Advertisement
Browse Subject Areas
?

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

Discrimination Power of Polynomial-Based Descriptors for Graphs by Using Functional Matrices

  • Matthias Dehmer ,

    matthias.dehmer@umit.at (MD); shi@nankai.edu.cn (YS)

    Affiliations Department of Computer Science, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany, Department of Mechatronics and Biomedical Computer Science UMIT, Eduard Wallnoefer Zentrum 1, A-6060, Hall in Tyrol, Austria

  • Frank Emmert-Streib,

    Affiliation Computational Medicine and Statistical Learning Laboratory, Department of Signal Processing, Tampere University of Technology, Korkeakoulunkatu 1, FI-33720 Tampere, Finland

  • Yongtang Shi ,

    matthias.dehmer@umit.at (MD); shi@nankai.edu.cn (YS)

    Affiliation Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin, 300071, P.R. China

  • Monica Stefu,

    Affiliation Department of Mechatronics and Biomedical Computer Science UMIT, Eduard Wallnoefer Zentrum 1, A-6060, Hall in Tyrol, Austria

  • Shailesh Tripathi

    Affiliation School of Mathematics and Physics, Queen’s University Belfast, University Road, Belfast, BT7 1NN, United Kingdom

Discrimination Power of Polynomial-Based Descriptors for Graphs by Using Functional Matrices

  • Matthias Dehmer, 
  • Frank Emmert-Streib, 
  • Yongtang Shi, 
  • Monica Stefu, 
  • Shailesh Tripathi
PLOS
x

Abstract

In this paper, we study the discrimination power of graph measures that are based on graph-theoretical matrices. The paper generalizes the work of [M. Dehmer, M. Moosbrugger. Y. Shi, Encoding structural information uniquely with polynomial-based descriptors by employing the Randić matrix, Applied Mathematics and Computation, 268(2015), 164–168]. We demonstrate that by using the new functional matrix approach, exhaustively generated graphs can be discriminated more uniquely than shown in the mentioned previous work.

Introduction

Polynomial representations have been investigated extensively in several application areas such as mathematical chemistry, discrete mathematics etc., see [15]. Some of these polynomials have been used as counting polynomials [2]. Another idea has been to define structural network measures based on the eigenvalues of graph polynomials, see [4, 6, 7]. A well-known example thereof is the Hosoya index [8] that has been defined by the coefficient of the so-called matching polynomial [3]. Another example relates to define spectral-based graph measures [9, 10]. Dehmer et al. [11] also made a contribution in this area by studying graph measures based on the zeros of the so-called information polynomial. It has been also Dehmer et al. [11] who explored the discrimination power of these polynomial-based measures. This relates to study the ability of these descriptors to distinguish non-isomorphic graphs structurally.

The paper is a successor of [6]. In [6], Dehmer et al. explored the discrimination of quantitative graph measures which are based on the eigenvalues of the Randić matrix. In this paper, we define a functional matrix that is more general than the Randić matrix. Therefore we expect an impact on the results when evaluating the discrimination power of the indices already used in [6] on exhaustively generated graphs. In fact, we find that by choosing a different structural setting than the one encoded by the Randić matrix, the proposed measures are even more unique than in [6].

Methods and Results

Novel Graph Measures Based on Complex Zeros

We use the idea of defining a matrix where its elements encode structural information as much as possible. For this, we use the information functional approach due to Dehmer [12]. An information functional is a function that maps the vertices to the reals; note that several information functionals have been already defined and they have been proven useful for discriminating graphs uniquely or classifying graphs efficiently, see [12, 13]. Let G = (V, E) a graph. As in [6, 14], we define the graph descriptors based on the zeros of (1) where is the functional matrix defined by (2) As we can see, is based on using an information functional f and a function g composing f(vi) and f(vj), vi, vjV. The composition function g should be symmetrical, so the eigenvalues of Eq 1 are then real-valued. It is clear that is given by (3) We define the symmetric functions (4) (5) (6) and the information functionals [13]. (7) (8) and (9) σ(vi) is the eccentricity of a vertex viV [15], δ(vi) is the degree of viV [15], and f3 is based in vertex spheres [12]; we have chosen the coefficients ck linearly decreasing like (10) Note that in case of using the matrix , we obtain the well-known Randić matrix [16] (11) This matrix has already been used when evaluating the discrimination power of the graph measures representing Eqs 1214, see [6]. Based on Eq 2, we see that g(f(vj), f(vj)) is taken into account and, hence, Eq 2 generalizes the Randić matrix. Therefore the here defined functional matrix enables us searching for cases where the functional matrix can encode structural information of graphs more uniquely than in case using the Randić matrix only, see [6].

