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### Probabilities of Sentences about Very Sparse Random Graphs

James F. Lynch
1992 Random structures & algorithms (Print)
We prove that for every firstorder sentence, the probability that the sentence is true for the random graph has an asymptotic limit.  ...  W e consider random graphs with edge probability Pn-Q, where n is the number of vertices of the graph, / 3 > 0 is fixed, and a = 1 or a = (I + 1)/1 for some fixed positive integer I .  ...  Of course, this gives us much more expressibility, and there are many instances where secondorder sentences do not have asymptotic probabilities (see ), but for very sparse random graphs, limit laws  ...

### Probabilities of sentences about very sparse random graphs

J.F. Lynch
Proceedings  31st Annual Symposium on Foundations of Computer Science
We prove that for every firstorder sentence, the probability that the sentence is true for the random graph has an asymptotic limit.  ...  W e consider random graphs with edge probability Pn-Q, where n is the number of vertices of the graph, / 3 > 0 is fixed, and a = 1 or a = (I + 1)/1 for some fixed positive integer I .  ...  Of course, this gives us much more expressibility, and there are many instances where secondorder sentences do not have asymptotic probabilities (see ), but for very sparse random graphs, limit laws  ...

### Faster Clustering via Non-Backtracking Random Walks [article]

Brian Rappaport, Anuththari Gamage, Shuchin Aeron
2017 arXiv   pre-print
This paper presents VEC-NBT, a variation on the unsupervised graph clustering technique VEC, which improves upon the performance of the original algorithm significantly for sparse graphs.  ...  VEC employs a novel application of the state-of-the-art word2vec model to embed a graph in Euclidean space via random walks on the nodes of the graph.  ...  However, accuracy is still lower than desirable for very sparse graphs. In addition, although 60 is not an excessively long random walk, reducing it would speed up the algorithm.  ...

### The logic of random regular graphs

Simi Haber, Michael Krivelevich
2010 Journal of Combinatorics
The case of sparse random graphs has also been studied in detail for the binomial random graph model. We obtain results for random regular graphs that match the main results for G(n, p).  ...  The classic Zero-One law for random graphs states that if p is some constant probability then for every first order property the limiting probability of the binomial random graph G(n, p) having this property  ...  As mentioned earlier, the first order behavior of very sparse random regular graphs was studied by Lynch  .  ...

### The Consciousness Prior [article]

Yoshua Bengio
2019 arXiv   pre-print
A low-dimensional thought or conscious state is analogous to a sentence: it involves only a few variables and yet can make a statement with very high probability of being true.  ...  This is consistent with a joint distribution (over high-level concepts) which has the form of a sparse factor graph, i.e., where the dependencies captured by each factor of the factor graph involve only  ...  Acknowledgements The author wants to thank Philippe Beaudoin, Gerry (Tong) Che, William Fedus, Devon Hjelm and Anirudh Goyal for preliminary discussions about the consciousness prior, as well as funding  ...

### Patterns in syntactic dependency networks

Ramon Ferrer i Cancho, Ricard V. Solé, Reinhard Köhler
2004 Physical Review E
., the ability of combining words for forming sentences. The origin of such traits is an issue of open debate.  ...  Such previously unreported features of syntax organization are not a trivial consequence of the structure of sentences, but an emergent trait at the global scale.  ...  The proportion of links provided with regard to the theoretical maximum is about 0.16. The German corpus is the most sparse of them.  ...

### Enriching PubMed related article search with sentence level co-citations

Nam Tran, Pedro Alves, Shuangge Ma, Michael Krauthammer
2009 AMIA Annual Symposium Proceedings
In this paper, we propose to enrich PubMed with a new type of related article link based on citations within a single sentence (i.e. sentence level co-citations or SLCs).  ...  We also showed that only half of SLCs are found among PubMed related article links. Additionally, we discuss how the citing sentence of an SLC explains the connection between two articles.  ...  The PLCs extracted from the above mentioned 100,000 articles would form a very sparse citation graph.  ...

