Random Graph Theory

We show the number of triangles of G n,1/2 is almost uniformly distributed among residue classes modulo q, where q is a prime number bounded by (log n). This implies a consequence of a conjecture of Bollobás, Pebody and Riordan (that almost every random graph G n,1/2 is uniquely determined by its Tutte polynomial): almost every pair of independently chosen random graphs G n,1/2 has different Tutte polynomials.

We use Schnyder woods of 3-connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − ∆) × (n − 2 − ∆). The parameter ∆ ≥ 0 depends on the Schnyder wood used for the drawing. This parameter is in the range 0 ≤ ∆ ≤ n 2 − 2. The algorithm is a refinement of the face-counting-algorithm, thus, in particular, the size of the grid is at most

We review recent progress in the study of the vertex-cover problem (VC). VC belongs to the class of NP-complete graph theoretical problems, which plays a central role in theoretical computer science. On ensembles of random graphs, VC exhibits an coverable-uncoverable phase transition. Very close to this transition, depending on the solution algorithm, easy-hard transitions in the typical running time of the algorithms occur.

Probabilistic flooding (parameterized by a forwarding probability) has frequently been considered in the past, as a means of limiting the large message overhead associated with traditional (full) flooding approaches that are used to disseminate globally information in unstructured peer-topeer and other networks. A key challenge in using probabilistic flooding is the determination of the forwarding probability so that global network outreach is achieved while keeping the message overhead as low as possible. By showing that a probabilistic flooding network generated by applying probabilistic flooding to a connected random graph network can be bounded by properly parameterized random graph networks and invoking random graph theory results, bounds on the value of the forwarding probability are derived guaranteeing global network outreach with high probability, while significantly reducing the message overhead. Bounds on the average number of messages-as well as asymptotic expressions-and on the average time required to complete network outreach are also derived, illustrating the benefits of the properly parameterized probabilistic flooding scheme.

Brain graphs provide a relatively simple and increasingly popular way of modeling the human brain connectome, using graph theory to abstractly define a nervous system as a set of nodes (denoting anatomical regions or recording electrodes) and interconnecting edges (denoting structural or functional connections). Topological and geometrical properties of these graphs can be measured and compared to random graphs and to graphs derived from other neuroscience data or other (nonneural) complex systems. Both structural and functional human brain graphs have consistently demonstrated key topological properties such as small-worldness, modularity, and heterogeneous degree distributions. Brain graphs are also physically embedded so as to nearly minimize wiring cost, a key geometric property. Here we offer a conceptual review and methodological guide to graphical analysis of human neuroimaging data, with an emphasis on some of the key assumptions, issues, and trade-offs facing the investigator.

We introduce Chinese Whispers, a randomized graph-clustering algorithm, which is time-linear in the number of edges. After a detailed definition of the algorithm and a discussion of its strengths and weaknesses, the performance of Chinese Whispers is measured on Natural Language Processing (NLP) problems as diverse as language separation, acquisition of syntactic word classes and word sense disambiguation. At this, the fact is employed that the small-world property holds for many graphs in NLP.

The new higher order specifications for exponential random graph models introduced by Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology 36, 99-153] exhibit substantial improvements in model fit compared with the commonly used Markov random graph models. Snijders et al., however, concentrated on non-directed graphs, with only limited extensions to directed graphs. In particular, they presented a transitive closure parameter based on path shortening. In this paper, we explain the theoretical and empirical advantages in generalizing to additional closure effects. We propose three new triadic-based parameters to represent different versions of triadic closure: cyclic effects; transitivity based on shared choices of partners; and transitivity based on shared popularity. We interpret the last two effects as forms of structural homophily, where ties emerge because nodes share a form of localized structural equivalence. We show that, for some datasets, the path shortening parameter is insufficient for practical modeling, whereas the structural homophily parameters can produce useful models with distinctive interpretations. We also introduce corresponding lower order effects for multiple two-path connectivity. We show by example that the in-and out-degree distributions may be better modeled when star-based parameters are supplemented with parameters for the number of isolated nodes, sources (nodes with zero in-degrees) and sinks (nodes with zero out-degrees). Inclusion of a Markov mixed star parameter may also help model the correlation between in-and out-degrees. We select some 50 graph features to be investigated in goodness of fit diagnostics, covering a variety of important network properties including density, reciprocity, geodesic distributions, degree distributions, and various forms of closure. As empirical illustrations, we develop models for two sets of organizational network data: a trust network within a training group, and a work difficulty network within a government instrumentality.

