(466d) The Construction of Networks with Prescribed Collective Properties

Gounaris, C. E. - Presenter, Princeton University
Rajendran, K. - Presenter, Princeton University
Floudas, C. A. - Presenter, Princeton University
Kevrekidis, I. G. - Presenter, Princeton University

Porous media exhibit network topologies and attempts to study such materials have naturally focused on this structural feature [1,2]. Network theory has also been used extensively to model systems in other fields ranging from social sciences to biology [3-5]. Our understanding of the dynamic behavior of these systems relies on predicting the evolution of network topology [6-8]. A promising approach to developing models for network evolution relies on using the equation-free framework [9], which focuses on suitable coarse observables of the system and then identifies an appropriate way to lift, or transform, from the coarse description to the full description of the system. For certain problems with simple dynamical evolution, using the degree distribution and the mean clustering coefficient as the coarse variables successfully reduces the system dynamics. However, for other dynamical evolution problems like those of social networks [10], we find that relying merely on these metrics does not convey enough information about the state of the system. In such cases, one has to resort to using more detailed descriptions of the network, such as the degree-dependent clustering coefficient (or, equivalently, degree-dependent number of triangles), to be included in the set of coarse variables and, to do so, knowledge of sample graphs that possess given degree-dependent clustering is required. The necessity for constructing a network, or graph, with prescribed collective properties presents us with an interesting combinatorial problem. So far, methods for constructing such graphs [11-14] have focused on the randomness of their results and the adherence to certain distributions of interest, with no explicit guarantees of achieving the prescribed properties exactly or within bounded deviations. Moreover, none of the existing methods addresses the requirement for certain degree-dependent number of triangles. To that end, we have developed a novel approach based on a rigorous mathematical programming formulation that can construct graphs that are guaranteed to possess the desired properties. Properties supported by the formulation include the degree distribution, the overall clustering coefficient (global or mean), and the degree-dependent number of triangles. The method uses well-established Mixed-Integer Linear Programming (MILP) solution techniques [15,16] and is capable of identifying one, multiple or even all ?when reasonable to do so? networks that possess the desired properties. Additional properties, such as the similarity to a given network, can also be accommodated by the formulation. In cases when there is no guarantee that the prescription is feasible, it typically suffices to look for solutions that satisfy the desired within appropriate upper and lower bounds. One of the major challenges of a pure Mathematical Programming approach is the problem's large combinatorial complexity, which makes instances with even a small number of nodes to be intractable for off-the-shelf commercial optimization software. In order to address such a challenge, we have devised an iterative preprocessing step to rank-order the set of possible arcs in terms of their likelihood to participate in feasible solutions. Then, in a subsequent iterative scheme, we take decisions on which of those arcs we could a-priori postulate as part of the solution. Postulating the existence of certain arcs, that is, postulating a sub-graph to be part of our final solution, reduces the size of the MILP formulation and brings the remaining problem ?which is of significantly smaller complexity than the original? within the capabilities of state-of-the-art optimization software. The overall scheme inherits the exactness of the MILP solution algorithm and can thus guarantee the identification of a solution, when one exists, in all cases. Computational experiments demonstrate the effectiveness of the approach and its suitability to serve the various applications that require the knowledge of networks with given properties.

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