Conditional Reliability in Uncertain Graphs

Network reliability is a well-studied problem that requires to measure the probability that a target node is reachable from a source node in a probabilistic (or uncertain) graph, i.e., a graph where every edge is assigned a probability of existence. Many approaches and problem variants have been considered in the literature, all assuming that edge-existence probabilities are fixed. Nevertheless, in real-world graphs, edge probabilities typically depend on external conditions. In metabolic networks a protein can be converted into another protein with some probability depending on the presence of certain enzymes. In social influence networks the probability that a tweet of some user will be re-tweeted by her followers depends on whether the tweet contains specific hashtags. In transportation networks the probability that a network segment will work properly or not might depend on external conditions such as weather or time of the day. In this paper we overcome this limitation and focus on conditional reliability, that is assessing reliability when edge-existence probabilities depend on a set of conditions. In particular, we study the problem of determining the k conditions that maximize the reliability between two nodes. We deeply characterize our problem and show that, even employing polynomial-time reliability-estimation methods, it is NP-hard, does not admit any PTAS, and the underlying objective function is non-submodular. We then devise a practical method that targets both accuracy and efficiency. We also study natural generalizations of the problem with multiple source and target nodes. An extensive empirical evaluation on several large, real-life graphs demonstrates effectiveness and scalability of the proposed methods.Comment: 14 pages, 13 figure

Characterizing the community structure of complex networks

Community structure is one of the key properties of complex networks and plays a crucial role in their topology and function. While an impressive amount of work has been done on the issue of community detection, very little attention has been so far devoted to the investigation of communities in real networks. We present a systematic empirical analysis of the statistical properties of communities in large information, communication, technological, biological, and social networks. We find that the mesoscopic organization of networks of the same category is remarkably similar. This is reflected in several characteristics of community structure, which can be used as ``fingerprints'' of specific network categories. While community size distributions are always broad, certain categories of networks consist mainly of tree-like communities, while others have denser modules. Average path lengths within communities initially grow logarithmically with community size, but the growth saturates or slows down for communities larger than a characteristic size. This behaviour is related to the presence of hubs within communities, whose roles differ across categories. Also the community embeddedness of nodes, measured in terms of the fraction of links within their communities, has a characteristic distribution for each category. Our findings are verified by the use of two fundamentally different community detection methods.Comment: 15 pages, 20 figures, 4 table