A simple random walk on a graph is a sequence of movements from one vertex to another where at each step an edge is chosen uniformly at random from the set of edges incident on the current vertex, and then transitioned to next vertex. Central to this thesis is the cover time of the walk, that is, the expectation of the number of steps required to visit every vertex, maximised over all starting vertices. In our first contribution, we establish a relation between the cover times of a pair of graphs, and the cover time of their Cartesian product. This extends previous work on special cases of the Cartesian product, in particular, the square of a graph. We show that when one of the factors is in some sense larger than the other, its cover time dominates, and can become within a logarithmic factor of the cover time of the product as a whole. Our main theorem effectively gives conditions for when this holds. The techniques and lemmas we introduce may be of independent interest. In our sec...

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent τ > 2.

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there

Actual network size depends on the application and the protocols developed for the routing for this kind of
networks should be scalable and efficient. Each routing protocol should support small as well as large
scale networks very efficiently. As the number of node increase, it increases the management functionality
of the network. Graph theoretic approach traditionally was applied to networks where nodes are static or
fixed. In this paper, we have applied the graph theoretic routing to MANET where nodes are mobile. Here,
we designed all identical nodes in the cluster except the cluster head and this criterion reduces the
management burden on the network. Each cluster supports a few nodes with a cluster head. The intracluster
connectivity amongst the nodes within the cluster is supported by multi-hop connectivity to ensure
handling mobility in such a way that no service disruption can occur. The inter-cluster connectivity is also
achieved by multi-hop connectivity. However, for inter-cluster communications, only cluster heads are
connected. This paper demonstrates graph theoretic approach produces an optimum multi-hop connectivity
path based on cumulative minimum degree that minimizes the contention and scheduling delay end-toend.
It is applied to both intra-cluster communications as well as inter-cluster communications. The
performance shows that having a multi-hop connectivity for intra-cluster communications is more power
efficient compared to broadcast of information with maximum power coverage. We also showed the total
number of required intermediate nodes in the transmission from source to destination. However, dynamic
behavior of the nodes requires greater understanding of the node degree and mobility at each instance of
time in order to maintain end-to-end QoS for multi-service provisioning. Our simulation results show that
the proposed graph theoretic routing approach will reduce the overall delay and improves the physical
layer data frame transmission.

Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the network's geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs -including point process theory, percolation theory, and probabilistic combinatorics -have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.

In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one nontrivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the crossover from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Padé approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show t...

The stochastic block model (SBM) is a probabilistic model designed to describe heterogeneous directed and undirected graphs. In this paper, we address the asymptotic inference on SBM by use of maximumlikelihood and variational approaches. The identifiability of SBM is proved, while asymptotic properties of maximum-likelihood and variational estimators are provided. In particular, the consistency of these estimators is settled, which is, to the best of our knowledge, the first result of this type for variational estimators with random graphs. 0 imsart-generic ver.

Bachelor's thesis
January 2015
Technical University of Munich,
Department for Mathematical Statistics

The paper starts with the mathematical fundamentals for random graphs and models for random graphs. Over the course of the paper, spotlight is on the 'Preferential Attachment Model'. The importance of its scale-free degree distribution in various real-world phenomena is emphasized. We then proceed to discuss the findings of a ECB researcher team who discovered a scale-free degree distribution in the network of global Credit-Default-Swap trades. Finally, we run through computational simulations of preferential attachment networks and discuss similarities to the CDS network.

We discuss the use of a ferromagnetic spin model on a random graph to model granular compaction. A multi-spin interaction is used to capture the competition between local and global satisfaction of constraints characteristic for geometric frustration. We define an athermal dynamics designed to model repeated taps of a given strength. Amplitude cycling and the effect of permanently constraining a subset of the spins at a given amplitude is discussed. Finally we check the validity of Edwards' hypothesis for the athermal tapping dynamics.

Let G = (V, E) a graph and L (vi) a set of colors associated to every node vi�V. A list coloring of G is an assignment of a color c (vi) � L (vi) to every node of V so that no two adjacent nodes are assigned the same color. Significant theoretical research into this special case of the classic coloring problem started roughly the last decade. In the corresponding literature, algorithms are referred to for a particular class of graphs, e.g. trees. In this paper a heuristic algorithm is formulated in order to achieve a list coloring to a given random graph G with the smallest possible number of colors, whenever G has such a coloring. A small numerical example of the presented algorithm is given and the paper is incorporated with a relative computational experiment.

