Abstract

There is an age-old question in all branches of network analysis. What makes an actor in a network important, courted, or sought? Both Crossley and Bonacich contend that rather than its intrinsic wealth or value, an actor’s status lies in the structures of its interactions with other actors. Since pairwise relation data in a network can be stored in a two-dimensional array or matrix, graph theory and linear algebra lend themselves as great tools to gauge the centrality (interpreted as importance, power, or popularity, depending on the purpose of the network) of each actor. We express known and new centralities in terms of only two matrices associated with the network. We show that derivations of these expressions can be handled exclusively through the main eigenvectors (not orthogonal to the all-one vector) associated with the adjacency matrix. We also propose a centrality vector (SWIPD) which is a linear combination of the square, walk, power, and degree centrality vectors with weightings of the various centralities depending on the purpose of the network. By comparing actors’ scores for various weightings, a clear understanding of which actors are most central is obtained. Moreover, for threshold networks, the (SWIPD) measure turns out to be independent of the weightings.

1. Introduction

An actor is represented by a vertex in a relational network. Following the terminology used in social analysis, the power and importance of a specific actor (ego) does not result from its inherent properties but rather from its position with respect to others (alter) in the network. The extent of its influence depends on the links to first neighbours and the neighbours of these neighbours. What is relevant is the importance of the actors in the subnetwork focused around ego, mostly that including all actors to whom ego has a connection up to a prescribed path length, and all the connections among all of these actors.

To rank actors according to their importance, one can gauge the amount of their influence in the whole network. There are various types of centralities of an actor in a relational network. A centrality measure is meant to give the relative importance of in the network. It is usually taken to be intuitively a measure of the access of to sources and the power can wield to disseminate ideas and create awareness. We will also quantify a new aspect of power depending on the extent alter is constrained to depend on ego for the purpose of their membership in the network. For any specified centrality measure, the vector with the th entry equal to the centrality of the vertex is said to be the centrality vector of the network.

A network is represented by a graph with a set of vertices often labelled and an edge-set of pairs of distinct vertices describing an adjacency relation. Network systems are modelled as graphs whose vertices represent the dynamical units or actors and whose links stand for the interactions (collaborations and business links for instance) among them. Powerful graph theoretical techniques can then be applied to yield results that give meaningful information about a network.

The degree of a vertex is the number of edges incident to . A vertex of degree one is termed an end-vertex. In a -regular graph, has the same value for each vertex . A walk of length starting from a vertex is an alternating sequence of (not necessarily distinct) vertices and edges where and an edge has and as end vertices. A clique is a subset of the vertices that induces a complete subgraph. An independent set is a subset of the vertices that induces an empty subgraph (with no edges). The path of length is an alternating sequence of distinct vertices and distinct edges so that is an edge connecting to . The distance between two vertices and is the number of edges in the shortest path joining to . The maximum of the distances between all the vertex pairs in is called the diameter of .

The cycle is a connected graph on vertices each with degree two. The complete graph has vertices and there is an edge between every pair of vertices. It follows that the vertices of a complete graph form a clique. In a bipartite graph, the vertex set is partitioned into independent sets and .

The purpose of a network determines which centrality measures are meaningful. Information networks aim to reach as many actors as possible, in contrast with epidemiology networks whose objective is to contain the spread of a virus. In exchange networks, bargaining power is the goal and ego’s power is alter’s dependence. To strike a balance among the intended behaviour and the latent forces emanating from the network structure, we consider various centrality measures, namely,(i)the degree centrality,(ii)the square centrality,(iii)the eigenvector centrality,(iv)the walk probability vector,(v)the walk centrality,(vi)the irregular scaled-walk centrality,(vii)the power centrality.

Measures (i) and (iii) are standard centralities, whereas the rest are inspired by behavioural network expectations. The local statistics of a vertex are captured, for instance, by its degree or by the entry, corresponding to a particular vertex, of a specific eigenvector for some matrix encoding the graph adjacencies. We propose another general-purpose centrality termed which is a linear combination of the square, walk, power, and degree centrality vectors where weightings can be varied depending on the overarching aim of the network.

