The glass ceiling effect has been defined in a recent US Federal Commission report as "the unseen, yet unbreakable barrier that keeps minorities and women from rising to the upper rungs of the corporate ladder, regardless of their qualifications or achievements". It is well documented that many societies and organizations exhibit a glass ceiling. In this paper we formally define and study the glass ceiling effect in social networks and propose a natural mathematical model, called the biased preferential attachment model, that partially explains the causes of the glass ceiling effect. This model consists of a network composed of two types of vertices, representing two sub-populations, and accommodates three well known social phenomena: (i) the "rich get richer" mechanism, (ii) a minority-majority partition, and (iii) homophily. We prove that our model exhibits a strong moment glass ceiling effect and that all three conditions are necessary, i.e., removing any one of them will prevent the appearance of a glass ceiling effect. Additionally, we present empirical evidence taken from a mentor-student network of researchers (derived from the DBLP database) that exhibits both a glass ceiling effect and the above three phenomena.

To meet the stringent requirements on the maximally tolerable disruptions of traffic under link failures, many communication networks feature some sort of static failover mechanism for fast rerouting. However, configuring such static failover mechanisms to achieve a high degree of robustness is known to be challenging, in particular when packet tagging or dynamic node state cannot be used. This paper initiates the systematic study of such local fast failover mechanisms which not only provide connectivity guarantees, even under multiple link failures, but also account for the quality of the resulting failover routes, with respect to locality (i.e., route length) and congestion. Failover quality has received less attention in the literature so far, yet it is increasingly important to support emerging applications. We first show that there exists an inherent tradeoff in terms of achievable locality and congestion of failover routes. We then present CASA, an algorithm providing a high degree of robustness as well as a provable quality of fast rerouting. CASA combines two crucial static resilient routing techniques: combinatorial designs and arc-disjoint arborescences. We complement our formal analysis with a simulation study, in which we compare our algorithms with the state-of-the-art in different scenarios and show benefits in terms of stretch, load, and resilience.

Abstract-Reliable and highly available computer networks must implement resilient fast rerouting mechanisms: upon a link or node failure, an alternative route is determined quickly, without involving the network control plane. Designing such fast failover mechanisms capable of dealing with multiple concurrent failures however is challenging, as failover rules need to be installed proactively, i.e., ahead of time, without knowledge of the actual failures happening at runtime. Indeed, only little is known today about the design of resilient routing algorithms. This paper presents a deterministic local failover mechanism which we prove to result in a minimum network load for a wide range of communication patterns, solving an open problem. Our mechanism relies on the key insight that resilient routing essentially constitutes a distributed algorithm without coordination. Accordingly, we build upon the theory of combinatorial designs and develop a novel deterministic failover mechanism based on symmetric block design theory which tolerates a maximal number of Ω(n) link failures in an n-node network and in the worstcase, while always ensuring routing connectivity. In particular, we show that at least Ω(φ 2 ) link failures are needed to generate a maximum link load of at least φ, which matches an existing bound on the number of link failures needed for an optimal failover scheme. We complement our formal analysis with simulations, showing that our approach outperforms prior schemes not only in the worst-case.

Network failures are frequent and disruptive, and can significantly reduce the throughput even in highly connected and regular networks such as datacenters. While many modern networks support some kind of local fast failover to quickly reroute flows encountering link failures to new paths, employing such mechanisms is known to be non-trivial, as conditional failover rules can only depend on local failure information. While over the last years, important insights have been gained on how to design failover schemes providing high resiliency, existing approaches have the shortcoming that the resulting failover routes may be unnecessarily long, i.e., they have a large stretch compared to the original route length. This is a serious drawback, as long routes entail higher latencies and introduce loads, which may cause the rerouted flows to interfere with existing flows and harm throughput. This paper presents the first deterministic local fast failover algorithms providing provable resiliency and failover route lengths, even in the presence of many concurrent failures. We present stretch-optimal failover algorithms for different network topologies, including multi-dimensional grids, hypercubes and Clos networks, as they are frequently deployed in the context of HPC clusters and datacenters. We show that the computed failover routes are optimal in the sense that no failover algorithm can provide shorter paths for a given number of link failures.

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