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We study the statistical behavior under random sequential renormalization(RSR) of several network models including Erd"os R'enyi (ER) graphs, scale-free networks and an annealed model (AM) related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [C.Song et.al], RSR allows a more fine-grained analysis of the renormalization group (RG) flow, and unravels new features, that were not discussed in the previous analyses. In particular we find that all networks exhibit a second order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling laws seen there. For critical trees it happens as N/N0 -> 0 in the limit of large systems (where N0 is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N0 in sparse ER graphs and in the annealed model, while it happens for N/N0 -> 1 on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a star-like structure in agreement with the results of Radicchi et. al. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering 'supernodes' as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics.

We study the statistical behavior under random sequential renormalization(RSR) of several network models including Erd"os R'enyi (ER) graphs, scale-free networks and an annealed model (AM) related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [C.Song et.al], RSR allows a more fine-grained analysis of the renormalization group (RG) flow, and unravels new features, that were not discussed in the previous analyses. In particular we find that all networks exhibit a second order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling laws seen there. For critical trees it happens as N/N0 -> 0 in the limit of large systems (where N0 is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N0 in sparse ER graphs and in the annealed model, while it happens for N/N0 -> 1 on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a star-like structure in agreement with the results of Radicchi et. al. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering 'supernodes' as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics.

scan of booklet Handbook (2.1) MT365 Graphs, networks and design by The MT365 Course Team [2009] Open University course handbooks are sometimes allowed into the exams on the course. This means that many students cram extra notes into the text and hints as how to answer types of exam question. This is not an exceptional example. Considerable image processing has had to be done to pull the original print out from behind the annotations; the whiter the background the more heavily the processing applied. In one place the notes so obscured the text that the text has been reconstructed in a word processor and an image pasted into place. There have been corrections applied as though directed by course management. Again in certain places instead to leaving the correction as carried out by the student a word processor has been used to produce an image of the correction that has been pasted into place.

scan of booklet Audio Notes 2 (3.4) MT365 Graphs, networks and design by The MT365 Course Team [2001] These notes accompany the audio sequences of CDs CDA 5675 and CDA 5676

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face is to find the discrete protoforms of the building blocks of continuum physics and mathematics. In the following we embark on developing such concepts for irregular structures like (large) graphs or networks which are intended to emulate (some of) the generic properties of the presumed combinatorial substratum from which continuum physics is assumed to emerge as a coarse grained and secondary model theory. We briefly indicate how various concepts of discrete (functional) analysis and geometry can be naturally constructed within this framework, leaving a larger portion of the paper to the systematic developement of dimensional concepts and their properties, which may have a possible bearing on various branches of modern physics beyond quantum gravity.

The mechanisms by which modularity emerges in complex networks are not well understood but recent reports have suggested that modularity may arise from evolutionary selection. We show that finding the modularity of a network is analogous to finding the ground-state energy of a spin system. Moreover, we demonstrate that, due to fluctuations, stochastic network models give rise to modular networks. Specifically, we show both numerically and analytically that random graphs and scale-free networks have modularity. We argue that this fact must be taken into consideration to define statistically-significant modularity in complex networks.

The Erdos-Renyi classical random graph is characterized by a fixed linking probability for all pairs of vertices. Here, this concept is generalized by drawing the linking probability from a certain distribution. Such a procedure is found to lead to a static complex network with an arbitrary connectivity distribution. In particular, a scale-free network with the hierarchical organization is constructed without assuming any knowledge about the global linking structure, in contrast to the preferential attachment rule for a growing network. The hierarchical and mixing properties of the static scale-free network thus constructed are studied. The present approach establishes a bridge between a scalar characterization of individual vertices and topology of an emerging complex network. The result may offer a clue for understanding the origin of a few abundance of connectivity distributions in a wide variety of static real-world networks.

The Erdos-Renyi classical random graph is characterized by a fixed linking probability for all pairs of vertices. Here, this concept is generalized by drawing the linking probability from a certain distribution. Such a procedure is found to lead to a static complex network with an arbitrary connectivity distribution. In particular, a scale-free network with the hierarchical organization is constructed without assuming any knowledge about the global linking structure, in contrast to the preferential attachment rule for a growing network. The hierarchical and mixing properties of the static scale-free network thus constructed are studied. The present approach establishes a bridge between a scalar characterization of individual vertices and topology of an emerging complex network. The result may offer a clue for understanding the origin of a few abundance of connectivity distributions in a wide variety of static real-world networks.

