Power and loyalty defined by proximity to influential relations
- Dror Fidler^{1}Email author
DOI: 10.1186/s40649-014-0009-9
© Fidler; licensee Springer. 2015
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 FissionIntroduction
Networks are often modeled as a graph, which consists of set of nodes (V) and edges (E), such that E⊆V×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 _{1}…x _{ 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 e∈E, \(C_{\textit {EB}}(e)=\sum _{s\in v}\sum _{t\in v}\frac {\sigma _{s,t}(e)}{\sigma _{s,t}}\). 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
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 e∈E, 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,v∈V, 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 e∈P _{ 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
Eigenvectors in a weighted graph and scaling behaviour
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.
Predicting loyalty in a fission scenario
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
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 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’.
Collaboration networks
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
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
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
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Copyright
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