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Learning Probabilistic Networks from Data
Published in Takushi Tanaka, Setsuo Ohsuga, Moonis Ali, Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, 2022
In a Bayesian network, the structure of a domain is represented in terms of a directed acyclic graph in which the vertices correspond to the variables of the domain, and the edges represent the relationships between the variables. Each vertex in the graph is quantified with a conditional probability distribution specifying conditional probability of each value of the corresponding random variable given every possible combination of values of its parents. In a Markov network, the domain structure is represented by an undirected graph, where the vertices represent the random variables and the edges represent symmetric associations between the variables. The graph is quantified with numerical quantities called evidence potentials. In a decomposable network, the domain structure is represented in terms of a triangulated undirected graph which is quantified with the marginal probability distributions of the cliques in the graph. A triangulated graph is a graph containing no cycles of length ≥ 4 without a chord. A clique in a graph is a maximal subset of vertices in which each pair of vertices is connected by an edge.
Factor Graphs and Message Passing Algorithms
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Aitzaz Ahmad, Erchin Serpedin, Khalid A. Qaraqe
A set X is often called a clique, if every two nodes in X are connected by some edge. Such a clique is maximal if it is not contained in any other clique. The joint probability mass function of an MRF can be expressed as the product of a collection of clique potential functions3, defined on the set C of maximal cliques in the MRF, i.e., () p(v1,…,vn)=1Z∏E∈CψE(VE),
Signature Generation Algorithms for Polymorphic Worms
Published in Mohssen Mohammed, Al-Sakib Khan Pathan, Automatic Defense Against Zero-day Polymorphic Worms in Communication Networks, 2016
Mohssen Mohammed, Al-Sakib Khan Pathan
A clique is a fully connected subgraph of a graph. A maximal clique is not contained in any other clique of the graph. It turns out that the set of conditional independence relations implied by the separation properties in the graph are satisfied by probability distributions, which can be written as a normalized product of nonnegative functions over the variables in the maximal cliques of the graph (this is known as the Hammersley-Clifford Theorem [73]). In the example in Figure 7.14, this implies that the probability distribution over (A, B, C, D, E) can be written as
Uncovering illicit supply networks and their interfaces to licit counterparts through graph-theoretic algorithms
Published in IISE Transactions, 2023
Rashid Anzoom, Rakesh Nagi, Chrysafis Vogiatzis
We could also adopt another distinct approach to this problem. Since our objective is to find p distinct trees that are δ dissimilar from each other, it is natural to think of the problem in terms of cliques. Consider that we develop a graph where represent the set of nodes corresponding to the k-best trees, and AD is a set of edges where an edge exists between a pair of node iff One could find the maximum clique that includes the first node, repeat this for a few other nodes that were not present in this clique, or find the maximal cliques from each node in If this approach results in one or more cliques that are of size p or greater, we can pass them on to the analyst for further investigation. If the cliques are not adequate (for law enforcement purposes), one can expand the search on the k-best trees and repeat the process. The complexity of forming the graph GD is The well-known Bron–Kerbosch (or similar) algorithm can be applied to find the maximum clique. As δ increases and GD becomes sparser, we can explore more specialized algorithms for sparser graphs (see, e.g., Buchanan et al. (2014)).
On solving the densest k-subgraph problem on large graphs
Published in Optimization Methods and Software, 2020
A number of recent results have focused on recovering planted k-subgraphs by using convex relaxation techniques, see e.g. [1,2]. Ames and Vavasis [2] show that the maximum clique in a graph consisting of a single large clique can be identified from the minimum nuclear norm solution of a particular system of linear inequalities. Ames [1] establishes analogous recovery guarantees for a convex relaxation of the planted clique problem that is robust to noise. For a survey on the topic see Li et al. [39].
Bearing-only formation control of multi-agent systems in local reference frames
Published in International Journal of Control, 2021
Xiaoyuan Luo, Xianluo Li, Xiaolei Li, Xinping Guan
A clique in a graph is a complete subgraph, i.e. a subset of its nodes and edges such that every two nodes are adjacent. A maximal clique is the one that cannot be augmented by incorporating one more node.