Refinements of pair approximation for network analysis

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When considering interactions on networks, it is usually very hard to calculate the dynamics analytically, and approximations are employed. Mean-field approximations usually end up ignoring the network structure completely, and so are seldom a good approximation. A popular approximation is the pair approximation, which considers the correlations inherent between adjacent nodes (intuitively we can think of it as a type of mean-field approximation on edges).

The approximation is exact if we are considering Cayley graphs, and very good if we are looking at $k$-regular random graphs. In practice it also provides good approximations for cases when we have a random graph with average degree $k$ and a tight distribution of degree around $k$. Unfortunately, a lot of the networks and interactions that are of interest, are not well modeled by these sort of graphs. They are usually well-modeled by graphs with very different degree distributions (like scale-free networks, for instance), with specific (and high) clustering coefficients, or specific average shortest-path distance (for more, see Albert & Barabasi 2001). Are there refinements of pair approximation that work well for these types of networks? Or are there other analytic approximations available?

An example of interactions on networks

I thought I would give an example of what I mean by interactions on networks. I will include a relatively general example from evolutionary game theory.

You can think of each node as an agent (usually represented just by a strategy), that plays some fixed game pairwise with each other agent it has an edge to. Thus, a given network with a some assignment of strategy to each node produces a payoff for each node. We then use these payoffs and the network structure to determine the distribution of strategies among the nodes for the next iteration (A common example might be for each agent to copy the neighbor with the highest payoff, or some probabilistic variant of this). The questions we are usually interested in corresponding to knowing the numbers of agents of each strategy and how that changes overtime. Often we have stable distribution (which we then want to know, or approximate) or sometimes limit-cycles or even more exotic beasts.

If we do mean-field approximation on this sort of model, we use get the replicator equation as our dynamic, which blatantly ignores the network structure and is only accurate for complete graphs. If we use pair approximation (as Ohtsuki & Nowak 2006) we will get slightly different dynamics (it will actually be replicator dynamics with a modified payoff matrix, where the modification depends on the degree of the graph, and the specifics of the update step) which matches simulation well for random graphs, but not for other networks of interest.

If the game-theoretic example is too non-physicsy, then replace the agents by spins and the call the payoff matrix an interaction Hamiltonian, then cool your system while performing periodic random measurements, and you will get another example.

Note

Straightforward generalizations of pair approximation of the sort that consider a type of mean-field approximation on triples, or quadruples of nodes) are unwieldy and still don't take into account very different degree distributions or average shortest-path distance.

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retagged Apr 19, 2014
Could you clarify what you need the approximation for? I.e. in which properties of the network are you interested?

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@Piotr I am interested in tools that can be used for graphs with various degree distributions (but at least scale-free) and where the analysis explicitly takes into account clustering coefficient and average shortest-path distance between nodes. In particular, it is desired for the tool to depend on those parameters (most pair approximation only depends on average degree, and sometimes standard error of the degree-spread for tight distributions).

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@Artem: One method is to calculate graph spectrum (i.e. [spectrum of its Laplace matrix](http://mathworld.wolfram.com/GraphSpectrum.html)). The spectrum is related to the degree distribution, but also depends on clustering and (I guess) average shortest-path distance between nodes.

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@Artem: I'm not entirely clear on what you want to be able to calculate/approximate. Obviously any approximation will fail to accurately represent all aspects of the graph, so it is important to know what functions of the graph you care about. There are lots of CMP methods that can be brought to bare, but you can always construct a property for which they will fail.

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@JoeFitzsimons I see where I am missing the mark. I have taken for granted that people would guess what I mean by dynamics. I will make a quick note about the type of dynamics I am interested in.

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@Artem: Do you mean something like the tight-binding model? The dynamics are only defined once you have a Hamiltonian or probabilistic transition for the system, and without defining what that means, its hard to see how you'll get a meaningful answer.

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@JoeFitzsimons I gave an example. The goal isn't to solve a specific system, but to find out if there are better tools (than just pair approximation or mean-field sort of things) that would work in some cases. Unfortunately, the more I define the problem, the less physics-y it becomes, becomes my actual motivation comes from outside physics. However, most of the best approximation come directly from physics (such as pair approximation) so there is definitely theoretic methods that can be borrowed. Hopefully the example I gave will make the question easier to understand.

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@JoeFitzsimons I haven't had a chance to really immerse myself in TP.SE yet, so I might be a little off-keel on making my questions easy to understand. Any help is appreciated.

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@Artem: Don't be afraid to give an explicit example, even if it is outside of physics.

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