Preferential attachment in growing spatial networks

Analysis of Pan-omics Data in Human Interactome Network (APODHIN)

Analysis of Pan-omics Data in Human Interactome Network (APODHIN) is a platform for integrative analysis of transcriptomics, proteomics, genomics, and metabolomics data for identification of key molecular players and their interconnections exemplified in cancer scenario. APODHIN works on a meta-interactome network consisting of human protein-protein interactions (PPIs), miRNA-target gene regulatory interactions, and transcription factor-target gene regulatory relationships. In its first module, APODHIN maps proteins/genes/miRNAs from different omics data in its meta-interactome network and extracts the network of biomolecules that are differentially altered in the given scenario. Using this context specific, filtered interaction network, APODHIN identifies topologically important nodes (TINs) implementing graph theory based network topology analysis and further justifies their role via pathway and disease marker mapping. These TINs could be used as prospective diagnostic and/or prognostic biomarkers and/or potential therapeutic targets. In its second module, APODHIN attempts to identify cross pathway regulatory and PPI links connecting signaling proteins, transcription factors (TFs), and miRNAs to metabolic enzymes via utilization of single-omics and/or pan-omics data and implementation of mathematical modeling. Interconnections between regulatory components such as signaling proteins/TFs/miRNAs and metabolic pathways need to be elucidated more elaborately in order to understand the role of oncogene and tumor suppressors in regulation of metabolic reprogramming during cancer. APODHIN platform contains a web server component where users can upload single/multi omics data to identify TINs and cross-pathway links. Tabular, graphical and 3D network representations of the identified TINs and cross-pathway links are provided for better appreciation. Additionally, this platform also provides few example data analysis of cancer specific, single and/or multi omics dataset for cervical, ovarian, and breast cancers where meta-interactome networks, TINs, and cross-pathway links are provided. APODHIN platform is freely available at http://www.hpppi.iicb.res.in/APODHIN/home.html.

Spatial strength centrality and the effect of spatial embeddings on network architecture

For many networks, it is useful to think of their nodes as being embedded in a latent space, and such embeddings can affect the probabilities for nodes to be adjacent to each other. In this paper, we extend existing models of synthetic networks to spatial network models by first embedding nodes in Euclidean space and then modifying the models so that progressively longer edges occur with progressively smaller probabilities. We start by extending a geographical fitness model by employing Gaussian-distributed fitnesses, and we then develop spatial versions of preferential attachment and configuration models. We define a notion of "spatial strength centrality" to help characterize how strongly a spatial embedding affects network structure, and we examine spatial strength centrality on a variety of real and synthetic networks.

Geometric characterisation of disease modules

There is an increasing accumulation of evidence supporting the existence of a hyperbolic geometry underlying the network representation of complex systems. In particular, it has been shown that the latent geometry of the human protein network (hPIN) captures biologically relevant information, leading to a meaningful visual representation of protein-protein interactions and translating challenging systems biology problems into measuring distances between proteins. Moreover, proteins can efficiently communicate with each other, without global knowledge of the hPIN structure, via a greedy routing (GR) process in which hyperbolic distances guide biological signals from source to target proteins. It is thanks to this effective information routing throughout the hPIN that the cell operates, communicates with other cells and reacts to environmental changes. As a result, the malfunction of one or a few members of this intricate system can disturb its dynamics and derive in disease phenotypes. In fact, it is known that the proteins associated with a single disease agglomerate non-randomly in the same region of the hPIN, forming one or several connected components known as the disease module (DM). Here, we present a geometric characterisation of DMs. First, we found that DM positions on the two-dimensional hyperbolic plane reflect their fragmentation and functional heterogeneity, rendering an informative picture of the cellular processes that the disease is affecting. Second, we used a distance-based dissimilarity measure to cluster DMs with shared clinical features. Finally, we took advantage of the GR strategy to study how defective proteins affect the transduction of signals throughout the hPIN.

