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Paths and flows for centrality measures in networks

We consider the number of paths that must pass through a subset X of vertices of a capacitated network N in a maximum sequence of arc‐disjoint paths connecting two vertices y and z. We consider then the difference between the maximum flow value from y to z in N and the maximum flow value from y to z in the network obtained from N by setting to zero the capacities of arcs incident to X. When X is a singleton, those quantities are involved in defining and computing the flow betweenness centrality and are commonly identified without any rigorous proof justifying the identification. On the basis of a deep analysis of the interplay between paths and flows, we prove that, when X is a singleton, those quantities coincide. Moreover they are both equal to the global flow that must pass through X in any maximum flow from y to z. On the other hand, we prove that, when X has at least two elements, those quantities and the global flow that must pass through X in any maximum flow from y to z may be different from each other. We next show that, by means of the considered quantities, two conceptually different group centrality measures, based on paths and flows respectively, can be naturally defined. Such group centrality measures both extend the flow betweenness centrality to groups of vertices and are proved to satisfy a desirable form of monotonicity.


INTRODUCTION
The concept of flow is certainly one of the most fruitful concepts in network theory with a plethora of recent applications varying from transport engineering to social choice theory and financial networks (see, for instance, [6,9,22]). The powerful maxflow-mincut theorem by Ford and Fulkerson [12] immensely contributed to the success of that concept giving rise, among other things, to the manageable augmenting path algorithm for computing maximum flows. Flows and paths are also at the core of the well-known centrality measure called flow betweenness due to Freeman et al. [14]. That centrality measure, here denoted by Λ 1 , is defined, for a capacitated network N with vertex set V and x ∈ V, by where N yz (x) is introduced as "the maximum flow from y to z that passes through the vertex x" in [14, pp. 147-148]. Since a flow is a function on the arcs of the network, the description of N yz (x) is not perfectly clear. However, as one can understand from the examples in [14] and the comments to [14] by Borgatti and Everett [4], N yz (x) corresponds to the number of paths that surely pass through x in any maximum sequence of arc-disjoint paths in N connecting y and z. In fact, the flow betweenness was designed to solve some of the drawbacks of the betweenness centrality measure, proposed by Freeman [13] and based on geodesics, by considering not only the shortest paths but all the possible paths between two vertices. Thus, despite its name, the flow betweenness is conceived as a centrality measure based on paths. The flow betweenness, and some of its variations, are present in UCINET [3] and in the R sna package [20]. One of those variations, proposed by Borgatti and Everett [4] and here denoted by Λ 2 , is defined, for every capacitated network N with vertex set V and x ∈ V, by where N yz corresponds to the maximum number of arc-disjoint paths from y to z in N. 1 Borgatti and Everett [4, p. 475] observe, however, that, since there is not in general a unique maximum sequence of arc-disjoint paths between two vertices, "flow betweenness cannot be calculated directly by counting paths." Thus, they explain that in UCINET the number N yz (x) is computed as the difference N yz (x) = N yz − N x yz , where N yz is the maximum flow value from y to z in N and N x yz is the maximum flow value from y to z in the network N x obtained by N by setting to zero all the capacities on arcs incident to x. In other words, N yz (x) is replaced by the amount of flow that gets lost when all the communications through x are interrupted. 2 As a consequence, the centrality measures Λ 1 and Λ 2 are de facto replaced by the centrality measures Φ 1 and Φ 2 based on flows and defined, for every capacitated network N with vertex set V and x ∈ V, by 3 Note that in [4] the centrality measure Φ 2 is called flow centrality. Of course, the computation of N yz (x) is definitely much less expensive than the one of N yz (x), since the two numbers N yz and N x yz can be simply computed via the augmenting path algorithm. Thus, Φ N 1 (x) and Φ N 2 (x) can be computed much more easily than Λ N 1 (x) and Λ N 2 (x). On the other hand, Λ N 1 (x) and Φ N 1 (x) (resp. Λ N 2 (x) and Φ N 2 (x)) might be in principle different for some networks and some of their vertices. In other words, Λ 1 and Φ 1 (resp. Λ 2 and Φ 2 ) might be different centrality measures. As a consequence, an analysis of the relation between the two numbers N yz (x) and N yz (x) is certainly crucial in order to understand the relation among the considered centrality measures. At the best of our knowledge, however, no general result linking N yz (x) and N yz (x) is available in the literature. As the main result of the paper, we prove that the equality always holds true (Theorem 20). That immediately implies that actually Λ 1 = Φ 1 and Λ 2 = Φ 2 and shows then that replacing N yz (x) by N yz (x) is perfectly justified. We stress that, while proving the inequality N yz (x) ≥ N yz (x) is quite simple, the proof of the equality N yz (x) = N yz (x) requires a quite sophisticated argument involving some delicate aspects of flow theory and, in particular, the Flow Decomposition Theorem (Theorem 12). The heart of the matter is that, as shown in detail in Section 3, it is possible to reconstruct flows from the knowledge of paths and conversely to derive paths from the knowledge of flows. Hence, in many