An entropy-based framework to analyze structural power and power alliances in social networks

Power is a central phenomenon in societies. So for ages, numerous power perceptions in philosophy and sociology have existed. Measuring power of an actor in its social fabric is a difficult issue, however. After sketching first attempts for this in social network analyses, we develop a new power theory. To this end, we distinguish between vertices in the network and actors acting in vertices. Vertices get structural power potential from their position in the net. In an entropy-driven model such potential can be calculated for all vertices; for selected networks, the method is exemplified. Actors in vertices can deploy power potential once they have respective personal skills, and dominate actors in adjacent vertices. If chosen with suitable care, an alliance of actors can even dominate the whole network. The findings are applied to the famous 9/11-network with 34 vertices and 93 edges.

This distinction then adds up to different forms of social power: expert power, information power, power by pressure, power by reward.
In all aspects presented so far there was little attempt to measure power. Jakob Moreno in 1925 emigrated from Vienna to the US and wrote his pioneering article 6 "Who shall survive: a new approach to the problem of human interrelation". For the first time sociological relations between actors were illustrated by graphs. Further research of sociologists made graphs a successful tool to measure structural characteristics of the social fabric, like centrality, closeness, betweenness, etc. But only in the 1960s did the very question come up of how to measure power. power in social networks. Social Networks (SN) are sets of actors and their manifold relations. Graphs, hypergraphs and multigraphs are modern tools to illustrate such networks. A first introduction we find in the textbooks of Jansen 7 or Scott 8 ; the reader interested in more sophisticated mathematical models might tend to study the compendium of Newman 9 . Importance, prestige, reputation or roles of actors can be analysed in Mathematical model. Probabilistic conditionals and structural power. Rules 1. to 4. of "Narrative motivation" section result in the following mathematical framework. Consider an undirected graph with vertices V = {V } , |V| = n , and corresponding edges. Each vertex V i ∈ V is a boolean variable V i = 1 or V i = 0 . The semantics reads: For V i = 1 the vertex houses an actor with full personal power, for V i = 0 the actor is powerless. v = (V 1 = 0/1, V 2 = 0/1, . . . , V n = 0/1) are repsective 2 n configurations. On {v} we install probability distributions Q . They are the medium conveying power relations in the net. From all possible Q we choose the ones which obey probabilistic conditionals Q(V j = 0 | V i = 1) = 1. for all adjacent vertices V i , V j . | is the wellestablished conditional operator. The semantics of such a conditional reads: A conditional is of if-then-type; it does not imply facts. Postulating such conditionals for all adjacent vertices and in either direction illuminates the whole net's possible power patterns. The next section gives further details.
If an actor in V i had full personal power (V i = 1) and if it were able to exert this power fully on an actor in V j (1.), then the actor in V j would be absolutely powerless (V j = 0).  20 . For a more intuitive introduction also cf. Rödder et al. 21 . Q and all probabilistic structure therein is inferred from the given conditionals. This inference process is an established concept in artificial intelligence, see the fundamental work "Recall and Reasoning-an information theoretical model of cognitive processes" 22 . The restrictions are probabilistic conditionals. Q then is the distribution with maximal entropy among all Q feasible in (1). H(Q ) is the remaining uncertainty about (conditional) structural power relations in the net: If all vertices are isolated, i.e. for an empty set of restrictions, it counts − log 2 1/2 n = n . If only one configuration is feasible in (1), H vanishes; only one structural power pattern is left.
The probabilities Q(V i = 1) , for i = 1, . . . , n , allow for the calculation of all vertices' structural power. It is well known that − log 2 Q(V i = 1) is the information a system receives when V i = 1 becomes true. Any textbook on information theory relates to that 23,24 . In our context, this information gain is realized when an actor exerts its full personal power in the vertex and makes the probability Q(V i = 1) to 1. The information gain measures change of (conditional) power relations in the net 19 and our observations in "Deployment of structural power and dominance" section. The higher the change potential of a vertex, the more influence an actor would have in the net. This is a good reason for the following definition.
For a three-vertex-path network we exemplify. Figure 1 shows a three-vertex-path with undirected edges.

