Degree-based entropies of networks revisited

Applied Mathematics and Computation 261 (2015) 141–147

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Degree-based entropies of networks revisited Shujuan Cao a, Matthias Dehmer b,c,∗ a b c

School of Mathematical Sciences, Nankai University, Tianjin 300071, PR China Department of Computer Science, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, Neubiberg 85577, Germany Department of Biomedical Computer Science and Mechatronics, UMIT, Eduard Wallnoefer Zentrum 1, Hall in Tyrol, A-6060, Austria

a r t i c l e

i n f o

Keywords: Entropy Shannon’s entropy Graph entropy Degree powers

a b s t r a c t Studies on the information content of graphs and networks have been initiated in the late fifties based on the seminal work due to Shannon and Rashevsky. Various graph parameters have been used for the construction of entropy-based measures to characterize the structure of complex networks. Based on Shannon’s entropy, in Cao et al. (Extremality of degree-based graph entropies, Inform. Sci. 278 (2014) 22–33), we studied graph entropies which are based on vertex degrees by using so-called information functionals. As a matter of fact, there has been very little work to find their extremal values when considered Shannon entropy-based graph measures. We pursue with this line of research by proving further extremal properties of the degree-based graph entropies. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The four main types of complex networks include weighted digraphs (directed graphs), unweighted digraphs, weighted graphs and unweighted graphs. In this paper, we only consider unweighted undirected graphs. Studies on the information content of graphs have been initiated in the late fifties inspired by the seminal work due to Shannon [60] and Rashevsky [57]. The concept of graph entropy [17,22] introduced by Rashevsky [57] has been used to measure the structural complexity of graphs [18,19]. The entropy of a graph is an information-theoretic quantity that has been firstly introduced by Mowshowitz [51]. Here the complexity of a graph [23] is based on the well-known Shannon’s entropy [12,17,51,50]. Importantly, Mowshowitz interpreted his graph entropy measure as the structural information content of a graph and demonstrated that this quantity satisfies important properties when using product graphs etc, see, e.g., [52–55]. Note the Körner’s graph entropy [38] has been introduced from an information theory point of view and has not been used to characterize graphs quantitatively. An extensive overview on graph entropy measures can be found in [22]. A statistical analysis of topological graph measures has been performed by Emmert-Streib and Dehmer [27]. Several graph invariants, such as the number of vertices, the vertex degree sequences, extended degree sequences (i.e., the second neighbor, third neighbor, etc.), Eigenvalues, and connectivity information, have been used for developing entropy-based measures [17,20,22,25–26]. In [10], the authors study novel properties of graph entropies which are based on an information functional by using degree powers of graphs. The degree powers is one of the most important graph invariant, which has been proven useful in information theory, and for studying social networks, network reliability and problems in mathematical chemistry. For more results on properties of degree powers, we refer to [3,4,31–33,41–44,46,48,49,58]. In particular, they ∗ Corresponding author at: UMIT – The Health and Life Sciences University, Institute for Bioinformatics and Translational Research, Eduard Wallnoefer 1, Hall, Austria, Tel.: +49 6464 646; fax: +49 6151 16 3052. E-mail addresses: [email protected] (S. Cao), [email protected], [email protected] (M. Dehmer).

