Methods to solve a node discovery problem for a social network are presented. Covert nodes refer to the nodes which are not observable directly. They transmit the influence and affect the resulting collaborative activities among the persons in a social network, but do not appear in the surveillance logs which record the participants of the collaborative activities. Discovering the covert nodes is identifying the suspicious logs where the covert nodes would appear if the covert nodes became overt. The performance of the methods is demonstrated with a test dataset generated from computationally synthesized networks and a real organization.
Markov random network is a model of the joint probability distribution of random variables. It is an undirected graphical model similar to a Bayesian network. The Markov random network is used to learn the dependency between the links which shares a node. The Markov random network is one of the dependence graphs @cite_17 , which models the dependency between links. Extension to hierarchical models @cite_25 , multiple networks (treating different types of relationships) @cite_9 , valued networks (with nodal attributes) @cite_23 , higher order dependency between the links which share no nodes @cite_12 , and 2-block chain graphs (associating one set of explanatory variables with the other set of outcome variables) @cite_0 are studied. A family of such extensions and model elaborations is named the exponential random graph @cite_2 .
@cite_9 @cite_0 @cite_23 @cite_2 @cite_25 @cite_12 @cite_17
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This article provides an introductory summary to the formulation and application of exponential random graph models for social networks. The possible ties among nodes of a network are regarded as random variables, and assumptions about dependencies among these random tie variables determine the general form of the exponential random graph model for the network. Examples of different dependence assumptions and their associated models are given, including Bernoulli, dyad-independent and Markov random graph models. The incorporation of actor attributes in social selection models is also reviewed. Newer, more complex dependence assumptions are briefly outlined. Estimation procedures are discussed, including new methods for Monte Carlo maximum likelihood estimation. We foreshadow the discussion taken up in other papers in this special edition: that the homogeneous Markov random graph models of Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832–842] are not appropriate for many observed networks, whereas the new model specifications of [Snijders, T.A.B., Pattison, P., Robins, G.L., Handock, M. New specifications for exponential random graph One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent work has shown that while these generative models do have the right degree distribution, they are not good models for real-life networks due to their differences on other important metrics like conductance. We believe this is, in part, because many of these real-world networks have very different joint degree distributions, that is, the probability that a randomly selected edge will be between nodes of degree k and l. Assortativity is a sufficient statistic of the joint degree distribution, and it has been previously noted that social networks tend to be assortative, while biological and technological networks tend to be disassortative. We suggest understanding the relationship between network structure and the joint degree distribution of graphs is an interesting avenue of further research. An important tool for such studies are algorithms that can generate random instances of graphs with the same joint degree distribution. This is the main topic of this article, and we study the problem from both a theoretical and practical perspective. We provide an algorithm for constructing simple graphs from a given joint degree distribution, and a Monte Carlo Markov chain method for sampling them. We also show that the state space of simple graphs with a fixed degree distribution is connected via endpoint switches. We empirically evaluate the mixing time of this Markov chain by using experiments based on the autocorrelation of each edge. These experiments show that our Markov chain mixes quickly on these real graphs, allowing for utilization of our techniques in practice. This paper studies networks where all nodes are distributed on a unit square A=Δ [- [1 2], [1 2]]2 following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability grρ(x)=Δg(x rρ), independent of the event that any other pair of nodes are directly connected. Here, g:[0,∞)→ [0,1] satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and g(x)=o(1 (x2log2x)) ; further, rρ=√ (logρ+b) (Cρ) where C=∫ℜ2g(||x||)dx and b is a constant. Denote the aforementioned network by G(Xρ,grρ,A). We show that as ρ→ ∞, 1) the distribution of the number of isolated nodes in G(Xρ,grρ,A) converges to a Poisson distribution with mean e-b ; 2) asymptotically almost surely (a.a.s.) there is no component in G(Xρ,grρ,A) of fixed and finite order k >; 1; c) a.a.s. the number of components with an unbounded order is one. Therefore, as ρ→ ∞, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for G(Xρ,grρ,A) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b→ ∞ as ρ→ ∞. These results expand recent results obtained for connectivity of random geometric graphs from the unit disk model and the fewer results from the log-normal model to the more general and more practical random connection model. The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent β > 2, we show that the PoA is bounded by O(T2(β -1))), where T = γ α is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations.