Entropy based measures for graphs

Entropy based measures for graphs
Master in Web Science, Department of Mathematics,
Aristotle University of Thessaloniki
Giorgos Bamparopoulos, Nikos Papathanasiou

Introduction
In many scientific areas, e.g. sociology, psychology, physics, chemistry,
biology etc., systems can be described as interaction networks where the
interactions correspond to edges and the elements to vertices. A variety of
problems in those fields can deal with network comparison and
characterization.
The problem of comparing networks is the task of measuring their
structural similarity. Moreover, “characterize networks” means that we are
searching for network characteristics which capture structural
information of networks. To analyze complex networks, several methods
can be combined, such as graph theory, information theory, and statistics.
In this project, in order to characterize and compare network structures,
we describe methods for measuring Shannon’s entropy of graphs.
To introduce entropy based measures for graphs, we first mention some
basic graph-theoretical preliminaries:

Preliminaries
A graph is a non-empty finite set V of elements called vertices
together with a possibly empty set E of pairs of vertices called
edges. It is denoted G=(V,E)
The degree of a vertex u is the number of links plus twice the
number of loops incident with u. It is denoted d(u). The degree
distribution is the relative frequency of vertices with different
degrees. (P(k) = fraction of vertices with degree k)
The j-sphere of a vertex vi in G is defined by the set Sj(vi, G): =

Preliminaries
A path of length n , n ≥1 , from vertex a to vertex b is a sequence
of directed edges e1, e2, ...,en such that the initial vertex of e1 is a ,
and the terminal vertex of en is b, and for i=2, ..., n , the initial
vertex of ei is the terminal vertex of ei-1 . When it is more
convenient, vertices can be listed in order (rather than the
edges) to represent the path.
If the first and last nodes of a path coincide it is called cycle.
d(u,v) denotes the distance between u∈V and v ∈ V expressed
as the minimum length of a path between u,v.

Preliminaries
A coloring of a graph G(V,E) is an assignment of colors to the
vertices such that no two adjacent vertices have the same color
 An n-coloring of a graph G = (V, E) is a coloring with n colors.
More precisely, a mapping f of V onto the set {1, 2, . . . , n} such that
whenever [u, v]∈E, f(u)≠ f(v).
The chromatic number X(G) of a graph G is the minimum
number of colors needed to color the vertices of G so that no two
adjacent vertices have the same color.
An n-coloring is complete if, ∀ i, j with i≠j, there exist adjacent
vertices u and v such that f(u) = i and f(v) = j. The sets of vertices
𝑉𝑖 𝑖=1
𝑛
with the same color is called color classes.

Preliminaries
We call the quantity σ(v)= )max 𝑢𝑒𝑣 𝑑(𝑢, 𝑣 the eccentricity of
v∈V. The diameter ρ of a connected graph G is the maximum
eccentricity among all nodes of G.
 For a given path P = {u0, u1, . . . , un}, a downstream edge of a
vertex ut ∈ P is any edge (ut,w) ∈E with w≠us for s < t.
A downstream vertex of ut is any vertex w such that (ut ,w) is a
downstream edge.
The downstream degree, D(ut), of a vertex is the number of
downstream edges it has.

Basic concepts
Information content of graphs
Entropy of degree distribution
Entropy as a measure of centrality
Entropy as a measure of connectivity
Topology entropy of a network

Information content of graphs
Rashevsky, Trucco and Mowshowitz were the first researchers to
define and investigate the Shannon-entropy of graphs.
Rashevsky in 1955, defined the topological information content of
the graph, dealing with the complexity of organic molecules, where
vertices represent physically indistinguishable atoms and edges
represent chemical bonds.
His definition is based on the partitioning of the n vertices of a
given graph into k classes of equivalent vertices, according to their
degree dependence. Then, assigned a probability to each partition
obtained as the fraction of vertices in this partition divided by the
total number of vertices.