http://dx.doi.org/10.1016/j.amc.2015.03.046 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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determine the extremal values for the underlying graph entropy of certain families of graphs and find the connection between graph entropy and the sum of degree powers, which is well-studied in graph theory and some related disciplines. The main contribution of the paper is to extend the results performed in [10]. In this paper, we explore extremal values of a special graph entropy measure and the relations between this entropy and the sum of degree powers for different values of k. We demonstrate this by generating numerical results using trees with 11 vertices and connected graphs with 7 vertices, respectively, while in [10], some graph transformations are established to find extremal values of entropy for some classes of graphs when k = 1. In addition, we also make an effort on finding relations between the values of the graph entropy measure and k. The paper is organized as follows. In Section 2, some concepts and notation in graph theory are introduced. In Section 3, we introduce some results on the sum of degree powers. In Section 4, we state the definitions of graph entropies based on the given information functional by using degree powers. In Sections 5 and 6, extremal properties of graph entropies have been studied. Further, we express some conjectures to find extremal values of trees. 2. Preliminaries A graph G is an ordered pair of sets V (G) and E(G) such that the elements uv ∈ E(G) are a sub-collection of the unordered pairs of elements of V (G). For convenience, we denote a graph by G = (V, E) sometimes. The elements of V (G) are called vertices and the elements of E(G) are called edges. If e = uv is an edge, then we say vertices u and v are adjacent, and u, v are two endpoints (or ends) of e. A loop is an edge whose two endpoints are the same one. Two edges are called parallel, if both edges have the same endpoints. A simple graph is a graph containing no loops and parallel edges. If G is a graph with n vertices and m edges, then we say the order of G is n and the size of G is m. A graph of order n is addressed as an n-vertex graph, and a graph of order n and size m is addressed as an (n, m)-graph. A graph is connected if, for every partition of its vertex set into two nonempty sets X and Y, there is an edge with one end in X and one end in Y. Otherwise, the graph is disconnected. In other words, a graph is disconnected if its vertex set can be partitioned into two nonempty subsets X and Y so that no edge has one end in X and one end in Y. A path graph is a simple graph whose vertices can be arranged in a linear sequence in such a way that two vertices are adjacent if they are consecutive in the sequence, and are nonadjacent otherwise. Likewise, a cycle graph on three or more vertices is a simple graph whose vertices can be arranged in a cyclic sequence in such a way that two vertices are adjacent if they are consecutive in the sequence, and are nonadjacent otherwise. Denote by Pn and Cn the path graph and the cycle graph with n vertices, respectively. A connected graph without any cycle is a tree. Actually, the path Pn is a tree of order n with exactly two pendent vertices. The star of order n , denoted by Sn , is the tree with n − 1 pendent vertices. All vertices adjacent to vertex u are called neighbors of u. The neighborhood of u is the set of the neighbors of u. The number of edges adjacent to vertex u is the degree of u, denoted by d(u). Vertices of degrees 0 and 1 are said to be isolated and pendent vertices, respectively. A pendent vertex is also referred to as a leaf of the underlying graph. A vertex of degree i is also addressed as an i-degree vertex. The minimum and maximum degree of G is denoted by δ(G) and (G) , respectively. If G has ai vertices  of degree di (i = 1, 2, . . . , t), where (G) = d1 > d2 > · · · > dt = δ(G) and ti=1 ai = n, we define the degree sequence of G as ai a1 a2 at D(G) = [d1 , d2 , . . . , dt ]. If ai = 1, we use di instead of di for convenience. For terminology and notations not defined here, we refer the reader to [9]. 3. Degree-based graph entropies First, we state the definition of Shannon’s entropy [60]. Definition 1. Let p = (p1 , p2 , . . . , pn ) be a probability vector, namely, 0 ≤ pi ≤ 1 and defined as

I (p) = −

n 

n i=1

pi = 1. Shannon’s entropy of p is

pi log pi .

i=1

To define information-theoretic graph measures, we will often consider a tuple (λ1 , λ2 , . . . , λn ) of non-negative integers

λi ∈ N [17]. This tuple forms a probability distribution p = (p1 , p2 , . . . , pn ), where

λi

pi = n

j=1

λj

i = 1, 2, . . . , n.

Therefore, the entropy of tuple (λ1 , λ2 , . . . , λn ) is given by

I (λ1 , λ2 , . . . , λn ) = −

n  i=1



pi log pi = log

n  i=1



λi −

n  i=1

λi

n

j=1

λj

log λi .