Geometric evolution of complex networks with degree correlations

We present a general class of geometric network growth mechanisms by homogeneous attachment in which the links created at a given time t are distributed homogeneously between a new node and the existing nodes selected uniformly. This is achieved by creating links between nodes uniformly distributed in a homogeneous metric space according to a Fermi-Dirac connection probability with inverse temperature β and general time-dependent chemical potential μ(t). The chemical potential limits the spatial extent of newly created links. Using a hidden variable framework, we obtain an analytical expression for the degree sequence and show that μ(t) can be fixed to yield any given degree distributions, including a scale-free degree distribution. Additionally, we find that depending on the order in which nodes appear in the network-its history-the degree-degree correlations can be tuned to be assortative or disassortative. The effect of the geometry on the structure is investigated through the average clustering coefficient 〈c〉. In the thermodynamic limit, we identify a phase transition between a random regime where 〈c〉→0 when β<β_{c} and a geometric regime where 〈c〉>0 when β>β_{c}.

Latent geometry of bipartite networks

Despite the abundance of bipartite networked systems, their organizing principles are less studied compared to unipartite networks. Bipartite networks are often analyzed after projecting them onto one of the two sets of nodes. As a result of the projection, nodes of the same set are linked together if they have at least one neighbor in common in the bipartite network. Even though these projections allow one to study bipartite networks using tools developed for unipartite networks, one-mode projections lead to significant loss of information and artificial inflation of the projected network with fully connected subgraphs. Here we pursue a different approach for analyzing bipartite systems that is based on the observation that such systems have a latent metric structure: network nodes are points in a latent metric space, while connections are more likely to form between nodes separated by shorter distances. This approach has been developed for unipartite networks, and relatively little is known about its applicability to bipartite systems. Here, we fully analyze a simple latent-geometric model of bipartite networks and show that this model explains the peculiar structural properties of many real bipartite systems, including the distributions of common neighbors and bipartite clustering. We also analyze the geometric information loss in one-mode projections in this model and propose an efficient method to infer the latent pairwise distances between nodes. Uncovering the latent geometry underlying real bipartite networks can find applications in diverse domains, ranging from constructing efficient recommender systems to understanding cell metabolism.

Manifold learning and maximum likelihood estimation for hyperbolic network embedding

The Popularity-Similarity (PS) model sustains that clustering and hierarchy, properties common to most networks representing complex systems, are the result of an optimisation process in which nodes seek to form ties, not only with the most connected (popular) system components, but also with those that are similar to them. This model has a geometric interpretation in hyperbolic space, where distances between nodes abstract popularity-similarity trade-offs and the formation of scale-free and strongly clustered networks can be accurately described. Current methods for mapping networks to hyperbolic space are based on maximum likelihood estimations or manifold learning. The former approach is very accurate but slow; the latter improves efficiency at the cost of accuracy. Here, we analyse the strengths and limitations of both strategies and assess the advantages of combining them to efficiently embed big networks, allowing for their examination from a geometric perspective. Our evaluations in artificial and real networks support the idea that hyperbolic distance constraints play a significant role in the formation of edges between nodes. This means that challenging problems in network science, like link prediction or community detection, could be more easily addressed under this geometric framework.

Efficient embedding of complex networks to hyperbolic space via their Laplacian

The different factors involved in the growth process of complex networks imprint valuable information in their observable topologies. How to exploit this information to accurately predict structural network changes is the subject of active research. A recent model of network growth sustains that the emergence of properties common to most complex systems is the result of certain trade-offs between node birth-time and similarity. This model has a geometric interpretation in hyperbolic space, where distances between nodes abstract this optimisation process. Current methods for network hyperbolic embedding search for node coordinates that maximise the likelihood that the network was produced by the afore-mentioned model. Here, a different strategy is followed in the form of the Laplacian-based Network Embedding, a simple yet accurate, efficient and data driven manifold learning approach, which allows for the quick geometric analysis of big networks. Comparisons against existing embedding and prediction techniques highlight its applicability to network evolution and link prediction.

Phase Transitions for Random Geometric Preferential Attachment Graphs

Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied.

Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

Phase Transitions for Random Geometric Preferential Attachment Graphs

Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied.

Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

Identification of important interacting proteins (IIPs) in Plasmodium falciparum using large-scale interaction network analysis and in-silico knock-out studies

Background

Plasmodium falciparum causes the most severe form of malaria and affects 3.2 million people annually. Due to the increasing incidence of resistance to existing drugs, there is a growing need to discover new and more effective drugs against malaria. Despite the global importance of P. falciparum, vast majority of its proteins are uncharacterized experimentally. Application of newer approaches using several “omics” data has become successful for exploring the biological interactions underlying cellular processes. Till date not many system level study has been published using P. falciparum protein protein interaction. Hence, the purpose of this study is to develop a standardized pipeline for structural, functional, and topographical analysis of large scale protein protein interaction network (PPIN) in order to identify proteins important for network topology and integrity. Here, P. falciparum PPIN has been utilized as a model for better understanding of the molecular mechanisms of survival and pathogenesis of malaria parasite.

Methods

Various graph theoretical approaches were implemented to identify highly interacting hub and central proteins that are crucial for network integrity. Further, potential network perturbing proteins via an in-silico knock-out (KO) analysis to isolate important interacting proteins (IIPs), which in principle, can elicit significant impact on the global and local environments of the P. falciparum interaction network.

Results

177 hubs and 132 central proteins were identified from the malarial (proteins: 1607; interactions: 4750) PPI networks. Using the in-silico knock-out exercise 131 and 99 global and local network perturbing proteins were also identified. Finally, 271 proteins from P. falciparum were shortlisted as important interacting proteins (IIPs), which not only play crucial role in intra-pathogen network integrity, stage specificity but also interact with various human proteins involved in multiple metabolic pathways within the host cell. These IIPs could be used as potential drug targets in malarial research.

Conclusion

Graph theoretical analysis of PPIN can be a very useful approach to identify proteins that are important for regulation of the interactions required for an organism’s survival. Important interacting proteins (IIPs) identified using P. falciparum PPIN provides a useful dataset containing probable candidates for future drug target analysis.

Electronic supplementary material

The online version of this article (doi:10.1186/s12936-015-0562-1) contains supplementary material, which is available to authorized users.

A latent parameter node-centric model for spatial networks

Spatial networks, in which nodes and edges are embedded in space, play a vital role in the study of complex systems. For example, many social networks attach geo-location information to each user, allowing the study of not only topological interactions between users, but spatial interactions as well. The defining property of spatial networks is that edge distances are associated with a cost, which may subtly influence the topology of the network. However, the cost function over distance is rarely known, thus developing a model of connections in spatial networks is a difficult task. In this paper, we introduce a novel model for capturing the interaction between spatial effects and network structure. Our approach represents a unique combination of ideas from latent variable statistical models and spatial network modeling. In contrast to previous work, we view the ability to form long/short-distance connections to be dependent on the individual nodes involved. For example, a node's specific surroundings (e.g. network structure and node density) may make it more likely to form a long distance link than other nodes with the same degree. To capture this information, we attach a latent variable to each node which represents a node's spatial reach. These variables are inferred from the network structure using a Markov Chain Monte Carlo algorithm. We experimentally evaluate our proposed model on 4 different types of real-world spatial networks (e.g. transportation, biological, infrastructure, and social). We apply our model to the task of link prediction and achieve up to a 35% improvement over previous approaches in terms of the area under the ROC curve. Additionally, we show that our model is particularly helpful for predicting links between nodes with low degrees. In these cases, we see much larger improvements over previous models.

Features and heterogeneities in growing network models

Many complex networks from the World Wide Web to biological networks grow taking into account the heterogeneous features of the nodes. The feature of a node might be a discrete quantity such as a classification of a URL document such as personal page, thematic website, news, blog, search engine, social network, etc., or the classification of a gene in a functional module. Moreover the feature of a node can be a continuous variable such as the position of a node in the embedding space. In order to account for these properties, in this paper we provide a generalization of growing network models with preferential attachment that includes the effect of heterogeneous features of the nodes. The main effect of heterogeneity is the emergence of an "effective fitness" for each class of nodes, determining the rate at which nodes acquire new links. The degree distribution exhibits a multiscaling behavior analogous to the the fitness model. This property is robust with respect to variations in the model, as long as links are assigned through effective preferential attachment. Beyond the degree distribution, in this paper we give a full characterization of the other relevant properties of the model. We evaluate the clustering coefficient and show that it disappears for large network size, a property shared with the Barabási-Albert model. Negative degree correlations are also present in this class of models, along with nontrivial mixing patterns among features. We therefore conclude that both small clustering coefficients and disassortative mixing are outcomes of the preferential attachment mechanism in general growing networks.