We use (or just when the context is clear) to denote the 0-1-adjacency matrix of a graph , where the entry of the symmetric matrix is 1 if and 0 otherwise. We note that the graph is determined, up to isomorphism, by . For a real symmetric matrix , the real number is an eigenvalue of a matrix if there exists a nonzero vector (termed a -eigenvector) satisfying . The - is the subspace containing all the -eigenvectors. The nullity of is the multiplicity of the eigenvalue zero of . It can also be seen as the deficiency in the rank of .

The vector with each entry equal to one indicates that all vertices have identical weights. The eigenvalues of the adjacency matrix of a graph having some eigenvector not orthogonal to are said to be the main eigenvalues of . A regular graph has as an eigenvector and therefore it has only one main eigenvalue, namely, the maximum eigenvalue. The vector is used as an initial vertex status in a bias-free network.

By the Perron-Frobenius theorem on nonnegative matrices, the adjacency matrix of a connected network has an eigenvector each of whose entries is positive. This eigenvector is referred to as the Perron vector and is associated with the maximum eigenvalue of (or of ). Note that the maximum eigenvalue of is not exceeded by the absolute value of any other eigenvalue and is always a main eigenvalue.

The degree diagonal matrix has as the diagonal entry at position for and zero in all off-diagonal positions. A very useful representation of a graph is the Laplacian matrix () defined as . For a connected network, the Laplacian has a simple eigenvalue zero, with an associated eigenvector equal to .

An attractive feature in the mathematical treatment of the formulae for the centralities derived here is that(i)the main eigenvalues suffice in all the derivations and proofs;(ii)each centrality measure is expressed in terms of the matrices and only, which renders the computations tractable.

The rest of the paper will be organised as follows. In Section 2, we review the established degree and eigenvector centralities of a network and refine them by expressing them in terms of invariant vectors associated with . We also introduce the square centrality that focuses on the subnetwork of actors that are at a distance at most two from . Networks often model the spread of fluids, the diffusion of information, or virus propagation. Diffusion is governed by a differential equation that can be expressed in terms of the Laplacian. In algorithms that determine network kinetics, the Laplacian is a recurring theme. We show, in Section 3, how the Laplacian features in the diffusion of information, which is found to rank vertices as the degree centrality does. We then propose, in Section 4, the walk centrality vector, based on the main eigenspaces and, later, the power centrality that reduces the contribution to ego by the well connected first neighbours.

A split graph is a graph in which the vertex set can be partitioned into a clique and an independent set. A connected threshold graph is a split graph in which the independent subset (if not empty) is partitioned into one or more parts and the clique is partitioned into one or more parts where and if is even, whereas and if is odd, as shown in Figure 1. Note that and, for a unique representation , the size . Each vertex of a particular independent subset has the same neighbourhood . The distinct vertex degrees are . Moreover the closed neighbourhood of a vertex contains the neighbourhood of any vertex of lower degree, which is the reason why threshold graphs are also referred to as nested split graphs. The interactions within many real world networks approach those of threshold graphs [1, 2]. In the sequel, we will point out instances when threshold graphs show limiting behaviour for the centrality measures we consider. In Section 5, we establish that, for threshold graphs, even the expected discriminating centralities such as the eigenvector and the power centrality coincide with the degree centrality.

(a)

(b)

(a)
(b)

Figure 1 

The threshold graphs , , , for even and odd, respectively.

In Section 4.2 we discuss the graph parameter (SWIPD) that combines the centralities that tend to provide unique information on a network. Only for irregular nonbipartite graphs does the SWIPD centrality give a meaningful ranking of the vertices.

2. Local Centralities

The most central vertices in a network are expected to head the list in a valid ranking of the vertices. At a first level of ego’s exposure, one may look at the immediate neighbours, captured by the degree centrality measure. The status of these first neighbours is also thought to be of significance to ego’s ranking as they may bring their influence to bear on the rest of the network according to ego’s needs. We therefore consider three aspects of the influence of vertices at a distance up to two from ego. Firstly, the number of the immediate neighbours in the degree centrality, secondly the total number of first and second neighbours of ego will be considered in the square centrality, and thirdly the ripple effect of the first neighbours’ own centrality covered by the eigenvector centrality.

