connectedness index of uncertain graph

January 21, 2013 17:3 WSPC/118-IJUFKS

S0218488513500074

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 21, No. 1 (2013) 127–137 c World Scientific Publishing Company

DOI: 10.1142/S0218488513500074

Int. J. Unc. Fuzz. Knowl. Based Syst. 2013.21:127-137. Downloaded from www.worldscientific.com by 195.154.80.94 on 06/12/14. For personal use only.

CONNECTEDNESS INDEX OF UNCERTAIN GRAPH

XIULIAN GAO Department of Mathematics, Dezhou University, Dezhou, Shandong, 253023, P. R. China [email protected] YUAN GAO Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P. R. China [email protected] Received 27 January 2011 Revised 31 August 2012 In practical applications of graph theory, non-deterministic factors are frequently encountered. This paper employs uncertainty theory to deal with non-deterministic factors in problems of graph connectivity. The concepts of uncertain graph and connectedness index of uncertain graph are proposed in this paper. It presents two algorithms to calculate connectedness index of an uncertain graph. Keywords: Uncertain graph; connectedness index; uncertainty theory.

1. Introduction The connectivity of graph is a basic concept of graph theory. Since originated in seven bridges problem by Euler in 1736, the main study of graph theory unfolds on the assumption that the graph is connected. In classic graph theory,1–4 the edges and the vertices are all deterministic, and connectivity of the graph can be easily verified. However, in practical application, non-deterministic factors appear in graphs, which leads to new situations. Such as in shortest path problem, if the length of each arc is non-deterministic, the optimization objective can not simply expressed and conventional algorithms can not be used. Sometimes, whether two vertices are joined by an edge can not be completely determined. As a result, whether the graph is connected can not be completely determined. Then, in how much belief degree we can regard a non-deterministic graph is connected? This problem was first investigated by Erd¨ os and R´ enyi.5 They thought that whether two vertices are joined can be described as a random variable. Under the assumption, Erd¨ os did further research on random graph, and more details can be obtained in his works.6,7 127



128

S0218488513500074

X.-L. Gao & Y. Gao

Unfortunately, it is not suitable to regard every non-deterministic phenomenon as random phenomenon, especially when the non-deterministic phenomenon is caused by subjective judgement. In order to deal with these non-deterministic, Liu8,9 proposed uncertainty theory and refined it in 2010. Gao10,11 introduced uncertainty theory into graph theory. He investigated shortest path problem and location problem on networks. This provides us a motivation to introduce uncertain theory into the study of connectivity of graph. In this paper, a concept of connectedness index of uncertain graph is first proposed. The main goal of this paper is to calculate connectedness index of uncertain graph. With the framework of uncertainty theory, two algorithms to calculate connectedness index are given. The algorithms are variant of the famous Kruskal’s Algorithm and Prim’s Algorithm in finding minimum spanning tree. Naturally, the complexity of calculating connectedness index is the same to find minimum spanning tree. The remainder of this paper is organized as follows. In Sec. 2, uncertainty theory and graph theory are introduced. Basic theories and definitions used in this paper can be found in this section. Section 3 is the main part of this paper. The concepts of uncertain graph and connectedness index are proposed, and the algorithms to calculate connectedness index are given. Section 4 concludes this paper with a brief summary. 2. Preliminaries 2.1. Uncertainty theory Now, we introduce some concepts of uncertainty theory used throughout of this paper. Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ ∈ L is called an event. The set function M{Λ} is called an uncertain measure if it satisfies the following three axioms Axiom 1. (Normality) M{Γ} = 1. Axiom 2. (Self-Duality) M{Λ} + M{Λc } = 1 for any event Λ. Axiom 3. (Countable Subadditivity) For every countable sequence of events {Λi }, we have (∞ ) ∞ [ X M Λi ≤ M{Λi } . i=1

i=1

The triplet (Γ, L, M) is called an uncertainty space. An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers. Product uncertain measure was defined by Liu12 in 2009, producing the fourth axiom as follows Axiom 4. (Liu,12 Product Measure Axiom) Let (Γi , Li , Mi ) be uncertainty spaces


