Leaf to leaf path lengths in trees of given degree sequence

We study a scale‐free random graph process in which the number of edges added at each step increases. This differs from the standard model in which a fixed number,m, of edges are added at each step.Letf(t) be the number of edges added at stept. In the standard scale‐free model,f(t) =mconstant, whereas in this paper we considerf(t) = [tc],c> 0. Such a graph process, in which the number of edges grows non‐linearly with the number of vertices is said to have accelerating growth.We analyze both an undirected and a directed process. The power law of the degree sequence of these processes exhibits widely differing behavior.For the undirected process, the terminal vertex of each edge is chosen by preferential attachment based on vertex degree. Whenf(t) =mconstant, this is the standard scale‐free model, and the power law of the degree sequence is 3. Whenf(t) = [tc],c< 1, the degree sequence of the process exhibits a power law with parameterx= (3 −c)/(1 −c). Asc→ 0,x→ 3, which gives a value ofx= 3, as in standard scale‐free model. Thus no more slowly growing monotone functionf(t) alters the power law of this model away fromx= 3. Whenc= 1, so thatf(t) =t, the expected degree of all vertices ist, the vertex degree is concentrated, and the degree sequence does not have a power law.For the directed process, the terminal vertex is chosen proportional to in‐degree plus an additive constant, to allow the selection of vertices of in‐degree zero. For this process whenf(t) =mis constant, the power law of the degree sequence isx= 2 + 1/m. Whenf(t) = [tc],c> 0, the power law becomesx= 1 + 1/(1 +c), which naturally extends the power law to [1,2]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 396–421, 2011

Enumeration of graphs with a heavy-tailed degree sequence

A general critical condition for the emergence of a giant component in random graphs with given degrees

Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p.

The notion of hypergraph generalizes that of graph in the sense that each hyperedge is a non-void subset of the set of vertices, without constraints on its cardinality. A fundamental and widely investigated notion related both to graphs and to hypergraphs is the characterization of their degree sequences, that is the lists of their vertex degrees. Concerning graphs, this problem has been solved in a classical study by Erdi¾?s and Gallai, while no efficient solutions are known for hypergraphs. If we restrict the degree sequences characterization to uniform hypergraphs, several necessary conditions are provided in the literature, but only few sufficient ones: among the latter, a recent one requires to split a sequence into suitable subsequences whose graphicality has to be recursively tested. Unfortunately, such an approach does not allow a direct efficient implementation. We study this problem under a tomographical perspective by adapting an already known reconstruction algorithm that has been defined for regular h-uniform degree sequences to the proposed instances, providing efficiency to the sufficient condition. Furthermore, we extend the set of h-uniform degree sequences whose graphicality can be efficiently tested. This tomographical approach seems extremely promising for further developments.

Emergence of new diseases and elimination of existing diseases is a key public health issue. In mathematical models of epidemics, such phenomena involve the process of infections and recoveries passing through a critical threshold where the basic reproductive ratio is 1. In this paper, we study near-critical behaviour in the context of a susceptible-infective-recovered epidemic on a random (multi)graph on n vertices with a given degree sequence. We concentrate on the regime just above the threshold for the emergence of a large epidemic, where the basic reproductive ratio is 1 + omega (n) n^{-1/3}, with omega (n) tending to infinity slowly as the population size, n, tends to infinity. We determine the probability that a large epidemic occurs, and the size of a large epidemic. Our results require basic regularity conditions on the degree sequences, and the assumption that the third moment of the degree of a random susceptible vertex stays uniformly bounded as n rightarrow infty . As a corollary, we determine the probability and size of a large near-critical epidemic on a standard binomial random graph in the ‘sparse’ regime, where the average degree is constant. As a further consequence of our method, we obtain an improved result on the size of the giant component in a random graph with given degrees just above the critical window, proving a conjecture by Janson and Luczak.