Solving Eq 3, i.e., computing , we determine the non-zero roots .

In order to compare the new results with previous ones, we define as in [4] straightforwardly the following graph measures based on the zeros of : (12) (13) and (14)

Numerical Results

To interpret the numerical results, we start with giving some technical preliminaries. Here we describe how we generate exhaustively generated trees and graphs. We use the tree classes Ti, 14 ≤ i ≤ 19 containing all non-isomorphic tress with i vertices. Ni, 5 ≤ i ≤ 9 are the set of all non-isomorphic graphs with i vertices. The sizes of these classes are depicted in corresponding tables. As in [6], we generated the graphs by employing the package Nauty by McKay [17] and implemented the graph measures Mi, 1 ≤ i ≤ 3 in R.

Now we start interpreting and discussing concrete numerical results; to do so, we start with trees. Tables 16 show the results when using g = g1 and f = f1, f2, f3. We see that given by the sum of the square roots of the roots of is highly unique on exhaustively generated trees. In most of the cases, this measure is fully unique (ndv = 0 and S = 0). Also, it is evident that the case f = f2, g = g1 and corresponds to the Randić matrix (see Tables 2 and 5 and [6]). We see that the cases g = g1 and f = f1, f3 give better results than in [6]. An explanation for this could be the better coverage of the local topological neighborhood of a vertex by the information functional f = f1, f3. In case of f3, this seems very plausible as f3 captures the full neighborhood of each vertex by using j-spheres [12]. In summary, all graph measures are highly unique on the chosen graph classes by using g = g1 and f = f1, f2, f3. However the results are even more striking when considering Tables 718. We see that the measures are more unique when using g = g2 and f = f1, f2, f3. It seems that g = g2 encodes/spreads its values more efficiently. This corresponds to an earlier finding due to Balaban et al. [18] where the descriptor was found to be highly unique; I1, …, In are topological indices. To study further details, see [18]. In this light, the zeros of the graph polynomial given by Eq 1 can be interpreted as structural graph descriptors (or topological indices). In particular, we observe the high uniqueness of when considering g = g2, g = g3 and the graph classes T18 and T19. Also note that T19 contains more than 300.000 connected graphs with 19 vertices each. Clearly, these results outperform the ones obtained in [6].

thumbnail
Table 1. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f1, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t001

thumbnail
Table 2. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f2, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t002

thumbnail
Table 3. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f3, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t003

thumbnail
Table 4. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f1, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t004

thumbnail
Table 5. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f2, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t005

thumbnail
Table 6. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f3, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t006

thumbnail
Table 7. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f1, g = g2 and .

https://doi.org/10.1371/journal.pone.0139265.t007

thumbnail
Table 8. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f2, g = g2 and .

https://doi.org/10.1371/journal.pone.0139265.t008

thumbnail
Table 9. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f3, g = g3 and .

https://doi.org/10.1371/journal.pone.0139265.t009

thumbnail
Table 10. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f1, g = g2 and .

https://doi.org/10.1371/journal.pone.0139265.t010

thumbnail
Table 11. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f2, g = g2 and .

https://doi.org/10.1371/journal.pone.0139265.t011

thumbnail
Table 12. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f3, g = g2 and .

https://doi.org/10.1371/journal.pone.0139265.t012

thumbnail
Table 13. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f1, g = g3 and .

https://doi.org/10.1371/journal.pone.0139265.t013

thumbnail
Table 14. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f2, g = g3 and .

https://doi.org/10.1371/journal.pone.0139265.t014

thumbnail
Table 15. Exhaustively generated sets of non-isomorphic trees.