Many theorems in graph theory give simple characterizations for testing the existence of objects with certain properties, which can be translated into fast parallel algorithms. However, transforming these tests into algorithms for constructing such objects is often a real challenge. In this thesis we develop fast parallel (NC) algorithms for several such construction problems. The first part is about tournaments. (A tournament is a digraph in which there is precisely one arc between every two vertices.) Two classical results state that every tournament has a Hamiltonian path and every strongly connected tournament has a Hamiltonian cycle. We derive efficient parallel algorithms for finding these objects. Our algorithms yield new proofs of these theorems. In solving the cycle problem we also solve the problem of finding a Hamiltonian path with one fixed endpoint. Next we address the problem of constructing a tournament with a specified degree-sequence, and give an NC algorithm for it which achieves optimal speedup. The second part is concerned with making graphs strongly connected via orientation and augmentation. A graph is strongly orientable if its edges can be assigned orientations to yield a strongly connected digraph. Robbins' theorem states the conditions for a graph to be strongly orientable. His theorem was generalized for mixed graphs, i.e. ones that have both directed and undirected edges. We give a fast parallel algorithm for strongly orienting mixed graphs. We then solve the problem of adding a minimum number of arcs to a mixed graph to make it strongly orientable. This problem was not previously known to have even a polynomial time sequential solution \201a sequential algorithm was discovered independently by Gusfield\202. In the process of solving the general problem we derive solutions for the special cases of undirected graphs. The final part of the thesis describes a methodology which yields deterministic parallel algorithms.

In a graph, nodes can be characterized locally (with their degree $k$) or globally (e.g. with their average length path $\xi$ to other nodes). Here we investigate how $\xi$ depends on $k$. Our earlier algorithm of the construction of the distance matrix is applied to the random graphs. Numerical calculations are performed for the random graphs and the growing networks: the scale-free ones and the exponential ones. The results are relevant for search strategies in different networks.

This paper studies local reconfiguration of trees into arrays and vice versa. It also studies the construction of adjacency graphs and quadtrees for images stored in cellular array processors. (Author)

scan of book Physical networks (1.6)  Networks 4 MT365 Graphs, networks and design by The MT365 Course Team [2009] {511.5--oclc}

scan of book Network flows (2.1) Networks 1 MT365 Graphs, networks and design 2nd edition by The MT365 Course Team [2009] {511.5--oclc}

scan of book Network flows (2.1) Networks 1 MT365 Graphs, networks and design 2nd edition by The MT365 Course Team [2009] {511.5--oclc}

This project aims to develop a framework that blends constructs from system theory and theory of networks to address problems pertaining to security and robustness of networked dynamic systems. Such networks have the additional layer of having dynamics overlapping the interconnection geometry and as such, the necessary formalisms for examining them and useful methodologies for their analysis and synthesis, are only beginning to take shape. Yet as the next generation Air Force systems more and more consist of dynamics entities interacting over complex networks, the study of the security and robustness of networked systems from a control and system theoretic point of view becomes of paramount importance.

We study the inverse problem of determining the conductivity matrix of an electrical network from the prescribed knowledge of the magnitude of the induced current along the edges coupled with the imposed voltage or injected current on the boundary nodes. This problem leads to a weighted $l^1$ minimization problem for the corresponding voltage potential. We also investigate the problem of determining the transition probabilities of random walks on graphs from the prescribed net number of times the walker passes along the edges of the graph. We also show that a mass preserving flow $J=(J_{i.j})$ on a network can be uniquely recovered from the knowledge of $|J|=(|J_{i,j}|)$ and the flux of the flow on the boundary nodes, where $J_{i,j}$ is the flow from node $i$ to node $j$ and $J_{i,j}=-J_{j,i}$. Convergent numerical algorithms for solving such problems are also presented.

