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. We study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices. We find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices. The expressions obtained are checked by means of numerical simulations in a particular example. The general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure. We also present an extension of the class, to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system. We consider the issue of protection in very large networks displaying randomness in topology. We employ random graph models to describe such networks, and obtain probabilistic bounds on several parameters related to reliability. In particular, we take the case of random regular networks for simplicity and consider the length of primary and backup paths in terms of the number of hops. First, for a randomly picked pair of nodes, we derive a lower bound on the average distance between the pair and discuss the tightness of the bound. In addition, noting that primary and protection paths form cycles, we obtain a lower bound on the average length of the shortest cycle around the pair. Finally, we show that the protected connections of a given maximum finite length are rare. We then generalize our network model so that different degrees are allowed according to some arbitrary distribution, and show that the second moment of degree over the first moment is an important shorthand for behavior of a network. Notably, we show that most of the results in regular networks carry over with minor modifications, which significantly broadens the scope of networks to which our approach applies. We present as an example the case of networks with a power-law degree distribution. A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its ith predecessor of degree k i with a directed link of length l using a probability proportional to k i l α . Our numerical study indicates that the network is scale free for all values of α>α c and the degree distribution decays stretched exponentially for the other values of a. The link length distribution follows a power law: D(l)∼l δ , where δ is calculated exactly for the whole range of values of a. Although undirected cycles in directed graphs of Bayesian belief networks have been thoroughly studied, little attention has so far been given to a systematic analysis of directed (feedback) cycles. In this paper we propose a way of looking at those cycles; namely, we suggest that a feedback cycle represents a family of probabilistic distributions rather than a single distribution (as a regular Bayesian belief network does). A non-empty family of distributions can be explicitly represented by an ideal of conjunctions with interval estimates on the probabilities of its elements. This ideal can serve as a probabilistic model of an expert's uncertain knowledge pattern; such models are studied in the theory of algebraic Bayesian networks. The family of probabilistic distributions may also be empty; in this case, the probabilistic assignment over cycle nodes is inconsistent. We propose a simple way of explicating the probabilistic relationships an isolated directed cycle contains, give an algorithm (based on linear programming) of its consistency checking, and establish a lower bound of the complexity of this checking. †§ ¶ Most studies of networks have only looked at small subsets of the true network. Here, we discuss the sampling properties of a network’s degree distribution under the most parsimonious sampling scheme. Only if the degree distributions of the network and randomly sampled subnets belong to the same family of probability distributions is it possible to extrapolate from subnet data to properties of the global network. We show that this condition is indeed satisfied for some important classes of networks, notably classical random graphs and exponential random graphs. For scale-free degree distributions, however, this is not the case. Thus, inferences about the scale-free nature of a network may have to be treated with some caution. The work presented here has important implications for the analysis of molecular networks as well as for graph theory and the theory of networks in general. complex networks protein interaction networks random graphs sampling theory Networks form an indispensable part of our lives. In particular, computer networks have ranked amongst the most influential networks in recent times. In such an ever-evolving and fast growing network, the primary concern is to understand and analyse different aspects of the network behaviour, such as the quality of service and efficient information propagation. It is also desirable to predict the behaviour of a large computer network if, for example, one of the computers is infected by a virus. In all of the aforementioned cases, we need protocols that are able to make local decisions and handle the dynamic changes in the network topology. Here, randomised algorithms are preferred because many deterministic algorithms often require a central control. In this thesis, we investigate three network-based randomised algorithms, threshold load balancing with weighted tasks, the pull-Moran process and the coalescing-branching random walk. Each of these algorithms has extensive applicability within networks and computational complexity within computer science. In this thesis we investigate threshold-based load balancing protocols. We introduce a generalisation of protocols in [2, 3] to weighted tasks. This thesis also analyses an evolutionary-based process called the death-birth update, defined here as the Pull-Moran process. We show that a class of strong universal amplifiers does not exist for the Pull-Moran process. We show that any class of selective amplifiers in the (standard) Moran process is a class of selective suppressors under the Pull-Moran