^{1}

^{1}

^{2}

^{3}

^{4}

^{1}

^{1}

^{*}

OS is a PLOS ONE Editorial Board Member (Section Editor). This does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials.

Conceived and designed the experiments: JG AAK OS. Performed the experiments: JG AAK MVDH NVDM RB OS. Analyzed the data: JG AAK MVDH NVDM RB OS. Contributed reagents/materials/analysis tools: JG AAK MVDH NVDM RB OS. Wrote the paper: JG AAK OS.

Graph theoretical analysis has played a key role in characterizing global features of the topology of complex networks, describing diverse systems such as protein interactions, food webs, social relations and brain connectivity. How system elements communicate with each other depends not only on the structure of the network, but also on the nature of the system's dynamics which are constrained by the amount of knowledge and resources available for communication processes. Complementing widely used measures that capture efficiency under the assumption that communication preferentially follows shortest paths across the network (“routing”), we define analytic measures directed at characterizing network communication when signals flow in a random walk process (“diffusion”). The two dimensions of routing and diffusion efficiency define a morphospace for complex networks, with different network topologies characterized by different combinations of efficiency measures and thus occupying different regions of this space. We explore the relation of network topologies and efficiency measures by examining canonical network models, by evolving networks using a multi-objective optimization strategy, and by investigating real-world network data sets. Within the efficiency morphospace, specific aspects of network topology that differentially favor efficient communication for routing and diffusion processes are identified. Charting regions of the morphospace that are occupied by canonical, evolved or real networks allows inferences about the limits of communication efficiency imposed by connectivity and dynamics, as well as the underlying selection pressures that have shaped network topology.

Characterizing the communication efficiency of a complex network should take into account dual sets of constraints, imposed by the topology and by the dynamical process operating on it. In this regard it is crucial to know whether communication is better described by

One of the most widely used efficiency measures is defined as the average of the inverse of shortest path lengths between every pair of nodes in a graph _{rout}_{rout}

In this paper we first introduce and define a set of interrelated measures of communication efficiency for systems whose dynamics are based on diffusion processes. These measures capture the probability with which single particles travel through shortest paths, their average propagation velocity across the network, and the degree to which additional resources help the system to approach optimal performance. These measures of diffusion efficiency complement the classical measure of routing efficiency

In this section we briefly define some basic graph theory concepts relevant to the work described in this article (see

A Markov chain

Diffusion in networks is generally modeled as a random walk process, which in the simplest case involves the use of only local information about connectivity. In a binary network, the probability to go from one state (corresponding to the

The next sections describe a set of measures characterizing communication efficiency. One measure has been introduced previously

The shortest path length between two nodes of an undirected binary graph is defined as the minimum number of edges (and thus steps) that separate two nodes within a graph. The set of shortest path lengths between all node pairs is denoted by the symmetric matrix _{global}_{rout}

Given a random walk process on a graph, an analytic expression can be derived that gives the probability that a single particle departing from a node

Let us denote by the matrix _{rout}_{diff}

Let us define _{rout}_{diff}

Graph measures are known to be strongly constrained by features such as network size or density _{rout}

The morphospace is a concept introduced originally in paleontology and evolutionary theory

Four canonical network models were used to capture the relationship between density, shortest-path probability and routing efficiency, selected because they capture relevant aspects of network organization encountered in a wide range of real-world systems. Models were Erdös-Rényi random graphs

The use of optimization algorithms operating on the efficiency morphospace allowed us to explore the commonalities and differences between

Simulations were carried out on populations (

At each epoch, the Pareto front concept was applied to define survival criteria by partitioning the population into non-dominated and dominated solutions

A total of 23 real-world networks were analyzed (see

First we examined how increasing density affects ^{−2} for networks of 50 nodes) but also tended to decrease as network density increases. Over all network densities and at a given level of

Scatter plot of

Descriptors shown are: number of nodes

The characterization of these idealized network topologies provided some intuition about which network architectures promote different aspects of communication efficiency. However, the cases listed in

Results shown are for evolutionary processes driven by network efficiency measures for networks with

Results shown are for networks with

To further explore the role of the degree distribution in these evolutionary experiments, additional evolutionary optimizations were carried out where all rewiring steps were performed such that the initial uniform degree sequence (

The figure shows a scatter plot of

The efficiency of communication in complex networks is of central interest across many disciplines studying physical, social, technological or biological systems. Here we explore the relation between different measures of communication efficiency based on routing or diffusion processes, and different aspects of network topology. This exploration is carried out within an efficiency morphospace whose two axes are defined by scaled measures of routing and diffusion efficiency. Complex networks are positioned in this space depending on the level of communication efficiency they support. We explore this space by adopting three different approaches. An examination of idealized topologies, evolved architectures and real-world networks reveals characteristic differences in the way different network architectures facilitate or impede communication via routing or diffusion.

Communication efficiency has been characterized in different ways, employing routing

Star-like topologies exhibit high

The association of certain network architectures with different levels of communication efficiency for routing and diffusion offers a new perspective on network performance. As in our multi-objective optimization experiments, key characteristics of network architecture encountered in real systems may represent the result of selection pressure on efficient communication given the constraints imposed by system dynamics and the cost of building and running the network's infrastructure

The apparent antagonism between dynamics dominated by diffusion versus routing can be reconciled by introducing the concept of resource efficiency. Even if the elements of a system cannot access global knowledge about network topology, this lack of knowledge can be partly overcome by multiplying the number of particles or messages (resources) used for communication. As more resources are deployed the probability that at least one particle travels along a shortest path increases. Hence, to achieve targeted (and less noisy) communication the addition of resources (involving an expense of material and/or energy) can compensate for a lack of knowledge about network structure. A different way of quantifying the knowledge needed to achieve shortest-path performance utilizes an information-theoretical approach

Our approach towards characterizing communication efficiency in networks can be extended in different directions. First, the measures and approaches introduced here could be fully explored for undirected weighted networks. Second, additional measures for characterizing diffusion in complex networks exist, including for instance the entropy rate of a diffusion process

Future applications and extensions of the framework for characterizing communication efficiency proposed in this article may offer new insights into how complex networks maximize performance when their elements operate with limited knowledge and resources. Such limits are prominently encountered in, for instance, neuronal networks, where the trade-off between cost and efficiency is a major driving force of brain organization

(DOCX)

(TIF)

(TIF)

(TIF)

(TIF)

(DOCX)

(AVI)

(AVI)

The authors appreciate fruitful discussions with Ricard V. Solé and Artemy Kolchinsky.