situations, one can conceptually interchange flows and paths but that interchange is not obvious at all, especially when the focus is on paths passing through a fixed vertex. Relying on that delicate interchange we also give a complete justification of the intuitive description of N yz (x) appearing in [14]. Indeed, we formally define the concept of global flow that must pass through a fixed vertex x in any maximum flow from y to z in the network N, denoted by N yz (x), and we prove that N yz (x) = N yz (x) (Proposition 23). In the paper, we also study the natural extensions of N yz (x) and N yz (x) to the case where groups of vertices are considered. Given a group X of vertices, we consider the number N yz (X) of paths that must pass through X in a maximum sequence of arc-disjoint paths connecting two distinct vertices y and z. Moreover, we consider the number N yz (X) = N yz − N X yz , where N X is the network obtained by N by setting to zero all the capacities related to arcs incident to X. On the basis of (1), one might expect that N yz (X) equals N yz (X), but that is not true, in general, when X is not a singleton (Proposition 21). Indeed, in general, we only have N yz (X) ≥ N yz (X) (Proposition 24). These facts make clear that N yz (X) and N yz (X) are in fact different concepts based on diverse rationales. Using those extensions, the centrality measures Λ 1 , Λ 2 , Φ 1 and Φ 2 can be immediately generalized to groups of vertices. Indeed, for a capacitated network N with vertex set V and X ⊆ V, one can set 1 The definition of Λ 2 is inspired to [13]. 2 Such extensions of centrality measures to groups of vertices are in line with the ideas by Everett and Borgatti [10], as pointed out by the fact that the sums only involve vertices not belonging to X. Of course, since N yz (X) and N yz (X) may be different, we have that Λ 1 and Φ 1 (resp. Λ 2 and Φ 2 ) do not coincide as group centrality measures. Moreover, as noticed in [4] for single vertices, Λ 1 and Λ 2 are hard to compute and thus their concrete application seems quite problematic.
In the last part of the paper, using the quantities N yz (X) and N yz (X), we propose two new group centrality measures (Section 5). The first one, based on N yz (X) and denoted by Λ, is called full-flow betweenness group centrality measure and is a variation of Λ 2 . The second one, based on N yz (X) and denoted by Φ, is called full-flow vitality group centrality measure and is a variation of Φ 2 . Precisely, for a capacitated network N with vertex set V and X ⊆ V, we define As evident, the new idea is just to take into account all the vertices of the network, including those belonging to X. On the one hand, the computational complexity of Λ (resp. Φ) is the same as the computational complexity of Λ 1 and Λ 2 (resp. Φ 1 and Φ 2 ). On the other hand, Λ and Φ satisfy a natural form of monotonicity which surely does not hold true for Λ 1 , . Such a form of monotonicity is invoked as a main property by Everett and Borgatti [10] and constitutes a basic requirement for other desirable properties for group centrality measures. A detailed analysis of the properties of Λ and Φ, especially in view of their applications, is an interesting topic for future research. We are particularly confident in the use of Φ that appears very clear in scope and, differently from Λ, tractable from the computational viewpoint. In fact, one of the objectives of our research is to support and promote the use of (group) centrality measures defined in terms of flows, like Φ. Indeed, it seems that flows had not played yet, in the context of centrality, the deep role that they would deserve. Flows allow us to investigate the characteristics of a network through a very complete and global approach that does not seem to be possible by making use of other concepts. Moreover, flows have a long consolidated mathematical history that could pave the road for a wide investigation of many interesting properties. There is some limited recent literature about new centrality measures based on flows. Remarkably, Gómez et al. [17] introduce the flow-cost closeness centrality measure and the flow-cost betweenness centrality measure. As declared by their names, those measures take into account not only the maximum flows but also the costs that can be associated to the paths of the network. The authors rely on ordered sets of the so-called nondominated vectors and use methods of linear programming. Since the definitions and the methods are very different from ours, a comparison with our centrality measures is not at hand but could certainly be of some interest.
The paper is organized as follows. In Section 2 some well-known concepts of network theory are recalled, among which are those of flow, generalized path, path, cycle, and sequence of arc-disjoint paths. We propose precise and formal definitions in order to fix notation and allow the proofs to run smoothly. We define then the two main concepts of our research, namely the numbers N yz (X) and N yz (X). In Section 3 we explain how to recover flows from the knowledge of sequences of arc-disjoint paths and conversely. Section 4 is about the analysis of the properties of N yz (X) and N yz (X). In particular, in Section 4.1 we prove our main result (1); in Section 4.2 we focus on the global flow that must pass through X in any maximum flow, showing that it coincides with N yz (X) when X is a singleton. In Section 5 we introduce the two flow group centrality measures Λ and Φ and we comment on them. Moreover, we show that they both satisfy the aforementioned type of monotonicity. The conclusions close the paper. For convenience of the reader, we collect in Table 1 the fundamental notation introduced in the paper. Let X be a (possibly empty) set and m ∈ N. A sequence of m elements in X is an element of the Cartesian product X m . Given

