The t-stability number of a random graph

Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets).Comment: 25 pages; v2 has 30 pages and is identical to the journal version apart from formatting and a minor amendment to Lemma 8 (and its proof on p. 21

Strongly reinforced P\'olya urns with graph-based competition

We introduce a class of reinforcement models where, at each time step $t$, one first chooses a random subset $A_t$ of colours (independent of the past) from $n$ colours of balls, and then chooses a colour $i$ from this subset with probability proportional to the number of balls of colour $i$ in the urn raised to the power $\alpha>1$. We consider stability of equilibria for such models and establish the existence of phase transitions in a number of examples, including when the colours are the edges of a graph, a context which is a toy model for the formation and reinforcement of neural connections.Comment: 32 pages, 5 figure

Combinatorics and Metric Geometry

This thesis consists of an introduction and seven chapters, each devoted to a different combinatorial problem. In Chapters 1 and 2, we consider the main subject of this thesis; the sharp stability of the Brunn-Minkowski inequality (BM). This celebrated theorem from the 19th century asserts that for bodies A,B $\subset$ $\mathbb{R}$$^{k}$, we have |A + B|$^{1/k}$ $\geq$ |A|$^{1/k}$ + |B|$^{1/k}$, where |$\cdot$| is the Lebesgue measure and A + B := {a + b : a $\in$ A, b $\in$ B} is the Minkowski sum. Moreover, we have equality if and only if A,B are homothetic convex sets. The stability question, studied in many papers, asks how the distance to equality in BM relates to the distance from A,B to homothetic convex sets. In particular, given Brunn-Minkowsi deficit $\delta$ := |A+B|$^{1/k}$ / |A|$^{1/k}$ + |B|$^{1/k}$ -1, and normalized volume ratio $\textit{t}$ := |A|$^{1/k}$ / |A|$^{1/k}$ + |B|$^{1/k}$, what is the best bound one can find on $\omega$ := |K$_{A}$ \ A| / |A| + |K$_{B}$ \ B| / |B|, where K$_{A}$ $\supset$ A, and K$_{B}$ $\supset$ B are homothetic convex sets of minimal size? In Chapter 2, we prove a conjecture by Figalli and Jerison establishing the sharp stability for homothetic sets. In particular, we show that for homothetic sets, we have $\omega$ = O$_{k}$($\delta$t$^{-1}$), for $\delta$ sufficiently small. In Chapter 3, we establish the sharp stability for planar sets, i.e. we show that for planar sets and $\delta$ sufficiently small, we have $\omega$ = O($\delta$$^{1/2}$t$^{-1/2}$). A crucial result in Chapter 3 shows that for any $\epsilon$ > 0, if $\delta$ is sufficiently small, then we have |co(A + B) \ (A + B)| $\leq$ (1 + $\epsilon$)(|co(A) \ A| + |co(B) \ B|). In Chapter 4, we consider a reconstruction problem for functions on graphs. Given a function $\textit{f}$:V(G) $\rightarrow$ [k] on the vertices of a graph G and a random walk (U$_{i}$)$^{{\infty}}_{i = 1}$ on that graph, can we reconstruct $\textit{f}$ (up to automorphisms) based on just ($\textit{f}$(U$_{i}$)$^{{\infty}}_{i = 1}$? Gross and Grupel showed this was not generally possible on the hypercube, by constructing non-isomorphic $\textit{locally p-biased}$ sets $\textit{X}$, so that for each vertex $\textit{v}$ the fraction of neighbours which is in $\textit{X}$ is exactly $\textit{p}$. Answering a question of Gross and Grupel, we construct uncountably many non-isomorphic partitions of $\mathbb{Z}$$^{k}$ into 2k parts such that every element of $\mathbb{Z}$$^{k}$ has exactly one neighbour in each part. As a result, we find $\textit{locally p-biased}$ sets for all $\textit{p = c/2n}$ with $\textit{c}$ $\in$ {0, ... , 2n}. In Chapter 5, we prove the complete graph case of the bunkbed conjecture. Given a graph G, let the bunkbed graph BB(G) be the graph G$\Box$K$_{2}$, i.e. the graph obtained from considering two copies of G and connecting equivalent vertices with an edge. The bunkbed conjecture posed by Kasteleyn in 1985 asserts the very intuitive statement that when considering percolation with uniform parameter p, we have $\mathbb{P}$(u$_{1}$ $\leftrightarrow$ v$_{1}$) $\geq$ $\mathbb{P}$(u$_{1}$ $\leftrightarrow$ v$_{2}$), i.e. a vertex has a higher probability of being connected to a vertex in the same copy of G than being connected to the equivalent vertex in the other copy of G. In Chapter 6, we consider the (t,r) broadcast domination number, a generalisation of the domination number in graphs. In this form of domination, we consider a set T $\subset$ V(G) of towers which broadcast at strength t, where broadcast strength decays linearly with distance in the graph. A set of towers is (t,r) broadcast dominating if every vertex in the graph receives at least r signal from all towers combined. More formally, the (t,r) broadcast domination number of a graph G is the minimal cardinality of a set T $\subset$ V(G) such that for every vertex v $\in$ V(G), we have $^{{\Sigma}}_{u {{\in}} T}$ max{t - d(u,v),0} $\geq$ r. Proving a conjecture by Drews, Harris, and Randolph, we establish that the minimal asymptotical density of (t,3) broadcasting subset of $\mathbb{Z}$$^{2}$ is the same as the minimal asymptotical density of a (t-1,1) broadcasting subset of $\mathbb{Z}$$^{2}$. In Chapter 7, we consider the eternal game chromatic number, a version of the game chromatic number in which the game continues after all vertices have been coloured. We show that with high probability $\chi$$^\infty_g$ (G$_{n,p}$) = (p/2 + o(1))n for odd n, and also for even n when p = 1/k for some k $\in$ $\mathbb{N}$. The upper bound applies for even n and any other value of p as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed by Klostermeyer and Mendoza. In Chapter 8, we consider the bridge-burning cops and robbers game, a version of the game where after a robber moves over an edge, the edge is removed from the graph. Proving a generalization of a conjecture by Kinnersley and Peterson, we establish the asymptotically maximal capture time in this game for graphs with bridge-burning cops number at least three. In particular, we show that this maximal capture time grows as k$^{-O(k)}$n$^{k+2}$, where k $\geq$ 3 is the bridge burning cop number and n is the number of vertices of the graph