The zero-temperature Glauber dynamics is used to investigate the persistence probability $P(t)$ in the Potts model with $Q=3,4,5,7,9,12,24,64, 128$, $256, 512, 1024,4096,16384 $,..., $2^{30}$ states on {\it directed} and {\it undirected} Barab\'asi-Albert networks and Erd\"os-R\'enyi random graphs. In this model it is found that $P(t)$ decays exponentially to zero in short times for {\it directed} and {\it undirected} Erd\"os-R\'enyi random graphs. For {\it directed} and {\it undirected} Barab\'asi-Albert networks, in contrast it decays exponentially to a constant value for long times, i.e, $P(\infty)$ is different from zero for all $Q$ values (here studied) from $Q=3,4,5,..., 2^{30}$; this shows "blocking" for all these $Q$ values. Except that for $Q=2^{30}$ in the {\it undirected} case $P(t)$ tends exponentially to zero; this could be just a finite-size effect since in the other "blocking" cases you may have only a few unchanged spins.

In infrastructure-less highly dynamic networks, computing and performing even basic tasks (such as routing and broadcasting) is a very challenging activity due to the fact that connectivity does not necessarily hold, and the network may actually be disconnected at every time instant. Clearly the task of designing protocols for these networks is less difficult if the environment allows waiting (i.e., it provides the nodes with store-carry-forward-like mechanisms such as local buffering) than if waiting is not feasible. No quantitative corroborations of this fact exist (e.g., no answer to the question: how much easier?). In this paper, we consider these qualitative questions about dynamic networks, modeled as time-varying (or evolving) graphs, where edges exist only at some times. We examine the difficulty of the environment in terms of the expressivity of the corresponding time-varying graph; that is in terms of the language generated by the feasible journeys in the graph. We prove that the set of languages $L_{nowait}$ when no waiting is allowed contains all computable languages. On the other end, using algebraic properties of quasi-orders, we prove that $L_{wait}$ is just the family of regular languages. In other words, we prove that, when waiting is no longer forbidden, the power of the accepting automaton (difficulty of the environment) drops drastically from being as powerful as a Turing machine, to becoming that of a Finite-State machine. This (perhaps surprisingly large) gap is a measure of the computational power of waiting. We also study bounded waiting; that is when waiting is allowed at a node only for at most $d$ time units. We prove the negative result that $L_{wait[d]} = L_{nowait}$; that is, the expressivity decreases only if the waiting is finite but unpredictable (i.e., under the control of the protocol designer and not of the environment).

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps. We also introduce related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we discuss two different dynamical feautures of networks: synchronizability and flow of random walkers and so that they are optimized if the corresponding Laplacian matrix have a large spectral gap. From this, we show, by developing a numerical optimization algorithm that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks. These turn out to be related to Ramanujan and Cage graphs. We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide almost or almost optimal solutions to many other problems as, for instance, searchability in the presence of congestion or performance of neural networks. Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case. Finally, a critical discussion of the limitations and possible extensions of this work is presented.

rip of cd to WAV files Audio CDA 5676 MT365 Graphs, networks and design by The MT365 Course Team [2006] Audio Notes 2 of course MT365 Graphs networks and design accompanies these audio sequences

Can one hear the shape of a graph? This is a modification of the famous question of Mark Kac "Can one hear the shape of a drum?" which can be asked in the case of scattering systems such as quantum graphs and microwave networks. It addresses an important mathematical problem whether scattering properties of such systems are uniquely connected to their shapes? Recent experimental results based on a characteristics of graphs such as the cumulative phase of the determinant of the scattering matrices indicate a negative answer to this question (O. Hul, M. Lawniczak, S. Bauch, A. Sawicki, M. Kus, L. Sirko, Phys. Rev. Lett 109, 040402 (2012).). In this paper we consider important local characteristics of graphs such as structures of resonances and poles of the determinant of the scattering matrices. Using these characteristics we study experimentally and theoretically properties of graphs and directly confirm that the pair of graphs considered in the cited paper is isoscattering. The experimental results are compared to the theoretical ones for a broad frequency range from 0.01 to 3 GHz. In the numerical calculations of the resonances of the graphs absorption present in the experimental networks is taken into account.

