Public Goods in Networks with Constraints on Sharing

This paper considers incentives to provide goods that are partially excludable along social links. We introduce a model in which each individual in a networked society makes a two-pronged decision: (i) decide how much of the good to provide, and (ii) decide which subset of neighbours to nominate as co-beneficiaries. An outcome specifies an endogenous subnetwork generated by nominations and a public goods game occurring over the realised subnetwork. We show the existence of specialised pure strategy Nash equilibria: those in which some individuals (the Drivers) contribute while the remaining individuals (the Passengers) free ride. We then consider how the set of efficient specialised equilibria vary as the constraints on sharing are relaxed and we show a monotonicity result. Finally, we introduce dynamics and show that only specialised equilibria can be stable against individuals unilaterally changing their provision level

The asymptotic number of prefix normal words

We show that the number of prefix normal binary words of length $n$ is $2^{n-\Theta((\log n)^2)}$. We also show that the maximum number of binary words of length $n$ with a given fixed prefix normal form is $2^{n-O(\sqrt{n\log n})}$.Comment: 9 page

Connectivity of the Uniform Random Intersection Graph

A \emph{uniform random intersection graph} $G(n,m,k)$ is a random graph constructed as follows. Label each of $n$ nodes by a randomly chosen set of $k$ distinct colours taken from some finite set of possible colours of size $m$. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks. Such graphs arise in particular when modelling the network graph of the well known key predistribution technique due to Eschenauer and Gligor. The paper determines the threshold for connectivity of the graph $G(n,m,k)$ when $n\to \infty$ with $k$ a function of $n$ such that $k\geq 2$ and $m=\lfloor n^\alpha\rfloor$ for some fixed positive real number $\alpha$. In this situation, $G(n,m,k)$ is almost surely connected when $$\liminf k^2n/m\log n>1,$$ and $G(n,m,k)$ is almost surely disconnected when $$\limsup k^2n/m\log n<1.$$Comment: 19 pages New version with rewritten intro, and a discussion section added. The results and proofs are unchanged from the previous versio

Pegging Graphs Yields a Small Diameter

We consider the following process for generating large random cubic graphs. Starting with a given graph, repeatedly add edges that join the midpoints of two randomly chosen edges. We show that the growing graph asymptotically almost surely has logarithmic diameter. This process is motivated by a particular type of peer-to-peer network. Our method extends to similar processes that generate regular graphs of higher degre

Die zeitgenössische Ruine.

Stefanie Gerke untersucht in der vorliegenden Dissertation die Frage, in welcher Weise sich Künstler*innen mit verfallender Architektur der 1960er und 1970er Jahre beschäftigen. Sie analysiert Fotografie-, Film- und Videoarbeiten seit 1990, die gezielt ikonografische Traditionen des Topos Ruine erweitern, aktualisieren und unter den Bedingungen ihres Mediums neu orientieren. War die Ruinenikonografie bislang durch die Ästhetik der Romantik sowie Georg Simmels geprägt, der den Reiz verfallender Bauten in der Rückeroberung durch die Natur sah, bestehen die modernen Ruinen aus industriellen, schwer vergänglichen Baustoffen und sind häufig die Folge menschlicher statt natürlicher Zerstörung. Die Analyse der Werke zeitgenössischer Künstler*innen, die sich maßgeblich mit dem Verfall von Nachkriegsbauten beschäftigt haben, führt in der Dissertation daher zu einer Aktualisierung der für den Ruinenbegriff prägenden ästhetischen Kategorien des Pittoresken und des Erhabenen für das 21. Jahrhundert. Einige der analysierten Arbeiten nutzen Nachkriegsarchitektur als zeitgemäßes Symbol der Vergänglichkeit und widmen sich in pittoresker Tradition vor allem Wahrnehmungsfragen, um die Konstruktionsmechanismen ihrer eigenen Medien zu reflektieren, wie etwa die 16-mm-Filme Tacita Deans (*1965), die Polaroid-Serie Cyprien Gaillards (*1980) und die Fotomontagen Beate Gütschows (*1970). Andere Künstler gewinnen den radikalen Bildern abgerissener oder dem Abbruch geweihter, gescheiterter Nachkriegsarchitektur und den damit einhergehenden sozialen Folgen eine ästhetische Komponente ab, die in der Tradition des Erhabenen Lust und Schrecken miteinander verbindet, wie etwa Julian Rosefeldt (*1965), Clemens von Wedemeyer (*1974) und Tobias Zielony (*1973). Die Dissertation verortet sich nicht nur innerhalb der Diskussion um das Erbe der Nachkriegsmoderne, sondern steckt systematisch ein Feld ab, in dem sich zeitgenössische Kunst momentan bewegt.In this dissertation, Stefanie Gerke investigates the question of how artists deal with decaying architecture of the 1960s and 1970s. She analyzes photographic, film, and video works since 1990 that specifically expand the iconographic tradition of the ruin, update it, and rearticulate it within the specific conditions of new media. While the iconography of ruins was previously influenced by the aesthetics of Romanticism as well as Georg Simmel, who saw the appeal of decaying buildings in their reconquest by nature, modern ruins are made of industrial building materials that are difficult to decompose and are often the result of human rather than natural destruction. In the dissertation, the analysis of the works of contemporary artists who are dealing with the decay of post-war buildings therefore leads to an update of two aesthetic categories that characterize the concept of ruins, for the 21st century: the picturesque and the sublime. Some of the works analyzed use postwar architecture as a contemporary symbol of transience and, in the picturesque tradition, devote themselves primarily to questions of perception in order to reflect on the construction mechanisms of their own media, such as Tacita Dean's (*1965) 16mm films, Cyprien Gaillard's (*1980) Polaroid series, and Beate Gütschow's (*1970) photomontages. Other artists extract an aesthetic component from the radical images of demolished or condemned, failed postwar architecture and the accompanying social consequences, which combines pleasure and horror in the tradition of the sublime, such as Julian Rosefeldt and Piero Steinle (*1965/1959), Clemens von Wedemeyer (*1974), and Tobias Zielony (*1973). The dissertation not only locates itself within the discussion about the legacy of postwar architecture, but also systematically delineates a field in which contemporary art is currently active

Successive shortest paths in complete graphs with random edge weights

Consider a complete graph Kn with edge weights drawn independently from a uniform distribution U(0,1). The weight of the shortest (minimum-weight) path P1 between two given vertices is known to be ln n/n, asymptotically. Define a second-shortest path P2 to be the shortest path edge-disjoint from P1, and consider more generally the shortest path Pk edge-disjoint from all earlier paths. We show that the cost Xk of Pk converges in probability to 2k/n + ln n/n uniformly for all k ≤ n − 1. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterize the collectively cheapest k edge-disjoint paths, that is, a minimum-cost k-flow. We also obtain the expectation of Xk conditioned on the existence of Pk

Proximity and Remoteness in Directed and Undirected Graphs

Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well

Proper orientation number of triangle-free bridgeless outerplanar graphs

An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation \emph{proper} if neighbouring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $\vec{\chi}(G)$, is the minimum maximum in-degree of a proper orientation of G. Araujo et al. (Theor. Comput. Sci. 639 (2016) 14--25) asked whether there is a constant $c$ such that $\vec{\chi}(G)\leq c$ for every outerplanar graph $G$ and showed that $\vec{\chi}(G)\leq 7$ for every cactus $G.$ We prove that $\vec{\chi}(G)\leq 3$ if $G$ is a triangle-free $2$-connected outerplanar graph and $\vec{\chi}(G)\leq 4$ if $G$ is a triangle-free bridgeless outerplanar graph