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Category: Graph Theory

Directed vs. Undirected Graphs in NNs: The (Surprising!) Implications

Directed vs. Undirected Graphs in NNs: The (Surprising!) Implications

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Most of us don’t always use graph language to describe neural networks, but if we dig into the implications of graph theory language, we get some surprising (and very useful) insights! We probably all know that a typical feedforward neural network can be described as a “directed graph.” Many of us also know that a restricted Boltzmann machine (RBM) is an “undirected graph.” In this little difference of terms, there is a wealth of meaning. Salakhutdinov, Mnih, and Hinton (2007;…

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The 1-D Cluster Variation Method (CVM) – Simple Application

The 1-D Cluster Variation Method (CVM) – Simple Application

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The 1-D Cluster Variation Method – Application to Text Mining and Data Mining There are three particularly good reasons for us to look at the Cluster Variation Method (CVM) as an alternative means of understanding the information in a system: The CVM captures local pattern distributions (for an equilibrium state), When the system is made up of equal numbers of units in each of two states, and the enthalpy for each state is the same (the simple unit activation energy…

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Big Data, Big Graphs, and Graph Theory: Tools and Methods

Big Data, Big Graphs, and Graph Theory: Tools and Methods

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Big Graphs Need Specialized Data Storage and Computational Methods {A Working Blogpost – Notes for research & study} Processing large-scale graph data: A guide to current technology, by Sherif Sakr (ssakr@cse.unsw.edu.au), IBM Developer Works (10 June 2013). Note: Dr. Sherif Sakr is a senior research scientist in the Software Systems Group at National ICT Australia (NICTA), Sydney, Australia. He is also a conjoint senior lecturer in the School of Computer Science and Engineering at University of New South Wales. He…

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Modeling Nonlinear Phenomena

Modeling Nonlinear Phenomena

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Modeling Nonlinear Phenomena – What is “X”? Many of us grew up hating word problems in algebra. (Some of us found them interesting, sometimes easy, and sometimes fun. We were the minority.) For most of us, even if we understood the mathematical formulas, there was a big “gap” in our understanding and intuition when it came to applying the formulas to some real-world situation. In the problem, we’d be given a set of statements, and then told to find “something.”…

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Community Detection in Graphs

Community Detection in Graphs

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Complexity and Graph Theory: A Brief Note Santo Fortunato has published an interesting and densly rich article, Community Detection in Graphs, in  Complexity (Inter-Wiley). This article is over 100 pages long, it is relatively complete, with numerous references and excellent figures. It is a bit surprising, however, that this extensive discussion misses one of the things that would seem to be most important in discussing graphs, and particularly, clusters within graphs: the stability of these clusters. That is; the theoretical basis for cluster…

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Chapter 2 (Part 3), Sennelart & Blondel – Automatic Discovery of Similar Words

Chapter 2 (Part 3), Sennelart & Blondel – Automatic Discovery of Similar Words

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In Section 2.3, we get to the meat of Sennelart & Blondel’s work, which is a graph-based method for determining similar words, using a dictionary as source. Their method uses a vXv matrix, where each v is a word in the dictionary. They compare their method and results with that of Kleinberg, who proposes a method for determining good Web hubs and authorities, and with the ArcRank and WordNet methods. They test the four methods on four words: disappear, parallelogram,…

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"Automatic Discovery of Similar Words" – Chapter 2 in Survey of Text Mining II

"Automatic Discovery of Similar Words" – Chapter 2 in Survey of Text Mining II

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This post begins a review of “Automatic Discovery of Similar Words,” by Pierre Senellart and Vincent D. Blondel, published as Chapter 2 in Berry and Castellanos’ Survey of Text Mining II. This is an excellent and useful chapter, in that it:1) Addresses the broad issue of computational methods for discovering “similar words” (including synonyms, near-synonyms, and thesauri-generating techniques) from large data corpora,2) Illustrates the different leading mathematical methods, giving an excellent overview of the SoA,3) Competently discusses how different methods…

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Graph Theory — Becoming "Organizing Framework"

Graph Theory — Becoming "Organizing Framework"

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Something I’ve been noting — both on my own, and in conversations with Jenn Sleeman , who’s in touch with the academic world at UMBC — Graph theory is growing in centrality as a fundamental organizing framework for many current and emerging computational processes. Specifically, anything more complex than a simple “tuple” (RDF or OWL, etc.), needs to be matched against a graph or partial graph. One good “integrative” paper is Understanding Belief Propagation and its Generalizations by J.S. Yedidia,…

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