Structural Complexity of One-Dimensional Random Geometric Graphs

We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> nodes randomly scattered in [0, 1] that c...

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Veröffentlicht in:IEEE transactions on information theory Jg. 69; H. 2; S. 794 - 812
Hauptverfasser: Badiu, Mihai-Alin, Coon, Justin P.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.02.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Biological system modeling
Channel coding
Complexity theory
Computational modeling
Data models
Entropy
graph compression
graph isomorphism
graphical structure
Graphs
Lower bounds
maximal cliques
Nodes
Permutations
Random geometric graph
Upper bound
Upper bounds
ISSN:0018-9448, 1557-9654
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Zusammenfassung:We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> nodes randomly scattered in [0, 1] that connect if they are within the connection range <inline-formula> <tex-math notation="LaTeX">r\in [{0,1}] </tex-math></inline-formula>. We provide bounds on the number of possible structures which give universal upper bounds on the structural entropy that hold for any <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> and distribution of the node locations. For fixed <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>, the number of structures is <inline-formula> <tex-math notation="LaTeX">\Theta (a^{2n}) </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">a=a(r)=2 \cos {\left ({\frac {\pi }{\lceil 1/r \rceil +2}}\right)} </tex-math></inline-formula>, and therefore the structural entropy is upper bounded by <inline-formula> <tex-math notation="LaTeX">2n\log _{2} a(r) + O(1) </tex-math></inline-formula>. For large <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, we derive bounds on the structural entropy normalized by <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, and evaluate them for independent and uniformly distributed node locations. When the connection range <inline-formula> <tex-math notation="LaTeX">r_{n} </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">O(1/n) </tex-math></inline-formula>, the obtained upper bound is given in terms of a function that increases with <inline-formula> <tex-math notation="LaTeX">n r_{n} </tex-math></inline-formula> and asymptotically attains 2 bits per node. If the connection range is bounded away from zero and one, the upper and lower bounds decrease linearly with <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>, as <inline-formula> <tex-math notation="LaTeX">2(1-r) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">(1-r)\log _{2} e </tex-math></inline-formula>, respectively. When <inline-formula> <tex-math notation="LaTeX">r_{n} </tex-math></inline-formula> is vanishing but dominates <inline-formula> <tex-math notation="LaTeX">1/n </tex-math></inline-formula> (e.g., <inline-formula> <tex-math notation="LaTeX">r_{n} \propto \ln n / n </tex-math></inline-formula>), the normalized entropy is between <inline-formula> <tex-math notation="LaTeX">\log _{2} e \approx 1.44 </tex-math></inline-formula> and 2 bits per node. We also give a simple encoding scheme for random structures that requires 2 bits per node. The upper bounds in this paper easily extend to the entropy of the labeled random graph model, since this is given by the structural entropy plus a term that accounts for all the permutations of node labels that are possible for a given structure, which is no larger than <inline-formula> <tex-math notation="LaTeX">\log _{2}(n!) = n \log _{2} n {-} n + O(\log _{2} n) </tex-math></inline-formula>.
Bibliographie:ObjectType-Article-1
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ObjectType-Feature-2
content type line 14
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3207819