Abstract
Decompositions of networks are useful not only for structural exploration. They also have implications and use in analysis and computational solution of processes (such as the Ising model, percolation, SIR model) running on a given network. Tree and branch decompositions considered here directly represent network structure as trees for recursive computation of network properties. Unlike coarse-graining approximations in terms of community structure or metapopulations, tree decompositions of sufficiently small width allow for exact results on equilibrium processes. Here we use simulated annealing to find tree decompositions of narrow width for a set of medium-size empirical networks. Rather than optimizing tree decompositions directly, we employ a search space constituted by so-called elimination orders being permutations on the network’s node set. For each in a database of empirical networks with up to 1000 edges, we find a tree decomposition of low width.
Highlights
The analysis and modelling of complex systems involves complex computational tasks
As a first step in exploring elimination orders by simulated annealing, we compare the effects of the choice of cost function, see section 2.4
We find tree decompositions with w2 < 10 and w∞ 13, except for the networks College football with w2 ≈ 30.6 and David Copperfield (w2 ≈ 23.9)
Summary
The analysis and modelling of complex systems involves complex computational tasks. In comparing empirical network structures with network models, for instance, one asks if they share common macroscopic behaviour under a percolation process or spin kinetics. This makes exact computation feasible as long as there is a tree decomposition of sufficiently low width k This suggests a two-step process as a general modus operandi for solving a computational problem on a given network. Elimination order Finding a tree decomposition of low width for a given graph is a hard problem in itself. An elimination order π for a graph G = (V, E) gives rise to a tree decomposition (B, T) of G in the following way: the set of bags is B = {Bi : i ∈ {1, . Some model networks grow incrementally by attaching a new node to an existing clique of given size m [32,33,34] In this case considering the nodes in reverse order of addition yields a perfect elimination order whose application does not involve edge addition. Note that wη has the tree-width w∞ as a limit η → ∞ if there is a unique maximum bag size
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