Moment-closure approximations for discrete adaptive networks
Introduction
Complex networks have been ubiquitously used to model problems from various disciplines [1], [2], [3], [4], [5], [6]. Treating a complex system as a network, a set of discrete nodes and links, leads to a conceptual simplification that often allows subsequent analytical insight that provides a deep understanding.
For many questions the networks of interest are not static entities but change in time due to the dynamics of and on the network. In the dynamics of networks, the network itself is regarded as a dynamical system. Prominent examples are network growth models leading to specific topologies such as scale-free [7] and small-world networks [8]. The dynamics on networks concerns dynamical processes such as epidemic spreading [9] that occur on a given fixed network, where each node carries a state which evolves through interactions with its neighbors.
If the dynamics on and of networks occur simultaneously and interdependently then the network topology coevolves with the states of the nodes and an adaptive network is formed [10], [11]. Adaptive networks have been used to model problems of opinion formation [12], [13], [14], [15], [16], [17], [18], [19], epidemic spreading [20], [21], [22], [23], [24], [25], [26], [27], [28], evolution of cooperation [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], synchronization [42], [43], [44], [45], [46], neuronal activity [47], [48], [49], [50], [51], [52], [53], [54], [55], collective motion [56], [57], cartelization of markets [58], and particle diffusion [59] among others.
Network models in general and adaptive networks in particular provide a powerful framework to model, analyze, and eventually understand a wide range of self-organization phenomena. For instance Tomita et al. [60] showed that a very simple adaptive network model can be used to produce a huge variety of different self-organizing structures including a self-replicating Turing machine.
While specific models can be studied by agent-based simulation, the numerical performance scales badly with the complexity of update rules in the model, which makes exploration of a wider range of models difficult. In particular those models where the update of the state or neighborhood of nodes depends on the states of multiple other neighboring nodes pose strong numerical demands. Additionally the general bad data locality of network simulations precludes efficient parallelization. This defines a strong need for analytical approaches, and, based on recent successes, highlights the exploration of dynamic networks with complex update rules as an area where analytical work could outpace and guide numerical exploration.
A direct microscopic description of dynamical networks generally constitutes a very high-dimensional dynamical system. While in some cases exact analytical results were nevertheless obtained (e.g. [36], [61], [62]), there are presently no approaches that are generally applicable. Much of the theoretical progress therefore relies on the derivation of reasonably low-dimensional coarse-grained approximations to the full microscopic model.
For networks in which the node can only assume states from a (small) discrete set of possibilities, approximation schemes are well established. These schemes are deeply rooted in physics and can be traced back to early work on the Ising model [63], [64]. In the networks literature there is presently a veritable zoo of different approximation schemes that build on similar principles but take different information into account. In the following we refer to these approaches as moment-closure approximations. The common idea in all of these approaches is to derive evolution equations for the abundance of certain subgraphs in the network. One starts with writing an evolution equation for small subgraphs, such as single nodes, before writing equations for larger motifs—a process that is reminiscent of classical moment expansions. The system of equations that is thus obtained generally depends on the abundance of other, typically larger, subgraphs that are not captured, and thus needs to be closed by estimating the abundance of these subgraphs—the actual moment-closure approximation.
Despite their underlying similarity moment-closure approximations proposed in the recent literature differ widely by the type and number of the subgraphs they capture. Generally speaking, capturing the dynamics of more subgraphs leads to better approximations at the cost of having to deal with a larger system of equations (see [15], [24], [65], [66], [67], [68]). In practice some recently proposed schemes are successfully applied which only capture one or two subgraphs, while others capture thousands or millions of subgraphs.
For the analysis of adaptive networks, but also certain types of dynamics on static networks, moment-closure approximations are presently the most commonly applied theoretical tool. In adaptive networks they were used for instance to study epidemics[20], [21], [22], [23], [24], [25], [27], [28], [69], [70], [71], [72], [73], collective motion [56], [57], evolution of cooperation [37], [74], [75], and social opinion formation [13], [15], [19], [76], [77].
