Recent Developments in Quantitative Graph Theory: Information Inequalities for Networks

In this article, we tackle a challenging problem in quantitative graph theory. We establish relations between graph entropy measures representing the structural information content of networks. In particular, we prove formal relations between quantitative network measures based on Shannon's entropy to study the relatedness of those measures. In order to establish such information inequalities for graphs, we focus on graph entropy measures based on information functionals. To prove such relations, we use known graph classes whose instances have been proven useful in various scientific areas. Our results extend the foregoing work on information inequalities for graphs.


Introduction
Complexity is an intricate and versatile concept that is associated with the design and configuration of any system [1,2]. For example, complexity can be measured and characterized by quantitative measures often called indices [3][4][5]. When studying the concept of complexity, information theory has been playing a pioneering and leading role. Prominent examples are the theory of communication and applied physics where the famous Shannon entropy [6] has extensively been used. To study issues of complexity in natural sciences and, in particular, the influence and use of information theory, see [7].
In this paper, we deal with an important aspect when studying the complexity of network-based systems. In particular, we establish relations between information-theoretic complexity measures [3,[8][9][10][11]. Recall that such entropic measures have been used to quantify the information content of the underlying networks [8,12]. Generally, this relates to exploring the complexity of a graph by taking its structural features into account. Note that numerous measures have been developed to study the structural complexity of graphs [5,8,[13][14][15][16][17][18][19][20][21][22]. Further, the use and ability of the measures has been demonstrated by solving interdisciplinary problems. As a result, such studies have led to a vast number of contributions dealing with the analysis of complex systems by means of information-theoretic measures, see, e.g., [8,[13][14][15][16][17][18][19][20][21][22]. Figure 1 shows a classification scheme of quantitative network measures exemplarily.
The main contribution of this paper is to study relations between entropy measures. We will tackle this problem by means of inequalities involving network information measures. In particular, we study so-called implicit information inequalities which have been introduced by Dehmer et al. [23,24] for studying graph entropies using information functionals. Generally, an implicit information inequality involves information measures which are present on either side of the inequality. It is important to emphasize that relatively little work has been done to investigate relations between network measures. A classical contribution in this area is due to Bonchev et al. [25]. Here, the relatedness between information-theoretic network measures has been investigated to detect branching in chemical networks. Further, implicit information inequalities have been studied for hierarchical graphs which turned out to be useful in network biology [26].
We first present closed form expressions of graph entropies using the graph classes, stars and path graphs. Further, we infer novel information inequalities for the measures based on the jsphere functional. The section ''Implicit Information Inequalities'' presents our main results on novel implicit inequalities for networks. We conclude the paper with a summary and some open problems. Before discussing our results, we will first present the information-theoretic measures that we want to investigate in this paper.

Methods
In this section, we briefly state the concrete definitions of the information-theoretic complexity measures that are used for characterizing complex network structures [3,6,9,27]. Here we state measures based on two major classifications namely partitionbased and partition-independent measures and deal mainly with the latter.
Given a simple, undirected graph G~(V ,E), let d(u,v) denote the distance between two vertices u and v, and let r(G)~maxfd(u,v) : u,v [ V g. Let S j (u; G) denote the j-sphere of a vertex u defined as S j (u; G)~fx [ V : d(u,x)~jg. Throughout this article, a graph G represents a simple undirected graph.
Definition 1 Let G~(V ,E) be a graph on n vertices and let X be a graph invariant of G. Let a be an equivalence relation that partitions X into k subsets X 1 ,X 2 , . . . X k , with cardinality jX i j for 1ƒiƒk. The total structural information content of G is given by Definition 2 Let G~(V ,E) be a graph on n vertices and let p i~j X i j=jX j, for 1ƒiƒk be the probability value for each partition. The mean information content of G is In the context of theory of communication, the above equation is called as Shannon equation of information [28].
Definition 3 Let G~(V ,E) be a graph on n vertices. The quantity is a probability value of v i [ V . f : V ?R z is an arbitrary information functional that maps a set of vertices to the non-negative real numbers. Remark 1 Observe that, p( : ) defines a probability distribution over the set of vertices as it satisfies 0ƒp(v i )v1, for every vertex v i , 1ƒiƒn and P n i~1 p(v i )~1.
Using the resulting probability distribution associated with G leads to families of network information measures [3,9].
Definition 4 The graph entropy of G given representing its structural information content: In order to define concrete graph entropies, we reproduce the definitions of some information functionals based on metrical properties of graphs [3,9,27].