2.1. Degree and Square Centrality Measures

An accepted premise, in propagation networks, is that the popularity of an actor increases with the number of links to others in the network. The degree centrality of a vertex is its degree , that is, the total number of its immediate neighbours. The vector with entry equal to the degree centrality of is said to be the degree centrality vector of the network. Recall that the vector gives equal importance to all the vertices.

Proposition 1. Let be a network with adjacency matrix . The degree centrality vector is .

Proof. Let be the degree of vertex . Then , as required.

The degree centrality is a main contributor to vertex centrality. In Section 3 we will see that there are other centrality measures which are also proportional to , for any graph. One of them, in particular, ranks vertices according to the likelihood that a random walk ends at a particular vertex.

The degree sequence is only the first level of understanding of ego’s status. The next step is to consider implicit interactions of alter with ego. One may ask whether the number of actors at distance two or more have a significant effect on ego. The rationale for this approach is that the formation of new links in a network, where dissemination of information is a priority, tends to be influenced by the choices of the first neighbours. We expect links with “friends of friends” to be more likely. The sum of the first and second neighbours from each vertex is given by the vector .

Large distances between two actors in a network do not necessarily exclude mutual influence. Exposure to information received by different actors even if not directly from ego may have a significant impact on them. The same goes for “similar” information sent by different actors at various distances from ego and received by ego through interaction, which often leads to progressing stages of empathy, a sense of familiarity with the information and acceptance, possibly leading to a proper understanding of (or yearning for) it. Seen in another light, consensus for the acceptance of a new product by many actors in a network is necessarily reached by means of such interactions. The centrality measures we will now study take into consideration distance of alter from ego throughout the network.

2.2. Eigenvector Centrality

We now look at the second aspect of the influence of neighbours, that is, the ripple effect of first neighbours’ own centrality on ego’s ranking. Effective measures aimed at increasing ego’s status are important in marketing. Wasserman and Faust [3] discuss what they call prestige measures of centrality, that is, measures in which the centralities (statuses) of positions are recursively related to the centralities of the positions to which they are connected. Instead of just looking at the vertex degree of a specific actor and of its immediate neighbours as an indication of ego’s importance in a network, the influence of its neighbours is also considered. The interpretation is that having a neighbour who has power over others adds to ego’s importance. Moreover, links are often made with actors recommended by a neighbour. A measure of centrality based on the centralities of neighbours is achieved by assigning a weight to each vertex equal to its interim centrality in an iteration converging to . Whereas degree centrality gives every contact the same weight, the eigenvector centrality weights link with others according to their centralities, thus taking into account the entire pattern in the network. Since repeated application of the adjacency matrix on a vector increases the value exponentially when the maximum eigenvalue of exceeds 1, control is achieved by scaling to .

Definition 2. Let , , and let . Then the eigenvector centrality is the unit vector along .

Starting with the vector ensures a bias-free process. The eigenvalues of a diagonalizable matrix whose eigenvectors span are said to be main. If is real and symmetric and has distinct eigenvalues , then diagonalization leads to the spectral decomposition , where is the projection onto the eigenspace of , for .

Lemma 3 (see [4]). If are the projections onto the -eigenspaces of the distinct eigenvalues , respectively, of , then is the identity operator .

Let be the main eigenvalues of , written in monotonic decreasing order, where . From Lemma 3 and the definition of main eigenvalues, we can write . Thus can be expressed as the sum of orthonormal eigenvectors of associated with the main eigenvalues , respectively.

Lemma 4. If are the unit vectors along , respectively, then where for each , .

The unique vectors of Lemma 4 are referred to as the main eigenvectors of [5, 6].

Lemma 5 (see [7]). Let be a graph with main eigenvalues . The total number of walks of length is given by , where for , the scalars are independent of .

The iteration defining the eigenvector centrality in Definition 2 is known as the power method. It converges provided the network is not bipartite (that is provided has an odd cycle) (see [8], e.g.). We give a proof using only the main eigenvalues.

