Open Access

Power and loyalty defined by proximity to influential relations

Computational Social Networks20152:2

https://doi.org/10.1186/s40649-014-0009-9

Received: 10 July 2014

Accepted: 30 December 2014

Published: 24 February 2015

Abstract

This paper examines a simple definition of power as a composite centrality being the composition of eigenvector centrality and edge betweeness. Various centralities related to the composition are compared on social and collaboration networks. A derived defection score for social fission scenarios is introduced and is demonstrated in Zachary’s Karate club to predict the sole defection in terms of network measures rather than psychological factors. In a network of political power in Mexico across various periods, the two definitions of power serve to shed light on a political power transition between two groups.

Keywords

Composite centrality Power Loyalty Eigenvector Betweeness Fission

Introduction

Networks are often modeled as a graph, which consists of set of nodes (V) and edges (E), such that EV×V. If E is a symmetric relation, then G is called an undirected graph. A network centrality is a function defined on V which assigns importance to nodes according to certain criteria.

Various feedback centralities have been introduced (Seeley [1], Hubbell [2], Katz [3], Bonacich [4]), which share the common objective of measuring a node’s importance while taking into account the importance of its neighbors. The simplified form of a feedback centrality termed eigenvector centrality is based on the Perron-Frobenius theorem which ensures that for a strongly connected graph, the leading eigenvector of the adjacency matrix contains only real positive values ([5]). Let X=(x 1x n ) be the eigenvector of the largest eigenvalue of the adjacency matrix A G of G, and λ 1 is the largest eigenvalue. Then, the eigenvector centrality of node i is \(C_{\textit {EV}}(i) = \frac {1}{\lambda _{1}}x_{i}\). Informally, C EV will find a set of nodes which are more densely connected (clique-like) than other subsets of V. A node with a high C EV score would have relatively more edges between its neighbors.

Betweeness centrality, which was introduced by Freeman in [6] and Anthonisse in [7], measures the proportion of shortest paths passing by a given node. Formally, let σ s,t be the number of shortest paths between nodes s,t, and σ s,t (v) be the number of shortest paths between s,t that pass through v; then, the betweeness of v is defined as \(C_{B}(v)=\sum _{s\neq v}\sum _{t\neq v}\frac {\sigma _{s,t}(v)}{\sigma _{s,t}}\). In [7], betweeness is also defined for edges, for an edge eE, \(C_{\textit {EB}}(e)=\sum _{s\in v}\sum _{t\in v}\frac {\sigma _{s,t}(e)}{\sigma _{s,t}}\). In a social network, an edge with high betweeness would mean that the relation between the represented actors is important in the sense that it is expected to be used more by other actors in the network. Edge betweeness has also been used to detect community structure ([8]).

Definitions and properties

Composite centralities have been suggested based on statistical measures ([9]); here, the natural composition is taken. Let C 1 be a node centrality, and C 2 be an edge centrality whose values are non negative. The fact that C 1 is node based and C 2 is edge based suggests a natural function composition, define:
$$ A_{C_{2}}\left[i,j\right] = \left\{ \begin{array}{ll} C_{2}((i,j)) & \quad (i,j)\in E \\ 0 & \quad (i,j)\not\in E \end{array} \right. $$
(1)

And notate C 1(C 2)(v) as the value for v when C 1 is computed on \(A_{C_{2}}\). A matrix is said to be irreducible if its interpretation as a graph adjacency matrix produces a strongly connected graph. If G is an edge weighted graph, it may be that C EB (e)=0 for eE, while for non-weighted graphs, this is not the case; since every edge would be on the shortest path between its endpoints. Thus, for a weighted graph, using Equation 1 may produce a reducible matrix, since some edges may have zero betweeness.

Proposition 0.1.

Let G be an positive edge weighted undirected connected graph, then \(A_{C_{\textit {EB}}}\) is irreducible.

Proof.

Let u,vV, since G is connected, there exists at least one shortest path P uv connecting u and v. From the definition of C EB , for any edge eP uv , C EB (e)>0. Therefore, in the graph defined by \(A_{C_{\textit {EB}}}\), there exists a positively weighted path connecting u and v.