Example 1
Corresponding restrictions in Eq. (1) read Conditionals in parentheses are redundant as they follow from the left ones. If any vertex houses a powerful actor (V = 1) , and if this actor fully dominates the adjacent vertices' actors, then these are powerless (V = 0) ; see also our narrative explanations in the previous section. Structural power of nodes V 1 , V 2 , V 3 counts sp 1 = −log 2 1/5 = 2.322 , sp 2 = −log 2 2/5 = 1.322 , sp 3 = −log 2 2/5 = 1.322 . The results confirm our intuition: V 1 has greatest structural power, V 2 and V 3 are next. ⋄ The following section presents structural power for a set of selected networks.
Structural power in selected networks. For all nets from Figs. 2 and 3, we now determine structural power for all vertices and compare the results with those of other methods. Figure 2a, b name and visualize the nets, Table 2 gives all results. The leading column indicates nets, the headline vertices, the entries in the matrix are sp-values and rankings. To solve (1) for all nets, we use the optimization software SPIRIT 25 . After presenting the data, results of the new method are compared with those of Cook et al. 13 , Easley and Kleinberg 18 , as well as Bonacich 14 .
The nets 1, 2, 6 are complete graphs whose vertices have equal structural power, see Table 2 Scientific RepoRtS | (2020) 10:10697 | https://doi.org/10.1038/s41598-020-67542-0 www.nature.com/scientificreports/ type vertex-path with 3, 4 and 5 vertices. In the first net, V 1 has highest structural power, in the second one V 2 , V 3 are best and in the five-vertex-path V 2 , V 3 outplay V 1 , and V 4 , V 5 are last. Easley and Kleinberg 18 confirm these results on page 345 and so do Cook et al. 13 on p. 287 ff. We note that in the five-vertex-path centrality and power definitely differ. Power in the nets 10, 11, 13 was determined by simulation instead of laboratory experiments. For nets 11, 13 the sp-method shows matchable results, not so for net 10. Here the results of the new method match those of Bonachic but not those of computer simulations. For nets 7 and 8, also Easley and Kleinberg 18 confirm our results. Net no. 13-the locomotive-impressively highlights the difference between centrality and power. V 3 has C D = 4 , C B = 12 , C C = 0.1 and hence is "pretty central". Its power sp 3 = 2.43 is significantly smaller than that of vertex V 4 , however, and even than that of V 6 , cf. numbers and ranking in Table 2.
The consistency between results in experimental exchange nets and the sp-method only at a first glance is surprising. Exchange networks determine power by disposable force of transactions upon actors whereas the sp-method focuses on suppression as the driving force of power. Apparently, the vehicle "exchange" very consistently detects power structures in networks, but unfortunately is restricted to very small nets.
The sp-method measures structural power in vertices, but how can an actor deploy this power? The next section gives the answer.

Deployment of structural power and dominance
Deployment of structural power. Once power of vertices is calculated, all classical methods sketched so far end in these results. Not so for the new sp-method. Because of the separation of vertices and actors, housed in vertices, the analysis can and must proceed: What happens when an actor deploys the structural power of a vertex? And if it does, how does this deployment alter the network? How does it alter the remaining structural power in the vertices?
• Increasing the probability Q(V i 0 = 1) to 1. means deployment of structural power in vertex V i 0 and its exertion on actors in adjacent vertices.  To realize this deployment, solve The following example shows respective results for the locomotive network. Corresponding Eqs. (1) and (2) Table 2. sp-method, structural power and rankings.