Theorem 6. For a nonbipartite graph, the eigenvector centrality is the unit Perron vector of .

Proof. Consider the iteration where . From Lemma 4, where are orthonormal eigenvectors belonging to the main eigenvalues . For a nonbipartite connected graph, is the maximum eigenvalue of and is larger than the absolute value of all the other eigenvalues. From the eigenvector equations and , we obtain . As , for and tends to a limit proportional to the Perron vector .

We note that for bipartite graphs, however, the minimum eigenvalue of is and if it happens to be main, then the iteration oscillates as .

For many graphs, the eigenvector centrality captures properties of second neighbours and gives a vertex ranking often different from the degree centrality ranking [9]. This is not the case for threshold graphs.

Proposition 7 (see [10]). For threshold graphs, the ranking of vertices according to the eigenvector centrality is equal to that according to the degree centrality.

3. The Discrete Laplacian and Walks

Understanding the specific details of interactions with ego is an essential part of figuring out the factors that may increase ego’s status. When direct methods prove difficult, a complementary approach is provided by discovering network substructures and invariants that also play a key role.

A graph invariant that goes beyond the immediate neighbourhood of ego is distance between ego and a vertex in alter. The investigation of random movement along a network often involves the Laplacian matrix. In the physical theory of diffusion, the Laplacian arises naturally in the mathematical analysis leading to the equilibrium state.

3.1. Propagation

Diffusion can be seen as the random motion of fluid particles from regions of higher concentration to regions of lower concentration. The same terminology is borrowed for the spread of a commodity such as information or disease. The rate of diffusion from a vertex in a network depends on the difference in the amounts of commodity between and its neighbours. Thus , for a diffusion constant .

Definition 8. If the amount of diffusing commodity at a vertex in a network is , then the column vector is said to be the diffusion vector.

We give a proof of the following well known result that links the Laplacian with the discrete Laplacian.

Proposition 9. The differential equation regulating diffusion in a network is .

Proof. Let be the Kronecker delta which is equal to one if and zero otherwise. Since , then

The well known diffusion equation is where is the Laplacian operator. For this reason, the matrix is referred to as the discrete Laplacian of the graph. The solution of (1) is obtained by expressing as a linear combination of the orthonormal eigenvectors corresponding to the eigenvalues of the real and symmetric Laplacian . Thus . Substituting in (1), . Solving gives the exponential decay for initial commodity amounts .

3.2. The Walk Probability Vector

In a random walk, or Markov chain, along the edges of a connected graph with adjacency matrix , starting from a particular vertex, the probability that a walker is at vertex , after traversing edges (or at time ), the sum over all the neighbours of of where is the probability that the walker moves along edge , given that it is at after time . The unbiased probability is . The adjacency matrix entries are used to select only the neighbours of . Therefore which can be expressed in terms of the Laplacian, as will be shown in Corollary 12.

Definition 10. The column vector is said to be the walk probability vector at time . In the limit, as , the iteration converges and approaches the walk probability vector .

The diagonal matrix with as the diagonal entry at position is . The entries of the th column of are obtained by dividing the entries of the th column of by . Expression (2) can be simplified immediately as in the following result.

Lemma 11. (i) The walk probability vector .
(ii) In the limit, as , and .

The following results follow immediately.

Corollary 12. Consider the following:
(i)   ;
(ii)   .

Let be the vector with the vertex degree as the entry at position . We observe that(i)in a directed graph, the matrix used in the iteration is that for PageRank where is obtained from modified by replacing each zero entries on its diagonal by one, so that it is invertible;(ii)from Corollary 12, is in the nullspace of the Laplacian Lap and ;(iii) and the matrix is referred to as the transition matrix for a random walk [11];(iv)the Perron vector of is for the maximum eigenvalue of , which is 1.

Theorem 13. The walk probability vector , for a connected network with edges, is .

Proof. Since , from Corollary 12(i), . For a connected network, the Laplacian is singular of nullity one, with the eigenvector generating the zero-eigenspace. Hence is a multiple of . Thus for . If , since is the diagonal matrix with as the diagonal entry at position and , then . Therefore .

