Society is often polarized by controversial issues, that split the population into groups of opposing views. When such issues emerge on social media, we often observe the creation of 'echo chambers', i.e., situations where like-minded people reinforce each other's opinion, but do not get exposed to the views of the opposing side. In this paper we study algorithmic techniques for bridging these chambers, and thus, reducing controversy. Specifically, we represent the discussion on a controversial issue with an endorsement graph, and cast our problem as an edge-recommendation problem on this graph. The goal of the recommendation is to reduce the controversy score of the graph, which is measured by a recently-developed metric based on random walks. At the same time, we take into account the acceptance probability of the recommended edge, which represents how likely the edge is to materialize in the endorsement graph. We propose a simple model based on a recently-developed user-level controversy score, that is competitive with state-of- the-art link-prediction algorithms. We thus aim at finding the edges that produce the largest reduction in the controversy score, in expectation. To solve this problem, we propose an efficient algorithm, which considers only a fraction of all the combinations of possible edges. Experimental results show that our algorithm is more efficient than a simple greedy heuristic, while producing comparable score reduction. Finally, a comparison with other state-of-the-art edge-addition algorithms shows that this problem is fundamentally different from what has been studied in the literature.
Adding edges to modify the graph structure. In addition to the work on explicitly reducing polarization in social media, there are many papers aiming to make a network more cohesive by edge additions, where cohesiveness is quantified using graph-theoretic properties, such as shortest paths @cite_3 @cite_32 , closeness centrality @cite_7 , diameter @cite_0 , eccentricity @cite_11 , communicability @cite_20 @cite_25 , synchronizability @cite_35 , and natural connectivity @cite_17 .
@cite_35 @cite_7 @cite_32 @cite_3 @cite_17 @cite_0 @cite_25 @cite_20 @cite_11
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In this paper, we develop a novel paradigm, namely hypergraph shift, to find robust graph modes by probabilistic voting strategy, which are semantically sound besides the self-cohesiveness requirement in forming graph modes. Unlike the existing techniques to seek graph modes by shifting vertices based on pair-wise edges (i.e, an edge with 2 ends), our paradigm is based on shifting high-order edges (hyperedges) to deliver graph modes. Specifically, we convert the problem of seeking graph modes as the problem of seeking maximizers of a novel objective function with the aim to generate good graph modes based on sifting edges in hypergraphs. As a result, the generated graph modes based on dense subhypergraphs may more accurately capture the object semantics besides the self-cohesiveness requirement. We also formally prove that our technique is always convergent. Extensive empirical studies on synthetic and real world data sets are conducted on clustering and graph matching. They demonstrate that our techniques significantly outperform the existing techniques. Given multimillion-node graphs such as "who-follows-whom", "patent-cites-patent", "user-likes-page" and "actor director-makes-movie" networks, how can we find unexpected behaviors? When companies operate on the graphs with monetary incentives to sell Twitter "Followers" and Facebook page "Likes", the graphs show strange connectivity patterns. In this paper, we study a complete graph from a large Twitter-style social network, spanning up to 3.33 billion edges. We report strange deviations from typical patterns like smooth degree distributions. We find that such deviations are often due to "lockstep behavior" that large groups of followers connect to the same groups of followees. Our first contribution is that we study strange patterns on the adjacency matrix and in the spectral subspaces with respect to several flavors of lockstep. We discover that (a) the lockstep behaviors on the graph shape dense "block" in its adjacency matrix and creates "rays" in spectral subspaces, and (b) partially overlapping of the behaviors shape "staircase" in its adjacency matrix and creates "pearls" in spectral subspaces. The second contribution is that we provide a fast algorithm, using the discovery as a guide for practitioners, to detect users who offer the lockstep behaviors in undirected directed bipartite graphs. We carry out extensive experiments on both synthetic and real datasets, as well as public datasets from IMDb and US Patent. The results demonstrate the scalability and effectiveness of our proposed algorithm. In practical military or first responder deployment scenarios, information flows need to adhere to specified policies regardless of the physical connectivity of nodes. Nodes in such networks are associated with various levels in a command-and-control hierarchy, and therefore typically form a logical hierarchical tree network that is used to route both command and data traffic. Associated with this logical hierarchical network is a communication network that represents the connectivity of these nodes in the deployed scenario. Such composite networks introduce constraints that can result in information flows having to traverse much longer paths in the underlying communication network. In this paper, we look at the problem of adding edges to a logical hierarchical network (or any other social network) so as to minimize the number of hops required to route data traffic in the underlying communication network from a node to other specified nodes. The edges added are a subset of all possible edges in the complementary logical hierarchical graph and have to satisfy specified hierarchical constraints. First, we consider the general problem of minimizing the eccentricity of a source node 's' (where eccentricity of 's' is the maximum of the shortest paths from 's' to all other nodes) in a metric graph on adding upto 'B' unequal cost metric edges from the set of all edges in the complementary graph. We develop an efficient constant factor approximation algorithm for this case that outperforms existing constant factor algorithms for eccentricity minimization. Here the added edge metric cost as well as the graph edge metric cost correspond to the number of hops in the shortest path required to route traffic in the actual deployed topology (i.e., underlying communication network). Next, we consider the case where the set of possible added edges is a specified subset of the edges in the complementary graph and the set of destinations is a subset of the graph nodes. For this case, we develop heuristic algorithms based on the previous eccentricity minimizing algorithm that show good performance. We validate our algorithms using two realistic military deployment scenarios. We find that adding even a low number of hierarchically constrained edges (of the order of 10) can cause a significant decrease (around 50 ) in the eccentricity of a node in the logical hierarchical network and thus can reduce the number of hops required for data traffic traversal. We study the response of complex networks subject to attacks on vertices and edges. Several existing complex network models as well as real-world networks of scientific collaborations and Internet traffic are numerically investigated, and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph. For each case of attacks on vertices and edges, four different attacking strategies are used: removals by the descending order of the degree and the betweenness centrality, calculated for either the initial network or the current network during the removal procedure. It is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network, suggesting that the network structure changes as important vertices or edges are removed. Furthermore, the correlation between the betweenness centrality and the degree in complex networks is studied.