S0218488513500074

Connectedness Index of Uncertain Graph

129

for i = 1, 2, . . . , n. Then the product uncertain measure M is an uncertain measure on the product σ-algebra L = L1 × L2 × · · · × Ln satisfying (n ) Y M Λi = min M{Λi }. 1≤i≤n


i=1

That is, for each event Λ ∈ L, we have  sup min M{Λi },    Λ1 ×Λ2 ×···×Λn ⊂Λ 1≤i≤n     if sup min M{Λi } > 0.5   Λ1 ×Λ2 ×···×Λn ⊂Λ 1≤i≤n      M {Λ} = 1 − sup min M{Λi },  Λ1 ×Λ2 ×···×Λn ⊂Λc 1≤i≤n     if sup min M{Λi } > 0.5    Λ1 ×Λ2 ×···×Λn ⊂Λc 1≤i≤n       0.5, otherwise.

A function is said to be Boolean if it is a mapping from {0, 1}n to {0, 1}. An uncertain variable is said to be Boolean if it takes values either 0 or 1. The following theorem is important to this paper. Theorem 1. Assume that ξ1 , ξ2 , . . . , ξn are independent Boolean uncertain variables, i.e., 1 with uncertain measure ai ξi = 0 with uncertain measure 1 − ai for i = 1, 2, . . . , n. If f is an increasing Boolean function, then ξ = f (ξ1 , ξ2 , . . . , ξn ) is a Boolean uncertain variable such that M{ξ = 1} =

sup

min M{ξi ∈ Bi } ,

f (B1 ,B2 ,...,Bn )=1 1≤i≤n

where Bi are subsets of {0, 1}, i = 1, 2, . . . , n. Proof: According to Axiom 4, we have  sup min M{ξi ∈ Bi } ,    f (B1 ,B2 ,...,Bn )=1 1≤i≤n     if sup min M{ξi ∈ Bi } > 0.5    f (B1 ,B2 ,...,Bn )=1 1≤i≤n      M {ξ = 1} = 1 − sup min M{ξi ∈ Bi } ,  f (B1 ,B2 ,...,Bn )=0 1≤i≤n     if sup min M{ξi ∈ Bi } > 0.5    f (B1 ,B2 ,...,Bn )=0 1≤i≤n        0.5, otherwise.

Obviously, when M {ξ = 1} = 1 or M {ξ = 1} = 0, Theorem 1 holds. We only need to prove that Theorem 1 holds when 0 < M {ξ = 1} < 1.


130

S0218488513500074

X.-L. Gao & Y. Gao

Since f is an increasing Boolean function, if f (B1 , B2 , . . . , {0}, . . . , Bn ) = 1, holds, the following equation also holds f (B1 , B2 , . . . , {0, 1}, . . . , Bn ) = 1.


î }n , taking values of {1} or {0, 1}, satisfying Thus, there must exist a series of {B i=1 î }. min M{ξi ∈ Bi } = min M{ξi ∈ B

sup

f (B1 ,B2 ,...,Bn )=1 1≤i≤n

1≤i≤n

¯i }n , taking values of {0} or {0, 1}, Similarly, there must exist a series of {B i=1 satisfying ¯i } . min M{ξi ∈ Bi } = min M{ξi ∈ B

sup

f (B1 ,B2 ,...,Bn )=0 1≤i≤n

1≤i≤n

Without loss of generality, we assume 1 = a0 ≥ a1 ≥ a2 ≥ · · · ≥ an ≥ an+1 = 0. Then, the following equation always holds sup

î } = M{ξ ∈ B ˆ p } = ap , min M{ξi ∈ Bi } = min M{ξi ∈ B

f (B1 ,B2 ,...,Bn )=1 1≤i≤n

1≤i≤n

(1)

ˆp = {1} and B î = {0, 1} for all i > p. Similarly, where B ¯i } = M{ξ ∈ B ¯q } = 1−aq , (2) min M{ξi ∈ Bi } = min M{ξi ∈ B

sup

f (B1 ,B2 ,...,Bn )=0 1≤i≤n

1≤i≤n

¯q = {0} and B ¯i = {0, 1} for all i < q. where B We will prove that p = q. Assume p > q. Choose B1 = B2 = · · · = Bq = · · · = Bp = {1} and Bp+1 = · · · = Bn = {0, 1}, according to Eq. (1), we have f (B1 , B2 , . . . , Bp , . . . , Bn ) = 1 . However, if choose B1 = B2 = · · · = Bq = · · · = Bp−1 = {1}, Bp = {0} and Bp+1 = · · · = Bn = {0, 1}, we have f (B1 , B2 , . . . , Bp , . . . , Bn ) = 0. This leads to sup