DVD ripped to M4V DVD00408 Video material in support of MT365 Graphs, Networks and Design by The MT365 Course Team [2007] {511.5--oclc} An additional written note in the packaging reads:- 'Errors aren't forever' Band 6 on course software DVD n the 3-circles animation, the pink circles 6 and 7 should be interchanged 'Television Notes' accompany these video sequences for the course MT365 Graphs, networks and design

We formulate a simple edge generation rule based on an inverse like mass action principle for random graphs over a structured vertex set. We show that under very weak assumptions on the structure generating distribution we obtain a scale free distribution for the degree. We furthermore introduce and study a "my friends are your friends" local search principle which makes the clustering coefficient large.

rip of cd to WAV files Audio CDA 5674 MT365 Graphs, networks and design by The MT365 Course Team [2006] Audio Notes 1 accompany this cd of audio sequences

The sensitivity of networks regarding the removal of vertices has been studied extensively within the last 15 years. A common approach to measure this sensitivity is (i) removing successively vertices by following a specific removal strategy and (ii) comparing the original and the modified network using a specific comparison method. In this paper we apply a wide range of removal strategies and comparison methods in order to study the sensitivity of medium-sized networks from real world and randomly generated networks. In the first part of our study we observe that social networks and web graphs differ in sensitivity. When removing vertices, social networks are robust, web graphs are not. This effect is conclusive with the work of Boldi et al. who analyzed very large networks. For similarly generated random graphs we find that the sensitivity highly depends on the comparison method. The choice of the removal strategy has surprisingly marginal impact on the sensitivity as long as we consider removal strategies implied by common centrality measures. However, it has a strong effect when removing the vertices in random order.

A finite Frobenius group is a permutation group which is transitive but not regular such that only the identity element can fix two points. Such a group can be expressed as a semidirect product $G = K \rtimes H$, where $K$ is a nilpotent normal subgroup. A first-kind $G$-Frobenius graph is a Cayley graph on $K$ whose connection set is an $H$-orbit $S$ on $K$ that generates $K$, where $H$ is of even order or $S$ consists of involutions. In this paper we classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel $K$ is cyclic. We give optimal gossiping, routing and broadcasting algorithms for such circulants and compute their forwarding indices, Wiener indices and minimum gossip time. We also prove that the broadcasting time of such a circulant is equal to its diameter plus two or three, indicating that it is efficient for broadcasting. We prove that all 6-valent first-kind Frobenius circulants with cyclic kernels are Eisenstein-Jacobi graphs, the latter being Cayley graphs on quotient rings of the ring of Eisenstein-Jacobi integers. We also prove that larger Eisenstein-Jacobi graphs can be constructed from smaller ones as topological covers, and a similar result holds for the family of 6-valent first-kind Frobenius circulants. As a corollary we show that any Eisenstein-Jacobi graph with order congruent to 1 modulo 6 and underlying Eisenstein-Jacobi integer not an associate of a real integer, is a cover of a 6-valent first-kind Frobenius circulant. We notice that a distributed real-time computing architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind Frobenius circulant.

scan of book Conclusion (1.5) MT365 Graphs, networks and design by James Donald et al. [2009] {302.30941--oclc}

We propose that hypergraphs can be used to model social networks with overlapping communities. The nodes of the hypergraphs represent the communities. The hyperlinks of the hypergraphs denote the individuals who may participate in multiple communities. The hypergraphs are not easy to analyze, however, the line graphs of hypergraphs are simple graphs or weighted graphs, so that the network theory can be applied. We define the overlapping depth $k$ of an individual by the number of communities that overlap in that individual, and we prove that the minimum adjacency eigenvalue of the corresponding line graph is not smaller than $-k_{max}$, which is the maximum overlapping depth of the whole network. Based on hypergraphs with preferential attachment, we establish a network model which incorporates overlapping communities with tunable overlapping parameters $k$ and $w$. By comparing with the Hyves social network, we show that our social network model possesses high clustering, assortative mixing, power-law degree distribution and short average path length.