The thesis consists of three chapters. The first chapter introduces the basic notions of graph theory and defines vertex-reconstruction and edge-reconstruction problem. The second chapter and third chapter are devoted to the edge-reconstruction of bi-degreed graphs and bipartite graphs respectively. A bi-degreed graph G is a graph with two degrees d > δ. By elementary arguments we can assume d = δ + 1 and there are at least two vertices of degree δ. Call vertices of degree d "big" vertex and degree δ "small" vertex. Define "symmetric" path of length p Sp to be one with both ends small vertices and all other internal vertices big vertices; define "asymmetric" path of length p Ap to be one with one end a small vertex and all others big vertices. If s(G) is the minimum distance between two small vertices in G, we can show that s(G) is "independent" of G (i.e. it is edge-reconstructable), and that G has at most one nonisomorphic edge-reconstruction H. From this, the concept of "forced move" posed by Dr. Swart is obvious. Using the principle of forced move (and sometimes also "forced edge" posed by Dr. Swart as well), it's easy to derive a few interesting properties, like say G is edge-reconstructable if s(G) is even or if two Ss(G)'s intersect at an internal vertex, etc. Write s for s(G). When s is odd, consider the concept of s - n-chain, which is n Ss's following from end to end. We can show first s - 3-chain and then s - 2-chain cannot exist. Hence Ss's are disjoint. Think of Ss's as "lines" in some geometry. Define two more "distance" functions s1 and s2 such that s1 "represents" the distance from a point to a line and s2 means the distance between to "skew" lines. With the aid of forced move principle again, we can at last prove every bi-degreed graph with at least four edges is edge-reconstructable. A bipartite graph G is a graph whose vertex set V is the disjoint union of two sets v1 and v2 such that every edge joins v1 and v2. By elementary reduction we can assume G to be connected. We define special chains inductively so that it starts at a vertex of minimum degree and always goes to a neighbor or minimum degree. Special chains will be the main tool to prove edge-reconstructability. By G's finiteness, we note they will "terminate" somewhere, and we have three types of termination for them. Let condition A•s be that degree sequence of special chain is edge-reconstructable, condition Bi's be that number of special chains is edge-reconstructable(and some more general variations); condition P's be that the "last vertices" of two special chains be not adjacent; we can prove that all A, Bi and P's should hold inductively in an interlocked way. (This is a big task). Then condition P's can be used to prove G's edge-reconstructability for all three types of termination. We can then prove every bipartite graph with at least four edges is edge-reconstructable.

Sampling from large graphs is an area which is of great interest, particularly with the recent emergence of huge structures such as Online Social Networks. These often contain hundreds of millions of vertices and billions of edges. The large size of these networks makes it computationally expensive to obtain structural properties of the underlying graph by exhaustive search. If we can estimate these properties by taking small but representative samples from the network, then size is no longer a problem.In this paper we develop an analysis of random walks, a commonly used method of sampling from networks. We present a method of biassing the random walk to acquire a complete sample of high degree vertices of social networks, or similar graphs. The preferential attachment model is a common method to generate graphs with a power law degree sequence. For this model, we prove that this sampling method is successful with high probability. We also make experimental studies of the method on various real world networks.For t-vertex graphs G(t) generated by a preferential attachment process, we analyze a biassed random walk which makes transitions along undirected edges {x,y} proportional to [d(x)d(y)]b, where d(x) is the degree of vertex x and b > 0 is a constant parameter. Let S(a) be the set of all vertices of degree at least ta in G(t). We show that for some b approx 2/3, if the biassed random walk starts at an arbitrary vertex of S(a), then with high probability the set S(a) can be discovered completely in ~O(t1-(4/3)a+d) steps, where d is a very small positive constant. The notation ~O ignores poly-log t factors.The preferential attachment process generates graphs with power law 3, so the above example is a special case of this result. For graphs with degree sequence power law c>2 generated by a generalized preferential attachment process, a random walk with transitions along undirected edges {x,y} proportional to (d(x)d(y))(c-2)/2, discovers the set S(a) completely in ~O(t1-a(c-2)+d) steps with high probability. The cover time of the graph is ~O(t).Our results say that if we search preferential attachment graphs with a bias b=(c-2)/2 proportional to the power law c then, (i) we can find all high degree vertices quickly, and (ii) the time to discover all vertices is not much higher than in the case of a simple random walk. We conduct experimental tests on generated networks and real-world networks, which confirm these two properties.

Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λi n vertices of degree i. Essentially, we show that if Σ i(i ‐ 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i ‐2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well‐known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

Let $G=(V,E)$ be a simple connected graph with $n$ vertices, $m$ edges and sequence of vertex degrees $d_1 \ge d_2 \ge \cdots \ge d_n>0$, $d_i=d(v_i)$, where $v_i\in V$. With $i\sim j$ we denote adjacency of vertices $v_i$ and $v_j$. The general sum--connectivity index of graph is defined as $\chi_{\alpha}(G)=\sum_{i\sim j}(d_i+d_j)^{\alpha}$, where $\alpha$ is an arbitrary real number. In this paper we determine relations between $\chi_{\alpha+\beta}(G)$ and $\chi_{\alpha+\beta-1}(G)$, where $\alpha$ and $\beta$ are arbitrary real numbers, and obtain new bounds for $\chi_{\alpha}(G)$. Also, by the appropriate choice of parameters $\alpha$ and $\beta$, we obtain a number of old/new inequalities for different vertex--degree--based topological indices.

Reconstructing the degree sequence and the number of components of an infinite graph

It is a well established fact, that—in the case of classical random graphs like (variants of) G n,p or random regular graphs-spectral methods yield efficient algorithms for clustering (e. g. coiouring or bisection) problems. The theory of large networks emerging recently provides convincing evidence that such networks, albeit looking random in some sense, cannot sensibly be described by classical random graphs. A variety of new types of random graphs have been introduced. One of these types is characterized by the fact that we have a fixed expected degree sequence, that is for each vertex its expected degree is given. Recent theoretical work confirms that spectral methods can be successfully applied to clustering problems for such random graphs, too—provided that the expected degrees are not too small, in fact≥log6 n. In this case however the degree of each vertex is concentrated about its expectation. We show how to remove this restriction and apply spectral methods when the expected degrees are bounded below just by a suitable constant. Our results rely on the observation that techniques developed for the classical sparse G n,p random graph (that is p=c/n) can be transferred to the present situation, when we consider a suitably normalized adjacency matrix: We divide each entry of the adjacency matrix by the product of the expected degrees of the incident vertices. Given the host of spectral techniques developed for G n,p this observation should be of independent interest.

We consider t-ary trees characterized by their numbers of nodes and their total path length. When t=2 these are called binary trees, and in such trees a parent node may have up to t child nodes. We give asymptotic expansions for the total number of trees with nodes and path length p, when n and p are large. We consider several different ranges of n and p. For n→∞ and p=O(n^{3/2}) we recover the Airy distribution for the path length in trees with many nodes, and also obtain higher order asymptotic results. For p→∞ and an appropriate range of n we obtain a limiting Gaussian distribution for the number of nodes in trees with large path lengths. The mean and variance are expressed in terms of the maximal root of the Airy function. Singular perturbation methods, such as asymptotic matching and WKB type expansions, are used throughout, and they are combined with more standard methods of analytic combinatorics, such as generating functions, singularity analysis, saddle point method, etc. The results are applicable to problems in information theory, that involve data compression schemes which parse long sequence into shorter phrases. Numerical studies show the accuracy of the various asymptotic approximations. Key Words: Trees; Universal Types; Asymptotics; Path Length; Singular Perturbations

On connected graphs and trees with maximal inverse sum indeg index

We discuss sequential stub matching for directed graphs and show that this process can be used to sample simple digraphs with asymptotically equal probability. The process starts with an empty edge set and repeatedly adds edges to it with a certain state-dependent bias until the desired degree sequence is fulfilled, whilst avoiding placing a double edge or self-loop. We show that uniform sampling is achieved in the sparse regime when the maximum degree dmax is asymptotically dominated by m1/4, where m is the number of edges. The proof is based on deriving various combinatorial estimates related to the number of digraphs with a given degree sequence and controlling concentration of these estimates in large digraphs. This suggests that sequential stub matching can be viewed as a practical algorithm for almost uniform sampling of digraphs. We show that this algorithm can be implemented to feature a linear expected runtime O(m).