Definition 5 Parameterized exponential information functional using jspheres: where aw0 and c k w0 for 1ƒkƒr(G). Definition 6 Parameterized linear information functional using jspheres: where c k w0 for 1ƒkƒr(G). Remark 2 Observe that, when either a~1 or the c k are all equal, the functional f P and f P' becomes a constant function and, hence, the probability on all the vertices are equal. That is Thus, the value of the entropy attains its maximum value, I f (G)~log 2 (n). Thus, in all our proofs, we only consider the non-trivial case, namely a=1 and/or at least for two coefficients holds c j =c k . Next, we will define the local information graph to use local centrality measures from [9]. Let L G (v,j) be the subgraph induced by the shortest path starting from the vertex v to all the vertices at distance j in G. Then, L G (v; j) is called the local information graph regarding v with respect to j, see [9]. A local centrality measure that can be applied to determine the structural information content of a network [9] is then defined as follows.
Definition 7 The closeness centrality of the local information graph is defined by Remark 3 Note that centrality is an important concept that has been introduced for analyzing social networks [29,30]. Many centrality measures have been contributed [30], and in particular, various definitions for closeness centrality [30][31][32]. We remark that the above definition has been firstly defined by Sabidussi [31] for arbitrary graphs. However, we use the measure as a local invariant defined on the subgraphs induced by the local information graph [9].
Similar to the j-sphere functionals, we define further functionals based on the local centrality measure as follows.
Definition 8 Parameterized exponential information functional using local centrality measure: where aw0, c k w0 for 1ƒkƒr(G). Definition 9 Parameterized linear information functional using local centrality measure: where c k w0, for 1ƒkƒr(G).
Note that the coefficients c k can be chosen arbitrarily. However, the functionals become more meaningful when we choose the coefficients to emphasize certain structural characteristics of the underlying graphs. Also, this remark implies that the notion of graph entropy is not unique because each measure takes different structural features into account. Further, this can be understood by the fact that a vast number of entropy measures have been developed so far. Importantly, we point out that the measures we explore in this paper are notably different to the notion of graph entropy introduced by Körner [21]. The graph entropy due to Körner [21] is rooted in information theory and based on the known stable set problem. To study more related work, survey papers on graph entropy measures have been authored by Dehmer et al. [3] and Simonyi [33].

Closed Form Expressions and Explicit Information Inequalities
When calculating the structural information content of graphs, it is evident that the determination of closed form expressions using arbitrary networks is critical. In this section, we consider simple graphs namely trees with smallest and largest diameter and compute the measures defined in the previous section. By using arbitrary connected graphs, we also derive explicit information inequalities using the measures based on information functionals (stated in the previous section).
Stars. Star graphs, S(n), have been of considerable interest because they represent trees with smallest possible diameter (r(S(n))~2) among all trees on n vertices. Now, we present closed form expressions for the graph entropy by using star graphs. For this, we apply the information-theoretic measures based on information functionals defined in the preliminaries section.
Theorem 4 Let S(n) be a star on n vertices. Let f [ ff P ,f ' P ,f C ,f ' C g be the information functionals as defined before. The information measure is given by where x is the probability of the central vertex of S(n): if f~f P .
if f~f C .
We get, Therefore, Hence, By substituting the value of p f (v) in I f (S(n)) and simplifying, we get where c k w0 for 1ƒkƒr(S(n)).
We get, if v is the central vertex, Therefore, Hence, ifv is the central vertex, , otherwise: By substituting the value of p f (v) in I f (S(n)) and simplifying, we get denotes the closeness centrality measure.
Then, we yield Therefore, Hence, , otherwise: By substituting the value of p f (v) in I f (S(n)) and simplifying, we obtain .
1ƒkƒr(S(n)). b is defined via Equation (18). We get, if v is the central vertex, ), otherwise: Therefore, Thus, , otherwise: By substituting the value of p f (v) in I f (S(n)) and simplifying, we get where x~c .
By choosing particular values for the parameters involved, we get concrete measures using the above stated functionals. For example, consider the functional f~f P' and set c 1 :~r(S(n))~2 and c 2 :~r(S(n)){1~1: If we plug in those values in Equations (10) and (11), we easily derive Paths. Let P n be the path graph on n vertices. Path graphs are the only trees with maximum diameter among all the trees on n vertices, i.e., r(P n )~n{1. We remark that to compute a closed form expression even for path graphs, is not always simple. To illustrate this, we present the concrete information measure I fP' (P n ) by choosing particular values for its coefficients.
