Subgraph fluctuations in random graphs

Local topological features of robust supply networks

The design of robust supply and distribution systems is one of the fundamental challenges at the interface of network science and logistics. Given the multitude of performance criteria, real-world constraints, and external influences acting upon such a system, even formulating an appropriate research question to address this topic is non-trivial. Here we present an abstraction of a supply and distribution system leading to a minimal model, which only retains stylized facts of the systemic function and, in this way, allows us to investigate the generic properties of robust supply networks. On this level of abstraction, a supply and distribution system is the strategic use of transportation to eliminate mismatches between production patterns (i.e., the amounts of goods produced at each production site of a company) and demand patterns (i.e., the amount of goods consumed at each location). When creating networks based on this paradigm and furthermore requiring the robustness of the system with respect to the loss of transportation routes (edge of the network) we see that robust networks are built from specific sets of subgraphs, while vulnerable networks display a markedly different subgraph composition. Our findings confirm a long-standing hypothesis in the field of network science, namely, that network motifs-statistically over-represented small subgraphs-are informative about the robust functioning of a network. Also, our findings offer a blueprint for enhancing the robustness of real-world supply and distribution systems.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s41109-022-00470-2.

The economy of chromosomal distances in bacterial gene regulation

In the transcriptional regulatory network (TRN) of a bacterium, the nodes are genes and a directed edge represents the action of a transcription factor (TF), encoded by the source gene, on the target gene. It is a condensed representation of a large number of biological observations and facts. Nonrandom features of the network are structural evidence of requirements for a reliable systemic function. For the bacterium Escherichia coli we here investigate the (Euclidean) distances covered by the edges in the TRN when its nodes are embedded in the real space of the circular chromosome. Our work is motivated by 'wiring economy' research in Computational Neuroscience and starts from two contradictory hypotheses: (1) TFs are predominantly employed for long-distance regulation, while local regulation is exerted by chromosomal structure, locally coordinated by the action of structural proteins. Hence long distances should often occur. (2) A large distance between the regulator gene and its target requires a higher expression level of the regulator gene due to longer reaching times and ensuing increased degradation (proteolysis) of the TF and hence will be evolutionarily reduced. Our analysis supports the latter hypothesis.

Topological reinforcement as a principle of modularity emergence in brain networks

Modularity is a ubiquitous topological feature of structural brain networks at various scales. Although a variety of potential mechanisms have been proposed, the fundamental principles by which modularity emerges in neural networks remain elusive. We tackle this question with a plasticity model of neural networks derived from a purely topological perspective. Our topological reinforcement model acts enhancing the topological overlap between nodes, that is, iteratively allowing connections between non-neighbor nodes with high neighborhood similarity. This rule reliably evolves synthetic random networks toward a modular architecture. Such final modular structure reflects initial "proto-modules," thus allowing to predict the modules of the evolved graph. Subsequently, we show that this topological selection principle might be biologically implemented as a Hebbian rule. Concretely, we explore a simple model of excitable dynamics, where the plasticity rule acts based on the functional connectivity (co-activations) between nodes. Results produced by the activity-based model are consistent with the ones from the purely topological rule in terms of the final network configuration and modules composition. Our findings suggest that the selective reinforcement of topological overlap may be a fundamental mechanism contributing to modularity emergence in brain networks.

Motif-based success scores in coauthorship networks are highly sensitive to author name disambiguation

Following the work of Krumov et al. [Eur. Phys. J. B 84, 535 (2011)] we revisit the question whether the usage of large citation datasets allows for the quantitative assessment of social (by means of coauthorship of publications) influence on the progression of science. Applying a more comprehensive and well-curated dataset containing the publications in the journals of the American Physical Society during the whole 20th century we find that the measure chosen in the original study, a score based on small induced subgraphs, has to be used with caution, since the obtained results are highly sensitive to the exact implementation of the author disambiguation task.

Identifying emerging motif in growing networks

As function units, network motifs have been detected to reveal evolutionary mechanisms of complex systems, such as biological networks, food webs, engineering networks and social networks. However, emergence of motifs in growing networks may be problematic due to large fluctuation of subgraph frequency in the initial stage. This paper contributes to present a method which can identify the emergence of motif in growing networks. Based on the Erdös-Rényi(E-R) random null model, the variation rate of expected frequency of subgraph at adjacent time points was used to define the suitable detection range for motif identification. Upper and lower boundaries of the range were obtained in analytical form according to a chosen risk level. Then, the statistical metric Z-score was extended to a new one,, which effectively reveals the statistical significance of subgraph in a continuous period of time. In this paper, a novel research framework of motif identification was proposed, defining critical boundaries for the evolutionary process of networks and a significance metric of time scale. Finally, an industrial ecosystem at Kalundborg was adopted as a case study to illustrate the effectiveness and convenience of the proposed methodology.

Motifs in triadic random graphs based on Steiner triple systems

Conventionally, pairwise relationships between nodes are considered to be the fundamental building blocks of complex networks. However, over the last decade, the overabundance of certain subnetwork patterns, i.e., the so-called motifs, has attracted much attention. It has been hypothesized that these motifs, instead of links, serve as the building blocks of network structures. Although the relation between a network's topology and the general properties of the system, such as its function, its robustness against perturbations, or its efficiency in spreading information, is the central theme of network science, there is still a lack of sound generative models needed for testing the functional role of subgraph motifs. Our work aims to overcome this limitation. We employ the framework of exponential random graph models (ERGMs) to define models based on triadic substructures. The fact that only a small portion of triads can actually be set independently poses a challenge for the formulation of such models. To overcome this obstacle, we use Steiner triple systems (STSs). These are partitions of sets of nodes into pair-disjoint triads, which thus can be specified independently. Combining the concepts of ERGMs and STSs, we suggest generative models capable of generating ensembles of networks with nontrivial triadic Z-score profiles. Further, we discover inevitable correlations between the abundance of triad patterns, which occur solely for statistical reasons and need to be taken into account when discussing the functional implications of motif statistics. Moreover, we calculate the degree distributions of our triadic random graphs analytically.

Artefacts in statistical analyses of network motifs: general framework and application to metabolic networks

Few-node subgraphs are the smallest collective units in a network that can be investigated. They are beyond the scale of individual nodes but more local than, for example, communities. When statistically over- or under-represented, they are called network motifs. Network motifs have been interpreted as building blocks that shape the dynamic behaviour of networks. It is this promise of potentially explaining emergent properties of complex systems with relatively simple structures that led to an interest in network motifs in an ever-growing number of studies and across disciplines. Here, we discuss artefacts in the analysis of network motifs arising from discrepancies between the network under investigation and the pool of random graphs serving as a null model. Our aim was to provide a clear and accessible catalogue of such incongruities and their effect on the motif signature. As a case study, we explore the metabolic network of Escherichia coli and show that only by excluding ever more artefacts from the motif signature a strong and plausible correlation with the essentiality profile of metabolic reactions emerges.