Min1≤i≤n M{ξi ∈ Bi } ≥ M{ξp ∈ {0}} = 1 − ap ≥ 1 − aq ,

f (B1 ,B2 ,...,Bn )=0

which conflicts with Eq. (2). Thus, p ≤ q. Similarly, we can prove that q ≤ p, that is, p = q. The proof is completed.


S0218488513500074


131

2.2. Connectivity of graphs In this section, we give a formal definition of graph and introduce some basic terminology of graph theory. Concepts and terminology in this section follow Ref. 4.


Definition 1. A graph G is a triple consisting of a vertex set V (G), an edge set E(G), and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints. A loop is an edge whose endpoints are equal. Multiple edges are edges having the same pair of endpoints. A simple graph is a graph having no loops or multiple edges. The number of vertices in G is often called the order of G. Every graph in this paper is simple graph with finite order. 1m

4m

2m

3m

Fig. 1.

A graph of order 4.

Let G be a graph with V (G) = {v1 , v2 , . . . , vn } and E(G) = {e1 , e2 , . . . , em }. Usually, we employ an adjacency matrix to describe a graph. The adjacency matrix of G is the n × n matrix   a11 a12 · · · a1n  a21 a22 · · · a2n    A= . .. ..   .. . ··· .  an1 an2 · · · ann where

aij =

1, 0,

if i and j are the endpoints of an edge otherwise.

Obviously, the adjacency matrix contains Fig. 1 and its adjacency matrix.  0 1 0 1 0 1 A= 0 1 0 1 0 1

all the information of a graph. See  1 0 . 1 0

Definition 2. In graph G, a walk is a list v0 , e1 , . . . , ek , vk of vertices and edges such that, for 1 ≤ i ≤ k, the edge ei has endpoints vi−1 and vi . A u–v walk has first vertex u and last vertex u. A u–v path is a walk with no repeated vertex.


132

S0218488513500074

X.-L. Gao & Y. Gao

Definition 3. A graph G is connected if there is a u–v path whenever u, v ∈ V (G). In the next section, we begin to investigate the connectivity of uncertain graph. 3. Uncertain Graph Int. J. Unc. Fuzz. Knowl. Based Syst. 2013.21:127-137. Downloaded from www.worldscientific.com by 195.154.80.94 on 06/12/14. For personal use only.

3.1. Concepts In this paper, the non-deterministic factor is that we are not sure whether an edge exists between two vertices. If there is no history data or experimental data, we can not employ random variable to describe this non-deterministic factor. Usually, we may ask experts to give a belief degree that the edge exists. This expert data is just the subject of uncertainty theory. Thus, in this paper, we employ uncertain variable to describe the non-deterministic factor. Definition 4. A graph G of order n is said to be an uncertain graph if its adjacency matrix is   0 α12 · · · α1n  α21 0 · · · α2n    A= . ..  ..  .. . ··· .  αn1 αn2 · · · 0

where αij represent that the edges between vertices i and j exist with uncertain measure αij , i, j = 1, 2, . . . , n, respectively. Obviously, the adjacency matrix contains all the information of an uncertain graph. Example: Figure 2 is an uncertain graph of order 4. Its adjacency matrix is   0 1 0 0.8  1 0 0.3 0   A=  0 0.3 0 0.4  0.8 0 0.4 0 1m

0.8

1

0.4 2m

Fig. 2.

4m

0.3

3m

An uncertain graph of order 4.