Notate C EVB (v)=C EV (C EB )(v). From 0.1, C EVB is well defined, as the Perron-Frobenius theorem holds the same way as for C EV . In this case, it is assumed that high edge betweeness indicates a potentially important relation, and that an actor is more powerful if it participates in important relations, either directly, or its neighbors have important relations between themselves.

An artificial example

A simple example in Figure 1 demonstrates that C EVB may differ from both C EV , C B or any linear combination of them. The example consists of two small complete graphs which are connected by one node (node 11) and another node (node 12) connected to node 11. Clearly, node 12 has zero C B and low C EV , but it has the second highest C EVB . Node 12 is accessible only via node 11, the most powerful node in all three measures, so if we assume that all nodes are initially accessed at a similar rate, the relation between 11 and 12 will be the most used, while the nodes within the cliques would have many relations that are used only between the two endpoints (see Figure 1). This shows that C EVB may detect a ‘behind the scenes’ player like node 12, while C EV and C B would assign it low scores.
Figure 1

A simple example.

Eigenvectors in a weighted graph and scaling behaviour

The justification of using C EV on an edge weighted network is, as explained by Newman in [10], if X=(x 1x n ) is the leading eigenvector, then
$$ C_{EV}(i)=x_{i}=\frac{1}{\lambda_{1}}\sum\limits_{j} A_{ij}x_{j} $$
(2)

hence multiplying the weight of an edge by a positive factor will adjust the contribution of the neighbour incident on that edge to the eigenvector centralities of its incident nodes by the same factor, i.e. if the weight of (v,u) is 3 then the contribution of u to C EV (v) is multiplied by a factor of 3. Thus, calculating the eigenvector centrality of an edge weighted network would score nodes according to the weighted density of their neighborhood.

An informal scaling argument regarding C EVB is shown as follows. It is proven in [11] that for a graph G=(V,E) and any node vV,
$$ C_{B}(v) = \sum\limits_{(u,v) \in E}C_{EB}((u,v))-(n-1) $$
(3)
Furthermore, it is numerically demonstrated in [12] that if G is a node degree scale free network with exponent between 2 and 3, then C B follows a power law distribution with exponent approximately 2.2. So, if C EV scales ‘nicely’ in relation to a node degree power law exponent, that would mean that the row sums of the original adjacency matrix are related to the scaling behaviour of C EV . Since node betweeness is distributed as a power law as mentioned, then Equation 3 implies that C EVB will scale in a similar way in relation to C B , as the row sums in the edge betweeness weighted adjacency matrix are proportional to C B . Indeed, visually inspecting the scatterplots of log-transformed C B and C EVB (Figure 2) demonstrates that for nodes that do have zero C B may still have significant values of C EVB in a similar way as in the artificial example; as for nodes that do not have zero C B , there is a linear or ‘cone’-shaped relation, which provides some evidence of a power scaling relation between C B and C EVB .
Figure 2

The Log/log relation of C B and C EVB . Clockwise from top left : Zachary’s Karate club, Mexican political elite, Astrophysics collaboration and Newmans network science collaboration network. Negligible values were added to C B to allow log of zero values. Both functions are translated to the origin (0,0).

Predicting loyalty in a fission scenario

Let V=S 1S 2 be a disjoint partition of V. Assume w.l.o.g that vS 1. Let X=(x 1,...,x n ) be the leading eigenvector, then the contribution of S 2 to C EVB (v) is:
$$ C_{EVB}^{S_{2}}(v)=\sum\limits_{(v,v_{i})\in E, v_{i}\in S_{2}}C_{EB}((v,v_{i}))x_{i} $$
(4)
By the definition of C EVB as power, Equation 4 describes the proportion of power of v that comes from direct links to the opposing group members. In a social fission situation, it may be the case that members of one group defect to the other as in [13]. Motivated by defection prediction, define the defection score as:
$$ D_{EVB}(v)=C_{EVB}^{S_{2}}(v)-C_{EVB}^{S_{1}}(v) $$
(5)

For a node v, D EVB (v) is simply the difference between the power of v that comes from links to the opposing group and the power that comes from links to its own group. It is hypothesized that a high positive D EVB would mean a higher temptation to defect, while a more negative D EVB would mean a greater tendency to stay put.