Examples:
Graph has 3 vertices of degree two (1, 3, and 4), and two vertices (2 and
5) of degree three. However, vertex 1 is topologically different from
vertices 3 and 4. Vertex 1 is adjacent to two vertices of degree three,
while each of the vertices 3 and 4 is adjacent to one vertex of degree
three and one of degree two. Hence we have classes of vertices, 1 with
probability
1
5
, 2 and 5 with probability
2
5
, 3 and 4 with probability
2
5
.
The information content of is:
By adding the vertex 6, vertices 2 and 5 have different degrees.
Moreover vertex 3 is adjacent to one vertex of degree four and to
one of degree two, while the vertex 4 is adjacent to one vertex of
degree three and one of degree two. So, we have six distinct classes
each of them have probability
1
6
. Hence, the information content of
is:
𝑰 𝒄 = −(
𝟏
𝟓
𝒍𝒅
𝟏
𝟓
+
𝟐
𝟓
𝒍𝒅
𝟐
𝟓
+
𝟐
𝟓
𝒍𝒅
𝟐
𝟓
) =
𝟏
𝟓
𝒍𝒅𝟓 +
𝟐
𝟓
𝒍𝒅
𝟓
𝟐
+
𝟐
𝟓
𝒍𝒅
𝟓
𝟐
= 𝟏. 𝟓𝟑
𝑰 𝒅 = 𝒍𝒅𝟔 = 𝟐. 𝟔

Trucco in 1956 made this definition more precision. He considered
that two vertices are equivalent if they belong to the same orbit of this
group, i. e., if they can interchange preserving the adjacency of the
graph.
We denote the group of all automorphisms of a graph X by G(X).
X is a graph with 𝑉(𝑋) = {𝑥1, . . . 𝑥 𝑛 . If f∈G(X), 𝑓(𝑥𝑖) = 𝑥𝑗 for 1 < i < n.
Hence, to each f∈G(X), there corresponds a unique permutation of the
elements 1,2,...,n. We can regard G(X) as a subgroup of 𝑆𝑛, the
symmetric group of degree n.
Suppose K is a subgroup of 𝑆𝑛 and 1 ≤ 𝑖 ≤ 𝑛. Then 𝑓(𝑖 ) 𝑓 ∈ 𝐾 is
called an orbit of K. Let 𝐴𝑖, 1 ≤ 𝑖 ≤ 𝑛 be the distinct orbits of K. Then
𝐴𝑖 ∩ 𝐴𝑗 = ∅ , if i≠j and ∪
𝑖=1
𝑛
𝐴𝑖 = 1, . . . , 𝑛 , hence the orbits form a
partition of the set {1, 2 , . . . , n}. We assigned probability 𝑝𝑖 =
|𝐴 𝑖|
𝑛
to
each orbit, where |𝐴𝑖| is the cardinality of 𝐴𝑖.

Examples:
)𝐺(𝑋1 ={e, (12)}, Orbits of )𝐺(𝑋1 : {1, 2}, {3},{4}. Hence,
)𝐺(𝑋2 ={e, (23), (24), (34), (234), (243)}, Orbits of )𝐺(𝑋2 :
{2,3,4},{1} Hence,
𝑰 𝒈(𝑿 𝟏) = −(
𝟐
𝟒
𝒍𝒅
𝟑
𝟒
+
𝟏
𝟒
𝒍𝒅
𝟏
𝟒
+
𝟏
𝟒
𝒍𝒅
𝟏
𝟒
) =
𝟑
𝟐
𝑰 𝒈(𝑿 𝟐) = −(
𝟑
𝟒
𝒍𝒅
𝟑
𝟒
+
𝟏
𝟒
𝒍𝒅
𝟏
𝟒
) = 𝟐 −
𝟑
𝟒
𝒍𝒅𝟑

