One could check social network and political science literature for tipping point political influence models of this type of minority influence. Besides Mark Kibanov's cite to Michael Kearns, do any other articles specifically directed to social network modeling of this type of political situation come to mind to SOCNET participants?
In Granovetter's well-known (1978) "Threshold Models of Collective Behavior" AJS 83(6): 1420 an influence network may need a very considerable "push" to tip. He refers to voting as one area of application. Thus, in a very stable network, the network doesn't tip as long as some minority are unswayed. [albeit oversimplifying]
For a similar example, let Di signify a +1/-1 decision by a House member indexed " i " to express support (and vote) for a "clean" (Di = +1) continuing resolution (CR) or express support (and vote) for a "non-clean" CR (Di= -1). Di =0 represents as-yet undecided or abstention. Collectively, the decisions Di form a vector.
The choice of signs is arbitrary, and the decisions D can be indexed independently on an index "i" or an index "j".
In such a tipping point model, Di = SIGN(Ui + Wij Dj). [Wij Dj signifies sum over the repeated index j.]
Ui is a vector of plus or minus numbers each indicating a combination of constituent pressure, pressure group pressure and motivation by each given House member to make a decision Di = +1 or Di= -1 respectively. Wij is an influence weight matrix of plus, minus or zero numbers each representing influence of House members on each other.
Even if the House Speaker or other House leadership persons have relatively high network out-degree, their behavior and decisions are subject to influence by the members too. Also, non-leader House members who might want to support a clean CR, but face a risk of primary challenge in their districts if they do, may have a net motivation to support a "non-clean" CR instead. [An example of such net motivation is: U25 (motivation of 25th Republican House member in a numbered list of House members) = +2 (personal preference) -3 (risk of primary challenge) results in a total U25= -1 (net motivation to support a "non-clean" CR).]
To the extent that House members might cluster in their numerical situations from a modeling point of view, the model could probably be boiled down from hundreds of agents down to perhaps ten or fewer groups of them as nodes, while retaining the form of the model. Note also, of course, that the initial state of the network matters--passing a CR of either type takes actual action by the House, unlike mandatory expenditures that go along on "autopilot" unless action is taken to terminate them.
One can specify what "fear" is going to correspond to, numerically. For example, one way would define a kind of agent-specific fear, tension or dissonance "Fear1" as the personal preference magnitude when total Ui +SumWijDj has a sign opposite to the personal preference, else zero.
A tipping point model with the right parameters can describe a network in which a minority can control a majority. We see this in repressive regime situations around the world. The methodological footwork to parameterize such a model is probably comparable in difficulty to challenges that political scientists deal with already in their other studies. One could consider more elaborate versions of this type of model, but would need to justify departing from parsimony.
(Texas Instruments, retired)
Little Rock, Arkansas