Technical Comments

Response to Comment on “Control profiles of complex networks”

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Science  31 Oct 2014:
Vol. 346, Issue 6209, pp. 561
DOI: 10.1126/science.1256714

Abstract

Campbell, Shea, and Albert propose an adaptation of the Barabási-Albert model of network formation that permits a level of tuning of the control profiles of these networks. We point out some limitations and generalizations of this method as well as highlight opportunities for future work to refine formation mechanisms to provide control profile tuning in synthetic networks.

In our recent work, we document that current synthetic network models largely lack the ability to customize, or tune, their control profiles to any arbitrary set of values (1). Campbell, Shea, and Albert propose an adaptation of the Barabási-Albert (BA) model that provides some capacity to tune the control profile of generated networks by reversing the conventional orientation of the edges with probability (1 − p), 0 ≤ p ≤1. (2).

We find it compelling that the method can produce networks with a spectrum of control profiles that largely keeps to the edges of the ternary plot—that is, exhibiting control profiles dominated by one or two types of controlled nodes (i.e., source nodes, external dilation points, and internal dilation points). This roughly tracks the control profile distributions observed in our analysis of 70 real-world networks.

At the same time, the parameter p, since it is a single parameter, necessarily parameterizes only a curve through (ηs, ηe, ηi) space–the space of control profiles, where ηs, ηe, ηi are the respective fractions of controlled source nodes, external dilation points, and internal dilation points. Choosing a value of p maps to a point along this curve and, therefore, a control profile. In addition, it appears that the relationship between the parameter p and the control profile is highly sensitive to the value of the other parameters of the system (see Fig. 1). Ideally, a more general model would incorporate at least two parameters so that complete tuning could be achieved—i.e., accessibility to any control profile by parameterizing the entire surface in (ηs, ηe, ηi) given by ηs + ηe + ηi = 1—and so that these tuning parameters would be as independent as possible from the values of other parameters. Notably, the independence criteria is more than a theoretical concern: From a practical perspective, a control profile tuning parameter that depends on other parameter choices will prove difficult to use without extensive experimentation.

Fig. 1 The control profiles of the modified Barabási-Albert model demonstrating the dependence of control profile tuning on other network parameters.

Each plot shows 1000 network instances with N = 10,000 nodes; m = 3, 6, 9, and 12 edges to be connected to each new node; and p sampled randomly from [0,1]. This larger network size was chosen to ensure that measurements are consistent with the network structures present at steady state.

On a more conceptual level, modified models that incorporate control profile tuning should retain the formation mechanisms that inspired the original model. One concerning aspect of the proposed formation mechanism is that, by randomly selecting an edge orientation in the BA model, some of the original motivation behind this model is less clear (3). The directed BA model is typically described by adding directed edges that point from a new node to an existing node. This is done to model a process in which new nodes seek connections to existing nodes with relatively high authority or popularity. The reverse orientation interpretation (new nodes connect with edges pointing from the existing node to the new node) can be used to indicate the direction that information or advice might flow under such a regime. A mixed orientation model, however, departs from this intuition because new nodes establish connections to existing nodes in different capacities. An argument is needed for how the modified model should be considered to be a preferential attachment model of the same kind as the original.

An attractive aspect of the modification suggested by Campbell, Shea, and Albert is that it generalizes to many current models that introduce nodes incrementally, although they all have the same restrictions as noted above. For example, local attachment networks admit a very similar curve if each new edge is added from the selected existing node to the new node with probability p and added in the opposite orientation with probability (1 – p) (see Fig. 2) (4).

Fig. 2 The control profiles of the modified local attachment model showing 1000 network instances with N = 1000, m = 3, mr = 2 (the number of attachments made randomly), and p sampled randomly from [0,1].

Campbell, Shea, and Albert have provided a viable extension to incorporate some tuning of the control profiles of Barabási-Albert networks. An important purpose of synthetic network models is to identify the fundamental aspects of network formation that lead to properties of interest. We anticipate that further extensions along these lines, such as the incorporation of additional parameters for complete tuning that are independent of other model parameters, will continue to explore and clarify the effect that network formation mechanisms have on the control properties of complex networks.

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