)𝐺(𝑋3 ={e, (13), (24), (13) (24)}. Orbits of )𝐺(𝑋3 :
{1, 3},{2, 4}. Hence,
G(𝑋4)= {(1234), (13), (24), (13) (24), (12) (34),
(14) (23) (1432)}. Orbits of G(𝑋4): {1,2,3,4}
•
𝑰 𝒈(𝑿 𝟑) = −(
𝟐
𝟒
𝒍𝒅
𝟐
𝟒
+
𝟐
𝟒
𝒍𝒅
𝟐
𝟒
) = −𝒍𝒅
𝟐
𝟒
= 𝒍𝒅𝟒 − 𝒍𝒅𝟐 = 𝟏
𝑰 𝒈(𝑿 𝟒) = −(
𝟒
𝟒
𝒍𝒅
𝟒
𝟒
) = −𝒍𝒅𝟏 = 𝟎

In 1968 Mowshovitz has studied in detail the information
content of the graphs and has formulated, on the basis of the
chromatic properties of the graphs.
 A decomposition 𝑉𝑖 𝑖=1
𝑛
of the set of vertices V is called a
chromatic decomposition of G if u, v ∈ Vi imply that [u, v]
∉ E. If f is an n-coloring, the collection of sets { 𝑣 ∈

For the graph G(V,E), |𝑉| = 6, X(G)=3
Chromatic decomposition is :
A:{1,3}, {2,5}, {4,6}
B: {2}, {1,3,} {4,5,6}
C:{2}, {3}, {1,4,5,6}

Information content base on functionals
M. Dehmer and F. Emmert-Streib (2007) presented a method to
determine the structural information content which is not based on the
problem of finding partitions of the vertices, and the overall time
complexity is polynomial.
They assigned a probability to each vertex 𝒗𝒊 as,
where f represents an arbitrary information functional.
In this case 𝒇 𝒗𝒊 = 𝒂 𝒄 𝟏|𝒔 𝟏 𝒗 𝒊,𝑮 |+...+𝒄 𝝆|𝒔 𝝆 𝒗 𝒊,𝑮 |
, 1 ≤ k ≤ ρ, α >0, 𝑐 𝑘 are real
positive coefficients and they are chosen in such a way to emphasize
certain structural characteristics e.g. high vertex degrees.
𝒑(𝒗𝒊) =
)𝒇(𝒗 𝒊
𝒋=𝟏
|𝑽|
𝒇(𝒗 𝒋

The process of defining vertex probabilities using graph-theoretical
quantities is not unique. Each such quantity captures different
structural information of a graph. In that way, M. Dehmer induced
another functional.
The associated paths of the j-sphere 𝑺𝒋(𝒗𝒊, 𝑮) = {𝒗 𝒖 𝒋
, 𝒗 𝒘 𝒋
, . . . , 𝒗 𝒙 𝒋
are
and their edge sets :
𝐸1 = {{𝑣𝑖, 𝑣 𝑢1
, . . . , {𝑣 𝑢 𝑗−1
, 𝑣 𝑢 𝑗
𝐸2 = {{𝑣𝑖, 𝑣 𝑤1
, . . . , {𝑣 𝑤 𝑗−1
, 𝑣 𝑤 𝑗
⋮
𝐸 𝑘 𝑗
= {{𝑣𝑖, 𝑣 𝑥1
, . . . , {𝑣 𝑥 𝑗−1
, 𝑣 𝑥 𝑗

For each vertex 𝑣𝑖∈V he determined the local information graph
𝐿 𝐺(𝑣𝑖, 𝑗) = (𝑉𝜑, 𝐸 𝜑 which is induced by the paths 𝑃1
𝑗
(𝑣𝑖), . . , 𝑃𝑘 𝑗
𝑗
(𝑣𝑖
where 𝛦 𝜑 = 𝛦1 ∪. . .∪ 𝛦 𝑘 𝑗
and 𝑉𝜑 = {𝑣𝑖, 𝑣 𝑢1
, . . . , 𝑣 𝑢 𝑗
∪. . .∪

