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The authors have declared that no competing interests exist.

Conceived and designed the experiments: HD JB. Performed the experiments: HD JB. Analyzed the data: HD JB. Wrote the paper: HD JB MA.

In many types of network, the relationship between structure and function is of great significance. We are particularly interested in community structures, which arise in a wide variety of domains. We apply a simple oscillator model to networks with community structures and show that waves of regular oscillation are caused by synchronised clusters of nodes. Moreover, we show that such global oscillations may arise as a direct result of network topology. We also observe that additional modes of oscillation (as detected through frequency analysis) occur in networks with additional levels of topological hierarchy and that such modes may be directly related to network structure. We apply the method in two specific domains (metabolic networks and metropolitan transport) demonstrating the robustness of our results when applied to real world systems. We conclude that (where the distribution of oscillator frequencies and the interactions between them are known to be unimodal) our observations may be applicable to the detection of underlying community structure in networks, shedding further light on the general relationship between structure and function in complex systems.

The problem of relating the structure of a network to the dynamical behaviour it supports is of significant interest in a large number of domains. Many different systems may be represented as networks of connected entities, from friends communicating via social media

Early work in this area focused on the development of model systems, which were used to analytically study the onset of certain behaviours (such as oscillations)

Network topology has a strong effect on the observed dynamics of oscillator networks

Many complex networks have been shown to demonstrate periodic dynamics. Neural systems, for example, display modes of oscillation at particular frequencies and this has in turn been linked to the hierarchical organisation of the brain network itself

In coupled oscillator networks with all-to-all coupling, oscillating waves of synchronization have been observed in systems with bimodal and trimodal frequencies

In this paper we show how the community structure of a complex network may actively drive periodic dynamics and that such periodic dynamics occur in real world networks. The remainder of this paper describes our methodology in detail, showing how a simple model system is capable of a variety of dynamical behaviours. We then give the results of experimental investigations into the effect of network topology on oscillatory dynamics and how the latter may be used to detect the former. In particular, we demonstrate how our methodology may be applied to two real world networks. We conclude with a discussion and suggestions for future work.

In order to rigorously establish the relationship between network structure and dynamics, we require a model system that is broadly applicable, but which supports a wide range of dynamical behaviours. We also need to be able to measure the global network dynamics in a way that readily admits analysis. The well-established

The model describes a system of coupled oscillators described by ordinary differential equations (ODEs) where interaction terms between oscillators are connected according to the specific network topology:

This original model of Kuramoto assumes mean-field interactions. In the absence of any external noise, the global dynamics are determined by the coupling strength

Many variations of the original Kuramoto model have been developed; of particular interest is the introduction of a

Chimera states can arise as a direct result of network topology; specifically, the existence of community structure

Motivated, in part, by the realisation that many naturally-occurring networks have complex topologies, recent studies have been extended to systems where the pattern of connections is local but not necessarily regular

We use a global order parameter

We now present the results of our experimental investigations. The over-arching aim is to show how global oscillatory behaviour may be related directly to the community structure of the underlying complex network.

We first study two classes of graph; those with and those without any community structure. For example, consider the typical community structured graph in

For such a network there exist parameter regimes where the smaller, globally connected sub-graphs may synchronise but the network as a whole does not (partial synchronisation or clustering).

Time series for order parameter,

In order to demonstrate that the oscillating dynamics shown above are not simply an artefact of network symmetry, we perturb the original network by repeatedly adding random connections.

Time series of global order parameter,

Time series for global order parameter,

To further develop the study of non-symmetric networks, we consider a large, idealised network of oscillators arranged such that three highly coupled sub-networks of oscillators are connected via a sparse network of random connections. We report the results of simulations for subgraphs of

We first investigate the effect of varying coupling strength,

Each network contains

In common with networks lacking community structure, these networks synchronise above a critical coupling strength; for small values of coupling strength, the oscillators are incoherent. In the first example there exists a specific region for

Time series for global order parameter, with

We now consider a more complex network, which displays an additional level of hierarchy (

For such a network there exist parameter regimes where the smaller, globally connected sub-graphs may synchronise but the network as a whole does not (partial synchronisation or clustering).