(1)

In the literature, there are various ways to obtain the tuple (λ1 , λ2 , . . . , λn ), like the so-called magnitude-based information measures introduced by Bonchev and Trinajstic´ [5], or partition-independent graph entropies, introduced by Dehmer [17,24], which are based on information functionals. We are now ready to define the entropy of a graph due to Dehmer [17] by using information functionals.

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Definition 2. Let G = (V, E) be a connected graph. For a vertex vi ∈ V, we define

f (vi ) p(vi ) := |V | , f (vj ) j=1 where f represents an arbitrary information functional. |V | Observe that i=1 p(vi ) = 1. Hence, we can interpret the quantities p(vi ) as vertex probabilities. Now we immediately obtain one definition of graph entropy for graph G. Definition 3. Let G = (V, E) be a connected graph and f be an arbitrary information functional. The entropy of G is defined as |V | 

⎛

f (vi )

⎞

f (vi )

⎠ log ⎝ |V | f ( v ) f ( v ) j j j=1 j=1 ⎞ ⎛ |V | |V |   f (vi ) f (vi )⎠ − log f (vi ). = log ⎝ | V | f (vj ) i=1 i=1 j=1

If (G) = −

i=1

| V |

(2)

In [10], the authors defined a novel information functional which is based on degree powers of graphs.

If (G, k) = −

n  i=1



dki

n

k j=1 dj

dki

n

log

k j=1 dj



= log

 n 



dki

−

n 

dki n

i=1

j=1

i=1

dkj

log dki .

(3)

In this paper, we will study further properties of the above graph entropy based on degree powers. 4. Properties of If (G, k) revisited Let D(k) :=

n j=1

dkj . Then we have

If (G, k) = log(D(k)) −

1  k kdi log di . D(k) n

i=1

It is interesting to study relations between D(k) and If . In Section 5, we consider the trees with 11 vertices and connected graphs with 7 vertices. By computing the values of D(k) and If for k = 2, 3, 4 and k = 0.2, 0.5, 0.8, respectively, the data shows that the curves If and D(k) are very similar but somewhat shifted. Theorem 1. Let G be a graph with n vertices. Denote by δ and  the minimum degree and maximum degree of G, respectively. Then we have

 



log D(k) − k log  ≤ If (G, k) ≤ log D(k) − k log δ .

Proof. From Eq. (3), we have



If (G, k) = log D(k) −

k  k di log di . D(k) n

(4)

i=1

Therefore,

 If (G, k) ≥ log D(k) −

n

 k  k di log  = log D(k) − k log , D(k) i=1

and



If (G, k) ≤ log D(k) −

n 

k  k di log δ = log D(k) − k log δ . D(k) i=1

From the above theorem, we easily obtain the following corollary.



Corollary 1. If G is a d-regular graph, then If (G, k) = log n for any k. Observe that if G is regular, then If is a function only on n. Numerical results support that if k > 0, then If is a monotonously increasing function on k for connected graphs. So, we state this as a conjecture. Conjecture 1. For k > 0, If is a monotonously increasing function on k for connected graphs. When k > 0 is large enough, log(D(k)) is the main part of If . Therefore, for connected graphs, If is a monotonously increasing function on k when k is large enough. In Section 5, we consider the trees with 11 vertices and connected graphs with 7 vertices. By computing the values of If for k = 2, 3, 4 and k = 0.2, 0.5, 0.8, respectively, we see that the data supports our conjecture.

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Fig. 1. The values of Dk (red) and If (blue) for k = 2 (left), k = 3 (middle) and k = 4 (right). Y axis denotes the values of Dk and If . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. The values of Dk (red) and If (blue) for k = 0.2 (left), k = 0.5 (middle) and k = 0.8 (right). Y axis denotes the values of Dk and If . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