|T14| = 3159, |T15| = 7741, |T16| = 19320, |T17| = 48629. Here we used f = f3, g = g3 and .

https://doi.org/10.1371/journal.pone.0139265.t015

thumbnail
Table 16. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f1, g = g3 and .

https://doi.org/10.1371/journal.pone.0139265.t016

thumbnail
Table 17. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f2, g = g3 and .

https://doi.org/10.1371/journal.pone.0139265.t017

thumbnail
Table 18. Exhaustively generated sets of non-isomorphic trees.

|T18| = 123867, |T19| = 317955. Here we used f = f3, g = g3 and .

https://doi.org/10.1371/journal.pone.0139265.t018

As to the graphs, we look at Tables 1921. We see that the results are shown for g = g1 and f = f1, f2, f3. The results for g = g2 and g = g3 are very similar and therefore they are not shown. We emphasize that most of the obtained results are much better than the ones in [6]. In this paper, the lowest ndv-value for N9 achieved by using equals 8 (see Table 20); that means 99.9996% out of 261080 graphs could be discriminated uniquely. This is a striking result and outperforms all earlier results done by Dehmer et al. and co-workers, see, [4, 6, 1922].

thumbnail
Table 19. Exhaustively generated sets of non-isomorphic graphs.

N6∣ = 112, ∣N7∣ = 853, ∣N8∣ = 11117, ∣N9∣ = 261080. Here we used f = f1, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t019

thumbnail
Table 20. Exhaustively generated sets of non-isomorphic graphs.

N6∣ = 112, ∣N7∣ = 853, ∣N8∣ = 11117, ∣N9∣ = 261080. Here we used f = f2, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t020

thumbnail
Table 21. Exhaustively generated sets of non-isomorphic graphs.

N6∣ = 112, ∣N7∣ = 853, ∣N8∣ = 11117, ∣N9∣ = 261080. Here we used f = f3, g = g1 and .

https://doi.org/10.1371/journal.pone.0139265.t021

In comparison, the lowest ndv-value for N9 equals 126 which has been achieved in [6] by using the entropy-like measure . Finally, we see that the results we have obtained in this paper are in parts slightly better and outperform the previous approach by using the Randić matrix.

Conclusion

In this paper, we generalized the work done by Dehmer et al. [6]. In [6], the eigenvalues of the Randić matrix have been used to define new quantitative network measures. A result of this work [6] was the high discrimination power of the measures on exhaustively generated networks. Note that the definition of Randić matrix comes from the famous Randić index [23, 24], from which some new invariants have been introduced, such as the Randić spectral [25] and the Randić energy [26].

Based on the construction of the functional matrix (see Eq 2), we have expected that the here proposed approach may be useful to discriminate graphs more efficiently than by using earlier methods, see, e.g., [4, 6]. From a mathematical point of view, this seems plausible as the involved information functionals f1, …, f3 encode structural information differently. In particular, f3 captures the full topological neighborhood of a vertex and, hence, it encodes structural information more efficiently than f1. This effect can be seen by the Tables 121. On top of that, the composite functions g1, …, g3 may optimize the spread of values of the involved information functionals. We therefore conclude that the ability how the final graph measure can discriminate graphs structurally also depends on the composite function g. In this case, evidence of this statement follows as we defined settings different from the Randić matrix, i.e., f = f2, g = g1 and finally .

Acknowledgments

Matthias Dehmer thank the Austrian Science Funds for supporting this work (project P26142). Yongtang Shi thank NSFC, PCSIRT, China Postdoctoral Science Foundation.

Author Contributions

Analyzed the data: MD FES MS YS ST. Wrote the paper: MD FES MS YS ST.

