I'm struggling with calculating the expectation of this event. Say we're studying a graph where each nodes has an associated random variable $X_v$, representing the time it takes for node $v$ to "fail". When a node fails, it disappears from the graph. We're interested in failure events that disconnect the graph.
We look at the example in the attached graph, which is the wheel graph on 6 graph. This graph is 3-vertex-connected, which means we need at least 3 nodes to fail to increase its number of connected components. The failure events involving exactly 3 vertices in this particular graph are exactly: [0, 2, 5], [0, 3, 5], [1, 3, 5], [1, 4, 5], and [2, 4, 5]. Assuming we know the distributions $X_v$ for each vertex of the graph, how would we go about calculating the distribution/expectation of the failure time for the graph (i.e. the time it takes for the graph to get disconnected? Mathematically, you could phrase the failure event here as $$min \{max\{X_0, X_2, X_5\}, max\{X_0, X_3, X_5\}, max\{X_1, X_3, X_5\}, max\{X_1, X_4, X_5\}, max\{X_2, X_4, X_5\} \}$$, but the events that make up this larger event are pretty dependent on one another. More generally, how do we account for all possible failure events?