenario of visually mining academic citation networks.Comment: to appear in IEEE International Conference on Data Mining (ICDM), Shen Zhen, China, December 201

Enhancing community detection using a network weighting strategy

A community within a network is a group of vertices densely connected to each other but less connected to the vertices outside. The problem of detecting communities in large networks plays a key role in a wide range of research areas, e.g. Computer Science, Biology and Sociology. Most of the existing algorithms to find communities count on the topological features of the network and often do not scale well on large, real-life instances. In this article we propose a strategy to enhance existing community detection algorithms by adding a pre-processing step in which edges are weighted according to their centrality w.r.t. the network topology. In our approach, the centrality of an edge reflects its contribute to making arbitrary graph tranversals, i.e., spreading messages over the network, as short as possible. Our strategy is able to effectively complements information about network topology and it can be used as an additional tool to enhance community detection. The computation of edge centralities is carried out by performing multiple random walks of bounded length on the network. Our method makes the computation of edge centralities feasible also on large-scale networks. It has been tested in conjunction with three state-of-the-art community detection algorithms, namely the Louvain method, COPRA and OSLOM. Experimental results show that our method raises the accuracy of existing algorithms both on synthetic and real-life datasets.Comment: 28 pages, 2 figure

Community Detection in Quantum Complex Networks

Determining community structure is a central topic in the study of complex networks, be it technological, social, biological or chemical, in static or interacting systems. In this paper, we extend the concept of community detection from classical to quantum systems---a crucial missing component of a theory of complex networks based on quantum mechanics. We demonstrate that certain quantum mechanical effects cannot be captured using current classical complex network tools and provide new methods that overcome these problems. Our approaches are based on defining closeness measures between nodes, and then maximizing modularity with hierarchical clustering. Our closeness functions are based on quantum transport probability and state fidelity, two important quantities in quantum information theory. To illustrate the effectiveness of our approach in detecting community structure in quantum systems, we provide several examples, including a naturally occurring light-harvesting complex, LHCII. The prediction of our simplest algorithm, semiclassical in nature, mostly agrees with a proposed partitioning for the LHCII found in quantum chemistry literature, whereas our fully quantum treatment of the problem uncovers a new, consistent, and appropriately quantum community structure.Comment: 16 pages, 4 figures, 1 tabl