Lemma 5 Let P n be a path graph and consider the functional f~f P' defined by Equation (6). We set c 1 :~r(P n )~n{1, c 2 :r (P n ){1, . . . ,c r :~1. We yield Proof: Let P n be a path graph trivially labeled by v 1 , v 2 , . . . ,v n (from left to right). Given By computing f , when v [ fv r ,v nz1{r g, for 1ƒrƒq n 2 r, we infer Therefore, and, hence, where v [ fv r ,v nz1{r g, for 1ƒrƒq n 2 r. By substituting these quantities into I f (P n ) yields the desired result. Note that when using the same measure with arbitrary coefficients, its computation is intricate. In this regard, we present explicit bounds or information inequalities for any connected graph if the measure is based on the information functional using j-spheres. That is, either f~f P or f~f P' .
General connected graphs. Theorem 6 Given any connected graph G~(V ,E) on n vertices and let f~f P given by Equation (5). Then, we infer the following bounds: ð37Þ where with c max~m axfc j : and c min~m infc j : 1ƒjƒr(G)g: Proof: where aw0 and c k w0 for 1ƒkƒr(G). Let c max~m axfc j : 1ƒjƒr(G)g and c min~m infc j : 1ƒjƒr(G)g. Recall (see Remark (2)) that, when either a~1 or when all the coefficients (c k ) are equal, the information functional becomes constant and, hence, the value of I f (G) equals log 2 n. In the following, we will discuss the cases aw1 and av1, and we also assume that not all c k are equal.
Case 1: aw1: We first construct the bounds for p f (v) as shown below: Similarly, Therefore, from the Equations (43) and (44), we get Hence, Let X~(n{1)½c max {c min . Then, the last inequality can be rewritten as, Upper bound for I f (G): Since X w0 and aw1, we have 1 n : a X v1. Hence, we have { log 2 1 n : a X §0 and 0v{ log 2 p f (v)ƒ{ log 2 1 n : a X . Thus we get, By adding over all the vertices of V , we obtain I f (G)ƒ{a X log 2 1 n : a X~a X log 2 (n : a X ): Lower bound for I f (G): We have to distinguish two cases, either a X vn or a X §n.
By adding over all the vertices of V , we get Case 1.2: a §n 1=X . In this case, we obtain log 2 a X n §0 and log 2 p f (v)v0ƒ log 2 a X n .
Therefore, by using these bounds in Equation (4), we infer I f (G)w0. Case 2: av1: Consider Equation (42). We get the following bounds for f (v): Therefore, Hence, Set X~(n{1)½c max {c min . Then, the last inequality can be rewritten as, Upper bound for I f (G): Since X w0 and av1, we have a X n ƒ1. Hence, we have { log 2 a X n §0 and 0v{ log 2 p f (v)ƒ{ log 2 a X n . Thus, we obtain, By adding over all the vertices of V , we get Lower bound for I f (G): Again, we consider two cases, either a X ƒ 1 n or a X w 1 n .
In this case, we have log 2 1 n : a X §0 and log 2 p f (v)v0ƒ log 2 1 n : a X . Therefore, by substituting these bounds in the Equation (4), we obtain I f (G)w0.
We have { log 2 p f (v) §{ log 2 1 na X w0. Therefore, By adding over all the vertices of V , we get Hence, the theorem follows.
In the next theorem, we obtain explicit bounds when using the information functional given by Equation (6).
Theorem 7 Given any connected graph G~(V ,E) on n vertices and let f~f ' P be given as in Equation (6). We yield with c max~m axfc j : 1ƒjƒr(G)g, and c min~m infc j : 1ƒjƒr(G)g: Proof: for 1ƒkƒr(G). Let c max~m axfc j : 1ƒjƒr(G)g and c minm infc j : 1ƒjƒr(G)g.
We have, Similarly, Therefore, from the Equations (64) and (65), we get Hence, Upper bound for I f (G): Since c min n : c max ƒ1, we have { log 2 c min n : c max §0 and 0v { log 2 p f (v)ƒ{ log 2 c min n : c max . Hence, By adding over all the vertices of V , we obtain Lower bound for I f (G): Let us distinguish two cases: Case 1: c max §n : c min .
We have log 2 c max n : c min §0 and log 2 p f (v)v0ƒ log 2 c max n : c min .
Therefore, by applying these bounds to Equation (4), we obtain I f (G)w0. Case 2: c max vn : c min . In this case, we have { log 2 p f (v) §{ log 2 c max n : c min w0. Therefore, By adding over all the vertices of V , we obtain the lower bound for I f (G) given by Hence, the theorem follows.