Notation and preliminary definitions
, we say that x j ∈ X is the jth component of x. Of course, different components of the same sequence can be equal. Note that X m ≠ ∅ if and only if X ≠ ∅ so that there are sequences of m elements in X if  and only if X ≠ ∅. We also set X 0 = {()} and call the symbol () the sequence of 0 elements of X. In order to have a uniform notation for sequences of 0 elements of X and sequences of m ≥ 1 elements of X, we will always interpret as () any writing of the type (x j ) j∈ [0] . Finally, given two sequences of elements of X, we say that they are equivalent if they both have 0 elements or if they have the same number of elements and one can be obtained from the other by a permutation of the components. Let V be a finite set with |V| ≥ 2. The complete digraph on V is the digraph K V = (V, A) with vertex set V and arc set A = {(x, y) ∈ V 2 ∶ x ≠ y}. Note that in a complete digraph the set A of arcs is completely determined by the choice of the set V of vertices. The set of the complete digraphs is denoted by . A network is a pair and c is a function from A to N 0 called capacity. If convenient we will also indicate a network in a more detailed way by N = (V, A, c). The set of networks is denoted by . 4 An important family of networks, often considered in network literature, are the 0-1 networks. Recall that a network N = (V, A, c) is called 0-1 if c(a) ∈ {0, 1} for all a ∈ A. Of course, there is a natural bijection between 0-1 networks on V and digraphs on V. Thus, the concept of network can be seen as an extension of the one of digraph.

Flows in a network
be fixed in the rest of this section. If a = (x, y) ∈ A we call x and y the endpoints of a. Moreover, we say that a exits from x and enters in y. Let X ⊆ V. An arc a ∈ A is called incident to X if at least one of its endpoints belongs to X. We define Note that A X is the set of arcs in A with a unique endpoint belonging to X. 5 We define by N X the network (K V , c X ) ∈ where, for every a ∈ A, Within flow theory, the capacity of X is defined by and, for every The function f 0 ∶ A → N 0 defined by f 0 (a) = 0 for all a ∈ A is a flow, called the null flow. We denote the set of flows from y to z in N by  (N, y, z). When we represent networks and flows via a figure, we are going to use some standard conventions: a single number attached to an arc represents the capacity of that arc; two numbers attached to an arc, respectively, represent the flow and the capacity of that arc; if an arc is not drawn, then its capacity is zero.
Recall that, The number We denote the set of maximum flows from y to z in N by (N, y, z).
and N′ yz ≤ N yz .
Let us introduce now an important definition.
and we call f (X) the flow that passes through X in the flow f .

The number N yz (X)
Let us introduce now the first main concept of our research, namely the number N yz (X). 5 Throughout the paper, in all the writings involving a subset X of the set of vertices of a network, we write x instead of X when X = {x}, for some vertex x.
The number N yz (X) represents the reduction of maximum flow value from y to z in N when the capacity of all the arcs incident to X are set to zero. Note that X ∩ {y, z} ≠ ∅ implies N yz (X) = N yz . The next proposition states a monotonicity property of N yz (X).

Generalized paths and cycles in a complete digraph
are called the vertices of , a 1 , … , a m−1 ∈ A are called the arcs of . The set of vertices of is denoted by V( ) and the set of arcs by A( ). Given X ⊆ V, we say that passes through X if X ∩ V( ) ≠ ∅. We are interested in the following specifications for .
) be a path or a cycle in K V . Then is completely determined by its vertices and thus we usually write = x 1 · · · x m . Of course, the same simple notation is not possible for generalized paths that are not paths.

Sequences of arc-disjoint paths in a network
The set of paths from y to z in N is denoted by P N yz . The set of cycles in N is denoted by C N .
The above definition agrees with the common definition of m arc-disjoint paths used for 0-1 networks. Indeed, if N is a 0-1 network then = ( j ) j∈[m] ∈ (P N yz ) m is a sequence of m arc-disjoint paths in N if and only if, for every j ∈ [m], all the arcs of j have capacity 1 and every arc with capacity 1 in the network is used at most once. In this case, the j are necessarily distinct. On the contrary, for a generic capacitated network, repetitions of paths in a sequence of m arc-disjoint paths are surely possible.
Note that, trivially, if is a sequence of m arc-disjoint paths and ′ is equivalent to , then ′ is a sequence of m arc-disjoint paths too. We denote the set of sequences of m arc-disjoint paths from y to z in N by The set of sequences of arc-disjoint paths from y to z in N is defined by yz , we say that the length of is m and we write l( ) = m. Observe that, l( ) = 0 if and only if = ( ). We also set Note that N yz is the maximum length of a sequence of arc-disjoint paths from y to z in N and that N yz = 0 if and only if P N yz = ∅. The set of sequences of arc-disjoint paths from y to z in N having maximum length is defined by Hence, we also have In other words, one cannot generally add paths to a sequence of arc-disjoint paths to get a new sequence of arc-disjoint paths of higher length. That fact surely introduces an element of complexity in treating the sequences of arc-disjoint paths. We will discuss in more detail that issue after having presented the Flow Decomposition Theorem (Theorem 12) in Section 3.2.
Given ∈  N yz and X ⊆ V, we denote now by l X ( ) the number of components of passing through X.

