Graph Theory Probability Graph Theory

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If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < p < 1, then we get an object G called an infinite random graph. Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property:

It turns out that if the vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.

Once we have a model of random graphs, every function on graphs, becomes a random variable. The study of this model is to determine if, or at least estimate the probability that, a property may occur.[4]

Random graphs are widely used in the probabilistic method, where one tries to prove the existence of graphs with certain properties. The existence of a property on a random graph can often imply, via the Szemerédi regularity lemma, the existence of that property on almost all graphs.

Properties of random graph may change or remain invariant under graph transformations. Mashaghi A. et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.[9]

Given a random graph G of order n with the vertex V(G) = {1, ..., n}, by the greedy algorithm on the number of colors, the vertices can be colored with colors 1, 2, ... (vertex 1 is colored 1, vertex 2 is colored 1 if it is not adjacent to vertex 1, otherwise it is colored 2, etc.).[3]The number of proper colorings of random graphs given a number of q colors, called its chromatic polynomial, remains unknown so far. The scaling of zeros of the chromatic polynomial of random graphs with parameters n and the number of edges m or the connection probability p has been studied empirically using an algorithm based on symbolic pattern matching.[10]

A random tree is a tree or arborescence that is formed by a stochastic process. In a large range of random graphs of order n and size M(n) the distribution of the number of tree components of order k is asymptotically Poisson. Types of random trees include uniform spanning tree, random minimal spanning tree, random binary tree, treap, rapidly exploring random tree, Brownian tree, and random forest.

The earliest use of a random graph model was by Helen Hall Jennings and Jacob Moreno in 1938 where a \"chance sociogram\" (a directed Erdős-Rényi model) was considered in studying comparing the fraction of reciprocated links in their network data with the random model.[11] Another use, under the name \"random net\", was by Ray Solomonoff and Anatol Rapoport in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.[12]

Suppose we wish to construct a graph in the following manner: Denote the vertices of the graph as $1,2,\\dots,n$. For every pair $\\{i,j\\}$, we flip a fair coin. If it comes up tails, $\\{i,j\\}$ is an edge of the graph; if it comes up heads, $\\{i,j\\}$ is not an edge of the graph. Please answer the following:

c. Notice that the expected degrees of vertices $2,\\dots,n$ should also be the same as that of vertex $1$. Given this observation, and the handshaking lemma, what is the expected number of edges in the graph

I get the probability theory part but I have trouble understanding where exactly graph theory fits in. What insights from graph theory have helped deepen our understanding of probability distributions and decision making under uncertainty

There is very little true mathematical graph theory in probabilistic graphical models, where by true mathematical graph theory I mean proofs about cliques, vertex orders, max-flow min-cut theorems, and so on. Even something as fundamental as Euler's Theorem and Handshaking Lemma are not used, though I suppose one might invoke them to check some property of computer code used to update probabilistic estimates. Moreover, probabilist graphical models rarely use more than a subset of the classes of graphs, such as multi-graphs. Theorems about flows in graphs are not used in probabilistic graphical models.

If student A were an expert in probability but knew nothing about graph theory, and student B were an expert in graph theory but knew nothing about probability, then A would certainly learn and understand probabilistic graphical models faster than would B.

References. [Murphy, 2012, 22.6.3] covers graph cuts usage for MAP inference. See also [Kolmogorom and Zabih, 2004; Boykov et al., PAMI 2001], which cover optimization rather than modelling.

There has been some work investigating the link between the ease of decoding of Low Density Parity Check codes (which gets excellent results when you consider it a probablistic graph and apply Loopy Belief Propagation), and the girth of the graph formed by the parity check matrix. This link to girth goes right the way back to when LDPCs were invented[1] but there's been further work in the last decade or so[2][3] after the were separately rediscovery by Mackay et al [4] and their properties noticed.

I often see pearl's comment on the convergence time of belief propagation depending on the diameter of the graph being cited. But I don't know of any work looking at graph diameters in non-tree graphs and what effect that has.