This research aims to identify strong structural features of real-world complex networks, sufficient to enable a host of graph algorithms that are much more efficient than what is possible for general graphs (and currently used for network analysis). Specifically, we study the property of bounded expansion. This is the strongest formalization of the well-observed notion of "sparsity" that might possibly apply to real-world networks and allow many previously intractable problems to become tractable. On the theoretical side, we analyze many previously proposed models for random networks and characterize which ones have bounded expansion. We show that, w.h.p., (1) graphs sampled with either the Molloy-Reed configuration model (including a variation of the model which achieves high clustering) or the Chung-Lu model with a prescribed sparse degree sequence (including heavy-tailed degree distributions); (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. We also prove that the Kleinberg model and the Barabasi-Albert model, in typical setups, contain large one-subdivisions of cliques and thus do not result in graphs of bounded expansion. On the practical side, we give experimental evidence that many complex networks have bounded expansion, by measuring the closely related "low treedepth coloring number" on a corpus of real-world data. On the algorithmic side, we show how tools provided by the bounded expansion framework can be used to efficiently solve the following common network analysis problems: for a fixed graph H, we obtain the fastest-known algorithm for counting the number of induced H-subgraphs and the number of H-homomorphisms; and we design linear algorithms for computing several centrality measures.

This research aims to identify strong structural features of real-world complex networks, sufficient to enable a host of graph algorithms that are much more efficient than what is possible for general graphs (and currently used for network analysis). Specifically, we study the property of bounded expansion. This is the strongest formalization of the well-observed notion of "sparsity" that might possibly apply to real-world networks and allow many previously intractable problems to become tractable. On the theoretical side, we analyze many previously proposed models for random networks and characterize which ones have bounded expansion. We show that, w.h.p., (1) graphs sampled with either the Molloy-Reed configuration model (including a variation of the model which achieves high clustering) or the Chung-Lu model with a prescribed sparse degree sequence (including heavy-tailed degree distributions); (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. We also prove that the Kleinberg model and the Barabasi-Albert model, in typical setups, contain large one-subdivisions of cliques and thus do not result in graphs of bounded expansion. On the practical side, we give experimental evidence that many complex networks have bounded expansion, by measuring the closely related "low treedepth coloring number" on a corpus of real-world data. On the algorithmic side, we show how tools provided by the bounded expansion framework can be used to efficiently solve the following common network analysis problems: for a fixed graph H, we obtain the fastest-known algorithm for counting the number of induced H-subgraphs and the number of H-homomorphisms; and we design linear algorithms for computing several centrality measures.

This research aims to identify strong structural features of real-world complex networks, sufficient to enable a host of graph algorithms that are much more efficient than what is possible for general graphs (and currently used for network analysis). Specifically, we study the property of bounded expansion. This is the strongest formalization of the well-observed notion of "sparsity" that might possibly apply to real-world networks and allow many previously intractable problems to become tractable. On the theoretical side, we analyze many previously proposed models for random networks and characterize which ones have bounded expansion. We show that, w.h.p., (1) graphs sampled with either the Molloy-Reed configuration model (including a variation of the model which achieves high clustering) or the Chung-Lu model with a prescribed sparse degree sequence (including heavy-tailed degree distributions); (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. We also prove that the Kleinberg model and the Barabasi-Albert model, in typical setups, contain large one-subdivisions of cliques and thus do not result in graphs of bounded expansion. On the practical side, we give experimental evidence that many complex networks have bounded expansion, by measuring the closely related "low treedepth coloring number" on a corpus of real-world data. On the algorithmic side, we show how tools provided by the bounded expansion framework can be used to efficiently solve the following common network analysis problems: for a fixed graph H, we obtain the fastest-known algorithm for counting the number of induced H-subgraphs and the number of H-homomorphisms; and we design linear algorithms for computing several centrality measures.

Graphs and networks provide a canonical representation of relational data, with massive network data sets becoming increasingly prevalent across a variety of scientific fields. Although tools from mathematics and computer science have been eagerly adopted by practitioners in the service of network inference, they do not yet comprise a unified and coherent framework for the statistical analysis of large-scale network data. This paper serves as both an introduction to the topic and a first step toward formal inference procedures. We develop and illustrate our arguments using the example of hypothesis testing for network structure. We invoke a generalized likelihood ratio framework and use it to highlight the growing number of topics in this area that require strong contributions from statistical science. We frame our discussion in the context of previous work from across a variety of disciplines, and conclude by outlining fundamental statistical challenges whose solutions will in turn serve to advance the science of network inference.