Implicit Information Inequalities
Information inequalities describe relations between information measures for graphs. An implicit information inequality is a special type of an information inequality where the entropy of the graph is estimated by a quantity that contains another graph entropy expression. In this section, we will present some implicit information inequalities for entropy measures based on information functionals. In this direction, a first attempt has been done by Dehmer et al. [23,24,26]. Note that Dehmer et al. [23,26] started from certain conditions on the probabilities when two different information functionals f and f Ã are given. In contrast, we start from certain assumptions which the functionals themselves should satisfy and, finally, derive novel implicit inequalities. Now, given any graph G~(V ,E),jV j~n. Let I f1 (G) and I f2 (G) be two mean information measures of G defined using the information functionals f 1 and f 2 respectively. Let us further define another functional f (v)~c 1 In the following, we will study the relation between the information measure I f (G) and the measures I f1 (G) and I f2 (G).
Theorem 8 Suppose f 1 (v)ƒf 2 (v), for all v [ V , then the information measure I f (G) can be bounded by I f1 (G) and I f2 (G) as follows: , ð72Þ where Proof: The information measures of G with respect to f 1 and f 2 are given by where where Now consider the probabilities, ƒ (c 1 zc 2 )A 2 : Using Equation (77) and based on the fact that p f (v)ƒ1, we get Thus, and Since the last term in the above inequality is positive, we get By adding up the above inequalities over all the vertices of V , we get the desired upper bound. From Equation (77), we also get a lower bound for p f (v), given by Now proceeding as before with the above inequality for p f (v), we obtain By using the concavity property of the logarithm, that is, By adding the above inequality over all the vertices of V , we get the desired lower bound. This proves the theorem.

Corollary 9
The information measure I f (G), for f~f 1 zf 2 , is bounded by I f1 (G) and I f2 (G) as follows: , ð87Þ Proof: Set c 1~c2 in Theorem (8), then the corollary follows. Corollary 10 Given two information functionals, Proof: Follows from Corollary (9). The next theorem gives another bound for I f in terms of both I f1 and I f2 by using the concavity property of the logarithmic function.
Theorem 11 Let f 1 (v) and f 2 (v) be two arbitrary functionals defined on a graph G. If f (v)~c 1 and where Proof: Starting from the quantities for p f (v) based on Equation (77), we obtain Since each of the last two terms in Equation (95) is positive, we get a lower bound for p f (v) log 2 p f (v), given by Applying the last inequality to Equation (4), we get the upper bound as given in Equation (91). By further applying the inequality to Equation (95), we get an upper bound given by Therefore, : Finally, we now apply this inequality to Equation (4) and get the lower bound as given in Equation (90). The next theorem is a straightforward extension of the previous statement. Here, an information functional is expressed as a linear combination of k arbitrary information functionals.
Theorem 12 Let k §2 and f 1 (v),f 2 (v), . . . ,f k (v) be arbitrary functionals defined on a graph G. I f1 (G),I f2 (G), . . . ,I f k (G) are the corresponding information contents. If f (v)~c 1 and where Union of Graphs. In this section, we determine the entropy of the union of two graphs. Let G 1~( V 1 ,E 1 ) and G 2~( V 2 ,E 2 ) be two arbitrary connected graphs on n 1 and n 2 vertices, respectively. Let f be an information functional defined on these graphs denoted by f G1 , f G2 and let I f (G 1 ) and I f (G 2 ) be the information measures on G 1 and G 2 respectively.
Theorem 13 Let G~(V ,E)~G 1 |G 2 be the disjoint union of the graphs G 1 and G 2 . Let f be an arbitrary information functional. The information measure I f (G) can be expressed in terms of I f (G 1 ) and I f (G 2 ) as follows: where .
Proof: Let f be the given information functional. Let . The information measures of G 1 and G 2 are given as follows: where , and where Hence, Using these quantities to determine I f G ð Þ, we obtain and Upon simplification, we get the desired result. Also, we immediately obtain a generalization of the previous theorem by taking k-disjoint graphs into account.