Theorem 13 gives a surprising result. It asserts that the walk probability vector that is designed to take into account the limiting behaviour of even the remote actors into consideration gives the same ranking of the vertices as the degree centrality.

4. Walk and Power Centralities

Although local properties are mostly influenced by immediate neighbourhoods, the relative importance of all actors in a network that can affect ego must be considered. The questions we wish to answer are as follows.(i)What is the extent of influence on of remote actors?(ii)Which centrality measures ensure that the impact on ’s centrality, of the actors at a large distance from , is not ignored?

To answer these questions, we discuss a centrality that includes all actors in a network, based on the number of walks. However, the more remote actors are made to exert less influence on ego in this measure by applying a geometric progression scaling factor (see Definition 14).

4.1. Walk Centrality

In marketing, it is a common belief that a certain threshold density of exposure to a message is required to achieve effective communication. The larger the number of walks of various lengths from a vertex is, the more the possibilities has to be influenced by actors that can reach it by traversing edges (possibly repeated).

The th entries of the vectors, , give the number of walks of length , respectively, starting at vertex . For an attenuation or damping factor , consider , where the number of walks of length are scaled down by . Similar measures have been proposed in [12, 13] by Bonacich and Katz, respectively.

Definition 14. Let be the maximum eigenvalue of the adjacency matrix of a graph . For , the unit vector along is said to be the walk centrality vector.

Lemma 15. If , then is invertible.

Proof. Consider the determinant of the matrix . We can write as . By Perron-Frobenius theorem for a nonnegative matrix , the absolute value of each eigenvalue of does not exceed the maximum eigenvalue of . The characteristic polynomial is of degree and is positive for . Thus for , and therefore is invertible.

The walk centrality vector is defined for all except at the eigenvalues of . The operator is referred to as the resolvent, usually used in the study of the spectrum of operators on Hilbert spaces and applied to solve the inhomogeneous Fredholm integral equations via the Liouville-Neumann series. We now present new formulae for the walk centrality by considering only the main eigenvalues and eigenvectors of the adjacency matrix.

Theorem 16. Let be the maximum eigenvalue of the adjacency matrix with corresponding Perron vector . Then the walk centrality vector is the unit vector along which is

Proof. Let (=, be the main eigenvalues of with corresponding eigenvectors as in Lemma 4. Then, Since , for , converges absolutely.
It follows that As ,  . Since for , then

For any damping factor , the main eigenvalues and eigenvectors of a graph suffice to determine the walk centrality vector.

Theorem 17. The walk centrality vector of a graph is given by the unit vector along

Proof. Spectral decomposition for a matrix with distinct eigenvalues gives . If is the adjacency matrix of a graph , then , since exactly for the nonmain eigenvalues of .

By Theorem 16, the walk centrality vector depends on and, as tends to , it approaches the eigenvector centrality. Estrada and Rodríguez-Velázquez suggested a damping factor of for . The centrality of vertex is then defined as the diagonal entry at position of and is referred to as the Estrada index [14]. It has also been used as a community detection tool.

4.2. Irregularity Scaled-Walk Vertex Representations

In our quest to capture further the realistic possible influences of remote actors, we now consider another vertex ranking parameter, also based on the number of walks of different lengths, which is very sensitive to graph structure. The number of walks of length in the range 0 to from a vertex of a -vertex graph can form a row vector representing .

For the graph in Figure 2, . The same representation may be shared by different vertices as is the case for vertices 8 and 10.

Figure 2 

Neighbourhoods of different type for the vertices of highest degree in on 14 vertices.

Note that the column vector , giving the number of walks of a particular length from each vertex is . We note that for a regular graph and a specific length , is a constant for all vertices and . Therefore vertices of a regular graph are equivalent with respect to the number of walks. We observe also that, for irregular graphs, it is not always the case that the number of walks from the vertices, as increases, ranks the vertices according to the vertex degrees.

