Combinatorial study of degree assortativity in networks

Apolipoprotein-E deficiency leads to brain network alteration characterized by diffusion MRI and graph theory

Late-onset Alzheimer's disease (LOAD) is a major health concern for senior citizens, characterized by memory loss, confusion, and impaired cognitive abilities. Apolipoprotein-E (ApoE) is a well-known risk factor for LOAD, though exactly how ApoE affects LOAD risks is unknown. We hypothesize that ApoE attenuation of LOAD resiliency or vulnerability has a neurodevelopmental origin via changing brain network architecture. We investigated the brain network structure in adult ApoE knock out (ApoE KO) and wild-type (WT) mice with diffusion tensor imaging (DTI) followed by graph theory to delineate brain network topology. Left and right hemisphere connectivity revealed significant differences in number of connections between the hippocampus, amygdala, caudate putamen and other brain regions. Network topology based on the graph theory of ApoE KO demonstrated decreased functional integration, network efficiency, and network segregation between the hippocampus and amygdala and the rest of the brain, compared to those in WT counterparts. Our data show that brain network developed differently in ApoE KO and WT mice at 5 months of age, especially in the network reflected in the hippocampus, amygdala, and caudate putamen. This indicates that ApoE is involved in brain network development which might modulate LOAD risks via changing brain network structures.

Topologically induced suppression of explosive synchronization

Nowadays, explosive synchronization is a well-documented phenomenon consisting in a first-order transition that may coexist with classical synchronization. Typically, explosive synchronization occurs when the network structure is represented by the classical graph Laplacian, and the node frequency and its degree are correlated. Here, we answer the question on whether this phenomenon can be observed in networks when the oscillators are coupled via degree-biased Laplacian operators. We not only observe that this is the case but also that this new representation naturally controls the transition from explosive to standard synchronization in a network. We prove analytically that explosive synchronization emerges when using this theoretical setting in star-like networks. As soon as this star-like network is topologically converted into a network containing cycles, the explosive synchronization gives rise to classical synchronization. Finally, we hypothesize that this mechanism may play a role in switching from normal to explosive states in the brain, where explosive synchronization has been proposed to be related to some pathologies like epilepsy and fibromyalgia.

Network constraints on the mixing patterns of binary node metadata

We consider the network constraints on the bounds of the assortativity coefficient, which aims to quantify the tendency of nodes with the same attribute values to be connected. The assortativity coefficient can be considered as the Pearson's correlation coefficient of node metadata values across network edges and lies in the interval [-1,1]. However, properties of the network, such as degree distribution and the distribution of node metadata values, place constraints upon the attainable values of the assortativity coefficient. This is important as a particular value of assortativity may say as much about the network topology as about how the metadata are distributed over the network-a fact often overlooked in literature where the interpretation tends to focus simply on the propensity of similar nodes to link to each other, without any regard on the constraints posed by the topology. In this paper we quantify the effect that the topology has on the assortativity coefficient in the case of binary node metadata. Specifically, we look at the effect that the degree distribution, or the full topology, and the proportion of each metadata value has on the extremal values of the assortativity coefficient. We provide the means for obtaining bounds on the extremal values of assortativity for different settings and demonstrate that under certain conditions the maximum and minimum values of assortativity are severely limited, which may present issues in interpretation when these bounds are not considered.

Abnormal Topological Organization of Sulcal Depth-Based Structural Covariance Networks in Parkinson's Disease

Recent research on Parkinson's disease (PD) has demonstrated the topological abnormalities of structural covariance networks (SCNs) using various morphometric features from structural magnetic resonance images (sMRI). However, the sulcal depth (SD)-based SCNs have not been investigated. In this study, we used SD to investigate the topological alterations of SCNs in 60 PD patients and 56 age- and gender-matched healthy controls (HC). SCNs were constructed by thresholding SD correlation matrices of 68 regions and analyzed using graph theoretical approaches. Compared with HC, PD patients showed increased normalized clustering coefficient and normalized path length, as well as a reorganization of degree-based and betweenness-based hubs (i.e., less frontal hubs). Moreover, the degree distribution analysis showed more high-degree nodes in PD patients. In addition, we also found the increased assortativity and reduced robustness under a random attack in PD patients compared to HC. Taken together, these findings indicated an abnormal topological organization of SD-based SCNs in PD patients, which may contribute in understanding the pathophysiology of PD at the network level.

Transitivity and degree assortativity explained: The bipartite structure of social networks

Dynamical processes, such as the diffusion of knowledge, opinions, pathogens, "fake news," innovation, and others, are highly dependent on the structure of the social network in which they occur. However, questions on why most social networks present some particular structural features, namely, high levels of transitivity and degree assortativity, when compared to other types of networks remain open. First, we argue that every one-mode network can be regarded as a projection of a bipartite network, and we show that this is the case using two simple examples solved with the generating functions formalism. Second, using synthetic and empirical data, we reveal how the combination of the degree distribution of both sets of nodes of the bipartite network-together with the presence of cycles of lengths four and six-explain the observed values of transitivity and degree assortativity coefficients in the one-mode projected network. Bipartite networks with top node degrees that display a more right-skewed distribution than the bottom nodes result in highly transitive and degree assortative projections, especially if a large number of small cycles are present in the bipartite structure.