(

In the previous section, we established the feasibility of using a global order measure to detect community structure in artificial networks. We now validate this approach against two classes of ‘real world’ network, both of which present examples that may or may not possess community structure.

In order to provide a metric for comparison, we use the standard measure of

The metabolic network of a cell or microorganism describes the connections between various cellular processes that are essential for sustaining function

We use metabolic pathway networks in SBML format

In order to establish a relationship between community structure and dynamics, we consider two versions of this network. The first comprises the global connectivity matrix of all chemical reactants in the cell, a connection being present if two or more components are involved in a known reaction (we exclude water and ATP, as these occur in almost all reactions). The second formulation of the metabolic network partitions reactions into sub-cellular networks, each representing different regions of the cell (nucleus, golgi bodies, etc.) which are connected in turn by reactions. Graphical representations of these networks are shown in

(

From a graph theoretical perspective, these two networks are very similar. Standard graph metrics such as the clustering coefficient, mean and maximum path length do not distinguish between the two. Furthermore, the eigenvalue spectrum (as described in

The main difference between these two networks lies in the values for

Simulations for optimised coupling strengths and frequency distributions are conducted on both forms of the metabolic network. For the non-partitioned network, we fail to observe multi-modal oscillations in the global order parameter. However, for the partitioned network we observe strong modal dynamics (See

(

(

We now investigate a completely different type of network; those describing mass transit systems in major cities. Specifically, we compare the network of the London Underground and the New York Subway systems, as both are large enough to be interesting, but they have very different underlying geographical structures. In particular, stations on the London Underground are more evenly distributed than in New York, where the presence of islands in the geography of the city gives rise to clusters of stations, particularly in South Manhattan and Brooklyn (

The London Underground (

Structurally, these networks are significantly different from the previous examples. Notably, there exist many long chains, the overall graph connectivity is low and there exists very few ‘small world’ effects. As such, we are confident that these networks present a novel challenge, over and above that offered by both the artificially-generated networks and the metabolic networks.

As before, we run numerical simulations in order to optimise model parameters, in an attempt to maximise any oscillatory dynamics. On the London network, we observe a small amount of oscillatory behaviour, although the amplitude of such oscillation is small - the maximum observed oscillation has an amplitude of

(

On the other hand, experiments on the New York network yield a

(

In order to demonstrate that this oscillating behaviour is indeed caused by the underlying hierarchy of the network, the New York subway network was rewired using the Xswap algorithm previously described. We observe that as the network is rewired and the modularity reduced to below

In this paper, we have demonstrated a robust and structurally stable relationship between form and function in complex networks whereby global oscillations are shown to be a factor of network topology. We observe modal oscillations in a measure of global synchronization which can be directly related to the community structure of the network itself.

By applying the method to two types of real world networks - whereby examples exist with significantly different community structures but with similar underlying topology, we show that this method also works on realistic, more irregular structures. We demonstrate the breakdown in oscillatory behaviour when networks are rewired (with the degree of each node remaining constant). This confirms that network modularity drives oscillations, as reducing the degree of modularity causes these oscillations to break down. We should note, however, that for the real world examples given, the underlying dynamics of the nodes on the network (chemical reactions and subway trains) are considerably more complex than the simple Kuramoto oscillators used to demonstrate the principle. As such, it is not possible to directly attribute any observed oscillatory dynamics in such systems to the network structure alone.

Many real world networks (e.g. transport, the brain) are examples of pseudo-hierarchical networks, in that their structure is not fully hierarchical

We thank the editor Petter Holme and anonymous reviewers for their comments and suggestions. We would also particularly like to thank Kieran Smallbone for providing the metabolic network data in a usable format.