References

  1. 1. Cvetković DM, Doob M, Sachs H. Spectra of Graphs. Theory and Application. Deutscher Verlag der Wissenschaften; 1980. Berlin, Germany.
  2. 2. Hosoya H. On Some Counting Polynomials. Discrete Applied Mathematics. 1988;19:239–257.
  3. 3. Gutman I. Polynomials in Graph Theory. In: Bonchev D, Rouvray DH, editors. Chemical Graph Theory. Introduction and Fundamentals. Abacus Press; 1991. p. 133–176. New York, NY, USA.
  4. 4. Dehmer M, Shi Y, Mowshowitz A. Discrimination power of graph measures based on complex zeros of the partial Hosoya polynomial. Applied Mathematics and Computation. 2015;250:352–355.
  5. 5. Ellis-Monaghan JA, Merino C. Graph Polynomials and Their Applications I: The Tutte Polynomial. In: Dehmer M, editor. Structural Analysis of Complex Networks. Boston/Basel: Birkhäuser; 2010. p. 219–255.
  6. 6. Dehmer M, Moosbrugger M, Shi Y. Structural Differentiation of Graphs Using Hosoya-Based Indices. Applied Mathematics and Computation, accepted. 2015;268:164–168.
  7. 7. Holzinger A, Dehmer M, Jurisica I. Knowledge Discovery and interactive Data Mining in Bioinformatics-State-of-the-Art, future challenges and research directions. BMC Bioinformatics. 2014;15(Suppl. 6: I1):1–9.
  8. 8. Hosoya H. Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons. Bull Chem Soc Jpn. 1971;44:2332–2339.
  9. 9. Randić M, Vracko M, Novic M. Eigenvalues as Molecular Descriptors. In: Diudea MV, editor. QSPR/QSAR Studies by Molecular Descriptors. Nova Publishing; 2001. p. 93–120. Huntington, NY, USA.
  10. 10. Todeschini R, Consonni V. Handbook of Molecular Descriptors. Wiley-VCH; 2002. Weinheim, Germany.
  11. 11. Dehmer M, Müller L, Graber A. New Polynomial-Based Molecular Descriptors with Low Degeneracy. PLoS ONE. 2010;5(7).
  12. 12. Dehmer M. Information Processing in Complex Networks: Graph Entropy and Information Functionals. Appl Math Comput. 2008;201:82–94.
  13. 13. Dehmer M, Mowshowitz A. A History of Graph Entropy Measures. Information Sciences. 2011;1:57–78.
  14. 14. Dehmer M, Shi Y, Emmert-Streib F. Structural Differentiation of Graphs Using Hosoya-Based Indices. PLoS ONE. 2014;9:e102459. pmid:25019933
  15. 15. Harary F. Graph Theory. Addison Wesley Publishing Company; 1969. Reading, MA, USA.
  16. 16. Liu B, Huang Y, Feng J. A Note on the Randić Spectral Radius. MATCH Communications in Mathematical and in Computer Chemistry. 2012;68:913–916.
  17. 17. McKay BD. Graph isomorphisms. Congressus Numerantium. 1981;730:45–87.
  18. 18. Balaban AT, Mills D, Kodali V, Basak SC. Complexity of Chemical Graphs in terms of size, branching and cyclicity. SAR and QSAR in Enviromental Research. 2006;17(4):429–450.
  19. 19. Dehmer M, Grabner M, Varmuza K. Information Indices with High Discriminative Power for Graphs. PLoS ONE. 2012;7:e31214. pmid:22393358
  20. 20. Dehmer M, Grabner M. The Discrimination Power of Molecular Identification Numbers Revisited. MATCH Commun Math Comput Chem. 2013;69(3):785–794.
  21. 21. Bonchev D, Mekenyan O, Trinajstić N. Isomer discrimination by topological information approach. J Comp Chem. 1981;2(2):127–148.
  22. 22. Konstantinova EV. The Discrimination Ability of Some Topological and Information Distance Indices for Graphs of Unbranched Hexagonal Systems. J Chem Inf Comput Sci. 1996;36:54–57.
  23. 23. Randić M. On characterization of molecular branching. J Amer Chem Soc. 1975;97:6609–6615.
  24. 24. Li X, Shi Y. A survey on the Randić index. MATCH Commun Math Comput Chem. 2008;59(1):127–156.
  25. 25. Gomes H, Gutman I, Martins E, Robbiano M, Martin BS. On Randić spread. MATCH Commun Math Comput Chem. 2014;72:249–266.
  26. 26. Das KC, Sorgun S. On Randić Energy of Graphs. MATCH Commun Math Comput Chem. 2014;72:227–238.