We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Anomaly detection is an important problem with multiple applications, and thus has been studied for decades in various research domains. In the past decade there has been a growing interest in anomaly detection in data represented as networks, or graphs, largely because of their robust expressiveness and their natural ability to represent complex relationships. Originally, techniques focused on anomaly detection in static graphs, which do not change and are capable of representing only a single snapshot of data. As real-world networks are constantly changing, there has been a shift in focus to dynamic graphs, which evolve over time.

Anomaly detection in dynamic networks: a survey

Are biological networks different from other large complex networks? Both large biological and non-biological networks exhibit power-law graphs (number of nodes with degree k, N(k) ~ k-b) yet the exponents, b, fall into different ranges. This may be because duplication of the information in the genome is a dominant evolutionary force in shaping biological networks (like gene regulatory networks and protein-protein interaction networks), and is fundamentally different from the mechanisms thought to dominate the growth of most non-biological networks (such as the internet [1-4]). The preferential choice models non-biological networks like web graphs can only produce power-law graphs with exponents greater than 2 [1-4,8]. We use combinatorial probabilistic methods to examine the evolution of graphs by duplication processes and derive exact analytical relationships between the exponent of the power law and the parameters of the model. Both full duplication of nodes (with all their connections) as well as partial duplication (with only some connections) are analyzed. We demonstrate that partial duplication can produce power-law graphs with exponents less than 2, consistent with current data on biological networks. The power-law exponent for large graphs depends only on the growth process, not on the starting graph.

Duplication Models for Biological Networks

We study spatial networks that are designed to distribute or collect a commodity, such as gas pipelines or train tracks. We focus on the cost of a network, as represented by the total length of all its edges, and its efficiency in terms of the directness of routes from point to point. Using data for several real-world examples, we find that distribution networks appear remarkably close to optimal where both these properties are concerned. We propose two models of network growth that offer explanations of how this situation might arise.

Shape and efficiency in spatial distribution networks

In recent years, neuromorphic hardware systems have significantly grown in size. With more and more neurons and synapses integrated in such systems, the neural connectivity and its configurability have become crucial design constraints. To tackle this problem, we introduce a generic extended graph description of connection topologies that allows a systematical analysis of connectivity in both neuromorphic hardware and neural network models. The unifying nature of our approach enables a close exchange between hardware and models. For an existing hardware system, the optimally matched network model can be extracted. Inversely, a hardware architecture may be fitted to a particular model network topology with our description method. As a further strength, the extended graph can be used to quantify the amount of configurability for a certain network topology. This is a hardware design variable that has widely been neglected, mainly because of a missing analysis method. To condense our analysis results, we develop a classification for the scaling complexity of network models and neuromorphic hardware, based on the total number of connections and the configurability. We find a gap between several models and existing hardware, making these hardware systems either impossible or inefficient to use for scaled-up network models. In this respect, our analysis results suggest models with locality in their connections as promising approach for tackling this scaling gap.

Analyzing the Scaling of Connectivity in Neuromorphic Hardware and in Models of Neural Networks

The task of model-based diagnosis is NP-complete, but it is not known whether it is computationally difficult for the "average" real-world system. There has been no systematic study of the complexity of diagnosing real-world problems, and few good benchmarks exist to test this. Real-world-graphs, a mathematical framework that has been proposed as a model for complex systems, have empirically been shown to capture several topological properties of real-world systems. We describe the adequacy with which a real-world-graph can characterise the complexity of model-based diagnostic inference on real-world systems. We empirically compare the inference complexity of diagnosing models automatically generated using the real-world-graph framework with comparable models from well-known ISCAS circuit benchmarks. We identify parameters necessary for the real-world-graph framework to generate benchmark diagnosis circuit models with realistic properties.

/pdf/automated-benchmark-model-generators-for-model-based-6gfjhrir52.pdf

Automated benchmark model generators for model-based diagnostic inference

Based on analyses of specific spatial networks, we compare the accuracy of three models in capturing topologies of two types of spatial networks: electronic circuits and brain networks. The models analyzed are an optimization model trading off multiple-objective constraints, an extended preferential attachment model with spatial constraints, and the generalized random graph model. First, we find that the optimization model and the spatial preferential attachment model can generate similar topological structures under appropriate parameters. Second, our experiments surprisingly show that the simple generalized random graph model outperforms the two proposed models. Third, we find that a series of spatial networks under global optimization of wire length, including the electronic circuits, brain networks, neuronal networks and transportation networks, have high s-metric values close to those of the corresponding generalized random graph models. These s-metric observations explain why the generalized random graph model can match the electronic circuits and the brain networks well from a probabilistic viewpoint, and distinguish their structures from self-organized spatial networks, such as the Internet.

Topological analysis of specific spatial complex networks

Although recent research has shown that the complexity of a network depends on its structural organization, which is linked to the functional constraints the network must satisfy, there is still no systematic study on how to distinguish topological structure and measure the corresponding structural complexity of complex networks. In this paper, we propose the first consistent framework for distinguishing and measuring the structural complexity of real-world complex networks. In terms of the smallest d of the dK model with high-order constraints necessary for fitting real networks, we can classify real-world networks into different structural complexity levels. We demonstrate the approach by measuring and classifying a variety of real-world networks, including biological and technological networks, small-world and non-small-world networks, and spatial and non-spatial networks.

/pdf/characterizing-the-structural-complexity-of-real-world-pg7wme7mug.pdf

Characterizing the Structural Complexity of Real-World Complex Networks

Although the emergence of motifs certainly reflects functional constraints and design principles underlying complex networks, our studies on digital circuits and a transcriptional regulatory network indicate that motifs are not “basic” functional building blocks as many claimed before, since most motif instances do not represent independent units that are functionally separable from the rest of the network. We also argue that the over-representation of motifs arises not because motifs are fundamental network fragments, but as a result of (1) predesigned basic functional modules consisting of motif-clusters, and (2) cross-module connections involving buses in the circuits and global regulators in the transcriptional regulatory network.

On motifs and functional modules in complex networks

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Papers

Automated benchmark model generators for model-based diagnostic inference

Topological analysis of specific spatial complex networks

Characterizing the Structural Complexity of Real-World Complex Networks

Characterizing the Structural Complexity of Real-World Complex Networks

On motifs and functional modules in complex networks