Despite the abundance of examples there is so far little intuition on when particular approximation schemes work and when they fail. This is most notable when considering the adaptive SIS model [20] and the adaptive voter model [13], [14], [15]. Both of these models are adaptive network models of similar complexity, and, depending on personal taste, either can be considered as the most simple non-trivial adaptive network. However, for the adaptive SIS model, the dynamics can be faithfully captured already by simple approximation schemes [20], with more sophisticated approaches leading expectedly to a further improvement [24], [71], [72]. By contrast, for the adaptive voter model, simple approximation schemes only provide unsatisfactory results [13], [15] and, as we show here, more sophisticated approaches can actually perform worse.
In the present paper, we aim to offer an in-depth analysis of the performance and the failure of different approximation schemes. For the purpose of illustration we focus on the adaptive voter model as it provides a mathematically simple, yet challenging example system. To this system we apply the major approximation schemes proposed in the recent literature. Thereby, we build up an intuition of the advantages and disadvantages of the respective schemes.
A second goal of the present paper is to provide researchers entering the field with “worked examples” for the major approximation schemes. To provide a comprehensive tutorial, we explain approximations that have already been developed in the literature for the adaptive voter model (Section 3), as well as new approximations that are being developed here for the first time (Sections 4 Heterogeneous approximations, 5 Active motif approach). While space constraints in the printed version of this paper force us to present these examples in strongly abbreviated form, an extended version showing all calculations in full detail is made available on arXiv [78].
The paper is structured as follows: We start in Section 2 by introducing the adaptive voter model. In Section 3 we study the so-called homogeneous approximations, which result in relatively low dimensional equation systems. For these we explore in particular the effect of the order of approximation. In Section 4 we discuss different heterogeneous approximations, which yield high-dimensional equation systems, but surprisingly do not significantly improve the performance of the approximation. Finally, in Section 5 we introduce a slightly different expansion that captures very similar information but works exactly in those parameter ranges where other approximations fail. A summary and discussion in Section 6 concludes the paper.
Section snippets
Adaptive voter model
The voter model considers the competition of equally attractive and mutually exclusive opinions (say A and B) in a population of interacting agents. The agents are represented by nodes that have an internal binary state variable, indicating the opinion held by the corresponding agents. The state is updated dynamically in time due to social interactions, occurring between linked agents.
In the original non-adaptive voter model [79] the underlying interaction topology is static. At each time
Homogeneous approximations
For classifying the different approximation schemes that have been proposed, it is useful to distinguish between homogeneous and heterogeneous approximations. While all approximations attempt to capture the dynamics of certain subgraphs, they differ in the way in which subgraphs are identified: Homogeneous approximations [13], [14], [15], [20], [21], [37], [69], [77], [82], [97] classify subgraphs according to states of the nodes and the internal topology in the subgraph, whereas heterogeneous
Heterogeneous approximations
Presently it is widely believed that quite universally better results can be obtained by heterogeneous approximations that capture information on the degree of the nodes. Indeed, such approaches have yielded an improvement in several example systems [17], [24], [100], [102].
In this section, we investigate two prominent heterogeneous moment-closure approximations. In the heterogeneous pair approximation [100], links are grouped according to the state and the degree of the nodes at their ends.
Active motif approach
In conventional moment expansions of previous sections, moments are taken as densities of regular subgraphs, where subgraphs were characterized by a given number of links and prescribed node states, and degrees in case of heterogeneous moments. While such bases provide reasonable general purpose approaches, they do not take into account the specific dynamics of the system. The considerations presented in Section 3 and the failure of the heterogeneous approximations in Section 4 convey a clear
Summary and discussion
In this paper we investigated the performance of moment-closure approximations for discrete adaptive networks. In particular we used the adaptive voter model as a benchmark model to assess different approaches. The comparison with agent-based simulations revealed that both homogeneous and heterogeneous moment-closure approximations capture qualitative properties of the fragmentation transition, but fail to provide good quantitative estimates close to the fragmentation point. Remarkably, even
Acknowledgment
The authors thank D. Kimura for the insightful discussions.
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Current address: Instituto de Fisica de Liquidos y Sistemas Biologicos (CCT-CONICET-La Plata, UNLP), Calle 59 Nro 789, 1900 La Plata, Argentina.