S0218488513500074


133


From the definition of uncertain graph, we can conclude that the edge set of uncertain graph G is a set of uncertain Boolean variable E(G) = {ξ12 , ξ13 , . . . , ξ1n , ξ23 , . . . , ξ2n , . . . , ξ(n−1)n }, where M{ξij = 1} = αij , for any 1 ≤ i < j ≤ n. For simplicity, remove the edges ξij satisfying M{ξij = 1} = 0, and denote E(G) = {ξ1 , ξ2 , . . . , ξm }. We give a concept of connectedness function. Definition 5. Assume that G is an uncertain graph with edge set E(G) = {ξ1 , ξ2 , . . . , ξm }. The connectedness function of G is denoted as: 1, if graph G is connected C(E(G)) = 0, otherwise. Obviously, C(E(G)) is an increasing Boolean function. For an uncertain graph G with E(G) = {ξ1 , ξ2 , . . . , ξm }, the connectedness index is defined as ρ = M{C(E(G)) = 1}, that is, connectedness index is the uncertain measure that the graph is connected. 3.2. Algorithms In above subsection, the concept of connectedness index was given. According to the definition of connectedness index, Theorem 1 directly leads to Theorem 2. Assume that G is an uncertain graph with edge set E(G) = {ξ1 , ξ2 , . . . , ξm }. The connectedness index of G is ρ(G) =

sup

min M{ξi ∈ Bi },

C(E(G))=1 1≤i≤m

where Bi are subsets of {0, 1}, i = 1, 2, . . . , m. According to Theorem 2, we can easily obtain Corollary 1. Assume G is an uncertain tree with edge set E(G) = {ξ1 , ξ2 , . . . , ξm }. Then the connectedness index of G is the smallest value of M{ξi = 1}, i = 1, 2, . . . , m. However, for other more complex graph, the connectedness index is not easy to obtain from Theorem 2. We will give two algorithms to calculus connectedness index. First, we introduce the maximum connectedness index spanning tree. Definition 6. Assume that an uncertain graph G has vertex set V (G) = {v1 , v2 , · · · , vn } and edge set E(G) = {ξ1 , ξ2 , . . . , ξm }. The spanning tree of G is a graph with V (T ) = {v1 , v2 , . . . , vn } and E(T ) = {ξt1 , ξt2 , . . . , ξtn−1 }. The maximum connectedness index spanning tree is a spanning tree with the maximum connectedness index.


134

S0218488513500074

X.-L. Gao & Y. Gao

In fact, the connectedness index of G is equal to the connectedness index of maximum connectedness index spanning tree of G.


Theorem 3. Assume that an uncertain graph G has vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G) = {ξ1 , ξ2 , . . . , ξm }, and T ∗ is the maximum connectedness index spanning tree with V (T ∗ ) = {v1 , v2 , . . . , vn } and E(T ∗ ) = {ξt1 , ξt2 , . . . , ξtn−1 }. The following equation holds ρ(G) = ρ(T ∗ ). Proof: Obviously, ρ(G) ≥ ρ(T ∗ ). We only need to prove that ρ(G) ≤ ρ(T ∗ ) when ρ(G) > 0. î }m , taking values As in the proof of Theorem 1, there must exist a series of {B i=1 of {1} or {0, 1}, satisfying ρ(G) =

sup

î } = min M{ξi = 1} > 0, min M{ξi ∈ Bi } = min M{ξi ∈ B k

C(E(G))=1 1≤i≤m

1≤i≤m

1≤k≤h

î }h takes values of {1}. where h is a positive number and the subseries {B k k=1 î }h such that h = n − 1. Obviously, h ≥ n − 1. In fact, we can choose {B k k=1 If h = n − 1, we are done. If h > n − 1, the subgraph S with vertex set V (S) = {v1 , v2 , . . . , vn } and edge set E(S) = {ξi1 , ξi2 , . . . , ξih } must contain at least a spanning tree and a circle C. Choose any edge of circle C, such as ξih , and remove it. We obtain a new subgraph S1 with vertex set V (S1 ) = {v1 , v2 , . . . , vn } and edge set E(S1 ) = {ξi1 , ξi2 , . . . , ξih−1 }. What is more, S1 must contain a spanning tree and ρ(G) ≥ ρ(S1 ) ≥

min

1≤k≤h−1

M{ξik = 1} ≥ min M{ξik = 1} = ρ(G), 1≤k≤h

that is ρ(G) = ρ(S1 ). Repeating this argument until there is no circle. That is, we obtain a spanning tree T such that ρ(G) = ρ(T ). Since T ∗ is the maximum connectedness index spanning tree, we have ρ(G) = ρ(T ) ≤ ρ(T ∗ ). Thus, we prove the theorem.