Graphs and networks provide a canonical representation of relational data, with massive network data sets becoming increasingly prevalent across a variety of scientific fields. Although tools from mathematics and computer science have been eagerly adopted by practitioners in the service of network inference, they do not yet comprise a unified and coherent framework for the statistical analysis of large-scale network data. This paper serves as both an introduction to the topic and a first step toward formal inference procedures. We develop and illustrate our arguments using the example of hypothesis testing for network structure. We invoke a generalized likelihood ratio framework and use it to highlight the growing number of topics in this area that require strong contributions from statistical science. We frame our discussion in the context of previous work from across a variety of disciplines, and conclude by outlining fundamental statistical challenges whose solutions will in turn serve to advance the science of network inference.

Graphs and networks provide a canonical representation of relational data, with massive network data sets becoming increasingly prevalent across a variety of scientific fields. Although tools from mathematics and computer science have been eagerly adopted by practitioners in the service of network inference, they do not yet comprise a unified and coherent framework for the statistical analysis of large-scale network data. This paper serves as both an introduction to the topic and a first step toward formal inference procedures. We develop and illustrate our arguments using the example of hypothesis testing for network structure. We invoke a generalized likelihood ratio framework and use it to highlight the growing number of topics in this area that require strong contributions from statistical science. We frame our discussion in the context of previous work from across a variety of disciplines, and conclude by outlining fundamental statistical challenges whose solutions will in turn serve to advance the science of network inference.

When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations. A simple idea to account for multiple constraints consists in taking the intersection of random graphs. In this paper we initiate the study of random graphs so obtained through a simple example. We examine the intersection of an Erdos-Renyi graph and of one-dimensional geometric random graphs. We investigate the zero-one laws for the property that there are no isolated nodes. When the geometric component is defined on the unit circle, a full zero-one law is established and we determine its critical scaling. When the geometric component lies in the unit interval, there is a gap in that the obtained zero and one laws are found to express deviations from different critical scalings. In particular, the first moment method requires a larger critical scaling than in the unit circle case in order to obtain the one law. This discrepancy is somewhat surprising given that the zero-one laws for the absence of isolated nodes are identical in the geometric random graphs on both the unit interval and unit circle.

Entity Disambiguation aims to link mentions of ambiguous entities to a knowledge base (e.g., Wikipedia). Modeling topical coherence is crucial for this task based on the assumption that information from the same semantic context tends to belong to the same topic. This paper presents a novel deep semantic relatedness model (DSRM) based on deep neural networks (DNN) and semantic knowledge graphs (KGs) to measure entity semantic relatedness for topical coherence modeling. The DSRM is directly trained on large-scale KGs and it maps heterogeneous types of knowledge of an entity from KGs to numerical feature vectors in a latent space such that the distance between two semantically-related entities is minimized. Compared with the state-of-the-art relatedness approach proposed by (Milne and Witten, 2008a), the DSRM obtains 19.4% and 24.5% reductions in entity disambiguation errors on two publicly available datasets respectively.

Degree ssortativity is the tendency for nodes of high degree (resp.low degree) in a graph to be connected to high degree nodes (resp. to low degree ones). It is sually quantified by the Pearson correlation coefficient of the degree-degree correlation. Here we extend this concept to account for the effect of second neighbours to a given node in a graph. That is, we consider the two-walks degree of a node as the sum of all the degrees of its adjacent nodes. The two-walks degree assortativity of a graph is then the Pearson correlation coefficient of the two-walks degree-degree correlation. We found here analytical expression for this two-walks degree assortativity index as a function of contributing subgraphs. We then study all the 261,000 connected graphs with 9 nodes and observe the existence of assortative-assortative and disassortative-disassortative graphs according to degree and two-walks degree, respectively. More surprinsingly, we observe a class of graphs which are degree disassortative and two-walks degree assortative. We explain the existence of some of these graphs due to the presence of certain topological features, such as a node of low-degree connected to high-degree ones. More importantly, we study a series of 49 real-world networks, where we observe the existence of the disassortative-assortative class in several of them. In particular, all biological networks studied here were in this class. We also conclude that no graphs/networks are possible with assortative-disassortative structure.

We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers $\mathbb{N}$ via a Poisson point process with mean measure $\Lambda$ on the power set of $\mathbb{N}$. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, and every monotone subset maps to an undirected graph $G\in\catg$, the space of undirected graphs with vertex set $\mathbb{N}$. We show a natural mapping $\cate\rightarrow\catf\rightarrow\catg$ which induces an infinitely exchangeable measure on the projective system $\catg^{\rest}$ of graphs $\catg$ under permutation and restriction mappings given an infinitely exchangeable family of measures on the projective system $\cate^{\rest}$ of subsets with permutation and restriction maps. We show potential connections of this process to applications in cluster analysis, machine learning, classification and Bayesian inference.