The number N yz (X)
We now have all the tools for providing the definition of the other main concept of our research, namely the number N yz (X). 6 The number N yz (X) represents the number of paths that must pass through X in a maximum sequence of arc-disjoint paths connecting the vertices y and z. Since  N yz ≠ ∅, N yz (X) is well defined. By (7) we have that Note that X ∩ {y, z} ≠ ∅ implies N yz (X) = N yz . Moreover, N yz (X) = 0 if and only if there exists ∈  N yz such that l X ( ) = 0, that is, none of the paths appearing as components of passes through X.
We also set In other words, the set  N yz (X) collects the maximum sequences of arc-disjoint paths from y to z in N minimally passing through X. Example 6. In order to clarify Definition 5, let us perform some explicit computations for the network N in Figure 1. First of all, we have that Up to a reordering of the components, the elements of  N,2 yz are given by while the elements of  N,3 yz are given by . 6 When X is a singleton, a definition similar to Definition 5 is proposed in [15].
Moreover, for every m ≥ 4,  N,m yz = ∅. As a consequence, N yz = 3 and  N yz =  N,3 yz . Considering now X = {x}, we have that l X ( ′ ) = 2 and l X ( ′′ ) = 2. Thus, and the elements of  N yz (X) are given by ′′ and all the sequences equivalent to ′′ .
We now prove a monotonicity property for N yz (X) similar to that proved for N yz (X) in Proposition 3.
. Since X ⊆ Y, the components of * passing through Y are at least as many as those passing through X, that is, l X ( * ) ≤ l Y ( * ). Thus,  [14] about their description of N yz (x), explains that, in order to take into account the fact that there is, in general, more than one maximum sequence of arc-disjoint paths from y to z in N, they consider "the maximum possible flow through x over all possible solutions to the yz maximum flow problem." Within our notation that means to consider the quantity max ∈ N yz l X ( ). It describes the flow that can pass through x, and not the one that must pass through x, in any maximum flow. 7 That object could be of some interest but surely it is not in line with the original idea in [14].

PATHS AND FLOWS
Once the formal definitions of N yz (X) and N yz (X) are given, our main purpose is to analyze the relation between those numbers. It turns out fundamental to deepen the link between paths and flows. A careful description of that link constitutes the indispensable tool for the proof of our main theorem, namely Theorem 20. First of all, let us introduce the concepts of generalized path function, path function, and cycle function, having in mind the definitions and comments in Section 2.4.
. If a ∈ A, the arc function associated with a is the function a ∶ A → N 0 defined by so that, if is a path, then If is a cycle in The functions (9), (10), (11) are, respectively, called the generalized path function, the path function and the cycle function associated with .
Note that the generalized path functions assume values in {−1, 0, 1}. In particular, given a generalized path , we have that

From paths to flows
The next result shows how every sequence of arc-disjoint paths can define a flow.
Proof. By (6), we immediately obtain that f satisfies the compatibility condition (2).
Indeed, let a, b ∈ A − x and assume by contradiction that there exists j ∈ U a ∩ U b . Then both a and b are arcs of the path j entering into its vertex x, contrary to the fact that in a path every vertex has at most one arc entering into it. The same argument applies to the case a, b ∈ A + x .
Then, using (14) and (15), we get which says that f satisfies the conservation law (3). Thus, we have proved that f is a flow. By (16), we also see that We next show that ⋃  (14) and (17), we get Now the equality f = ∑

j∈[m]
j is an immediate consequence of (12) and (10). Finally observe that the equality (13) holds also for x ∈ {y, z} because, by Definition 1 and by (18), The above proposition allows us to give an important definition. Note that if , ′ ∈  N yz are equivalent, then f = f ′ .

From flows to paths
In this section we present the well-known Flow Decomposition Theorem in a form that is useful for our purposes and explore its fundamental consequences for our research. We will make large use of generalized path functions and cycle functions (Definition 8).
We start recalling, within our notation, the well-known concept of augmenting path. A generalized path from y to z in K V is called an augmenting path for f in N if c(a) − f (a) ≥ 1 for all a ∈ A( ) + , and f (a) ≥ 1 for all a ∈ A( ) − . We denote by AP N yz (f ) the set of the augmenting paths from y to z for f in N. By the celebrated Ford and Fulkerson Theorem, f ∈  (N, y, z) if and only if AP N yz (f ) = ∅. The next proposition is a straightforward but useful interpretation of the flow augmenting path algorithm within our notation.
The following result is substantially a technical rephrase of the Flow Decomposition Theorem as it appears in [1, Chapter 3].