5. Numerical results In [10], the authors studied the extremal values of If for some special classes of graphs when k = 1. The case for k > 0 and k = 1 is much more complicated. This section presents the numerical results when applying the entropies to certain classes of graphs for k > 0 and k = 1, which will give some intuitionistic views. The data sets used are as follows: trees with 11 vertices and connected graphs with 7 vertices [59]. The number of non-isomorphic trees with 11 vertices is 235. Moreover, the number of distinct degree sequences of these trees is 32. In Fig. 1, the values of Dk (red) and If (blue) for k = 2 (left), k = 3 (middle) and k = 4 (right) are plotted, respectively. In Fig. 2, the values of Dk (red) and If (blue) for k = 0.2 (left), k = 0.5 (middle) and k = 0.8 (right) are plotted, respectively. From Figs. 1 and 2, we see that for a given k, the values of Dk and If have the same variation tendency. Moreover, for large k, the values of Dk and If is much closer. In Fig. 3, the values of If for k = 2 (left, red), k = 3 (left, blue) and k = 4 (left, green); for k = 0.2 (right, red), k = 0.5 (right, blue) and k = 0.8 (right, green) are plotted, respectively. From Fig. 3, we can see that for a given graph G, the values of If (G, k) is an increasing function on k. Moreover, for k = 2, k = 3 and k = 4, the maximum value and the minimum value of If are attained when T is the star graph and the path graph, respectively. However, for k = 0.2, k = 0.5 and k = 0.8, the graphs with the extremal values are completely different, and seems to be much complicated: the minimum value is attained when T is the star graph, while for k = 0.2 and k = 0.5, the maximum value is attained when T is the path graph, and for k = 0.8, the maximum value is attained when T is the tree T212 (p. 69 of [59]), which is obtained by attaching 3 and 5 leaves to the two ends of the path P3 , respectively. The number of distinct degree sequences of connected graphs with 7 vertices is 236. In Fig. 4, the values of Dk (red) and If (blue) for k = 2 (left), k = 3 (middle) and k = 4 (right) are plotted, respectively. In Fig. 5, the values of Dk (red) and If (blue) for k = 0.2 (left), k = 0.5 (middle) and k = 0.8 (right) are plotted, respectively. From Figs. 4 and 5, we can see that for a given k, the values of Dk and If have the same variation tendency. Moreover, for large k, the values of Dk and If is much closer. In Fig. 6, the values of If for k = 2 (left, red), k = 3 (left, blue) and k = 4 (left, green); for k = 0.2 (right, red), k = 0.5 (right, blue) and k = 0.8 (right, green) are plotted, respectively. From Fig. 6, we also observe that for a given graph G, the values of If (G, k) is an increasing

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Fig. 3. The values of If for k = 2 (left, red), k = 3 (left, blue) and k = 4 (left, green); for k = 0.2 (right, red), k = 0.5 (right, blue) and k = 0.8 (right, green). Y axis denotes the values of If . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

300

10000

1600

9000

1400 250

8000 1200 7000

200

6000

800

Y

150

Y

Y

1000

5000 4000

600 100

3000 400 2000

50 200 0

0

50

100

150

200

0

250

1000 0

50

Connected graphs

100

150

200

0

250

0

50

Connected graphs

100

150

200

250

Connected graphs

Fig. 4. The values of Dk (red) and If (blue) for k = 2 (left), k = 3 (middle) and k = 4 (right). Y axis denotes the values of Dk and If . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

10.5

18

30

17 10 16 9.5

25

15 14

8.5

20

13

Y

Y

Y

9

12

15

11

8

10

10

7.5 9 7

0

50

100 150 Connected graphs

200

250

8

0

50

100 150 Connected graphs

200

250

5

0

50

100 150 Connected graphs

200

250

Fig. 5. The values of Dk (red) and If (blue) for k = 0.2 (left), k = 0.5 (middle) and k = 0.8 (right). Y axis denotes the values of Dk and If . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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10000