process. We then introduce a class of selective amplifiers called Punk graphs. Finally, we improve the broadcasting time of the coalescing-branching (COBRA) walk analysed in [4], for random regular graphs. Here, we look into the COBRA approach as a randomised rumour spreading protocol. We propose and analyse a quasirandom analogue to the classical push model for disseminating information in networks ("randomized rumor spreading"). In the classical model, in each round each informed node chooses a neighbor at random and informs it. Results of Frieze and Grimmett (Discrete Appl. Math. 1985) show that this simple protocol succeeds in spreading a rumor from one node of a complete graph to all others within O(log n) rounds. For the network being a hypercube or a random graph G(n, p) with p ≥ (1 +e)(log n) n, also O(log n) rounds suffice (Feige, Peleg. Raghavan, and Upfal, Random Struct. Algorithms 1990). In the quasirandom model, we assume that each node has a (cyclic) list of its neighbors. Once informed, it starts at a random position of the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above mentioned bounds still hold. In addition, we also show a O(log n) bound for sparsely connected random graphs G(n, p) with p = (log n + f(n)) n, where f(n) → ∞ and f(n) = O(log log n). Here, the classical model needs Θ(log2(n)) rounds. Hence the quasirandom model achieves similar or better broadcasting times with a greatly reduced use of random bits. A class of statistical models is proposed for longitudinal network data. The dependent variable is the changing (or evolving) relation network, represented by two or more observations of a directed graph with a fixed set of actors. The network evolution is modeled as the consequence of the actors making new choices, or withdrawing existing choices, on the basis of functions, with fixed and random components, that the actors try to maximize. Individual and dyadic exogenous variables can be used as covariates. The change in the network is modeled as the stochastic result of network effects (reciprocity, transitivity, etc.) and these covariates. The existing network structure is a dynamic constraint for the evolution of the structure itself. The models are continuous-time Markov chain models that can be implemented as simulation models. The model parameters are estimated from observed data. For estimating and testing these models, statistical procedures are proposed that are based on the method of moments. The statistical procedures are implemented using a stochastic approximation algorithm based on computer simulations of the network evolution process. We propose a graphical model for representing networks of stochastic processes, the minimal generative model graph. It is based on reduced factorizations of the joint distribution over time. We show that under appropriate conditions, it is unique and consistent with another type of graphical model, the directed information graph, which is based on a generalization of Granger causality. We demonstrate how directed information quantifies Granger causality in a particular sequential prediction setting. We also develop efficient methods to estimate the topological structure from data that obviate estimating the joint statistics. One algorithm assumes upper bounds on the degrees and uses the minimal dimension statistics necessary. In the event that the upper bounds are not valid, the resulting graph is nonetheless an optimal approximation in terms of Kullback-Leibler (KL) divergence. Another algorithm uses near-minimal dimension statistics when no bounds are known, but the distribution satisfies a certain criterion. Analogous to how structure learning algorithms for undirected graphical models use mutual information estimates, these algorithms use directed information estimates. We characterize the sample-complexity of two plug-in directed information estimators and obtain confidence intervals. For the setting when point estimates are unreliable, we propose an algorithm that uses confidence intervals to identify the best approximation that is robust to estimation error. Last, we demonstrate the effectiveness of the proposed algorithms through the analysis of both synthetic data and real data from the Twitter network. In the latter case, we identify which news sources influence users in the network by merely analyzing tweet times. We present a distribution model for binary vectors, called the influence combination model and show how this model can be used as the basis for unsupervised learning algorithms for feature selection. The model can be represented by a particular type of Boltzmann machine with a bipartite graph structure that we call the combination machine. This machine is closely related to the Harmonium model defined by Smolensky. In the first part of the paper we analyze properties of this distribution representation scheme. We show that arbitrary distributions of binary vectors can be approximated by the combination model. We show how the weight vectors in the model can be interpreted as high order correlation patterns among the input bits, and how the combination machine can be used as a mechanism for detecting these patterns. We compare the combination model with the mixture model and with principle component analysis. In the second part of the paper we present two algorithms for learning the combination model from examples. The first learning algorithm is the standard gradient ascent heuristic for computing maximum likelihood estimates for the parameters of the model. Here we give a closed form for this gradient that is significantly easier to compute than the corresponding gradient for the general Boltzmann machine. The second learning algorithm is a greedy method that creates the hidden units and computes their weights one at a time. This method is a variant of