Example 13.
As an illustration of Theorem 12, consider the network N and the flow f from y to z in N described in Figure 2. Note that v(f ) = 2. A simple check shows that we have where (yvxz, yuz) is a sequence of two arc-disjoint paths from y to z in N and vxuv is a cycle in N. Moreover we also have where (yvxuz, yuvxz) is a sequence of two arc-disjoint paths from y to z in N and no cycle is involved. In other words, ((yvxz, yuz), (vxuv)) and ((yvxuz, yuvxz), ()) are two decompositions of f . That confirms the well-known fact that, in general, a flow can admit diverse decompositions.
By Theorem 12 we deduce that if there exists a flow of value m, then there also exists a flow of the same value of the type f , where ∈  N,m yz . Such a can be obtained by considering any decomposition ( * , w * ) of an arbitrarily chosen m-valued flow f * and setting ∶= * .
By Theorem 12 we can also better comment upon and comprehend the issue raised in Section 2.5. Let us consider = ( j ) j∈ [m] ∈  N,m yz , where m ∈ N and m < n ≤ N yz (so that has not maximum length). Then, by Proposition 9, f is not a maximum flow. Applying the flow augmenting path algorithm n − m times using suitable augmenting paths 1 , … , n−m , we find the floŵ f = ∑  Finally, Theorem 12 also allows us to naturally associate with every flow a set of sequences of arc-disjoint paths in the sense of the following definition.

Definition 14.
Let N = (K V , c) ∈ , y, z ∈ V be distinct and f ∈  (N, y, z) with v(f ) = m. We set and we call  N yz (f ) the set of sequences of arc-disjoint paths associated with f . We also set and we call  N,m yz the set of sequences of arc-disjoint paths for m-valued flows and  N yz the set of sequences of arc-disjoint paths for maximum flows.
Note that, by Theorem 12, if f ∈  (N, y, z) with We are now in position to clarify the link between sequences of arc-disjoint paths and flows in a network. Proposition 15 significantly extends (7) showing that, whatever is m, the sequences of m arc-disjoint paths are exactly those associated with the flows of value m, through the Flow Decomposition Theorem. Moreover, it shows that the set of sequences of arc-disjoint paths for maximum flows coincides with the set of maximum sequences of arc-disjoint paths. Proposition 15. Let N = (K V , c) ∈ , y, z ∈ V be distinct and m ∈ N 0 . Then the following facts hold: Proof.

Some technical lemmas
In this section we present some technical results to which we will appeal for the proof of the main theorem (Theorem 20). To start with, given a maximum flow f , we show an interesting inequality between f (x) and N yz (x). c) ∈ , x, y, z ∈ V with y, z distinct and f ∈  (N, y, z) with v(f ) = m ∈ N 0 . Then the following facts hold: Proof.
(ii) Assume that f ∈ (N, y, z) and pick ∈  N yz (f ). By (i) and by equality (13), we have that (iii) Let ∈  N yz (x). Then, by (13), we have that f (x) = l x ( ) = N yz (x). ▪ The next lemma establishes a natural bound for N yz (x) in terms of the outdegree and the indegree of x.
Proof. Consider ∈  N yz (x). By Lemma 16 (iii) and Definition 1, we have In the following two results we explain how some crucial objects of our research behave with respect to a decrease of capacity in the network.
and y, z ∈ V be distinct. Assume that c ′ ≤ c. Then, the following facts hold: Proof.
and thus ∈  N yz . Recall now that, by definition,  y, z). Then, by (4), f ∈  (N, y, z) and v(f ) = N ′ yz = N yz . Thus, f ∈  (N, y, z). FIGURE 3 The network N used in the proof of Lemma 18 FIGURE 4 The network N ′ used in the proof of Lemma 18 (iii) ⇒ (i) Assume that (N ′ , y, z) ⊆  (N, y, z). Let ∈  N ′ yz . Then l( ) = N ′ yz and, by Lemma 18 (i), ∈  N yz . Consider the flow f associated with , and recall that v(f ) = l( ) = N ′ yz . Hence f ∈ (N ′ , y, z) and thus f ∈  (N, y, z). Thus, l( ) = N yz , which gives ∈  N yz . ▪ 4 Properties of N yz (X) and N yz (X)