We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers $\mathbb{N}$ via a Poisson point process with mean measure $\Lambda$ on the power set of $\mathbb{N}$. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, and every monotone subset maps to an undirected graph $G\in\catg$, the space of undirected graphs with vertex set $\mathbb{N}$. We show a natural mapping $\cate\rightarrow\catf\rightarrow\catg$ which induces an infinitely exchangeable measure on the projective system $\catg^{\rest}$ of graphs $\catg$ under permutation and restriction mappings given an infinitely exchangeable family of measures on the projective system $\cate^{\rest}$ of subsets with permutation and restriction maps. We show potential connections of this process to applications in cluster analysis, machine learning, classification and Bayesian inference.

By applying a recently proposed mapping, we derive exactly the upper phase boundary of several Ising spin glass models defined over static graphs and random graphs, generalizing some known results and providing new ones.

On cover: Mathematics and Computing/Technology: A third level course

Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases - usually associated with topological restrictions- their clustering is low and they are almost planar. In this paper we introduce a family of graphs which share all these properties and are defined by two parameters. As their construction is deterministic, we obtain exact analytic expressions for relevant properties of the graphs including the degree distribution, degree correlation, diameter, and average distance, as a function of the two defining parameters. Thus, the graphs are useful to model some complex networks, in particular technological and biological networks.

The following is a short report about recent work on discrete physics/mathematics on the Planckscale and the use of the concept of ''random graphs'' in this business, appearing in the group21-proceedings (Gosslar 1996)

We consider from the localization perspective the new critical behavior discovered recently for the regular random graphs (RRG) and constrained Erd\H{o}s-Renyi networks (CERN). The diagonal disorder for standard models, we replace by the fugacity $\mu$ of triads in the RRG and CERN. At some critical value of $\mu$ the network decays into the maximally possible number of almost full graphs, and the adjacency matrix acquires the two-gapped structure. We find that the eigenvalue statistics corresponds to delocalized states in the central zone, and to the localized states in the side one. The mobility edge lies between zones. We apply these findings to the many-body localization assuming the approximation of the hierarchical structure of the Fock space (for some interacting many-body system) by the RGG and by CERN with some vertex degree. We allow the 3-cycles in the Fock space and identify particles in the many-body system above the phase transition with clusters in the RRG. We discuss the controversial issue of the additional phase transition between ergodic and non-ergodic regions in the delocalized phase in the Fock space and find the strong "`memory-dependence"' of the states in the delocalized phase.

In this paper we introduce k-laminar graphs a new class of graphs which extends the idea of Asteroidal triple free graphs. Indeed a graph is k-laminar if it admits a diametral path that is k-dominating. This bio-inspired class of graphs was motivated by a biological application dealing with sequence similarity networks of reads (called hereafter read networks for short). We briefly develop the context of the biological application in which this graph class appeared and then we consider the relationships of this new graph class among known graph classes and then we study its recognition problem. For the recognition of k-laminar graphs, we develop polynomial algorithms when k is fixed. For k=1, our algorithm improves a Deogun and Krastch's algorithm (1999). We finish by an NP-completeness result when k is unbounded.

scan of book Underground Maps and Neural Networks, The theory of graphs by Claudi Alsina [2017] {511.5--jtl} ISBN 9788447387731 Chapter Material:- Introduction to Graphs Graphs and Colours Graphs, Circuits and Optimization Graphs and Geometry Surprising Applications of Graphs Graphs, Sets and Relations

scan of book Geometric Design (1.5) Design 1 MT365 Graphs, networks and design by The MT365 Course Team [2009] {511.5--oclc}

Scan of book Graphs and digraphs (1.6) Graphs 1 MT365 Graphs, networks and design by The MT365 Course Team [2009] {511.5--oclc}

scans of the list, front and back Contents checklist MT365 Graphs, networks and design (2019J) by The MT365 Course Team [2019] The course mailing package contents list

scan of book Finite graphs and networks, an introduction with applications by Robert G Busacker & Thomas L Saaty [1965] {512.5--loc} Colouring Theorems for those in the UK