Synchronization in random networks with given expected degree sequences

Synchronization in random networks with given expected degree sequences is studied. We also investigate in details the synchronization in networks whose topology is described by classical random graphs, power-law random graphs and hybrid graphs when N goes to infinity. In particular, we show that random graphs almost surely synchronize. We also show that adding small number of global edges to a local graph makes the corresponding hybrid graph to synchroniz

Building Damage-Resilient Dominating Sets in Complex Networks against Random and Targeted Attacks

We study the vulnerability of dominating sets against random and targeted node removals in complex networks. While small, cost-efficient dominating sets play a significant role in controllability and observability of these networks, a fixed and intact network structure is always implicitly assumed. We find that cost-efficiency of dominating sets optimized for small size alone comes at a price of being vulnerable to damage; domination in the remaining network can be severely disrupted, even if a small fraction of dominator nodes are lost. We develop two new methods for finding flexible dominating sets, allowing either adjustable overall resilience, or dominating set size, while maximizing the dominated fraction of the remaining network after the attack. We analyze the efficiency of each method on synthetic scale-free networks, as well as real complex networks

On the Stability of Community Detection Algorithms on Longitudinal Citation Data

There are fundamental differences between citation networks and other classes of graphs. In particular, given that citation networks are directed and acyclic, methods developed primarily for use with undirected social network data may face obstacles. This is particularly true for the dynamic development of community structure in citation networks. Namely, it is neither clear when it is appropriate to employ existing community detection approaches nor is it clear how to choose among existing approaches. Using simulated data, we attempt to clarify the conditions under which one should use existing methods and which of these algorithms is appropriate in a given context. We hope this paper will serve as both a useful guidepost and an encouragement to those interested in the development of more targeted approaches for use with longitudinal citation data.Comment: 17 pages, 7 figures, presenting at Applications of Social Network Analysis 2009, ETH Zurich Edit, August 17, 2009: updated abstract, figures, text clarification

The stability of a graph partition: A dynamics-based framework for community detection

Recent years have seen a surge of interest in the analysis of complex networks, facilitated by the availability of relational data and the increasingly powerful computational resources that can be employed for their analysis. Naturally, the study of real-world systems leads to highly complex networks and a current challenge is to extract intelligible, simplified descriptions from the network in terms of relevant subgraphs, which can provide insight into the structure and function of the overall system. Sparked by seminal work by Newman and Girvan, an interesting line of research has been devoted to investigating modular community structure in networks, revitalising the classic problem of graph partitioning. However, modular or community structure in networks has notoriously evaded rigorous definition. The most accepted notion of community is perhaps that of a group of elements which exhibit a stronger level of interaction within themselves than with the elements outside the community. This concept has resulted in a plethora of computational methods and heuristics for community detection. Nevertheless a firm theoretical understanding of most of these methods, in terms of how they operate and what they are supposed to detect, is still lacking to date. Here, we will develop a dynamical perspective towards community detection enabling us to define a measure named the stability of a graph partition. It will be shown that a number of previously ad-hoc defined heuristics for community detection can be seen as particular cases of our method providing us with a dynamic reinterpretation of those measures. Our dynamics-based approach thus serves as a unifying framework to gain a deeper understanding of different aspects and problems associated with community detection and allows us to propose new dynamically-inspired criteria for community structure.Comment: 3 figures; published as book chapte