Definition 2 A set of vertices achieving all bullet points is called a minimal power alliance.
To find a good power alliance, we could proceed as follows: 1. Find a vertex with maximal sp. 2. Deploy structural power in such a vertex.
The following algorithm details steps 1. to 4. Determining minimal power alliances is equivalent to solving the so-called min#MIS problem in graph theory 26 . Here, min#MIS means minimal cardinality Maximal Independent Set. For such problems, classical optimization software is disposable, e.g. MATLAB or GAMS. Equation (4) shows the respective binary optimization problem.
The ã ij are entries of the adjacency matrix complemented by 1s in the diagonal. For an optimal solution to (4), a minimal power alliance then reads: If x i = 1 , make V i a dominant vertex and non-dominant, otherwise.
In the next section, we apply Algorithm 1 and (4) to selected networks.
(4) min n j=1 x j s.t.ã ij x i +ã ij x j ≦ 1 ∀ i � = j and adjacent n j=1ã ij x j ≧ 1 ∀ i x i ∈ {0, 1} ∀ i. www.nature.com/scientificreports/ Power alliances in selected networks. First, we study the undirected graph of the terrorism network as presented by Latora and Marchiori 27 . It counts 34 vertices and 93 edges. The edges represent relations between actors like "who lived with whom", "which hijackers ordered tickets at the same time", "who had joint flight training with whom", etc. Even if these relations are pretty inhomogenous, we follow earlier network analyses and consider respective edges as equal value. The network is shown in Fig. 8. Solving (1) for this 9/11-network results in structural power indices sp as in Table 3. Furthermore, the table shows centralities C D , C C , C B and rankings of all indices. Vertex V 1 is most central and has maximal structural power. Vertex V 2 has rank 7 for C D , rank 4 for C C and rank 6 for C B ; only sp-ranking is a poor 14. Further inspection of Table 3 indicates very clearly the difference between centrality and power. The names of terrorists in vertices are given in Table 4.
To vertex V 1 Mohammed Atta is assigned. Very likely, he was the head of all crash pilots. Power and centrality coincide. The actor in V 2 was Salem Alhazmi. Salem Alhazmi was subordinate to pilot Hani Hanjour 28 . However, his closeness to Hani Hanjour gives him a high centrality but by no means a high power, namely rank 14, see above. While classical rankings only take into account the graphical structure of vertices and edges, sp does something more. It perceives or feels an actor's powerlessness even when this actor is central in the social fabric.
For all networks from Table 2, the results of Algorithm 1 and (4) also coincide, except for network 10. Hence, Algorithm 1 not always yields optimality, but has the advantage of transparency: In the 9/11-example, the first actor to be selected is in V 1 , then the next in V 6 , V 16 , V 26 , V 28 , V 30 , V 34 , in this order. Knowledge about the importance of vertices in the net allows for a competent assignment of actors with personal skills. Hopefully, this eureka moment is present in any (non-)governmental organization. www.nature.com/scientificreports/ Structural power is a new concept in power theory, detached from any costly laboratory experiments. Identifying alliances then is a natural continuation of this concept. All these findings can be realized even for networks comprising umpteen vertices.

Resumé and the road ahead
Power is an omnipresent phenomenon in human societies and an ongoing concern for sociologists, politicians and economists. In this paper, we first give a short overview of power perceptions in history. Sociologists very early analyzed power relations and detected their central determinants: instruments of power, fiefdom, power resources, costs of power, etc. A general method to measure power was missing for a long time. Only from the 1960s attempts were made to fill this gap: It was the birth of exchange networks. An actor is powerful when it has many alternatives of action to negotiate with others.
In this paper, an abstract concept of measuring power in networks-beyond the exchange idea-is developed. The position of a vertex is the only determinant of its structural power. To realize this concept, we use a probabilistic-conditional framework. Elementary postulations concerning power relations lead to a mathematical optimization problem allowing for the calculation of all vertices' structural power. The findings are applied to numerous selected networks. Furthermore, we separate actors from vertices. How an actor housed in a vertex exerts influence on other actors is the next step of our research.
To find dominating alliances of actors in networks is a further topic of this paper. For the famous 9/11-network, we determine such alliance and analyze respective results.
There are open questions left for further research: • Can an actor housed in a vertex always fully deploy the vertex's structural power? And what if it cannot? Is the new method able to treat partial deployment? • Can positive and negative relations among actors be modeled in our probabilistic framwork? And if so, how to check for consistency in the net; is this consistency equivalent to Harary et al.'s balance structure 29 in networks? What about structural power of vertices in such signed networks?
These are promising issues for further research.