Example:
)𝜌(𝐺1 =4 and it is considered that: 𝑐1 = 𝑏1 = 4, 𝑐2 = 𝑏2 = 3,
𝑐3 = 𝑏3 = 2, 𝑐4 = 𝑏4 = 1
𝑓 𝑉(𝑣1) = 𝑓 𝑉(𝑣4) = 𝑓 𝑉(𝑣5) = 𝑓 𝑉(𝑣8) = 𝑎2𝑐1+2𝑐2+2𝑐3+𝑐4 = 𝑎19
𝑓 𝑉
(𝑣2) = 𝑓 𝑉
(𝑣7) = 𝑎2𝑐1+3𝑐2+2𝑐3 = 𝑎21
𝑓 𝑉
(𝑣3) = 𝑓 𝑉
(𝑣6) = 𝑎3𝑐1+3𝑐2+𝑐3 = 𝑎23
𝐼 𝑓 𝑉(𝐺1) = −
𝑖=1
8
)𝑝 𝑣
(𝑣𝑖)𝑙𝑑(𝑝 𝑣
(𝑣𝑖) =
= − 4
𝑎19
4𝑎19 + 2𝑎21 + 2𝑎23
𝑙𝑑 4
𝑎19
4𝑎19 + 2𝑎21 + 2𝑎23
+ 2
𝑎21
4𝑎19 + 2𝑎21 + 2𝑎23
𝑙𝑑
𝑎21
4𝑎19 + 2𝑎21 + 2𝑎23

𝒇 𝑷
(𝒗 𝟏) = 𝒇 𝑷
(𝒗 𝟒) = 𝒇 𝑷
(𝒗 𝟓) = 𝒇 𝑷
(𝒗 𝟖) = 𝒇 𝑷
(𝒗 𝟑) = 𝒇 𝑷
(𝒗 𝟔) = 𝒂 𝟐𝒄 𝟏+𝟒𝒄 𝟐+𝟔𝒄 𝟑+𝟒𝒄 𝟒 = 𝒂 𝟑𝟔
𝒇 𝑷
(𝒗 𝟐) = 𝒇 𝑷
(𝒗 𝟕) = 𝒂 𝟐𝒄 𝟏+𝟔𝒄 𝟐+𝟔𝒄 𝟑 = 𝒂 𝟑𝟖
𝐼 𝑓 𝑃(𝐺1) = −
𝑖=1
8
)𝑝 𝑃
(𝑣𝑖)𝑙𝑑(𝑝 𝑃
(𝑣𝑖) =
= 4
𝑎36
6𝑎36 + 2𝑎38
𝑙𝑑
𝑎36
6𝑎36 + 2𝑎38

The degree distribution network entropy
The entropy of the degree distribution had be defined as 𝑯 =
− 𝒌=𝟏
𝜨−𝟏
)𝒑(𝒌 𝒍𝒅𝒑 𝒌 , where p(k) is the probability that a node has
degree k an N is the number of nodes. The maximum value of
entropy is obtained for a uniform degree distribution and the
minimum value is achieved when all vertices have the same
degree.
Entropy provides an average measure of heterogeneity, since it
measures the diversity of the link distribution. Heterogeneity is in
relationship with the network’s resilience to attacks.
B.Wang, H.Tang, C. Guo and Z. Xiu studied the robustness of scale-
free networks to random failures ,with entropy of the degree
distribution. By maximizing the entropy of the degree distribution,
we get an optimal design of scale-free network’s robustness to
random failures.

Solé and Valverde used entropy of the remaining degree .
The remaining degree of a vertex at one end of an edge is
the number of edges connected to that vertex not counting
the original edge. The remaining degree distribution q(k)
is obtained from:
𝑞 𝑘 =
𝑘+1)𝑃(𝑘+1
<𝑘>
, 𝑤ℎ𝑒𝑟𝑒 < 𝑘 >= 𝑘 𝑘𝑃𝑘 .
Fig. Here two given, connected nodes 𝑠𝑖 , 𝑠𝑗are shown, displaying different degrees 𝑘𝑖 ,
𝑘𝑗. Since we are interested in the remaining degrees, a different value needs to be
considered (here indicated as 𝑞𝑖 , 𝑞𝑗)

Entropy as a measure of centrality in
networks
Frank Tutzauer (2007) has proposed a measure of centrality for
networks characterized by path-based transfer flows.
To model a path-transfer process, it is helpful to think of a specific
object being passed from one node to another. If the starting node
is chosen, the flow either stops, otherwise, the object passes to the
chosen node.
The next node then randomly chooses from among its neighbors
and again the flow either stops or continues. The object thus
traverses a path in the network, traveling along links, stopping
when a loop is chosen. Each neighbor is assumed to be chosen with
equal likelihood.