The effects of various population topologies on the particle swarm algorithm were systematically investigated. Random graphs were generated to specifications, and their performance on several criteria was compared. What makes a good population structure? We discovered that previous assumptions may not have been correct.

Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random gra...

We show that random graphs in the preferential connectivity model have constant conductance, and hence have worst-case routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the random walk matrix associated with these graphs. We also show that the expected frugality (overpayment in the Vickrey-Clarke-Groves mechanism for shortest paths) of a random graph is bounded by a small constant.

In this paper we develop the interpolating cavity field technique for the mean field ferromagnetic p-spin. The model we introduce is a natural extension of the diluted Curie-Weiss model to p>2 spin interactions. Several properties of the free energy are analyzed and, in particular, we show that it recovers the expressions already known for p=2 models and for p>2 fully connected models. Further, as the model lacks criticality, we present extensive numerical simulations to evidence the presence of a first order phase transition and deepen the behavior at the transition line. Overall, a good agreement is obtained among analytical results, numerics and previous works.

The problem of counting the number of s-t paths in a graph is #P-complete. We provide an algorithm to estimate the solution stochastically, using sequential im- portance sampling. We show that the method works eectiv ely for both graphs and digraphs. We also use the method to investigate the expected number of s-t paths in a random graph of size

We have studied an evolutionary prisoner's dilemma game with players located on two types of random regular graphs with a degree of 4. The analysis is focused on the effects of payoffs and noise (temperature) on the maintenance of cooperation. When varying the noise level and/or the highest payoff, the system exhibits a second order phase transition from a mixed state of cooperators and defectors to an absorbing state where only defectors remain alive. For the random regular graph (and Bethe lattice) the behavior of the system is similar to those found previously on the square lattice with nearest neighbor interactions, although the measure of cooperation is enhanced by the absence of loops in the connectivity structure. For low noises the optimal connectivity structure is built up from randomly connected triangles.

Let G = (V, E) a graph and L (vi) a set of colors associated to every node vi V. A list coloring of G is an assignment of a color c (vi )  L (vi ) to every node of V so that no two adjacent nodes are assigned the same color. Significant theoretical research into this special case of the classic coloring problem started roughly the last decade. In the corresponding literature, algorithms are referred to for a particular class of graphs, e.g. trees. In this paper a heuristic algorithm is formulated in order to achieve a list coloring to a given random graph G with the smallest possible number of colors, whenever G has such a coloring. A small numerical example of the presented algorithm is given and the paper is incorporated with a relative computational experiment.

The one-dimensional contact model for the spread of disease may be viewed as a directed percolation model on ℤ×ℝ in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the time-axis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a random-cluster model on ℤ×ℝ, with associated continuum Ising and Potts models. These models are of independent interest, in addition to providing a path-integral representation of the quantum Ising model with transverse field. This representation may be used to obtain a bound on the entanglement of a finite set of spins in the quantum Ising model on ℤ, where this entanglement is measured via the entropy of the reduced density matrix. The mean-field version of the quantum Ising model gives rise to a random-cluster model on K n ×ℝ, thereby extending the Erdős-Rényi random graph on the complete graph K n .

We analyze the properties of Small-World networks, where links are much more likely to connect "neighbor nodes" than distant nodes. In particular, our analysis provides new results for Kleinberg's Small-World model and its extensions. Kleinberg adds a number of directed long-range random links to an n × n lattice network (vertices as nodes of a grid, undirected edges between any two adjacent nodes). Links have a non-uniform distribution that favors arcs to close nodes over more distant ones. He shows that the following phenomenon occurs: between any two nodes a path with expected length O(log 2 n) can be found using a simple greedy algorithm which has no global knowledge of long-range links.

Missing data, such as non-response, is often problematic in social network analysis since what is missing may potentially alter the conclusions about what we have observed. This in the sense that individual tie-variables typically need to be interpreted in relation to their local neighbourhood and the global structure. Some ad-hoc methods for dealing with missing data in social networks have been proposed but here we consider a model-based approach. We discuss various aspects of fitting exponential family random graph (or p-star) models (ERGMs) to networks with missing data and present a Bayesian data augmentation algorithm for the purpose of estimation. This involves drawing from the fully conditional posterior distribution of the parameters, something which is made possible by a recently developed algorithm. With ERGMs already having complicated interdependencies, we argue that it is particularly important to provide inference that adequately describes the uncertainty, something that the Bayesian approach caters for. To the extent that we wish to explore the missing parts of the network, the posterior predictive distributions, immediately available at the termination of the algorithm, are at our disposal, which allows us to explore the distribution of what is missing unconditionally on any particular parameter values. Some important features of treating missing data and of the implementation of the algorithm are illustrated using a well known collaboration network.