According to Theorem 3, we can easily obtain Corollary 2. Assume G is an uncertain circle with edge set E(G) = {ξ1 , ξ2 , . . . , ξm }. Then the connectedness index of G is the second smallest value of M{ξi = 1}, i = 1, 2, . . . , m.


S0218488513500074


135

Now, we will give two algorithms to calculus the connectedness index, or say, find the maximum connectedness index spanning tree. Assume that uncertain graph G has vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G) = {ξ1 , ξ2 , . . . , ξm }. The first algorithm comes from Prim’s Algorithm.


Algorithm 1. Step 1. Set V = {v1 }, U = V (G) − V and E = ∅. Step 2. For all edges ξi = (s, t) satisfying s ∈ V and t ∈ U , choose the edge with biggest M{ξi = 1}, such as ξi = (s, t). Reset V = V ∪ {t}, U = U − {t} and E = E ∪ {ξi }. Step 3. Repeat Step 2 until V = V (G). The final E is the edge set of maximum connectedness index spanning tree. In each iteration, Step 2 creates a tree T with V (T ) = V and E(T ) = E. After the last iteration, V (T ) = V (G) and T is a spanning tree of G. Let T ∗ be a maximum connectedness index spanning tree. If T = T ∗ , we are done. Otherwise, choose an edge ξ ∈ T such that ξ 6∈ T ∗ . It is known that T = T1 ∪T2 ∪ξ, where T1 and T2 are both trees and T1 ∩T2 = ∅. What is more, Algorithm 1 says that for all edges in G joining V (T1 ) and V (T2 ), edge ξ has the biggest value of M{ξ = 1}. Add ξ to T ∗ , we can obtain a cycle C. There must exist an edge ξ ∗ in cycle C that is not in T , and ξ ∗ joins V (T1 ) and V (T2 ). Obviously, M{ξ = 1} ≥ M{ξ ∗ = 1}. According to Corollary 1, T1∗ = T ∗ + ξ − ξ ∗ is a spanning tree with ρ(T1∗ ) ≥ ρ(T ∗ ). Since T ∗ is the maximum connectedness index spanning tree, we have ρ(T1∗ ) = ρ(T ∗ ). That is, we obtain a maximum connectedness index spanning tree T1∗ that agrees more with T than T ∗ does. Repeating this argument, we will finally obtain a maximum connectedness index spanning tree that agrees completely with T . Thus, the spanning tree obtained by Algorithm 1 is the maximum connectedness index spanning tree. Obviously, Algorithm 1 has the same complexity with Prim’s Algorithm, that is, O(n2 ). Figure 3 gives an example. According to Corollary 1, the connectedness index of the graph in Fig. 3(1) is 0.8. The second algorithm comes from Kruskal’s Algorithm. Assume graph T is a subgraph of G. Algorithm 2. Step 1. Sort {ξ1 , ξ2 , . . . , ξm } by M{ξi = 1}. Set V (T ) = V (G), E(T ) = ∅ and E2 = {ξ1 , ξ2 , . . . , ξm }. Step 2. Choose the edge with biggest M{ξi = 1} in E2 . If ξ joins two components of T , reset E(T ) = E(T ) + {ξi } and E2 = E2 − {ξi }; otherwise, reset E2 = E2 − {ξi }. Step 3. Repeat Step 2 until E(T ) has n − 1 edges. The final T is a maximum connectedness index spanning tree.


136

S0218488513500074

X.-L. Gao & Y. Gao

1

1

1

0.6

0.7 1

4

0.8

2

3 0.8


2

3

0.9

6

6

5

0.7

(1)

(2)

1

1

3

(3)

1 4

0.8

2

3

0.8

6

1

3

0.9

6

5 (6)

(5)

(4) Fig. 3.

Maximum connectedness index spanning tree by Algorithm 1.

1

1

0.6

1

0.7 1 0.8

0.8

4

2 3

0.8

1

4

2 3

0.9

0.9

0.6

6

5

6

5

0.7

(1)

(2)

1

1 4

0.8

2

1 4

0.8

3

3 6 (4) Fig. 4.