30

9000 8000

25

7000 20

5000

Y

Y

6000

4000

15

3000 2000

10

1000 0

0

50

100 150 Connected graphs

200

250

5

0

50

100 150 Connected graphs

200

250

Fig. 6. The values of If for k = 2 (left, red), k = 3 (left, blue) and k = 4 (left, green); for k = 0.2 (right, red), k = 0.5 (right, blue) and k = 0.8 (right, green). Y axis denotes the values of If . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

function on k. Moreover, for k = 0.2, k = 0.5 and k = 0.8, the maximum value of If is attained when G is the complete graph, while the minimum value is attained when G is the star graph. For k = 2, k = 3 and k = 4, the maximum value of If is attained when G is the complete graph, while the minimum value is attained when G is the path graph. Notice that the number of distinct degree sequences with at least one member whose value is 1, is 106 and we list these sequences before others. So there is a separatrix at point 106 in Figs. 4–6, from which we can see the values of entropy on the two sides of this point. Also we see that both sides of this point have the same increasing tendency. Actually, from this, we obtain the extremal values of If among all graphs with one leaf, and all graphs without leaves, respectively. Hence, in the future, when we study the extremal values of graph entropy, a possible way is to divide the graphs into two classes: graphs with leaves and without leaves. We believe that this procedure may be useful to proceed investigating the problem. 6. Summary and conclusion In this paper, we studied again properties of graph entropies by using the degree– power information functional. As we have already outlined, proving extremal results of graph entropies analytically has been intricate. That is the reason why we have some supported our mathematical results by numerical results too. The generated numerical results strive the direction to study extremal problems. However, until now we did not find efficient methods to generalize these extremal results to general graphs with n vertices. It would be also interesting to investigate interrelations between graph entropies and some other graph invariants or topological indices, such as Randic´ index [41,46,58], various of Wiener indices [26,37,50,61], graph energies [14,15,34,35,45,47], Randic´ energy [6,16], incidence energy [7,8], matching energy [11,36], Laplacian energy [13], energy of matrices [30], HOMO–LUMO index [40], Zagreb index [29,56,62], ABC index [55], connective eccentricity index [63], Kirchhoff index [28,39], Harary index [2], Szeged index [1], chromatic number [44]. As there infinitely many graph invariants (indices) exist, this task is far from trivial and will be tackled as future work. Acknowledgments Matthias Dehmer thanks the Austrian Science Funds for supporting this work (project P26142). References [1] T. Al-Fozan, P. Manuel, I. Rajasingh, R.S. Rajan, Computing Szeged index of certain nanosheets using partition technique, MATCH Commun. Math. Comput. Chem. 72 (2014) 339–353. [2] M. Azari, A. Iranmanesh, Harary index of some nano-structures, MATCH Commun. Math. Comput. Chem. 71 (2014) 373–382. [3] B. Bollobás, V. Nikiforov, Degree powers in graphs with forbidden subgraphs, Electron. J. Comb. 11 (2004) R42. [4] B. Bollobás, V. Nikiforov, Degree powers in graphs: the Erdös– Stone theorem, Comb. Probab. Comput. 21 (2012) 89–105. ´ Information theory, distance matrix and molecular branching, J. Chem. Phys. 67 (1977) 4517–4533. [5] D. Bonchev, N. Trinajstic, [6] S.B. Bozkurt, D. Bozkurt, Sharp upper bounds for energy and Randic´ energy, MATCH Commun. Math. Comput. Chem. 70 (2013) 669–680. [7] C.B. Bozkurt, D. Bozkurt, On incidence energy, MATCH Commun. Math. Comput. Chem. 72 (2014) 215–225. [8] C.B. Bozkurt, I. Gutman, Estimating the incidence energy, MATCH Commun. Math. Comput. Chem. 70 (2013) 143–156. [9] J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, 2008. [10] S. Cao, M. Dehmer, Y. Shi, Extremality of degree-based graph entropies, Inform. Sci. 278 (2014) 22–33. [11] L. Chen, Y. Shi, Maximal matching energy of tricyclic graphs, MATCH Commun. Math. Comput. Chem. 73 (1) (2015) 105–119. [12] T.M. Cover, J.A. Thomas, Elements of Information Theory, Wiley, New York, 2006.

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