This chapter treats statistical methods for network evolution. It is argued that it is most fruitful to consider models where network evolution is represented as the result of many (usually non-observed) small changes occurring between the consecutively observed networks. Accordingly, the focus is on models where a continuous-time network evolution is assumed although the observations are made at discrete time points (two or more).

The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and maximum degree. As a corol- lary, we obtain the result that the domatic number of an r-regular graph is at least (r + 1)/(3ln(r + 1)).

We introduce a general stochastic model for the spread of rumours, and derive mean-field equations that describe the dynamics of the model on complex social networks (in particular, those mediated by the Internet). We use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks: random graphs, uncorrelated scale-free networks and scale-free networks with assortative degree correlations. We show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system. In the case of scale-free networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. We find that the initial rate at which a rumour spreads is much higher in scale-free networks than in random graphs, and that the rate at which the spreading proceeds on scale-free networks is further increased when assortative degree correlations are introduced. The impact of degree correlations on the final fraction of nodes that ever hears a rumour, however, depends on the interplay between network topology and the rumour spreading rate. Our results show that scale-free social networks are prone to the spreading of rumours, just as they are to the spreading of infections. They are relevant to the spreading dynamics of chain emails, viral advertising and large-scale information dissemination algorithms on the Internet. r

This paper focuses on the Internet IP-level routing topology and proposes relevant explanations to its apparent dynamics. We first represent this topology as a power-law random graph. Then, we incorporate to the graph two well known factors responsible for the observed dynamics, which are load balancing and route evolution. Finally, we simulate on the graph traceroute-like measurements. Repeating the process many times, we obtain several graph instances that we use to model the dynamics. Our results show that we are able to capture on power-law graphs the dynamic behaviors observed on the Internet. We find that the results on power-law graphs, while qualitatively similar to the one of Erdös-Rényi random graphs, highly differ quantitatively; for instance, the rate of discovery of new nodes in power-law graphs is extremely low compared to the rate in Erdös-Rényi graphs.

We present a decentralized cooperative exploration strategy for mobile robots. A roadmap of the explored area, with the associate safe region, is built in the form of a compact data structure, called Sensor-based Random Graph. This is incrementally expanded by the robots by using a randomized local planner which automatically realizes a trade-off between information gain and navigation cost. Connecting structures, called bridges, are incrementally added to the graph to create shortcuts and improve the connectivity of the roadmap. Decentralized cooperation and coordination mechanisms are used so as to guarantee exploration efficiency and avoid conflicts. Simulations are presented to show the performance of the proposed technique.

We study the distribution of optimal path lengths in random graphs with random weights associated with each link ͑"disorder"͒. With each link i we associate a weight i = exp͑ar i ͒, where r i is a random number taken from a uniform distribution between 0 and 1, and the parameter a controls the strength of the disorder. We suggest, in an analogy with the average length of the optimal path, that the distribution of optimal path lengths has a universal form that is controlled by the expression ͑1/ p c ͒͑ᐉ ϱ / a͒, where ᐉ ϱ is the optimal path length in strong disorder ͑a → ϱ͒ and p c is the percolation threshold. This relation is supported by numerical simulations for Erdős-Rényi and scale-free graphs. We explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network.

Abstract. We define a cellular assignment graph to model the channel assignment problem in a cellular network where overlapping cell segments are included in the model. Our main result is the Capacity-Demand Theorem which shows a channel assignment function is always possible unless there is a connected subregion of cells and overlap segments containing more channel requests then the total capacity of all transceivers within or on the boundary of the subregion and covering any part of the subregion with an overlapping segment. We further describe the simplicity and regularity of our proposed cellular assignment graphs and their accessibility for simulation and theoretical investigation without artifacts from the overall geographical region boundaries.

We consider the Internet at the level of its sub-networks (called Autonomous Systems, or ASes). Most previous studies have used the connection degree as the indicator variable to decompose the network into what one hopes will be nodes with distinct functions

We describe some new exactly solvable models of the structure of social networks, based on random graphs with arbitrary degree distributions. We give models both for simple unipartite networks, such as acquaintance networks, and bipartite networks, such as affiliation networks. We compare the predictions of our models to data for a number of real-world social networks and find that in some cases, the models are in remarkable agreement with the data, whereas in others the agreement is poorer, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents - crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. At least at equilibrium, dilution simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved.