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

Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.

Epidemic processes in complex networks

Many ``real-world'' networks are clearly defined while most ``social'' networks are to some extent subjective. Indeed, the accuracy of empirically-determined social networks is a question of some concern because individuals may have distinct perceptions of what constitutes a social link. One unambiguous type of connection is sexual contact. Here we analyze data on the sexual behavior of a random sample of individuals, and find that the cumulative distributions of the number of sexual partners during the twelve months prior to the survey decays as a power law with similar exponents $\alpha \approx 2.4$ for females and males. The scale-free nature of the web of human sexual contacts suggests that strategic interventions aimed at preventing the spread of sexually-transmitted diseases may be the most efficient approach.

The Web of Human Sexual Contacts

https://www.linkgroup.hu/docs/13PharmTher.pdf

Structure and dynamics of molecular networks: A novel paradigm of drug discovery: A comprehensive review

Terrorist attacks on transportation networks have traumatized modern societies. With a single blast, it has become possible to paralyze airline traffic, electric power supply, ground transportation or Internet communication. How and at which cost can one restructure the network such that it will become more robust against a malicious attack? We introduce a new measure for robustness and use it to devise a method to mitigate economically and efficiently this risk. We demonstrate its efficiency on the European electricity system and on the Internet as well as on complex networks models. We show that with small changes in the network structure (low cost) the robustness of diverse networks can be improved dramatically whereas their functionality remains unchanged. Our results are useful not only for improving significantly with low cost the robustness of existing infrastructures but also for designing economically robust network systems.

/pdf/mitigation-of-malicious-attacks-on-networks-3vgg1gyljk.pdf

Mitigation of malicious attacks on networks

An individual-based approach to SIR epidemics in contact networks

Intentional islanding of a power system can be an emergency response for isolating failures that might propagate and lead to major disturbances. Some of the islanding techniques suggested previously do not consider the power flow model; others are designed to minimize load shedding only within the islands. Often these techniques are computationally expensive. We aim to find approaches to partition power grids into islands to minimize the load shedding not only in the region where the failures start, but also in the topological complement of the region. We propose a new constraint programming formulation for optimal islanding in power grid networks. This technique works efficiently for small networks but becomes expensive as size increases. To address the scalability problem, we propose two grid partitioning methods based on modularity, properly modified to take into account the power flow model. They are modifications of the Fast Greedy algorithm and the Bloom algorithm, and are polynomial in running time. We tested these methods on the available IEEE test systems. The Bloom type method is faster than the Fast Greedy type, and can potentially provide results in networks with thousands of nodes. Our methods provide solutions which retain at least 40-50% of the system load. Overall, our methods efficiently balance load shedding and scalability.

/pdf/optimal-intentional-islanding-to-enhance-the-robustness-of-2zgv0mp83c.pdf

Optimal intentional islanding to enhance the robustness of power grid networks

With increasingly ambitious initiatives such as GENI and FIND that seek to design the future Internet, it becomes imperative to define the characteristics of robust topologies, and build future networks optimized for robustness. This paper investigates the characteristics of network topologies that maintain a high level of throughput in spite of multiple attacks. To this end, we select network topologies belonging to the main network models and some real world networks. We consider three types of attacks: removal of random nodes, high degree nodes, and high betweenness nodes. We use elasticity as our robustness measure and, through our analysis, illustrate that different topologies can have different degrees of robustness. In particular, elasticity can fall as low as 0.8% of the upper bound based on the attack employed. This result substantiates the need for optimized network topology design. Furthermore, we implement a tradeoff function that combines elasticity under the three attack strategies and considers the cost of the network. Our extensive simulations show that, for a given network density, regular and semi-regular topologies can have higher degrees of robustness than heterogeneous topologies, and that link redundancy is a sufficient but not necessary condition for robustness.

/pdf/characterizing-the-robustness-of-complex-networks-knxxpclm45.pdf

Characterizing the Robustness of Complex Networks

Viral conductance: Quantifying the robustness of networks with respect to spread of epidemics

This paper presents an optimal control problem formulation to minimize the total number of infection cases during the spread of susceptible-infected-recovered SIR epidemics in contact networks. In the new approach, contact weighted are reduced among nodes and a global minimum contact level is preserved in the network. In addition, the infection cost and the cost associated with the contact reduction are linearly combined in a single objective function. Hence, the optimal control formulation addresses the tradeoff between minimization of total infection cases and minimization of contact weights reduction. Using Pontryagin theorem, the obtained solution is a unique candidate representing the dynamical weighted contact network. To find the near-optimal solution in a decentralized way, we propose two heuristics based on Bang-Bang control function and on a piecewise nonlinear control function, respectively. We perform extensive simulations to evaluate the two heuristics on different networks. Our results show that the piecewise nonlinear control function outperforms the well-known Bang-Bang control function in minimizing both the total number of infection cases and the reduction of contact weights. Finally, our results show awareness of the infection level at which the mitigation strategies are effectively applied to the contact weights.

Mina Youssef

Papers

An individual-based approach to SIR epidemics in contact networks

Optimal intentional islanding to enhance the robustness of power grid networks

Characterizing the Robustness of Complex Networks

Viral conductance: Quantifying the robustness of networks with respect to spread of epidemics

Mitigation of epidemics in contact networks through optimal contact adaptation.