Computational complexity

The complexity of computing a composite function as defined here is simply the sum of the complexities of the underlying functions. An algorithm of O(|V||E|) for betweeness is described in [14]. Eigenvector centrality requires only the largest eigenvalue and the corresponding eigenvector. In practice, this is solvable in O(|V|+|E|) using an ARPACK eigenvector solver. Thus, the expected overall time is the same as for edge betweeness. The computational complexity for D EVB (v) is O(|V|) if C EVB (v) is already computed.

Case studies

Several social networks are examined, two ‘friendship’ networks: Zachary’s Karate club ([13]) and a network of the Mexican political elite in the twentieth century. In addition, two larger collaboration networks are studied, a collaboration network of researchers in Network Science (NS) taken from [15] and the collaboration network of preprints on the astrophysics archive at www.arxiv.org, 1995-1999, as compiled by Newman [16]. As can be seen in Table 1, C EVB and C B are more correlated in the friendship networks than the collaboration networks.
Table 1

Spearman rank correlation scores compared for various networks

Network

| V |

| E |

C EVB C B

C EVB C EV

C B C EV

Karate club

34

78

0.762

0.479

0.398

Mexican politicians

35

117

0.703

0.607

0.739

NS collaboration

374

914

0.427

0.504

0.049

Astro-ph

14,845

119,652

0.604

0.786

0.431

A network describing social fission

Zachary’s Karate club is one of the earliest social networks studied as a graph ([13]). The network consists of 34 actors whose common activity is a Karate club, edges are weighted by the level of acquaintance shared by actors beyond the club. The club underwent fission during the period of observation due to a long conflict between the club administrator and the Karate instructor. Zachary’s original analysis was based on network flow and minimal cuts, where the edge weights represented capacity. In this case, the reciprocal of the weights are taken, as the edge weights represent distance and not flow capacity. As seen in Figure 3, C EVB differs from C EV and C B with regard to key players; for instance, actor 34 is reduced in C EVB in comparison to other actors, while actors 3 and 9 have a relative increase in C EVB . By inspecting C EB visually in Figure 3, it indeed seems that actors 3 and 9 are better located within the network regarding proximity to edges with high betweeness. Zachary’s flow analysis managed to model and predict the group affiliation before and after the fissure with near perfection save one case.
Figure 3

The Karate club. Four centralities in Zachary’s network. Colors of nodes (blue or red) designate faction membership, while the larger the node is, the higher the centrality score. For edge betweeness (lower left), thicker edges have higher scores.

Zachary’s original explanation [13] was psychological, based on the temporal circumstance of individual 9:

‘This can be explained by noting that he was only three weeks away from a test for black belt (master status) when the split in the club occurred. Had he joined the officers’ club he would have had to give up his rank and begin again in a new style of karate with a white (beginner’s) belt, since the officers had decided to change the style of karate practiced in their new club’.

Here, an additional factor may be observed based on D EVB scores. D EV is defined in a similar way to D EVB , using the adjacency matrix of G without computing edge betweeness, and as visible in Figure 4, D EV predicts that actor 10 would have the highest incentive to defect while D EVB predicts actor 9 for defection (as indeed took place). In addition, D EVB predicts that actors 1 and 34, which are the leaders of the factions, would have the greatest tendency to stay put while D EV makes no such prediction. Unfortunately, no data exists as to the possible dilemmas of other actors such as nodes 10 and 33.
Figure 4

Loyalty or defection. Predicting the incentive to defect. Larger squares mean a higher positive defection score, while larger circles mean a more negative score, which represents the tendency to stay within the original group.

Collaboration networks

The NS network constructed by Newman ([15]) consists of 1,589 nodes; here, the largest connected component is studied, consisting of 379 nodes. Edges are weighted by collaboration strength as defined in [16], so reciprocals are taken here to represent distance between collaborators. The three centralities are shown in Figure 5, in which the five nodes with the highest scores are identified by their ID (specific names are available in [15] for readers interested). The occasional ‘local’ nature of C EV is apparent in Figure 5, nodes on one sole branch of the network receive the highest scores due to a higher density of edges. In this case, C EV may show isolation rather than power, a group with many links between themselves on an isolated branch. Examining the differences between C EVB and C B , it is clear that actors 51, 52 and 95 score well in both measures, but actor 26 loses power in C EVB , while actor 4 drops to the 19th place in the C EVB scores.
Figure 5

A collaboration network of network scientists. Newman’s NS network. The ID’s of the five nodes with the highest scores in each case are listed in descending order.