projection pursuit density estimation. In the third part of the paper we give experimental results for these learning methods on synthetic data and on natural data of handwritten digit images. Random key graphs form a class of random intersection graphs that are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much attention recently, owing in part to their wide applicability in various domains, including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of graph links is captured by independent Bernoulli random variables, rendering them to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdős–Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model’s parameters so that: 1) the minimum node degree in the graph is at least @math and 2) the graph is @math -connected, both with high probability as the number of nodes becomes large. The results are given in the form of zero-one laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on @math -connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on one-connectivity of random key graphs with unreliable links. Learning about the structure of hidden or covert networks is a major challenge in epidemiology, sociology, and intelligence analysis. Vertices in hidden networks usually cannot be enumerated or sampled in a systematic way; they can only be revealed by tracing links emanating from already-observed vertices. Observers sometimes cannot follow links directly, and instead must rely on passive observation of a dynamic process to reveal vertices and edges. This paper outlines a framework for estimating network structures from partial observation of information diffusion through the network. Diffusion is modeled by a continuous-time Markov epidemic model. Edges are revealed by transmission events and new vertices are uncovered when information is transmitted to them. The approach is a generalization of tools developed to reconstruct drug-user networks from respondent-driven sampling studies in epidemiology. The likelihood of the diffusion process can be interpreted as an exponential random graph model. A Bayesian method for probabilistic reconstruction of the transmission-induced subgraph is described. The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology. Consider the random graph on n vertices 1,...,n. Each vertex i is assigned a type x(i) with x(1),...,x(n) being independent identically distributed as a nonnegative random variable X. We assume that EX3 < infinity. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability min 1, x(i)x(j) n (1 + a n(1 3)) . We study the critical phase, which is known to take place when EX2 = 1. We prove that normalized by n(-2 3) the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient root EXEX3 and drift a - EX3 EX s. In particular, we conclude that the size of the largest connected component is of order n(2 3). (c) 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486-539, 2013 (Less) Bayesian networks, and especially their structures, are powerful tools for representing conditional independencies and dependencies between random variables. In applications where related variables form a priori known groups, chosen to represent different “views” to or aspects of the same entities, one may be more interested in modeling dependencies between groups of variables rather than between individual variables. Motivated by this, we study prospects of representing relationships between variable groups using Bayesian network structures. We show that for dependency structures between groups to be expressible exactly, the data have to satisfy the so-called groupwise faithfulness assumption. We also show that one cannot learn causal relations between groups using only groupwise conditional independencies, but also variable-wise relations are needed. Additionally, we present algorithms for finding the groupwise dependency structures. We propose in this paper a new approach to train the Generative Adversarial Nets (GANs) with a mixture of generators to overcome the mode collapsing problem. The main intuition is to employ multiple generators, instead of using a single one as in the original GAN. The idea is simple, yet proven to be extremely effective at covering diverse data modes, easily overcoming the mode collapsing problem and delivering state-of-the-art results. A minimax formulation was able to establish among a classifier, a discriminator, and a set of generators in a similar spirit with GAN. Generators create samples that are intended to come from the same distribution as the training data, whilst the discriminator determines whether samples are true data or generated by generators, and the classifier specifies which generator a sample comes from. The distinguishing feature is that internal samples are created from multiple generators, and then one of them will be randomly selected as final output similar to the mechanism of a probabilistic mixture model. We term our method Mixture Generative Adversarial Nets (MGAN). We develop theoretical analysis to prove that, at the equilibrium, the Jensen-Shannon divergence (JSD) between the mixture of generators’ distributions and the empirical data distribution is minimal, whilst the JSD among generators’ distributions is maximal, hence effectively avoiding the mode collapsing problem. By utilizing parameter sharing, our proposed model adds minimal computational cost to the standard GAN, and thus can also efficiently scale to large-scale datasets. We conduct extensive experiments on synthetic 2D data and natural image databases (CIFAR-10, STL-10 and ImageNet) to demonstrate the superior performance of our MGAN in achieving state-of-the-art Inception scores over latest baselines, generating diverse and appealing recognizable objects at different resolutions, and specializing in capturing different types of objects by the generators.