3.2 Methods and Basic Results
3.2.1 Equilibrium Analysis
As throughout the other chapters, one can use analytical means (“paper and pencil”) to classify networks into stable and unstable according to the notion of pairwise stability.
25
For the conciseness of the model, the function is not defined explicitly. 26
Instead of setting the slopes (λ and 1− λ) in relation to each other, we could also have defined them independently. Both notations allow us to represent any linear preferences and there is no difference when examining stability and efficiency. The relative notation is advantageous for comparative statics, because c then measures the costs in comparison to one unit of benefit.
Centrality 36
Analytical results on stability typically need the maximal incentive of any agent to sever a link and the maximal incentive of any two agents to add a link and compare them to linking costs c. Because benefits are based only on closeness and betweenness, the crucial aspects for a focal agent i are the change in distances (P
j∈Ndij(g) ˆ= non-normalized closeness) and the change
in the number of shortest paths he is on, which will be called “brokerage” (P
j<k(j6=i,k6=i) τi
jk(g)
τjk(g) =ˆ
non-normalized betweenness). Specifically, if a new link for some agent i in some network g means a decrease in distances of x and an increase in brokerage of y, then he is willing to form the link only if c ≤ (M −1)(n−1)(1−λ)[x] +
λ2[y]
(n−1)(n−2). As players compare marginal costs with marginal
benefits. Although deriving the changes in distances and brokerage for a given situation might be tedious, it is a straightforward task.
Let us first have a look at some prominent networks. The following Prop. 3.1 presents the parameter combinations for which five prominent network structures are pairwise stable.
Proposition 3.1. In the centrality model the following holds: (i) The complete network gN is stable if and only if c≤ 1−λ
(n−1)(M −1).
(ii) The empty network g∅ is stable if and only if c≥ 1−λ n−1.
(iii) A star network g? is stable if and only if 1−λ
(n−1)(M −1) ≤ c ≤ min{1+λn−1;
(1−λ)[M (n−1)−2n+3] (n−1)(M −1) }.
(iv) Let n be a multiple of 4. Then a circle network g is stable if and only if (1−λ)[18n2− 1 2n+1] (M −1)(n−1) + 2λ[18n2−3 4n+1] (n−1)(n−2) ≤ c ≤ (1−λ)[14n2−1 2n] (M −1)(n−1) + 2λ[18n2−1 2n+ 1 2] (n−1)(n−2) .
(v) A complete bipartite network gl:r with 2 ≤ r ≤ l (where l and r are the sizes of the two
groups) is pairwise stable if and only if 1−λ
(n−1)(M −1) ≤ c ≤
2(1−λ) (n−1)(M −1) +
2λ[r−1l ] (n−1)(n−2).
All proofs of this chapter can be found in Section 3.5. The first implication of Prop. 3.1 is that non-existence of stable networks is not an issue in this model.
Proposition 3.2. In the centrality model for any parameters (λ, c)∈ [0, 1] × R+ there exists
at least one stable network.
As for the model in Section 2.3, at least one of the trivial networks (empty network, complete network or the star) is stable. Figure 4 depicts the parameter space with weight λ on the horizontal axis and marginal cost on the vertical axis. It illustrates (among other results) the “regions” of the parameter space where the complete network, the star network, the balanced
complete bipartite network and the circle network are stable.27
Not illustrated is the region where the empty network is stable.
Closeness 0.5 Betweenness
c c
empty network unique
circle network
complete bipartite networks star network and trees
complete network unique
Figure 4: “Parameter map” with stability for some prominent networks.
It is intuitive that for sufficiently low c, the complete graph is stable, as long as there are some incentives for closeness. Above the upper bound for the stability of the complete network, complete bipartite networks can be stable. Among them is the star network that is stable for quite a range of the parameter space, but not for λ = 1. The figure indicates that the balanced complete bipartite network and the circle network, both can be stable for any weight λ. While the circle networks are high cost phenomena, the complete bipartite networks are low cost phenomena.
Some aspects deserve additional attention. The boundary between stability of the complete network and the complete bipartite networks marks a special border. At this border agents are indifferent between keeping and removing a link with the minimal possible benefits – that is a link that only serves to reducing the distance to one other agent by the amount of one,
27
Results look different for small network size and slightly different for networks with an odd number of players. For example, Figure 6 and Figure 7 qualitatively present how these regions look like for n = 8 and for n = 14.
Centrality 38
while it does not provide any brokerage. As a consequence, below this border the complete network is uniquely stable. A similar observation can be made for the upper boundary of the star network. This is the highest cost level, such that a network with loose ends can be stable. Finally, for high enough c, the empty network must be uniquely stable.
Proposition 3.3. In the centrality model the following holds:
(i) The complete network gN is uniquely stable if and only if c < 1−λ (n−1)(M −1).
(ii) If c > min{n−11+λ;(1−λ)[M (n−1)−2n+3](n−1)(M −1) }, any network with pendants (players at loose ends) is unstable.
(iii) For large enough c, the empty network is uniquely stable.
So, the first set of equilibrium analysis results structures the parameter space. Below the first frontier, the complete network is uniquely stable. As a second frontier one can consider the upper bound for the stability of the balanced complete bipartite network. It can be shown that any complete bipartite network with at least two agents in each group can only be stable below this frontier.28
The intuition for this result is simple: If a complete bipartite network is not balanced, there is one group of agents with less beneficial ties than in the balanced network, such that high costs c lead to a deletion of ties. The third frontier has a “roof”-like shape and restricts stability of networks with loose ends. Among them are all minimal networks (that are non-empty). Consequently, trees can only be stable in the parameter region where the star network is stable.
The last frontier is the lower boundary for the uniqueness of the empty network. Prop. 3.3 shows existence of such a boundary without answering where it is. We depicted it at the upper boundary for stability of the circle network. Our conjecture that this is the maximal cost level where networks with non-critical links can be stable.29
If this conjecture is true, it follows that above the third and the fourth frontier, neither a network with non-critical links nor a network with loose ends can be stable. Thus, only the empty network remains. Note that the empty
28
This statement includes complete bipartite networks (with more than one player in both groups) with isolates, which will be discussed in Subsection 3.3.2.
29
We argue that among all networks that contain non-critical links, the circle network has the highest marginal benefit for any of those (non-critical links). Define for ij∈ g the marginal benefit of agent i as βiij(g) := (1− λ)CLOSEi(g)+λBET Wi(g)−[(1−λ)CLOSEi(g\ij)+λBET Wi(g\ij)]. And let T be the maximal marginal benefit that a non-critical link can mean to both its owners, that is T := maxg∈ ˜Gmin{βiij(g), β
ij
j (g)}, where ˜
G:={g ∈ G : 1 < dij(g) < M}. The problem is to find the argmax of this expression. Our conjecture is that T = βiij(g ). For n odd this is (1− λ)
n−1 4(M−1) + λ
n2
−4n+3
4(n−1)(n−2). For n even the threshold is slightly smaller. If the conjecture holds, then for c > ub(g ) in any network with circles (circles consist of non-critical links) at least one agent is willing to cut a link. Thus ub(g ) is the maximal cost level where any network with circles can be stable.
graph is stable “far below” this frontier,30
and trivially stable if the importance of closeness is sufficiently low.
The equilibrium analysis is not only used to analyze for which parameter settings a particular network is stable, but also to characterize the stable networks by properties they must or must not satisfy. What is not possible by “paper and pencil”, however, is to find all pairwise stable networks.