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

The Internet has led to the creation of a digital society, where (almost) everything is connected and is accessible from anywhere. However, despite their widespread adoption, traditional IP networks are complex and very hard to manage. It is both difficult to configure the network according to predefined policies, and to reconfigure it to respond to faults, load, and changes. To make matters even more difficult, current networks are also vertically integrated: the control and data planes are bundled together. Software-defined networking (SDN) is an emerging paradigm that promises to change this state of affairs, by breaking vertical integration, separating the network's control logic from the underlying routers and switches, promoting (logical) centralization of network control, and introducing the ability to program the network. The separation of concerns, introduced between the definition of network policies, their implementation in switching hardware, and the forwarding of traffic, is key to the desired flexibility: by breaking the network control problem into tractable pieces, SDN makes it easier to create and introduce new abstractions in networking, simplifying network management and facilitating network evolution. In this paper, we present a comprehensive survey on SDN. We start by introducing the motivation for SDN, explain its main concepts and how it differs from traditional networking, its roots, and the standardization activities regarding this novel paradigm. Next, we present the key building blocks of an SDN infrastructure using a bottom-up, layered approach. We provide an in-depth analysis of the hardware infrastructure, southbound and northbound application programming interfaces (APIs), network virtualization layers, network operating systems (SDN controllers), network programming languages, and network applications. We also look at cross-layer problems such as debugging and troubleshooting. In an effort to anticipate the future evolution of this new paradigm, we discuss the main ongoing research efforts and challenges of SDN. In particular, we address the design of switches and control platforms—with a focus on aspects such as resiliency, scalability, performance, security, and dependability—as well as new opportunities for carrier transport networks and cloud providers. Last but not least, we analyze the position of SDN as a key enabler of a software-defined environment.

https://publications.uni.lu/bitstream/10993/20596/1/survey_on_SDN.pdf

Software-Defined Networking: A Comprehensive Survey

Software-Defined Networking (SDN) is an emerging paradigm that promises to change this state of affairs, by breaking vertical integration, separating the network's control logic from the underlying routers and switches, promoting (logical) centralization of network control, and introducing the ability to program the network. The separation of concerns introduced between the definition of network policies, their implementation in switching hardware, and the forwarding of traffic, is key to the desired flexibility: by breaking the network control problem into tractable pieces, SDN makes it easier to create and introduce new abstractions in networking, simplifying network management and facilitating network evolution. In this paper we present a comprehensive survey on SDN. We start by introducing the motivation for SDN, explain its main concepts and how it differs from traditional networking, its roots, and the standardization activities regarding this novel paradigm. Next, we present the key building blocks of an SDN infrastructure using a bottom-up, layered approach. We provide an in-depth analysis of the hardware infrastructure, southbound and northbound APIs, network virtualization layers, network operating systems (SDN controllers), network programming languages, and network applications. We also look at cross-layer problems such as debugging and troubleshooting. In an effort to anticipate the future evolution of this new paradigm, we discuss the main ongoing research efforts and challenges of SDN. In particular, we address the design of switches and control platforms -- with a focus on aspects such as resiliency, scalability, performance, security and dependability -- as well as new opportunities for carrier transport networks and cloud providers. Last but not least, we analyze the position of SDN as a key enabler of a software-defined environment.

Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be captured at the level of individual nodes and edges. However, higher-order organization of complex networks—at the level of small network subgraphs—remains largely unknown. Here, we develop a generalized framework for clustering networks on the basis of higher-order connectivity patterns. This framework provides mathematical guarantees on the optimality of obtained clusters and scales to networks with billions of edges. The framework reveals higher-order organization in a number of networks, including information propagation units in neuronal networks and hub structure in transportation networks. Results show that networks exhibit rich higher-order organizational structures that are exposed by clustering based on higher-order connectivity patterns.

/pdf/higher-order-organization-of-complex-networks-3n5pj1c6nh.pdf

Higher-order organization of complex networks

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Call detail records (CDRs) have recently been used in studying different aspects of human mobility. While CDRs provide a means of sampling user locations at large population scales, they may not sample all locations proportionate to the visitation frequency of a user, owing to sparsity in time and space of voice-calls, thereby introducing a bias. Also, as the rate of sampling is inherently dependent on the calling frequencies of an individual, high voice-call activity users are often chosen for conducting a meaningful study. Such a selection process can, inadvertently, lead to a biased view as high frequency callers may not always be representative of an entire population. With the advent of 3G technology and wide adoption of smart-phones, cellular devices have become versatile end-hosts. As the data accessed on these devices does not always require human initiation, it affords us with an unprecedented opportunity to validate the utility of CDRs for studying human mobility. In this work, we investigate various metrics for human mobility studied in literature for over a million cellular users in the San Francisco bay-area, for over a month. Our findings reveal that although the voice-call process does well to sample significant locations, such as home and work, it may in some cases incur biases in capturing the overall spatio-temporal characteristics of individual human mobility. Additionally, we motivate an "artificially" imposed sampling process, vis-a-vis the voice-call process with the same average intensity. We observe that in many cases such an imposed sampling process yields better performance results based on the usual metrics like entropies and marginal distributions used often in literature.

Are call detail records biased for sampling human mobility

https://www-users.cs.umn.edu/~boley/publications/papers/Laplacian10-LAA.pdf

Commute Times for a Directed Graph using an Asymmetric Laplacian

We present SAMPLES: Self Adaptive Mining of Persistent LExical Snippets; a systematic framework for classifying network traffic generated by mobile applications. SAMPLES constructs conjunctive rules, in an automated fashion, through a supervised methodology over a set of labeled flows (the training set). Each conjunctive rule corresponds to the lexical context, associated with an application identifier found in a snippet of the HTTP header, and is defined by: (a) the identifier type, (b) the HTTP header-field it occurs in, and (c) the prefix/suffix surrounding its occurrence. Subsequently, these conjunctive rules undergo an aggregate-and-validate step for improving accuracy and determining a priority order. The refined rule-set is then loaded into an application-identification engine where it operates at a per flow granularity, in an extract-and-lookup paradigm, to identify the application responsible for a given flow. Thus, SAMPLES can facilitate important network measurement and management tasks --- e.g. behavioral profiling [29], application-level firewalls [21,22] etc. --- which require a more detailed view of the underlying traffic than that afforded by traditional protocol/port based methods. We evaluate SAMPLES on a test set comprising 15 million flows (approx.) generated by over 700 K applications from the Android, iOS and Nokia market-places. SAMPLES successfully identifies over 90% of these applications with 99% accuracy on an average. This, in spite of the fact that fewer than 2% of the applications are required during the training phase, for each of the three market places. This is a testament to the universality and the scalability of our approach. We, therefore, expect SAMPLES to work with reasonable coverage and accuracy for other mobile platforms --- e.g. BlackBerry and Windows Mobile --- as well.

/pdf/samples-self-adaptive-mining-of-persistent-lexical-snippets-5blz2qrii4.pdf

SAMPLES: Self Adaptive Mining of Persistent LExical Snippets for Classifying Mobile Application Traffic

We explore the geometry of complex networks in terms of an n-dimensional Euclidean embedding represented by the Moore-Penrose pseudo-inverse of the graph Laplacian $(\bb L^+)$. The squared distance of a node $i$ to the origin in this n-dimensional space $(l^+_{ii})$, yields a topological centrality index $(\mathcal{C}^{*}(i) = 1/l^+_{ii})$ for node $i$. In turn, the sum of reciprocals of individual node structural centralities, $\sum_{i}1/\mathcal{C}^*(i) = \sum_{i} l^+_{ii}$, i.e. the trace of $\bb L^+$, yields the well-known Kirchhoff index $(\mathcal{K})$, an overall structural descriptor for the network. In addition to this geometric interpretation, we provide alternative interpretations of the proposed indices to reveal their true topological characteristics: first, in terms of forced detour overheads and frequency of recurrences in random walks that has an interesting analogy to voltage distributions in the equivalent electrical network; and then as the average connectedness of $i$ in all the bi-partitions of the graph. These interpretations respectively help establish the topological centrality $(\mathcal{C}^{*}(i))$ of node $i$ as a measure of its overall position as well as its overall connectedness in the network; thus reflecting the robustness of node $i$ to random multiple edge failures. Through empirical evaluations using synthetic and real world networks, we demonstrate how the topological centrality is better able to distinguish nodes in terms of their structural roles in the network and, along with Kirchhoff index, is appropriately sensitive to perturbations/rewirings in the network.

Geometry of Complex Networks and Topological Centrality

We explore the geometry of complex networks in terms of an n -dimensional Euclidean embedding represented by the Moore–Penrose pseudo-inverse of the graph Laplacian ( L + ) . The squared distance of a node i to the origin in this n -dimensional space ( l i i + ) , yields a topological centrality index, defined as C ∗ ( i ) = 1 / l i i + . In turn, the sum of reciprocals of individual node centralities, ∑ i 1 / C ∗ ( i ) = ∑ i l i i + , or the trace of L + , yields the well-known Kirchhoff index ( K ) , an overall structural descriptor for the network. To put into context this geometric definition of centrality, we provide alternative interpretations of the proposed indices that connect them to meaningful topological characteristics — first, as forced detour overheads and frequency of recurrences in random walks that has an interesting analogy to voltage distributions in the equivalent electrical network; and then as the average connectedness of i in all the bi-partitions of the graph. These interpretations respectively help establish the topological centrality ( C ∗ ( i ) ) of node i as a measure of its overall position as well as its overall connectedness in the network; thus reflecting the robustness of i to random multiple edge failures. Through empirical evaluations using synthetic and real world networks, we demonstrate how the topological centrality is better able to distinguish nodes in terms of their structural roles in the network and, along with Kirchhoff index, is appropriately sensitive to perturbations/re-wirings in the network.

Gyan Ranjan

Papers

Are call detail records biased for sampling human mobility

Commute Times for a Directed Graph using an Asymmetric Laplacian

SAMPLES: Self Adaptive Mining of Persistent LExical Snippets for Classifying Mobile Application Traffic

Geometry of Complex Networks and Topological Centrality

Geometry of Complex Networks and Topological Centrality