One successful application of graph algorithms to probabilistic graphical models is the Chow-Liu algorithm. It solves the problem of finding the optimum (tree) graph structure and is based on maximum spanning trees (MST) algorithm.

The default mode network (DMN) is related to brain functions and its abnormalities were associated with mental disorders' pathophysiology. To further understand the common and distinct DMN alterations across disorders, we capitalized on the probability tracing method and graph theory to analyze the role of DMN across three major mental disorders. A total of 399 participants (156 schizophrenia [SCZ], 90 bipolar disorder [BP], 58 major depression disorder [MDD], and 95 healthy controls [HC]) completed magnetic resonance imaging (MRI)-scanning, clinical, and cognitive assessment. The MRI preprocessing of diffusion-tensor-imaging was conducted in FMRIB Software Library and probabilistic fiber tracking was applied by PANDA. This study had three main findings. First, patient groups showed significantly lower cluster coefficient in whole-brain compared with HC. SCZ showed significantly longer characteristic path compared with HC. Second, patient groups showed inter-group specificity in abnormalities of DMN connections. Third, SCZ was sensitive to left_medial_superior_frontal_gyrus (L_SFGmed)-right_anterior_cingulate_gyrus (R_ACG) connection relating to positive symptoms; left_ACG-right_ACG connection was the mania's antagonistic factor in BP. This trans-diagnostic study found disorder-specific structural abnormalities in the fiber connection of R_SFGmed-L_SFGmed-R_ACG_L_ACG within DMN, where SCZ showed more disconnections compared with other disorders. And these connections are diagnosis-specifically correlated to phenotypes. The current study may provide further evidence of shared and distinct endo-phenotypes across psychopathology.

Population structure can affect evolutionary and ecological dynamics7,8,9,10,11,12,13,14,15,16. In evolutionary graph theory, the structure of a population is described by a graph17,18,19,20,21,22,23,24: each individual occupies a vertex; the edges mark the neighboring sites where a reproducing individual can place an offspring. The edge weights represent the proportional preference to make such a choice. If each neighbor is chosen uniformly at random, then the outgoing edges of every vertex have identical weights. This is modeled by an unweighted graph. A self-loop represents the possibility that an offspring does not migrate but instead replaces its parent25. The classical well-mixed population is described by an unweighted, complete graph with self-loops.

In general, the fixation probability depends not only on the graph, but also on the initial placement of the invading mutants26, 27. The two most natural cases are the following. First, mutation is independent of reproduction and occurs at all locations at a constant rate per unit time. Thus, mutants arise with equal probability in each location. This is called uniform initialization. Second, mutation happens during reproduction. In this case, mutants are more likely to occur in locations that have a higher turnover. This is called temperature initialization. Our approach also allows us to study any combination of the two cases: some mutants arise spontaneously while others occur during reproduction.

In this work we resolve several open questions regarding strong amplification under uniform and temperature initialization. First, we show that there exists a vast variety of graphs with self-loops and weighted edges that are arbitrarily strong amplifiers for both uniform and temperature initialization. Moreover, many of those strong amplifiers are structurally simple, therefore they might be realizable in natural or laboratory setting. Second, we show that both self-loops and weighted edges are key features of strong amplification. Namely, we show that without either self-loops or weighted edges, no graph is a strong amplifier under temperature initialization, and no simple graph is a strong amplifier under uniform initialization.

We prove that almost all families of connected graphs with self-loops can be turned into arbitrarily strong amplifiers of natural selection by assigning suitable edge weights. The resulting structures are arbitrarily strong amplifiers for both types of mutants: those that arise during reproduction and those that arise spontaneously, or any combination of the two. Our result proves not only the existence of those structures, but provides an explicit procedure for their construction. Note that by assigning small (or even zero) weight to an edge, we can effectively erase it. Hence, our construction is particularly interesting for sparse graphs. 59ce067264

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