Main theorem
We are finally ready to prove our main result. 8 We are going to prove the theorem showing, by induction on n, that (23) holds true for all n ∈ N 0 . Lemma 17,implies N yz (x) = 0 and, by (22), the statement holds.
For brevity, let us set N yz (x) = s and N yz = m. By (8), we have that 0 ≤ s ≤ m. If s = 0, then we again conclude by (22). Assume then s ≥ 1. As a consequence, we also have m ≥ 1. Choose among the sequences in  N yz (x) a sequence ∈  N yz (x) in which the components passing through x are the last s. Let f be the flow associated with , defined in (12). Recall that l( ) = m and l x ( ) = s.
We divide our argument into two cases. Case (I). Assume that there exists ã ∈ A x such that f (ã) < c(ã). Then, obviously, c(ã) ≥ 1. Consider the network Ñ = (K V ,c) wherec is defined, for every a ∈ A, as and note now that, for every a ∈ A,c(a) ≤ c(a).
It is immediate to observe that Ñ x = N x , so that We also have ∈  Ñ yz and then Ñ yz ≥ l( ) = m = N yz . Moreover, by (5), we also have Ñ yz ≤ N yz . Thus, Ñ yz = N yz and ∈  Ñ yz . As a consequence, N yz − yz . We are then left with proving that Ñ yz (x) = N yz (x). Note that Ñ yz (x) ≤ l x ( ) = s. Assume now, by contradiction, that there exists̃∈  Ñ yz such that l x (̃) < s. As proved before, Ñ yz = N yz and then, by Lemma 19, we have that̃∈  N yz and then N yz (x) ≤ l x (̃) < s, a contradiction. Case (II). Assume now that, for every a ∈ A x , we have By Lemma 16(iii), we then get The component m of passes through x and reaches z ≠ x. Thus there exists ã ∈ A + x ∩ A( m ) and, by (24), we have c(ã) ≥ 1.
Define the network Ñ = (K V ,c) ∈ by: and note now that, for every a ∈ A,c(a) ≤ c(a).
Hence, by the inductive assumption, we get Ñ yz (x) = Ñ yz − Ñ x yz . In order to complete the proof we show the following three equalities: Let us start by considering̃∈  N yz obtained by by deleting the component m . In other words,̃= (̃j) j∈[m −1] where, for every j ∈ [m − 1],̃j = j . By definition of Ñ, we surely havẽ∈  Ñ yz and thus Moreover, by Lemma 17 and (25), we have Let us now prove the equalities (a), (b) and (c). (a) Assume by contradiction that Ñ yz > N yz −1, that is, Ñ yz ≥ N yz . By (5), we then obtain N yz = Ñ yz . By Lemma 19, we also deduce that  Ñ yz ⊆  N yz so that Ñ yz (x) ≥ s. On the other hand, by (27), we also have Ñ yz (x) ≤ s − 1, a contradiction. As a consequence, Ñ yz ≤ N yz − 1. Using now (26), we conclude Ñ yz = N yz − 1, as desired.
(c) Clearly we have that N x = Ñ x and thus The next proposition shows that the equality N yz (X) = N yz (X) does not hold true in general when X is not a singleton. Proposition 21 follows by an example due to Bang-Jensen (private communication).

Proposition 21.
There exist N = (K V , c) ∈ , X ⊆ V and y, z ∈ V distinct such that N yz (X) > N yz (X).
Proof. Consider the network N in Figure 5 and X = {x 1 , x 2 }. It is immediately checked that N yz = 3 and Since l X ( ) = 2, we have that N yz (X) ≤ 2. Moreover, by (33), we know that N yz (X) ≥ N yz (X) = 1. Thus N yz (X) ∈ {1, 2}. Assume, by contradiction, that N yz (X) = 1. Then there exists = ( 1 , 2 , 3 ) ∈  N yz (X) such that only 3 passes through X. Thus,̃= ( 1 , 2 ) ∈  N X yz and, since N X yz = 2, we deduce that̃∈  N X yz . Hence, it is immediately observed that there are only two possibilities for a sequence of two arc-disjoint paths from y to z in N X (up to reordering of the components). More precisely, we havẽ 3 ). Assume first that 3 passes through x 1 . Then the only arc entering into x 1 and having capacity 1, that is (u 1 , x 1 ), must be an arc of 3 . That forces A( 3 ) to contain also the arc (y, u 1 ). On the other hand, that arc is also an arc of yu 1 v 1 z and we contradict the independence requirement. Similarly, if 3 passes through x 2 , then the only arc entering into x 2 and having capacity 1, that is (u 2 , x 2 ), must be an arc of 3 . That forces A( 3 ) to contain also the arc (y, u 2 ), which is an arc of the path yu 2 v 2 z, again contrary to the independence requirement.
If now̃= (yu 1 v 1 z, yv 2 z), then = (yu 1 v 1 z, yv 2 z, 3 ). As in the previous case, there is no way to include x 1 as a vertex of 3 . Moreover, if 3 passes through x 2 , then necessarily 3 = yu 2 x 2 v 1 z. Hence, A( 3 ) must contain the arc (v 1 , z), which is an arc of yu 1 v 1 z contrary to the independence requirement. ▪ Proposition 21 ultimately clarifies that the numbers N yz (X) and N yz (X) stem from different ideas and that Theorem 20 is a pure miracle happening when X is a singleton. 9

Global flow through a set of vertices
Let us introduce a new concept based on Definition 1.
The number N yz (X) is called the global flow that must pass through X in any maximum flow from y to z.
. Propositions 23 and 24 state some interesting links among N yz (X), N yz (X), and N yz (X). In particular, we show that when X = {x} is a singleton the global flow that must pass through x in any maximum flow coincides with N yz (x). That fact clarifies that the original intuitive definition of N yz (x) given by Freeman et al. [14, pp. 147-148] is completely sensible.