Theorem 14 Let G 1~( V 1 ,E 1 ), G 2~( V 2 ,E 2 ), . . . ,G k~( V k ,E k ) be k arbitrary connected graphs on n 1 ,n 2 , . . . ,n k vertices, respectively. Let f be an information functional defined on these graphs denoted by f G1 ,f G2 , . . . ,f G k . Let G~(V ,E)~G 1 |G 2 | Á Á Á |G k be the disjoint union of the graphs G 1 ,G 2 , . . . ,G k for k §2. The information measure I f (G) can be expressed in terms of I f (G 1 ),I f (G 2 ), . . ., I f (G k ) as follows: where Join of Graphs. Let G 1~( V 1 ,E 1 ) and G 2~( V 2 ,E 2 ) be two arbitrary connected graphs on n 1 and n 2 vertices, respectively. The join of the graphs G 1 zG 2 is defined as the graph G~(V ,E) with vertex set V~V 1 |V 2 and the edge set Let f~f P be the information functional (given by Equation (5)) based on the j-sphere functional (exponential) defined on these graphs and denoted by f G1 , f G2 . Let I f (G 1 ) and I f (G 2 ) be the information measures on G 1 and G 2 respectively.
Theorem 15 Let G~(V ,E)~G 1 zG 2 be the join of the graphs G 1~( V 1 ,E 1 ) and G 2~( V 2 ,E 2 ) with n 1 zn 2 vertices. The information measure I f (G) can then be expressed in terms of I f (G 1 ) and I f (G 2 ) as follows: where f H (v)~a Proof: Let G~G 1 zG 2 be the join of two connected graphs G 1 and G 2 . Here, r(G)~maxfr(G 1 ),r(G 2 )g. Let f H (v)ã P j~1 r(H)cj Sj (v;H) be the information functional defined by using the j-sphere functional on H [ fG,G 1 , . The information measures of G 1 and G 2 are given as follows: where , and where Hence, Using those entities to determine I f (G), we infer and Upon simplification, we get the desired result. If we consider the linear j-sphere functional f ' P (see Equation (6)), to infer an exact expression for the join of two graphs as in Theorem (15) is an intricate problem. By Theorem (16) and Theorem (17), we will now present different bounds in terms of I f ' P (G 1 ) and I f ' P (G 2 ).
Theorem 16 Let G~(V ,E)~G 1 zG 2 be the join of the graphs G 1~( V 1 ,E 1 ) and G 2~( V 2 ,E 2 ) on n 1 zn 2 vertices. Then, we yield where f H (v)~P r(H) j~1 c j S j (v; H) for H [ fG 1 ,G 2 ,Gg, c j w0 and A2 c 1 n 1 n 2 zA The information measures of G 1 and G 2 are given as follows: where , and where p G 2 (v)~f Putting Inequality (139) and Inequality (141) together finishes the proof of the theorem.

Summary and Conclusion
In this article, we have investigated a challenging problem in quantitative graph theory namely to establish relations between graph entropy measures. Among the existing graph entropy measures, we have considered those entropies which are based on information functionals. It turned out that these measures have widely been applicable and useful when measuring the complexity of networks [3].
In general, to find relations between quantitative network measures is a daunting problem. The results could be used in various branches of science including mathematics, statistics, information theory, biology, chemistry and social sciences. Further, the determination of analytical relations between measures is of great practical importance when dealing with large scale networks. Also, relations involving quantitative network measures could be fruitful when determining the information content of large complex networks.
Note that our proof technique follows the one proposed in [23]. It is based on three main steps: Firstly, we compute the information functionals and in turn, we calculate the probability values for every vertex of the graph in question. Secondly, we start with certain conditions for the computed functionals and arrive at a system of inequalities. Thirdly, by adding up the corresponding inequality system, we obtain the desired implicit information inequality. Using this approach, we have inferred novel bounds by assuming certain information functionals. It is evident that further bounds could be inferred by taking novel information functionals into account. Further, we explored relations between the involved information measures for general connected graphs and for special classes of graphs such as stars, path graphs, union and join of graphs.
At this juncture, it is also relevant to compare the results proved in this paper with those proved in [23]. While we derived the implicit information inequalities by assuming certain properties for the functionals, the implicit information inequalities derived in [23] are based on certain conditions for the calculated vertex probabilities. Interestingly, note that by using Theorem (11) and Theorem (17), the range of the corresponding bounds is very small. We inferred that the difference between the upper and lower bound equals log 2 e&1:442695.
As noted earlier, relations between entropy-based measures for graphs have not been extensively explored so far. Apart from the results we have gained in this paper, we therefore state a few open problems as future work: N To find relations between I f (G) and I f (H), when H is an induced subgraph of G and f is an arbitrary information functional.
N To find relations between I f (G) and fI f (T 1 ),I f (T 2 ), . . . , I f (T n )g, where T i , 1ƒiƒn are so-called generalized trees, see [34]. Note that it is always possible to decompose an arbitrary, undirected graph into a set of generalized trees [34].

Author Contributions
Wrote the paper: MD LS. Performed the mathematical analysis: MD LS.