What is the probability that a flow beginning at
vertex 5 ends at vertex 2, in the network below?
Consequently there is a 1/4 probability that
vertex 5 passes control to vertex 3 (because 5
chooses with equal likelihood from among nodes
3, 4, 5, and 6). Once node 3 receives control, the
flow will not pass back to node 5, so there is a 1/2
probability that node 3 stops the flow, and a 1/2
probability that control passes to node 2.
Likewise, vertex 2 chooses between stopping the
flow or continuing it to vertex 1. As a result, the
probability that a flow beginning at vertex 5 ends
at vertex 2 is (1/4)(1/2)(1/2) = 1/16
The transfer and the stopping probability of a vertex 𝑣 𝑡 ∈ 𝑃𝑘 is
given by:

To obtain the single path probability – i.e., the likelihood that a flow
beginning at 𝑖 = 𝑢0 ends at 𝑗 = 𝑢 𝑛 𝑘 by traveling along the path
𝑃𝑘 = { 𝑢0, . . . , 𝑢 𝑛 𝑘 } – simply multiply by the transfer probability of
each of the first 0, . . ., 𝑛 𝑘–1 vertices and the stopping probability of the
last vertex in the path.
Then the overall probability that a flow starting at i ends at j is given by
the combined path probability, which is simply the single path
probabilities summed across the K(i, j) paths from i to j:
𝑝𝑖𝑗 =
𝑘=1
)𝐾(𝑖,𝑗
𝜎 𝑘(𝑗) 𝑡=0
𝑛(𝑘)−1
𝜏 𝜅 𝑢 𝑡 =
𝑘=1
)𝐾(𝑖,𝑗
𝑡=0
𝑛(𝑘)−1
𝜏 𝜅 𝑢 𝑡
The path-transfer centrality of vertex i is then given by the entropy
𝐶 𝐻(𝑖) = −
𝑗=1
𝑁
𝑝𝑖𝑗 𝑙𝑑𝑝𝑖𝑗

Entropy as a connectivity measure
Entropy as a connectivity measure of a graph is defined as
𝐻 𝐺 = − 𝑣∈𝑉
)𝑝(𝑣)𝑙𝑑𝑝(𝑣 , where 𝑝(𝑣) =
)indeg(𝑣
2|𝐸| 𝑣∈𝑉
To obtain a measure with a [0, 1] range, we divide H(G) by
the maximum entropy given by ld|V|
High entropy indicates that many vertices are equally
important, whereas low entropy indicates that only a few
vertices are relevant.

Topology entropy of network
In order to describe the uncertainty of complex networks, we
proposed another network entropy concept based on the topology
configuration of network. For a complex networks generated
according to certain rules, in the given parameters, each test can
generate a specific network configuration. Repeated tests, a wide
range of configurations will be produced.
The topology entropy of network is defined as 𝑺(𝜴, 𝑷) =

The number of all possible edges is equal to M=3*2/2=3. Here 3
possible connected processes can be treated as 3 independent
random events.
The actual link edge number 𝑚 is a random variable, it will comply
the binomial distribution, 𝑃 𝑚 = 𝐶3
𝑚
0.2 𝑚
1 − 0.2 3−𝑚
, where
𝐶3
𝑚
=
3
𝑚
. The possible configurations are 𝛺(3,0.2) =
𝑚=0
3
𝐶3
𝑚
=8.
All the configurations in these 𝐶3
𝑚
different configurations appear
with probability 1 𝐶3
𝑚
. This is a conditional probability which
links numbers are 𝑚, so the realization of a G(3,0.02) random
network, here 𝑚 edges can be regard as a random incident A𝑖
which has occurrence probability: 𝑃𝑖 = 𝑃(𝑚 ) 𝐶3
𝑚
= 0.2 𝑚(1 −

As the probabilities of each 𝐶3
𝑚
configuration for certain 𝑚 are
the same, the entropy can be calculated for different 𝑚
respectively, so 𝑆 can be written as:

Example
For this network, we compute entropy as a connectivity and centrality measure
and entropy of degree and remaining degree distribution. Then we compare
them with some existing measures.

