Risk network structure in the early epidemic phase of HIV transmission in Colorado Springs

Potterat, John J.; Phillips-Plummer, L; Muth, Stephen Q.; Rothenberg, Richard; Woodhouse, Donald E.; Maldonado-Long, T S; Zimmerman, H P; Muth, John

doi:10.1136/sti.78.suppl_1.i159

Cited by 135 publications

(117 citation statements)

References 11 publications

(9 reference statements)

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“…Note that a small value of s e means that the network is very sparse-however, there are several sparse social networks [13,16,17]. Table 1 summarizes our results.…”

Section: Introductionsupporting

confidence: 54%

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

et al. 2012

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Abstract. For directed and undirected graphs, we study the problem to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph's feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters "feedback vertex set number" and "number of vertices to delete". For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the "number of vertices to delete". On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness, including a vertex-linear problem kernel with respect to the parameter "feedback edge set number". On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter "vertex cover number and number of vertices to delete", implying corresponding nonexistence results when replacing vertex cover number by treewidth or feedback vertex set number.

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“…Note that a small value of s e means that the network is very sparse-however, there are several sparse social networks [13,16,17]. Table 1 summarizes our results.…”

Section: Introductionsupporting

confidence: 54%

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

et al. 2012

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“…The feedback edge number of a graph is the size of a minimum feedback edge set. Graphs with small feedback edge set number are "almost trees"; such social networks occur in the context of sexually transmitted infections [50] and extremism propagation [24]. Note that the feedback edge …”

Section: Parameters Lower-bounded By Degeneracymentioning

confidence: 99%

Parameterized Aspects of Triangle Enumeration

Bentert

Fluschnik

Nichterlein

et al. 2017

Fundamentals of Computation Theory

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Listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration, providing both positive results (new enumerative kernelizations, "subcubic" parameterized solving algorithms) as well as negative results (likely uselessness in terms of "faster" parameterized algorithms of certain parameters such as graph diameter).

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“…[80]; Drugs: Social network of injecting drug users (IDUs) that have shared a needle in the last six months [82]; Zachary: Social network of friendship between members of the Zachary karate club [83]; College: Social network among college students in a course about leadership. The students choose which three members they wanted to have in a committee [84]; ColoSpring: The risk network of persons with HIV infection during its early epidemic phase in Colorado Spring, USA, using analysis of community wide HIV/AIDS contact tracing records (sexual and injecting drugs partners) from 1985-1999 [85]; Galesburg: Friendship ties among 31 physicians [64]; High_Tech: Friendship ties among the employees in a small high-tech computer firm which sells, installs, and maintain computer systems [64,86]; Saw Mills: Social communication network within a sawmill, where employees were asked to indicate the frequency with which they discussed work matters with each of their colleagues [64,87];…”

Section: Social and Economic Networkmentioning

confidence: 99%

Exploring the “Middle Earth” of network spectra via a Gaussian matrix function

Estrada

Alhomaidhi

Al-Thukair

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. In particular, we study the Gaussian Estrada index-an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdős-Rï¿oenyi random graphs as well as for the Barabï¿oesi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs. 2The spectrum of a network-the set of its eigenvalues-provides important information about the structural and dynamical properties of the corresponding system. Most of the functions used to study network spectra give more weight to the largest modular eigenvalues. Then, the information contained in the eigenvalues close to the centre of the spectra, i.e, those close to zero, has remained totally unexplored in the study of graph spectra. Here we study a Gaussian matrix function that gives more weights to the eigenvalues closest to the centre of the spectrum of a network. Using this function we extract important structural information hidden in the spectra of networks, such as emergence of complete bipartite subgraphs (bicliques) which appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. These bicliques are also ubiquitous in random networks generated by preferential attachment mechanisms, such as the Barabï¿oesi-Albert model. In this work we provide a series of analytical results that pave the way for further analysis and uses of this Gaussian matrix function to understand network structure and dynamics.3

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Risk network structure in the early epidemic phase of HIV transmission in Colorado Springs

Cited by 135 publications

References 11 publications

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

Parameterized Aspects of Triangle Enumeration

Exploring the “Middle Earth” of network spectra via a Gaussian matrix function

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