Estimating degree-degree correlation and network cores from the connectivity of high-degree nodes in complex networks

Many of the structural characteristics of a network depend on the connectivity with and within the hubs. These dependencies can be related to the degree of a node and the number of links that a node shares with nodes of higher degree. In here we revise and present new results showing how to construct network ensembles which give a good approximation to the degree-degree correlations, and hence to the projections of this correlation like the assortativity coefficient or the average neighbours degree. We present a new bound for the structural cut-off degree based on the connectivity within the hubs. Also we show that the connections with and within the hubs can be used to define different networks cores. Two of these cores are related to the spectral properties and walks of length one and two which contain at least on hub node, and they are related to the eigenvector centrality. We introduce a new centrality measured based on the connectivity with the hubs. In addition, as the ensembles and cores are related by the connectivity of the hubs, we show several examples how changes in the hubs linkage effects the degree-degree correlations and core properties.

Emergence of fractal scaling in complex networks

Some real-world networks are shown to be fractal or self-similar. It is widespread that such a phenomenon originates from the repulsion between hubs or disassortativity. Here we show that this common belief fails to capture the causality. Our key insight to address it is to pinpoint links critical to fractality. Those links with small edge betweenness centrality (BC) constitute a special architecture called fractal reference system, which gives birth to the fractal structure of those reported networks. In contrast, a small amount of links with high BC enable small-world effects, hiding the intrinsic fractality. With enough of such links removed, fractal scaling spontaneously arises from nonfractal networks. Our results provide a multiple-scale view on the structure and dynamics and place fractality as a generic organizing principle of complex networks on a firmer ground.

Wild cricket social networks show stability across generations

BACKGROUND

A central part of an animal's environment is its interactions with conspecifics. There has been growing interest in the potential to capture these interactions in the form of a social network. Such networks can then be used to examine how relationships among individuals affect ecological and evolutionary processes. However, in the context of selection and evolution, the utility of this approach relies on social network structures persisting across generations. This is an assumption that has been difficult to test because networks spanning multiple generations have not been available. We constructed social networks for six annual generations over a period of eight years for a wild population of the cricket Gryllus campestris.

RESULTS

Through the use of exponential random graph models (ERGMs), we found that the networks in any given year were able to predict the structure of networks in other years for some network characteristics. The capacity of a network model of any given year to predict the networks of other years did not depend on how far apart those other years were in time. Instead, the capacity of a network model to predict the structure of a network in another year depended on the similarity in population size between those years.

CONCLUSIONS

Our results indicate that cricket social network structure resists the turnover of individuals and is stable across generations. This would allow evolutionary processes that rely on network structure to take place. The influence of network size may indicate that scaling up findings on social behaviour from small populations to larger ones will be difficult. Our study also illustrates the utility of ERGMs for comparing networks, a task for which an effective approach has been elusive.

The Cognitive Social Network in Dreams: Transitivity, Assortativity, and Giant Component Proportion Are Monotonic

For five individuals, a social network was constructed from a series of his or her dreams. Three important network measures were calculated for each network: transitivity, assortativity, and giant component proportion. These were monotonically related; over the five networks as transitivity increased, assortativity increased and giant component proportion decreased. The relations indicate that characters appear in dreams systematically. Systematicity likely arises from the dreamer's memory of people and their relations, which is from the dreamer's cognitive social network. But the dream social network is not a copy of the cognitive social network. Waking life social networks tend to have positive assortativity; that is, people tend to be connected to others with similar connectivity. Instead, in our sample of dream social networks assortativity is more often negative or near 0, as in online social networks. We show that if characters appear via a random walk, negative assortativity can result, particularly if the random walk is biased as suggested by remote associations.

Reexamination of explosive synchronization in scale-free networks: the effect of disassortativity

Previous work [J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, Phys. Rev. Lett. 106, 128701 (2011)] has reported that explosive synchronization can be achieved in heterogeneous networks with a microscopic correlation between the structural and dynamical properties of the networks. This phenomenon, however, cannot be observed in all heterogeneous networks even if this structure-dynamics correlation is preserved. It is therefore of particular interest to identify the general topological factors that can induce the first order synchronization transition and to understand the underlying mechanisms. Here we investigate this issue using the scenario of the smooth transformation from homogeneous Erdös-Rènyi networks to heterogeneous Barabàsi-Albert networks. Specifically, we scrutinize how local and global properties of the network change during this process, and how these properties are associated with the emergence of explosive synchronization. We find that the local degree-degree correlation in the network contributes primarily to explosive synchronization, other than the global topological property or starlike subgraphs. We furthermore demonstrate that the degree of disassortative mixing also has a great effect in the presence of explosive synchronization.

Local motifs in proteins combine to generate global functional moves

Literature on the topological properties of folded proteins that has emerged as a field in its own right in the past decade is reviewed. Physics-based construction of coarse-grained models of proteins from knowledge of all-atom coordinates of the average structure is discussed. Once network is thus obtained with the node and link information, local motifs provide plethora of information on protein function. The hierarchical structure of the proteins manifested in the interrelations of local motifs is emphasized. Motifs are also related to modularity of the structure, and they quantify shifts in the landscapes upon conformational changes induced by, e.g. ligand binding. Redundancy emerges as a balance between local and global network descriptors and is related to the collectivity of the protein motions. Introducing weight on links followed by sequential removal of least cohesive contacts allows interactions in proteins to be represented as the superposition of essential and redundant sets. Lack of the former makes the network non-functional, while the latter ensures robust functioning under a wide range of perturbation scenarios.

Accuracy of mean-field theory for dynamics on real-world networks

Mean-field analysis is an important tool for understanding dynamics on complex networks. However, surprisingly little attention has been paid to the question of whether mean-field predictions are accurate, and this is particularly true for real-world networks with clustering and modular structure. In this paper, we compare mean-field predictions to numerical simulation results for dynamical processes running on 21 real-world networks and demonstrate that the accuracy of such theory depends not only on the mean degree of the networks but also on the mean first-neighbor degree. We show that mean-field theory can give (unexpectedly) accurate results for certain dynamics on disassortative real-world networks even when the mean degree is as low as 4.