As with the Karate club, the reason why actor 4 loses power according to C EVB is clearer by looking at Figure 5, the edges with higher betweeness form a path through the ‘middle’ while actor 4 is located on a lower scored subsidiary of the sub-network of edges with high betweeness. So, although actor 4 has high betweeness as a node centrality, C EVB is reduced due to a lower scoring edge betweeness neighborhood.

A transition of political power

A network of the Mexican political elite was described in [17] and compiled in [18]. The network consists of the core of the political elite and their collaborators across a time period stretching from the early to late twentieth century. The edges of the network are unweighted and represent close ties. During the time period examined, the PRI (Partido Revolucionario Institucional) was continuously in power, however there was an internal struggle between two main groups within the party, politicians associated with the military against ‘civilian’ politicians. During the period, there was a transition of power from the former to the latter. In this context, since the network spans most of the twentieth century, C EV would represent the amount of connections surrounding a politician during the height of his political activity but as already demonstrated in the NS network (Figure 5) that is not necessarily the same as a high C EVB score. On the other hand, C EVB would indicate the proximity of a politician to the relations that are expected to be significant throughout the era; therefore, C EVB is interpreted as political power. The defection scores, in this context, are interpreted as the level of political collaboration with members of the other side. In Figure 6, C EV and C EVB are examined in order to understand if both perspectives can illuminate the power transition purely by examining the network. In Figure 6, it can be seen that node 12 has the highest C EVB and C EV score. Node 12 represents Miguel Alemán Valdés, the 46th Mexican president whose reign marked the transition from military associated power to more ‘civilian’ rule. Interestingly, he also has the highest D EVB and D EV scores, meaning that the most powerful politician in the network (highest C EVB score) collaborated closely with ‘civilian’ politicians, both from a ‘local’ viewpoint (high D EV ) and from a ‘global power’ viewpoint (high D EVB ). Indeed, in 1952, he was succeeded by node 18, Adolfo Ruiz Cortines, a ‘civilian’ politician, which signified the beginning of the new era. A different observation from the D EVB chart is that a high level of collaboration with the opposing side is more expressed than in D EV when family ties are present. For instance, node 34 is Miguel Alemán Velasco who is the son of Miguel Alemán Valdés, and node 14 is Ramón Beteta who was the brother of node 13, major general Ignacio Beteta, a close associate of node 10, the powerful Lázaro Cárdenas, and both 34 and 14 score high on D EVB . To conclude, the point of political power transition is visible in Figure 6, and the idea that family ties may precede group affiliation in political power sharing is visible in D EVB .
Figure 6

An analysis of a network of the Mexican elite in the twentieth century. Viewing the different measures side by side enables to pinpoint the individual whose reign marked the power transition from ‘militarists’ to ‘civilians’.

Conclusions

The composition C EVB was shown to be well defined, and it was shown to differ in several aspects from C EV and C B in case studies. A node defection score based on C EVB and C EV was defined for two-fission situations, and D EVB was shown to better predict the sole defection in Zachary’s study than a similar defection score based purely on C EV . A significance threshold for D EVB could be useful (scores below the threshold would mean no defection) and may be worthwhile of further research. The empirical analysis suggests that C EVB balances the local properties of eigenvector centrality with the global properties of betweeness, giving a different perspective on power distribution. C EVB in combination with C EV and the defection scores were demonstrated to be useful tools in the analysis of the transition and sharing of power in twentieth century Mexican politics. Finally, the possibility of modeling k-fission scenarios (using a more general defection score) is a natural expansion but would need considerable supporting empirical data as to the behaviour of individuals in such situations.

Data accessibility

Newman’s NS network is available at [19], and Zachary’s Karate club data was accessed through [20].

Declarations

Acknowledgments

This paper was produced from research funded from the EPSRC Platform grant awarded to the Space Group at the Bartlett, Faculty of the Built Environment, University College London (grant reference EP/G02619X/1).

Authors’ Affiliations

(1)
The Bartlett, University College London

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© Fidler; licensee Springer. 2015

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