### HARPY: Hypernyms and Alignment of Relational Paraphrases

Adam Grycner, Gerhard Weikum
2014 International Conference on Computational Linguistics
To this end, we devise judicious features and develop a graph-based alignment algorithm by adapting and extending the SimRank random-walk method.  ...  The resulting taxonomy of relational phrases and verb senses, coined HARPY, contains 20,812 synsets organized into a Directed Acyclic Graph (DAG) with 616,792 hypernymy links.  ...  However, the subsumption hierarchy of Patty is very sparse.  ...

### Cognitive agents and machine learning by example: Representation with conceptual graphs

Alexandros Gkiokas, Alexandra I. Cristea
2018 Computational intelligence
The average length of a sentence increases from set to set, from 5 words per input up to 30 words per input. Figure 5 (Graph Example) shows an example of a simple CG. [ Figure 5 about here.]  ...  Continuous updating of the probabilities look-up table enabled us to create a very large training sample set for the ANNs.  ...

### Small Worlds of Concepts and Other Principles of Semantic Search [chapter]

Stefan Bordag, Gerhard Heyer, Uwe Quasthoff
2003 Lecture Notes in Computer Science
A combination of the strengths of both classic information retrieval with the distributed approach of P2P networks can avoid both their weaknesses: The organisation of document collections relevant for  ...  On the other hand, the clustering coefficient will always be very low for the random graph as opposed to the regular graph.  ...  If graphs are sparse graphs, then with growing graph size the path length will grow linearly for the regular graph and only logarithmically for the random graph because of the many possible shortcuts throughout  ...

### Zero-one laws for k-variable first-order logic of sparse random graphs [article]

A.S. Razafimahatratra, M. Zhukovskii
2019 arXiv   pre-print
In this paper, we prove that for every positive ε, there exists an α∈(1/(k-1),1/(k-1)+ε) such that the binomial random graph G(n,n^-α) does not obey 0-1 law w.r.t. first order sentences with k variables  ...  In this paper, we study asymptotical behavior of probabilities of truth of FO sentences from L k on the binomial random graph G(n, p).  ...  For α > 2, the random graph obeys the zero-one law w.r.t. L ω ∞,ω since a.a.s. (with asymptotical probability 1) this graph is empty.  ...

### Syntree2Vec - An algorithm to augment syntactic hierarchy into word embeddings [article]

Shubham Bhardwaj
2018 arXiv   pre-print
Word embeddings aims to map sense of the words into a lower dimensional vector space in order to reason over them.  ...  We propose a graph based embedding algorithm inspired from node2vec. Experimental results have shown that our algorithm improves the syntactic strength and gives robust performance on meagre data.  ...  This method in a way inhibits the movement of random walk from leaf nodes since there are less nodes connected to it, that lie at the end of parse trees, the tag transition matrix for them is sparse.  ...

### Convergence law for hyper-graphs with prescribed degree sequences [article]

Nans Lefebvre
2015 arXiv   pre-print
It defines a random hyper-multigraph specified by two distributions, one for the degrees of the vertices, and one for the sizes of the hyper-edges.  ...  We view hyper-graphs as incidence graphs, i.e. bipartite graphs with a set of nodes representing vertices and a set of nodes representing hyper-edges, with two nodes being adjacent if the corresponding  ...  Most natural graphs have a sparse structure yet are locally highly clustered, a property that classical random graphs fail to model.  ...

### Randomisation and Derandomisation in Descriptive Complexity Theory

Kord Eickmeyer, Martin Grohe, Anuj Dawar
2011 Logical Methods in Computer Science
We study probabilistic complexity classes and questions of derandomisation from a logical point of view.  ...  The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised.  ...  Acknowledgements We would like to thank Nicole Schweikardt and Dieter van Melkebeek for helpful comments on an earlier version of this paper.  ...

### Regular decomposition of large graphs and other structures: scalability and robustness towards missing data [article]

Hannu Reittu, Ilkka Norros, Fülöp Bazsó
2017 arXiv   pre-print
In this way it would be possible to find out a large scale structure of a huge graphs of certain type using only a tiny part of graph information and obtaining a compact representation of such graphs useful  ...  We continue our previous work on the subject, considering cases of missing data and scaling of algorithms to extremely large size of graphs.  ...  The work of the Finnish authors was supported by Academy of Finland project 294763 (Stomograph).  ...
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