Centrality
The table above depicts entropy-based centrality, degree centrality,
betweenness centrality and closeness centrality.
Node Entropy Degree Closeness Betweenness
11 3.407 2 (9-13) 0.319 (3) 50 (3-4)
7 3.239 3 (4-8) 0.341 (2) 54 (2)
10 3.211 4 (1-3) 0.349 (1) 57.5 (1)
6 3.131 2 (9-13) 0.313 (4) 50 (3-4)
8 3.061 3 (4-8) 0.306 (5) 3.5 (10)
12 3.003 4(1-3) 0.283 (6) 44 (6)
5 2.944 3 (4-8) 0.278 (7) 47 (5)
14 2.87 4 (1-3) 0.238 (9) 14 (8)
9 2.868 2 (9-13) 0.268 (8) 0 (11-16)
13 2.861 3 (4-8) 0.234 (10-12) 0 (11-16)
15 2.787 3 (4-8) 0.234 (10-12) 0 (11-16)
3 2.7 2 (9-13) 0.234 (10-12) 26 (7)
4 2.421 1 (14-16) 0.221 (13) 0 (11-16)
16 2.343 1 (14-16) 0.195 (15) 0 (11-16)
2 2.308 2 (9-13) 0.197 (14) 14 (8)
1 2.043 1 (14-16) 0.167 (16) 0 (11-16)

The table below portrays Spearman’s rank-order correlation:
If we are interesting about centrality as a score, rather than
simply the ranking, Pearson’s correlation coefficient is more
appropriate. So the table below illustrate the Pearson’s r
correlation:

Entropy, betweenness, and closeness all agree on the top four
nodes, though they rank them differently.
Measures are highly, but not perfectly correlated. Degree
centrality provides the minimum correlation with entropy
based centrality, in comparison with betweenness and
closeness centrality.
In contrast, closeness produces the rankings most similar to
the rankings of the entropy centrality.

Scatter plots and linear regression
Entropy-Closeness

Entropy-Betweenness

Entropy-Degree

Another visualization of the network:
The size of the nodes depends on the value of the entropy.

Connectivity
For the graph with loops the number of edges is: |E|=36. We
obtain H(G)=3,967. To normalize H(G), we divide it by the
maximum entropy given by ld(|V|)=ld(16)=4. So, we obtain
0,991.
If we simplify the loops, the number of edges is: |E|=20. Thus,
H(G)=3,877
Normalize: H(G)/ld(|V|)=H(G)/ld(16)=3,877/4=0,969.

Degree distribution
 Entropy of degree
distribution: H(p)=1,9544
 Entropy of the remaining
degree distribution:
H(q)=1,832
Remainin
g
degree
q(k)
0 0,075
1 0,25
2 0,375
3 0,3

Summary and conclusion
This project has attempted to demonstrate a variety of methods for
measuring the entropy of graphs. We started with a review of classical
measures for determining the structural information content of graphs.
Furthermore, we represented some other approaches characterized by
information functionals. Then we represented topological network
entropy. Moreover, we represented the entropy as centrality and
connectivity measure, the entropy of degree and remaining degree
distribution. Finally, we gave an example of a graph and computed the
relative measures.
Nowadays, graph-based models are applied in a wide range of disciplines.
Shannon’s entropy, as a measure of structural characteristics has been
proven very useful. Further development of that theory and more efficient
algorithms for computing entropy is needed.

References
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Entropy based measures for graphs

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