Proposition 23.
Let N = (K V , c) ∈ , y, z ∈ V be distinct and X ⊆ V. Then N yz (X) ≥ N yz (X) and equality holds when X is a singleton.
Proof. Assume first that X ∩ {y, z} ≠ ∅. Then we have N yz (X) = N yz ≤ N yz (X). If X = {x}, so that x = y or x = z, recalling Definition 1, we have that, for every f ∈ (N, y, z), f (x) = N yz and so also N yz (x) = N yz = N yz (x). Assume next that X ∩ {y, z} = ∅. Let f ∈  (N, y, z) and ( , w) be a decomposition of f . By Lemma 16 (i), we have 9 Note that the equality N yz (X) = N yz (X) trivially holds true also when X = ∅ or |V ⧵ X| ≤ 1. where, by Proposition 9, f ∈ (N, y, z) and ∈  N yz . We now observe that, by (13), we have where the inequality in the chain is an equality when X is a singleton. By (31) and (32) it then follows that and the inequality in the chain is an equality when X is a singleton. ▪ Proposition 24. Let N = (K V , c) ∈ , y, z ∈ V be distinct and X ⊆ V. Then Proof. Let us prove first that N yz (X) ≥ 0. Consider the network N X and observe that, for every a ∈ A, c X (a) ≤ c(a). Thus, by (5) . Then l( ) = N yz and l X ( ) = N yz (X). Let ′ be a sequence of arc-disjoint paths having as components those components of not passing through X. Thus, ′ ∈  N X yz and where the last equality follows from (7). As a consequence, N yz By Proposition 21 we already know that it may happen that N yz (X) > N yz (X). The next proposition shows that also the equality N yz (X) = N yz (X) does not hold true in general. That fact, taking into account Proposition 23, reinforces the idea that it is risky to immediately extend any intuition about paths and flows from single vertices to sets of vertices.
Proposition 25. There exist N = (K V , c) ∈ , X ⊆ V and y, z ∈ V distinct such that N yz (X) > N yz (X).

TWO FLOW GROUP CENTRALITY MEASURES
Consider the ordered pairs of the type ((K V , c), X), where (K V , c) ∈ and X ⊆ V and denote the set of such pairs by . A group centrality measure (GCM) is a function from to R. If is a group centrality measure, we denote the value of at (N, X) ∈ by N (X) and we interpret it as a measure of the importance of the set of vertices X in N. A variety of group centrality measures has been obtained by generalizing classic centrality measures [10,11].
By means of the numbers N yz (X) and N y,z (X), we define in this section two new group centrality measures. Recall that, by Propositions 3 and 7, given Definition 26. The full flow vitality GCM, denoted by Φ, is defined, for every (N, X) ∈ with N = (V, A, c), by Note that Φ is a vitality measure in the sense of Koschützki et al. [18]. Indeed, its value at a given set of vertices X takes into consideration how much eliminating the set X from the network impacts the flow of the network.
We emphasize that in (34) we are summing over the maximum set of arcs which makes the definition meaningful, that is over the set {(y, z) ∈ A ∶ N yz > 0}. That corresponds to the idea of a uniform treatment for the vertices in the network and it is what the adjective full in the name of Φ refers to. 10 10 In [18] it is called the max-flow betweenness vitality of the vertex x of an undirected connected network N, the number Note that Λ is a typical betweenness measure because it takes into considerations to what extent the paths in the network are forced to pass through a set of vertices. In line with (34), also in (35) we are summing over the maximum set of arcs which makes the definition meaningful, that is over the set {(y, z) ∈ A ∶ N yz > 0} and that explains the adjective full in the name of Λ. The fact that Λ is a betweenness measure makes Λ conceptually different from Φ, which is instead a typical vitality measure. Anyway some comparison is surely possible. First of all, it is immediate to check that Λ and Φ coincide when X = {x}. Indeed, by Theorem 20 we know that, for every y, z ∈ V distinct, we have N yz (x) = N yz (x). Thus, we also have Φ N (x) = Λ N (x). Moreover, by Proposition 24, it is immediately deduced that Φ N (X) ≤ Λ N (X). Finally, by Propositions 21 and 24 it immediately follows that, when |X| ≥ 2, we may have Φ N (X) ≠ Λ N (X).
Observe that, from the computational point of view, there is an important difference between Φ and Λ. Indeed, in order to compute the term We close this section showing that both Φ and Λ satisfy a relevant monotonicity property. It guarantees that enlarging the subset of vertices cannot determine a decrease in the centrality level and appears very desirable for the usual phenomena described through networks. That seems to be at least the opinion expressed by Everett and Borgatti [10] and we certainly agree.
Proposition 28. Let N = (K V , c) ∈ and X ⊆ Y ⊆ V. Then the following facts hold true:

Proof.
Let (y, z) ∈ A with N yz > 0. By Propositions 3 and 7, we have N yz (X) ≤ N yz (Y) and N yz (X) ≤ N yz (Y). Thus summing up over all the arcs (y, z) ∈ A such that yz > 0, we immediately get Φ N (X) ≤ Φ N (Y) and Λ N (X) ≤ Λ N (Y). ▪ There is certainly a simple link between Φ and Φ 2 as well as between Λ and Λ 2 , where Φ 2 and Λ 2 are considered in the introduction. Indeed, once we have defined the set In particular, the difference between Φ N (X) and Φ N 2 (X) and between Λ N (X) and Λ N 2 (X) is the same and is related to connectivity properties of the network N.
However, despite that link, Φ 2 and Λ 2 do not satisfy the monotonicity property considered in Proposition 28. Indeed, consider the 0-1 network N in Figure 7 and note that Φ N This failure is surely a consequence of the fact that only vertices not belonging to X are considered in the sums defining Φ 2 and Λ 2 so that, for every network, those measures assume value 0 when X is the whole set of vertices.
The lack of monotonicity of Φ 2 and Λ 2 is far from being their only deficiency. Indeed, consider again the 0-1 network N in Figure 7. A computation shows that On the other hand, we have The centrality values determined via Φ and Λ seem to better take into account the nature of the network under consideration, allowing a higher level of diversification among its vertices and groups of vertices. In particular, it is reasonable to think that vertex c has a higher level of centrality than vertices a and d in N. That fact is recognized by Φ and Λ but it is not recognized by Φ 2 and Λ 2 . Similarly, it is reasonable to think that {b, c} has a higher level of centrality than {a, b}, {a, } and {c, } in N. Again, that fact is recognized by Φ and Λ but it is not recognized by Φ 2 and Λ 2 .

CONCLUSIONS AND FURTHER RESEARCH
We have clarified the equality between two fundamental numbers used in network theory that surprisingly were identified without a solid formal argument: the minimum number N yz (x) of paths passing through the vertex x in a maximum sequence of arc-disjoint paths from y to z in the network N and the number N yz (x) expressing the reduction of maximum flow value from y to z in N when the capacity of all the arcs incident to x is reduced to zero. The proof of that fact has involved a tricky analysis of the relationship between paths and flows which goes beyond the original goal and is interesting in itself. We also proved that the natural generalizations of N yz (x) and N yz (x) to groups of vertices are not, in general, equal. Our analysis has led to the definition of two conceptually different centrality measures Φ and Λ both satisfying an important monotonicity property. The contexts in which they could be fruitfully applied are in principle many. Just to give an example, consider a scientific community V of scholars and, for every x, y ∈ V, the number c(x, y) of times that, within a certain fixed period of time, the researcher x ∈ V cited the researcher y ∈ V. Construct then the corresponding citation network N = (K V , c) ∈ . It is reasonable to think that, for x, y, z ∈ V, if x cited y and y cited z, then indirectly x cited z so that z gains prestige not only from y but also from x and thus, more generally, z gains prestige from any path in N having z as endpoint. 11 Consider, in particular, a situation in which two groups of researchers, say X 1 and X 2 , received the same total number of citations from the scholars outside the groups, that is, c(a). Suppose that you want to diversify X 1 and X 2 putting in evidence the quality of those citations. For X i , where i ∈ {1, 2}, that quality can be evaluated by looking at the set Y i of scholars who cited the scholars in X i and taking into account the number of citations that the scholars in Y i themselves received. On the other hand, also the quality of the citations received by the scholars in Y i is important and that can be in turn evaluated by looking at the set Z i of the scholars who cited the scholars in Y i . That reasoning can be continued so that the quality of the citations received by the scholars in X i can effectively emerge only by a global approach which takes into consideration the whole configuration and complexity of the citation network (see, for instance, [5]). That can be obtained by computing which between Λ N (X 1 ) and Λ N (X 2 ) is larger. Another reasonable idea could be instead to look at the impact on the amount of direct and indirect citations in the network caused by a hypothetical absence from the scientific scenario of the scholars in X i , that is, comparing Φ N (X 1 ) and Φ N (X 2 ). Indeed, if Φ N (X 1 ) > Φ N (X 2 ), then the absence of X 1 mostly damages the scientific community in its dynamic exchange of contacts. We also emphasize that the formal approach we used seems very promising for dealing with the properties of our centrality measures, in particular, with those invoked by Sabidussi [21] as the main desirable. As is well known, an axiomatic satisfactory definition of centrality is missing in the literature. The presence or absence of certain properties can though help in deciding which measure better fits in a certain application, as largely recognized in the bibliometric literature which is recently oriented in using methods from social choice theory (see, for instance, [7,8]) as well as in the game theoretic approach to centrality (see [16]). Thus, exploring the properties of Φ and Λ is surely an interesting research topic. However, from the practical point of view we already know that the computation of Λ appears very hard and thus, reasonably, the main focus in the future will be on Φ. Of course, a further crucial project is the implementation of Φ on concrete networks in order to discover empirically in which sense, and in which kind of networks, Φ behaves better than other classic centrality measures.