The entries of columns are the sequences of walks of length . Table 1 shows the entries from the 8th, 10th, and 11th vertices, respectively, according to the labelling of the graph in Figure 2. They demonstrate oscillating vertex priorities for small lengths, as shown in Table 1.

However we prove, again using the main eigensystem alone, that there exists a positive integer such that the ranking of the vertices according to the number of walks of length remains unchanged for all .

Theorem 18. Let be a connected nonbipartite graph and the number of walks of length from . Then there exists such that, for all , the ordering of the magnitudes of the number of walks of length from each vertex is independent of .

Proof. Let be the main eigenvalues of with corresponding orthonormal eigenvectors . If , then As , for , which is the case for all eigenvalues of a nonnegative matrix. Hence for all real , there exists such that Now Perron-Frobenius theorem guarantees that each entry of is positive. Hence if is chosen to be less than the minimum value of the entries of , then all the entries of will be positive for . Hence there exists such that, for all , the order of magnitude of the number of walks of length from each vertex is independent of .

An implicit result in the proof of Theorem 18 is that, as , is proportional to the eigenvector centrality. This suggests a vertex representation which we term the irregularity scaled-walk centrality, where a vertex is represented by , where , being the maximum vertex degree in the network. The vertex priority given by the second entry of is the degree centrality, whereas, by Theorem 18, the entries of for larger approach the eigenvector centrality, given in Definition 2.

The overall efforts we considered so far, to make sure that the contribution by distant actors is taken into consideration, involved the concept of distance and walks in graphs. For any network, the walk probability vector and the second entry of the irregularity scaled-walk vertex representation were shown to agree with the degree centrality vector. On the other hand, as , the walk centrality was shown to approach the eigenvector centrality. The vectors in the irregularity scaled-walk vertex representation also give rankings close to the eigenvector centrality for large . Therefore so far, the centralities that may give different vertex rankings turned out to be covered by the degree, the square, and the walk centralities.

4.3. Power Centrality

Now we present a very different concept of authoritative power, derived from everyday experience, which goes counter to that governing the spread of data. Dominance of ego on others may not depend solely on the number of direct subordinates but also on the extent to which the latter are dependent on ego for access to information. The larger the number of connections a subordinate has, the more independent of ego it tends to be, reducing ego’s power. This contrasts sharply with all the other centrality measures we have considered.

We choose to measure the importance of ego (vertex ) by considering , for adjacent to , which we term power centrality, denoted by or sometimes by . In this way, the power of ego’s neighbours is restricted rather than enhanced by the neighbours’ connections.

5. Threshold Graphs

In this section we focus on threshold graphs. For this class of graphs, the value of in Theorem 18 has been proved to be 0 in [10]. Therefore for threshold graphs the ranking of the vertices according to the number of walks is independent of the length of the walks.

Proposition 19 (see [10]). For threshold graphs, the ranking of the vertices according to the number of walks of any length is the same as that for the degree centrality.

Proposition 19 asserts that, for threshold graphs, the irregularity scaled-walk vertex representation gives the same vertex priorities as the degree centrality. Degree and eigenvector centralities usually differ as the latter is sensitive to the importance of second neighbours [9]. Moreover, we observe that according to Proposition 19, for threshold graphs, the eigenvector centrality does not add information to the degree centrality.

Corollary 20. For threshold graphs, the degree and eigenvector centralities rank the vertices in the same way.

Since the contribution to power centrality decreases with increasing degree of a neighbouring vertex, this measure is specifically designed to give a ranking of the vertices possibly different from other centralities. It is surprising that for threshold graphs this is not the case.

Theorem 21. For threshold graphs, the ranking of the vertices according to the power centrality is the same as that for the degree centrality.

Proof. Let , for and , be a connected threshold graph conformal with the notation used in Figure 1. For , let be a vertex lying in the group having degree . Recall that denotes the power centrality of a vertex .
Case 1. For odd , we show that agrees with the degree centrality.
We consider the independent subsets first. For , the neighbourhood of is . Hence For the cliques , , This completes the proof for odd .
Case 2. For even , we show that agrees with the degree centrality.
We consider the independent subsets first. For , the neighbourhood of is . Hence For , Thus the result follows.