6 (5)

4

0.8

3 0.9

5

1

2

0.8

0.9

5

6

5 (3)

1

1

2

1

4

3

0.5

0.8

0.8

6

5

4

0.8

2

0.8 0.9

5

6

5

1

4

0.8

2

4

0.8

0.6

1

2

4

3

0.8

0.5

5

1

1

2

0.8 0.9

0.8

6

5 (6)

Maximum connectedness index spanning tree by Algorithm 2.

The proof that T is a maximum connectedness index spanning tree is similar to that in Algorithm 1. Obviously, Algorithm 2 has the same complexity with Kruskal’s Algorithm, that is, O(m log m). Figure 4 gives an example. According to Corollary 1, the connectedness index of the graph in Fig. 4(1) is 0.8.


S0218488513500074


137

When we calculus connectedness index of uncertain graph, we can use either Algorithm 1 or Algorithm 2. However, taking into account of their complexity, it is better to use Algorithm 1 when the graph contains a large number of edges, for the complexity of Algorithm 1 has no relation to edge. When the graph contains a small number of edges, it is better to use Algorithm 2.


4. Conclusion This paper introduced uncertainty theory into the study of connectivity of graph. The concept of connectedness index of uncertain graph was proposed. The main result of this paper was proposing two algorithms to calculate the connectedness index. Acknowledgments This work was supported by National Natural Science Foundation of China Grant No.60874067 and No. 91024032. References 1. F. Harary, The maximum connectivity of a graph, in Proc. Nat. Acad. Sci. USA 48 (1962) 1142–1146. 2. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications (Macmillan London and Elsevier, New York, 1976). 3. W. T. Tutte, Graph Theory (Cambridge University Press, England, 2001). 4. D. B. West, Introduction to Graph Theory, 2nd edn. (Prentice-Hall, Inc., New Jersey, 2001). 5. P. Erd¨ os and A. Rényi, On the strength of connectedness of a random graph, Acta Math. Hungar. 12 (1961) 261–267. 6. P. Erd¨ os, Graph theory and probability, Canad. J. Math. 11 (1959) 34–38. 7. P. Erd¨ os, Graph theory and probability II, Canad. J. Math. 13 (1961) 346–352. 8. B. Liu, Uncertainty Theory, 2nd edn. (Springer-Verlag, Berlin, 2007). 9. B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty (Springer-Verlag, Berlin, 2010). 10. Y. Gao, Shortest path problem with uncertain arc lengths, Comput. Math. Appl. 62 (2011) 2591–2600. 11. Y. Gao, Uncertain models for single facility location problems on networks, Appl. Math. Model. 36 (2012) 2592–2599. 12. B. Liu, Some research problems in uncertainty theory, J. Uncertain Systems. 3 (2009) 3–10.

connectedness index of uncertain graph

connectedness index of uncertain graph

Suggest Documents

Matching Index of Uncertain Graph: Concept and Algorithm

DHL GLOBAL CONNECTEDNESS INDEX 2014

DHL Global Connectedness Index 2012

DHL Global Connectedness Index 2012

DHL Global Connectedness Index 2011

DHL Global Connectedness Index 2011

DHL GLOBAL CONNECTEDNESS INDEX 2014 [PDF]

Random geometric graph description of connectedness percolation in ...

Dynamic Uncertain Causality Graph for Knowledge Representation ...

DHL Global Connectedness Index 2016 â The State of Globalization ...

Graph in h-index

Wiener index and Steiner 3-Wiener index of a graph

The vertex-rainbow index of a graph

Connective eccentric index of some graph operations

ON THE HARMONIC INDEX OF GRAPH

The Hamiltonian Index of a Graph

DHL Global Connectedness Index 2016 â The State

Spirituality of Connectedness - Naadac

Index for Fast Retrieval of Uncertain Spatial Point Data - CiteSeerX

Uncertain Graph Classification Based on Extreme Learning Machine ...

OSGAR: A Scene Graph with Uncertain Transformations - CiteSeerX

Path connectedness, local path connectedness and contractibility of ...

Uncertain Times, uncertain measures

UPI: A Primary Index for Uncertain Databases - MIT Database Group

connectedness index of uncertain graph